abelard.flet.mita.keio.ac.jpabelard.flet.mita.keio.ac.jp/person/takemura/paper/18... · 2009. 10....

Proof-Theoretical Studies on

Diagrammatic Proofs and Focusing Proofs

A Thesis

Submitted to

Faculty of Letters

Keio University

In Partial Fulfillment

of the Requirements for the Degree of

Doctor of Philosophy

by

Ryo Takemura

Acknowledgments

First of all, I would like to express my sincere gratitude to my supervisor,Professor Mitsuhiro Okada, for a number of suggestions and constant en-couragements; without his advice and help, this thesis would never havebeen completed.

In writing Part II, the attentive guidance of Dr. Masahiro Hamano wasinvaluable, and I am profoundly grateful to him for his help.

I am grateful to Dr. Paul-Andre Mellies for his kind advices and discus-sions at many occasions on the preparation stage of this thesis.

I would like to thank Dr. Olivier Laurent who kindly read a preliminarydraft of Part II and gave me fruitful comments to improve the part.

I am also grateful to Professor Masahiko Sato for his helpful commentsand advices.

Part I resulted from the stimulating discussions that I had with Mr.Koji Mineshima on many occasions. I am grateful to him for his fruitfulsuggestions and advices.

2

Contents

Introduction 8

I Diagrammatic proofs with Euler circles 14

1 Introduction to Part I 16

2 A diagrammatic representation system (EUL) for Euler cir-cles and its set-theoretical semantics 212.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Diagrammatic syntax of EUL . . . . . . . . . . . . . . . . . . 252.3 Set-theoretical semantics of EUL . . . . . . . . . . . . . . . . 29

3 Diagrammatic inference system GDS 313.1 Introduction to unification . . . . . . . . . . . . . . . . . . . . 313.2 Generalized diagrammatic syllogistic inference system GDS . 333.3 Soundness and completeness of GDS . . . . . . . . . . . . . . 413.4 Some consequences of completeness of GDS . . . . . . . . . . 57

3.4.1 Unification of any (two) diagrams . . . . . . . . . . . 573.4.2 Decomposition set of an EUL-diagram . . . . . . . . . 573.4.3 On normal diagrammatic proofs . . . . . . . . . . . . 573.4.4 Structure of canonical diagrammatic proofs . . . . . . 59

4 EUL-structure 62

5 A relationship between EUL-diagrams and Venn diagrams 695.1 Syntax and semantics of Venn diagrams . . . . . . . . . . . . 695.2 Transformation of EUL-diagrams to Venn diagrams . . . . . . 71

4

6 Normal form of diagrammatic proofs of GDS and syllogisms 756.1 Syllogistic diagrams . . . . . . . . . . . . . . . . . . . . . . . 756.2 Syllogistic normal diagrammatic proofs and chains of Aris-

totelian categorical syllogisms . . . . . . . . . . . . . . . . . . 77

7 Some extensions and future work for Part I 82

II Focusing proofs in polarized linear logic 86

8 Introduction to Part II 88

9 Preliminary 1: Syntax of linear logic and polarized linearlogic 929.1 Second order linear logic LL2 . . . . . . . . . . . . . . . . . . 929.2 Second order polarized linear logic . . . . . . . . . . . . . . . 95

9.2.1 Second order polarized linear logic LLpol2 . . . . . . . 959.2.2 Second order multiplicative additive polarized linear

logic MALLP2 . . . . . . . . . . . . . . . . . . . . . . . 989.3 Focalization and polarization . . . . . . . . . . . . . . . . . . 99

9.3.1 Focalization of linear logic LL into focalized sequentcalculus LLfoc . . . . . . . . . . . . . . . . . . . . . . . 100

9.3.2 Polarization of linear logic LL into polarized linearlogic with shiftings LL↑↓

pol . . . . . . . . . . . . . . . . . 102

9.3.3 LLfoc and LL↑↓pol are almost isomorphic . . . . . . . . . 107

10 Preliminary 2: Categories for phase semantics of linear logic11310.1 ∗-autonomous category with products and phase space for

MALL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11410.2 Monad and comonad . . . . . . . . . . . . . . . . . . . . . . . 11710.3 Seely category and enriched phase space for LL . . . . . . . . 11910.4 Polarized ∗-autonomous category . . . . . . . . . . . . . . . . 121

11 A phase semantics for polarized linear logic 12411.1 Polarized phase space for MALLP . . . . . . . . . . . . . . . . 12511.2 Second order polarized phase model for MALLP2 and com-

pleteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13011.3 Enriched polarized phase space for LLpol . . . . . . . . . . . . 13411.4 Second order polarized phase model for LLpol2 and completeness135

5

11.5 An application of polarized phase semantics: First order con-servativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

12 Second order conservativity of linear logic over its polarizedfragment 13812.1 Second order conservativity . . . . . . . . . . . . . . . . . . . 139

12.1.1 LL2 is not conservative over LLpol2 . . . . . . . . . . . 13912.1.2 Second order η-expanded system LLη2 . . . . . . . . . 14012.1.3 Main proposition for LLη2: Polarization . . . . . . . . 14212.1.4 Additives . . . . . . . . . . . . . . . . . . . . . . . . . 14912.1.5 Weakening and contraction . . . . . . . . . . . . . . . 15012.1.6 LLη2 is conservative over LLη

pol2 . . . . . . . . . . . . . 15112.1.7 Some syntactical properties derived from Theorem 12.1.16151

12.2 Second order definability of restricted additives in polarizedlinear logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

13 Future work for Part II 154

Bibliography for Part I 156

Bibliography for Part II 162

6

Introduction

The purpose of this thesis is to conduct a proof-theoretical investigation intoconstructions of diagrammatic proofs and focusing proofs.

Part I of this thesis is concerned with Euler diagrammatic reasoning.Proof-theory has traditionally been developed based on linguistic (symbolic)representations of logical proofs. Recently, however, logical reasoning basedon diagrammatic or graphical representations has been investigated by manylogicians. Euler diagrams were introduced in the 18th century by LeonhardEuler [1768]. But it is quite recent (more precisely, in the 1990s) that logi-cians started to study them from a formal logical viewpoint, and there arestill only few proof-theoretical investigations that have been done. Accord-ingly, in order to fill this gap, we formalize an Euler diagrammatic inferencesystem and prove the soundness and completeness theorems with respectto a formal set-theoretical semantics. We further investigate structure andmanners of constructing diagrammatic proofs from a proof-theoretical view-point. We also introduce a notion of normal diagrammatic proof, whichcharacterizes basic logical reasoning such as Aristotelian categorical syllo-gisms.

In Part II of this thesis, we consider the construction of focusing proofswithin the framework of linear logical proof-theory. Linear logical struc-ture is often considered a refined basic logical structure for the traditionallogics such as classical logic and intuitionistic logic. In the study of linearlogical proof-theory, Jean-Marc Andreoli [1992] introduced focusing proofsto make proof-search or proof-construction effective. Focusing proofs definea complete subclass of the usual normal proofs of linear logic, where manyirrelevant choices in searching normal proofs are eliminated. In the study onfocusing proofs, the notion of polarity plays a central role. Since Jean-YvesGirard’s introduction of the notion of polarity in [Girard 1991], and sinceOlivier Laurent’s formalization of polarized linear logic in [Laurent 1999],polarity has emerged as an important parameter which controls linear logi-

8

cal proof-theory. In particular, it has been established that polarity providesa natural framework to construct focusing proofs for the first order linearlogic. However, it has been an open question whether polarity provides fo-cusing proofs for the second order linear logic. In this thesis we attempt toprovide an answer to this open question.

Euler diagrams were introduced by Leonhard Euler [1768] to representlogical relations among the terms of a syllogism by topological relations,inclusion and exclusion relations, among circles. For example, a categoricalstatement All A are B is represented by the inclusion relation between twocircles named A and B. Given two Euler diagrams which represent thepremises of a syllogism, the syllogistic inference can be naturally replaced bythe task of manipulating the diagrams, in particular of unifying the diagramsand extracting information from them.

Another well-known diagrammatic representation system for syllogisticreasoning is Venn diagrams, which were introduced by John Venn [1881] andmodified by Charles Peirce [1897] originally to overcome expressive limita-tions of Euler diagrams. In Venn diagrams a novel syntactic device, namelyshading, to represent emptiness plays a central role in place of the topolog-ical relations of Euler diagrams. Any claim in Venn diagrams is expressedin terms of shading, i.e., logical negation. For example, All A are B is ex-pressed by a Venn diagram through a translation to the statement There isnothing which is A but not B. (See Fig.1.2 of Chapter 1.)

Because of its expressive power and its uniformity in formalizing infer-ence rules, Venn diagrams have been studied thoroughly; formal semanticsand inference systems are given, and basic logical properties such as sound-ness, completeness, and decidability are shown. For a historical review, see[Hammer-Shin 1998], and for recent surveys, see [Stapleton 2005, Howse2008]. However, the development of systems of Venn diagrams is obtainedat the cost of clarity of the representations of Euler diagrams: In Venn dia-grams, logical relations among terms are represented not simply by topolog-ical relations, but by making use of shadings, which makes the translationsof categorical sentences, particularly those with several negations, uncom-fortably complex.

In contrast to the studies in the tradition of Venn diagrams, we intro-duce a diagrammatic representation system EUL for Euler circles from thefollowing standpoint:

• Our diagrammatic syntax and semantics are defined in terms of topo-logical relations between two diagrammatic objects (without using

9

shadings as in Venn diagrams). This formalization makes the transla-tions of categorical sentences natural and intuitive.

• In order to characterize the most basic diagrammatic system, we avoidintroducing auxiliary syntactic devices such as disjunctive linkings (be-tween points and between diagrams) which might require arbitraryconventions. Although our basic system EUL is weaker in its expres-sive power than usual Venn diagrammatic systems (e.g. Venn-II sys-tem of [Shin 1994] which is equivalent to the monadic first order logicin its expressive power), EUL is expressive enough to characterize basiclogical reasoning such as syllogistic reasoning.

Based on the representation system EUL, we formalize a diagrammaticinference system GDS, which consists of two kinds of inference rules: uni-fication and deletion. In order to keep our diagrams free from disjunctiveambiguity and from representation of conflicting graphical information ina single diagram, we impose certain constraints on unification rules. Wedefine the notion of diagrammatic proof, which is considered as a chain ofunification and deletion steps. The inference system GDS is shown to besound (Theorem 3.3.1) and complete (Theorem 3.3.10) with respect to ourformal set-theoretical semantics.

Our proof of completeness of GDS not only implies provability of anyvalid diagram, but also provides a canonical way of constructing a diagram-matic proof for the diagram. Roughly speaking, the canonical diagrammaticproof consists of the following two steps: Given premise diagrams,

1. first decompose the premise diagrams into minimal diagrams, whichconstitute general diagrams;

2. then construct the conclusion diagram from the minimal diagrams,while avoiding disjunctive ambiguities.

An examination of the canonical construction of diagrammatic proofs interms of Gentzen’s natural deduction system suggests that the structure ofeach canonical diagrammatic proof essentially corresponds to the structureof a normal (linguistic) natural deduction proof.

We also introduce a notion of normal diagrammatic proof, where a uni-fication and a deletion appear alternately. It is shown that for any (lin-guistic) chain of valid patterns of Aristotelian categorical syllogisms thereis a corresponding normal diagrammatic proof in GDS (Proposition 6.2.3).Conversely, it is also shown that, in syllogistic fragment of GDS, if there isa diagrammatic proof (not necessarily in normal form) of a diagram which

10

corresponds to a categorical sentence, then there is a normal diagrammaticproof for the diagram (Proposition 6.2.4). Hence, the syllogistic fragmentof GDS completely characterizes the chains of Aristotelian categorical syllo-gisms.

In Gentzen’s sequent calculus formulation ([Gentzen 1935]) of logicalproofs, the most important normal form is the cut-free normal proof, whichprovides, for example, a basis for the study of proof-construction or proof-search. Jean-Marc Andreoli [1992] introduced another important normalform, called focusing proof in the study of linear logical proof-theory. Lin-ear logical structure is considered a refined basic logical structure for thetraditional logics such as classical logic and intuitionistic logic. The focus-ing proofs form a subclass of cut-free normal proofs, where many irrelevantchoices in searching a cut-free proof are eliminated. Andreoli classified linearlogical connectives into two groups: Asynchronous, or reversible, connectivesare those whose (right) introduction rule is reversible; and synchronous,or focalized, connectives are those whose (right) introduction rule is notreversible. Thus, in proof-search, the reversible formulas are decomposedimmediately when they appear in a sequent. Although the dual focalizedconnectives are not reversible, Andreoli observed that they can be treated asa cluster in proof-search: Once a focalized formula is selected, i.e., focused,it is successively decomposed up to reversible subformulas. Andreoli formal-ized the idea, focalization, in his Triadic sequent system [Andreoli 1992]. Hissystem is shown to be equivalent to the usual linear logic, and the focusingproofs defined for his system form a complete subset of proofs of linear logic.

It has been pointed out that the dual proof-theoretical properties re-versibility/focalization of linear logical connectives can be captured by thenotion of polarity. Polarity was invented by [Girard 1991] in his work onLC (Logique Classique), which is a refinement of Gentzen’s standard for-mulation of classical logic LK. In LC, disjunction and existential quantifierare divided in terms of positive/negative polarities, and, for hereditary pos-itive formulas, intuitionistic disjunctive and existential properties hold inthe classical logic framework. There is a link between polarity and focaliza-tion studied by [Danos-Joinet-Schellinx 1997] and clarified by [Laurent 1999,2002]. Laurent’s formalization of polarized linear logic, in effect, providesa framework to construct focusing proofs in terms of the focalized sequentproperty. The focalized sequent property of a linear logical system, which isnot necessarily a polarized system, means that if a sequent is provable withonly polarized formulas, especially in polarized linear logic, it contains at

11

most one positive formula, in which case we call the sequent focalized. Sincethe positive formula is always focused, each proof of polarized linear logicgives a focusing proof in Andreoli’s sense. Laurent [2002] shows a first orderconservativity theorem of linear logic LL over its polarized fragment LLpol.That is, if a (polarized) focalized sequent is provable in LL, then it is alsoprovable in LLpol. Since all the proofs of LLpol are automatically focusing,it follows that any focalized sequent is provable with a focusing proof inLL. Combined with the focalized sequent property of LL, the conservativityneatly captures a main idea underlying polarity in linear logic: the “polar-ity” restriction on formulas leads naturally to focusing proofs. Moreover,seen from a logic programming viewpoint (cf. [Miller 2004]), the conserva-tivity is also important since we have only to work with focusing proofs.

In his proof of first order conservativity, Laurent made essential use ofthe subformula property of LL, which ensures that if a focalized sequentis provable then it is provable with only polarized formulas. When we tryto extend conservativity to second order linear logic LL2, we immediatelyencounter a difficulty with the second order ∃-rule, which results in the lossof the subformula property. For this reason Laurent [2002] has left openthe question of whether or not the conservativity result can be extended tosecond order.

In order to answer the above question, we introduce a phase semanticsfor second order polarized linear logic. The main feature of our polarizedphase semantics is its employment of a topological structure, which accom-modates the positive/negative polarities as openness/closedness. This inter-pretation is an algebraic instance of the categorical construction developed in[Hamano-Scott 2007] and is based upon the adjunction between interior andclosure operators for the topology. To the best of our knowledge, no formu-lation of phase semantics to completely characterize provability of polarizedlinear logic has previously appeared in the literature. We prove strong com-pleteness of polarized linear logic by revising Okada’s [1999] method, whichimplies second order cut-elimination.

Then we first show by using a counter model construction that LL2 is notconservative over LLpol2 (Proposition 12.1.1), which is a rather unexpectedby-product of our polarized phase semantics, and which is a “negative an-swer” to Laurent’s open question. With this result, it appears that LL2lacks the central idea of polarity in linear logic mentioned above, and thatit offers no bridge between polarity and focalization. In order to remedy thisshortcoming, we introduce an η-expanded fragment LLη2 of LL2, in whichatoms are exponential forms (i.e., !X⊥ (resp. ?X) for a positive (resp. neg-ative) atom). Such a slight restriction was also adopted by Laurent to show

12

a correspondence between polarized linear logic and Girard’s LC. Underthis slight restriction, the conservativity of LLη2 over its polarized fragmentLLη

pol2 (Theorem 12.1.16) is obtained, which is another “positive answer” toLaurent’s open question.

13

Part I

Diagrammatic proofswith Euler circles

14

Chapter 1

Introduction to Part I

Leonhard Euler [1768] introduced Euler diagrams to represent logical rela-tions among the terms of a syllogism by topological relations, inclusion andexclusion relations, among circles. Given two Euler diagrams which repre-sent the premises of a syllogism, the syllogistic inference can be naturallyreplaced by the task of manipulating the diagrams, in particular of unify-ing the diagrams and extracting information from them. For example, thewell-known syllogism named “Barbara,” i.e., All A are B and All B are C;therefore All A are C, can be represented diagrammatically as in Fig.1.1.

AB

D1 R

BC

D2

A

B

C

?

AC

E

Fig.1.1 Barbara with Euler diagrams

A B

Dv1 ?

B C

Dv2?

A B

CR

A B

C

A B

C?

A

CEv

Fig. 1.2 Barbara with Venn diagrams

Another well-known diagrammatic representation system for syllogisticreasoning is Venn diagrams. In Venn diagrams a novel syntactic device,namely shading, to represent emptiness plays a central role in place of the

16

topological relations of Euler diagrams. Because of their expressive powerand their uniformity in formalizing the manipulation of combining diagramssimply as the superposition of shadings, Venn diagrams have been very wellstudied. Cf. Venn-I, II systems of Shin [Shin 1994], Spider diagrams SD1and SD2 of [Howse-Molina-Taylor 2000], [Molina 2001], etc. For a recentsurvey, see [Stapleton 2005]. However, the development of systems of Venndiagrams is obtained at the cost of clarity of Euler diagrams. As Venn[Venn 1881] himself already pointed out, when more than three circles areinvolved, Venn diagrams fail in their main purpose of affording intuitiveand sensible illustration. (For some discussions on visual disadvantages ofVenn diagrams, see [Hammer-Shin 1998, Gil-Howse-Tulchinsky 2002]. Seealso [Sato-Mineshima-Takemura-Okada 2009] for our cognitive psychologicalexperiments comparing linguistic, Euler diagrammatic, and Venn diagram-matic representations.)

Recently, Euler diagrams with shading were introduced to make up forthe shortcoming of Venn diagrams: E.g., Euler/Venn diagrams of [Swoboda-Allwein 2004, 2005]; Spider diagrams ESD2 of [Molina 2001] and SD3 of[Howse-Stapleton-Taylor 2005]. However, their abstract syntax and seman-tics are still defined in terms of regions, where shaded regions of Venn dia-grams are considered as “missing” regions. That is, the idea of the region-based Euler diagrams is essentially along the same line as Venn diagrams.

We may point out the following complications of region-based formaliza-tion of diagrams:

1. In region-based diagrams, logical relations among circles are repre-sented not simply by topological relations, but by the use of shading ormissing regions, which makes the translations of categorical sentencesuncomfortably complex. For example, All A are B is expressed by aregion-based diagram through a translation to the statement There isnothing which is A but not B as seen in Dv

1 of Fig. 1.2.

2. The inference rule of unification, which plays a central role in Eu-ler diagrammatic reasoning, is defined by way of the superposition ofVenn diagrams. For example, when we unify two region-based Eulerdiagrams as in D1 and D2 of Fig. 1.1, they are first transformed intoVenn diagrams Dv

1 and Dv2 of Fig. 1.2, respectively; then, by superpos-

ing the shaded regions of Dv1 and Dv

2 , and by deleting the circle B, theVenn diagram Ev is obtained, which is transformed into the region-based Euler diagram E . In this way, processes of deriving conclusionsare often made complex, and hence less intuitive, in the region-basedframework.

17

In contrast to the studies in the tradition of region-based diagrams, weproposed a novel approach in [Mineshima-Okada-Sato-Takemura 2008] toformalize Euler diagrams in terms of topological relations. Our system hasthe following features and advantages:

1. Our diagrammatic syntax and semantics are defined in terms of topo-logical relations, inclusion and exclusion relations, between two dia-grammatic objects. This formalization makes the translations of cat-egorical sentences natural and intuitive. Furthermore, our formaliza-tion makes it possible to represent a diagram by a simple ordered (orgraph-theoretical) structure (cf. Chapter 4).

2. Our unification of two diagrams is formalized directly in terms of topo-logical relations without making a detour to Venn diagrams. Thus, itcan directly capture the inference process as illustrated in Fig. 1.1.We formalize the unification in the style of Gentzen’s natural de-duction, a well-known formalization of logical reasoning in symboliclogic, which is intended to be as close as possible to actual reason-ing ([Gentzen 1969]). This makes it possible to compare our Eulerdiagrammatic inference system directly with natural deduction sys-tem. Through such comparison, we can apply well-developed proof-theoretical approaches to diagrammatic reasoning.

From a perspective of proof-theory, the contrast between the standpointsof the region-based framework and the topological-relation-based frameworkcan be understood as follows: At the level of representation, the contrastis analogous to the one between disjunctive (dually, conjunctive) normalformulas and implicational formulas; at the level of reasoning, the contrastis analogous to the one between resolution calculus style proofs and naturaldeduction style proofs. The contrast between the two systems is summarizedin the following table:

Venn diagrams Our Euler diagramsRepresentation Region with shading Topological relation

Disjunctive normal formulas Implicational formulas¬∃x(A ∧ ¬B), ¬∃x(A ∧B) ∀x(A→ B), ∀x(A→ ¬B)

Reasoning Superposition of shadings UnificationResolution calculus style Natural deduction style

From a perspective of cognitive psychology, our system is designed notjust as an alternative of usual linguistic/symbolic representations; we make

18

the best use of advantages of diagrammatic representations so that inher-ent definiteness or specificity of diagrams can be exploited in actual rea-soning. See [Sato-Mineshima-Takemura-Okada 2009] for our experimentalresult, which shows that our Euler diagrams are more effective than Venndiagrams or linguistic representations in syllogism solving tasks.

The rest of this Part I is organized as follows. In Chapter 2, we introducea topological-relation-based Euler diagrammatic representation system EUL.We first roughly review previous work on Euler and Venn diagrammaticsystems, and we make clear our underlying conception for our formalizationin Section 2.1. We give a definition of an Euler diagrammatic syntax EULin Section 2.2 and a set-theoretical semantics for it in Section 2.3.

In Chapter 3, we formalize a diagrammatic inference system GDS. Weintroduce two kinds of inference rules: unification and deletion. We define inSection 3.2 the notion of diagrammatic proof (d-proof, in short), which is con-sidered as a (possibly long) chain of unification and deletion steps. The in-ference system GDS is shown in Section 3.3 to be sound (Theorem 3.3.1) andcomplete (Theorem 3.3.10) with respect to our formal set-theoretical seman-tics. In Section 3.4, we discuss some consequences of completeness of GDS.In particular, a class of ±-normal diagrammatic proofs of GDS is defined,and a normal form theorem (Theorem 3.4.3) of GDS is shown. Based onthe completeness and the normal form theorems, we give a proof-theoreticalanalysis on structure of diagrammatic proofs.

In Chapter 4, we give a graph theoretical representation of EUL-diagramsbased on a partially ordered structure, called an EUL-structure. Then, usingthe graphical representation, we discuss validity of our unification rules ofSection 3.2.

In Chapter 5, we investigate into a relationship between our EUL-diagramsand Venn diagrams. We first review informally syntax and semantics of Venndiagrams. Then we give a translation of each EUL-diagram into a semanti-cally equivalent Venn diagram.

In Chapter 6, we give a characterization of chains of Aristotelian cate-gorical syllogisms in our diagrammatic inference system GDS. For the char-acterization, an important subclass of the ±-normal diagrammatic proofs,called syllogistic normal diagrammatic proofs, are introduced. Then it isshown that each chain of Aristotelian categorical syllogisms corresponds toa syllogistic normal diagrammatic proof (Proposition 6.2.3). Furthermore,we also show that the syllogistic fragment of GDS completely characterizesthe chains of Aristotelian categorical syllogisms (Proposition 6.2.4).

Finally, in Chapter 7 we discuss some possible extensions of our system

19

and outline some future work.

20

Chapter 2

A diagrammaticrepresentation system (EUL)for Euler circles and itsset-theoretical semantics

In this chapter, we introduce a diagrammatic representation system, EUL.In order to characterize our system, we begin with a short review of Eulerand Venn diagrammatic representation systems in Section 2.1. Then weintroduce the syntax of EUL in Section 2.2, and its set-theoretical semanticsin Section 2.3.

2.1 Background

Euler diagrams were introduced by Leonhard Euler [1768] to illustrate syl-logistic reasoning. In Euler diagrams, logical relations among the terms ofa syllogism are simply represented by topological relations among circles. 1

With Euler diagrams the universal categorical statements of the forms AllA are B and No A are B are simply represented by the inclusion and theexclusion relations between circles, respectively, as follows:

AB

All A are B

A B

No A are B

1 Throughout this thesis, we mean by a circle a simple closed curve.

21

However, things become complicated when existential statements comeinto the picture. In Euler’s original system, any region in a diagram is as-sumed to represent a non-empty set, and this existential import destroysthe simple correspondence between Euler diagrams and categorical state-ments. For instance, the diagram of Fig.2.1 can be read as the followingfour categorical statements (1)–(4):

A B

Fig. 2.1

1. Some A are B,

2. Some B are A,

3. Some A are not B,

4. Some B are not A.

John Venn [1881] overcame this difficulty by removing the existentialimport from circles. Venn fixed such a diagram of Fig.2.1 as a so-called“primary diagram,” which does not convey any specific information aboutthe relation between A and B. Thus Venn diagrams can represent par-tial, not fully specified, information between circles. Meaningful relationsbetween circles are then expressed by specifying which regions are “empty”with the novel syntactic device of shading, which corresponds to logical nega-tion. Observe that All A are B is equivalent to There is nothing which isA but not B, and the statement is expressed as the following Venn diagramby making use of the shading:

A B

All A are B in Venn diagram(There is nothing which is A but not B)

In Venn diagrams, existential claims are expressed by using another syn-tactic device, “×,” which was introduced by Charles Peirce [1897], and whichrepresents non-emptiness of the corresponding region. Existential categor-ical statements of the forms Some A are B and Some A are not B arerepresented by using the symbol × as follows:

A×

B

Some A are B

A

×B

Some A are not B

22

In order to make Venn diagrams more expressive, Peirce further intro-duced another syntactic device: a linear symbol “−” which connects ×symbols to represent disjunctive information. 2 For example, the followingdiagram expresses that There is something A or B:

A B

× × ×

Based on Venn’s and Peirce’s work, Sun-Joo Shin [1994] formalized adiagrammatic reasoning system called the Venn-II system. The followingdevices were adopted, which with some modifications came to be regardedas the set of standard devices in subsequent studies:

1. Venn’s shading (for emptiness);

2. Primary diagrams (for non-specific information);

3. Peirce’s × (for non-emptiness);

4. Peirce’s linking between ×’s (for disjunctive information on objects);

5. Linking between diagrams (for disjunctive information on diagrams).

Furthermore, some options can also be considered:

6. Constant symbols (for existence of particular objects);

7. Linking between constants (for disjunctive information on particularobjects).

See, for example, [Stapleton 2005, Molina 2001] for surveys of various syn-tactic devices.

The Venn-II system has its own formal semantics and inference rules,and some basic logical properties such as soundness and completeness areproved. Furthermore, Venn-II system is shown equivalent to the monadicfirst order predicate logic in its expressive power. Because of its expressivepower and its uniformity in formalizing the inference rules, Venn diagramshave been very well-studied, and they have been developed into varioussystems, such as heterogeneous inference system, [Hammer 1994]; Spiderdiagrams SD1 and SD2, [Howse-Molina-Taylor 2000, Molina 2001]; etc. Forrecent surveys, see [Stapleton 2005, Howse 2008].

2Cf. [Hammer-Shin 1998] for the other device “o,” which Peirce introduced in place ofVenn’s shading.

23

Recently, Euler diagrams with shading were introduced: E.g., Euler/Venndiagrams of [Swoboda-Allwein 2004, Swoboda-Allwein 2005]; Spider diagramsESD2 of [Molina 2001] and SD3 of [Howse-Stapleton-Taylor 2005]. However,their abstract syntax and semantics are still defined in terms of regions,where shaded regions of Venn diagrams are considered as “missing” regions.That is, the idea of the region-based Euler diagrams is essentially along thesame line as Venn diagrams.

We may summarize the underlying conception in the literature of Venndiagrammatic systems as follows:

• Region-based formalization: Logical relations among terms are repre-sented by shading (or erasing) regions.

• Emphasis on expressive power: In order to make diagrams as expres-sive as possible, various syntactic devices are introduced.

In contrast to the studies in the tradition of Venn diagrams, we introduceour Euler diagrammatic system based on the following conception:

• Topological-relation-based formalization: Our diagrammatic syntaxand semantics are defined in terms of topological relations betweentwo diagrammatic objects (circles and points).

• Preservation of visual clarity of diagrams: In order to keep the inherentdefiniteness or specificity of diagrams, we avoid introducing auxiliarysyntactic devices such as shading and linking, which may require ar-bitrary conventions.

(See, for example, [Hammer-Shin 1998, Allwein-Barwise 1996] for somediscussion on the nature of diagrams.)

Based on the conception, we start our study on the relation-based Eulerdiagrammatic representation system EUL by concentrating on the followingbasic syntactic devices:

1. Inclusion and exclusion relations between two diagrammatic objects(circles and points).

2. Crossing relation between circles, which does not represent specificinformation between circles as it does in Venn diagrams.

3. Named points (constant symbols) to represent the existence of partic-ular objects.

24

Compared with Shin’s Venn-II system, our system lacks shading, linking be-tween points and linking between diagrams; and hence our system is weakerthan Venn-II in its expressive power. However, it is shown that EUL isexpressive enough to characterize syllogistic reasoning. Cf. Chapter 6.

2.2 Diagrammatic syntax of EUL

We introduce the diagrammatic syntax of EUL. Each EUL-diagram is definedas a set of named simple closed curves and named points in a plane. Wefurther consider some equivalence classes of concrete diagrams in terms oftopological binary relations, called EUL-relations, between pairs of diagram-matic objects.

Let us start by defining the diagrams of EUL.

Definition 2.2.1 (EUL-diagram) An EUL-diagram is a plane (R2) with afinite number, at least two, of named simple closed curves 3 (denoted byA,B, C, . . . ) and named points (denoted by a, b, c, . . . ), where

• each named simple closed curve and named point has exactly onename;

• any two distinct named simple closed curves and named points havedifferent names.

EUL-diagrams are denoted by D, E ,D1,D2, . . . .

In what follows, a named simple closed curve is sometimes called a namedcircle. Moreover, named circles and named points are collectively calledobjects, and denoted by s, t, u, . . . . We use a rectangle to represent a planefor an EUL-diagram. 4

3See [Blackett 1983, Stapleton-Rodges-Howse-Taylor 2007] for a formal definition ofsimple closed curve on R2.

4Several Euler diagrammatic representation systems impose some additional con-ditions for well-formed diagrams. E.g., at most two circles meet at a single point,no tangential meetings or concurrency etc. Cf. e.g., [Rodgers-Zhang-Fish 2008,Stapleton-Rodges-Howse-Taylor 2007]. However, for simplicity of the definition, thoseare all considered to be well-formed in our system EUL. (See also well-formed diagrams ofEUL in Fig.2.3 below.)

25

The following are typical examples of non well-formed diagrams of EUL:

D1 D2

•a

D3

A

D4

A B

D5

A A

D6

A •b•b

Fig. 2.2 Non well-formed diagrams of EUL

D1,D2 and D3 respectively consist of at most one object; a circle has twonames in D4; two distinct objects have the same name in D5 and D6.

Among EUL-diagrams, there are particularly simple diagrams which con-sist of only two objects:

Definition 2.2.2 (Minimal diagram) An EUL-diagram consisting of onlytwo objects is called a minimal diagram.Minimal diagrams are denoted by α, β, γ, . . . .

We study mathematical properties of EUL-diagrams in terms of the fol-lowing topological relations between two diagrammatic objects:

Definition 2.2.3 (EUL-relation) EUL-relations are the following binaryrelations between diagrammatic objects which have distinct names:

A @ B “the interior5of A is inside of the interior of B,”

A ⊢⊣ B “the interior of A is outside of the interior of B,”

A ◃▹ B “there is at least one crossing point between A and B,”

b @ A “b is inside of the interior of A,”

b ⊢⊣ A “b is outside of the interior of A,”

a ⊢⊣ b “a is outside of b (i.e. a is not located at the point of b).”

Observe that EUL-relations ⊢⊣ and ◃▹ are symmetric, while @ is not. Notealso that all EUL-relations are irreflexive.

Each of the EUL-relations is illustrated in the following EUL-diagrams ofFig.2.3:

5Here, the interior of a named circle A means the region strictly inside of A. Cf.[Blackett 1983].

26

AB A B A B A B A B BA A

B

A @ B A ⊢⊣ B A ◃▹ B

A

•bA

•bA •b •a •b

b @ A b ⊢⊣ A a ⊢⊣ b

Fig. 2.3 EUL-relations

Proposition 2.2.4 Let D be an EUL-diagram. For any distinct objects sand t of D, exactly one of the EUL-relations s @ t, t @ s, s ⊢⊣ t, s ◃▹ t holds.More precisely,

1. for any distinct named simple closed curves A and B, exactly one ofA @ B,B @ A,A ⊢⊣ B, and A ◃▹ B holds;

2. for any named point b and any named simple closed curve A, exactlyone of b @ A and b ⊢⊣ A holds;

3. for any distinct named points a and b, a ⊢⊣ b holds.

Observe that, by Proposition 2.2.4, for a given EUL-diagram D, the setof EUL-relations holding on D is uniquely determined. We denote the setby rel(D).

For example, consider the EUL-diagram D1 below, composed of namedcircles A,B, C, and a named point a.

A B•a

C

D1

The set of EUL-relations rel(D1) is {A ◃▹ B, A ◃▹ C, B ◃▹ C, a ⊢⊣ A, a @ B, a ⊢⊣ C}.

The following properties, as well as Proposition 2.2.4, characterize EUL-diagrams. (See also Chapter 4.)

27

Lemma 2.2.5 Let D be an EUL-diagram. Then for any objects (namedcircles or points) s, t, u ∈ ob(D), we have the following:

1. (Transitivity) If s @ t, t @ u ∈ rel(D), then s @ u ∈ rel(D).

2. (⊢⊣-downward closedness) If s ⊢⊣ t, u @ s ∈ rel(D), then u ⊢⊣ t ∈ rel(D).

3. (Point determinacy) For any point x of D, exactly one of x @ s andx ⊢⊣ s is in rel(D).

4. (Point minimality) For any point x of D, s @ x ∈ rel(D).

In order to develop our study on mathematical properties of our dia-grammatic system, it is convenient to talk about equivalence classes (ortypes) of diagrams rather than drawn tokens of diagrams. We first identifyobjects (named circles or points) which have the same name. For example, ifs is a circle named by A in one diagram and t is a circle also named by A inanother diagram, then s and t are identified up to topological isomorphism.Intuitively, the circles (resp. points) s and t are intended to represent thesame set (resp. element). For diagrams, we define their equivalence in termsof the EUL-relations:

Definition 2.2.6 (Equivalence of EUL-diagrams)

• When any two objects of the same name appear in different diagrams(planes), we identify them up to isomorphism.

• Any EUL-diagrams D and E such that ob(D) = ob(E) are syntacticallyequivalent when rel(D) = rel(E), that is, the following condition holds:For any objects s, t ∈ ob(D) and any ∗ ∈ {@,A,⊢⊣, ◃▹},

s ∗ t holds on D iff s ∗ t holds on E .

For example, the following diagramsD1, D2, andD3 of Fig.2.4 are equiva-lent since exactly the same EUL-relations A ◃▹ B, A ◃▹ C, B ◃▹ C, a ⊢⊣ A, a @B, and a ⊢⊣ C hold on them. (Cf. also Chapter 7 (1-1) for an extension ofour representation system EUL, where D1,D2, and D3 are distinguished.)On the other hand, D1 and D4 (resp. D1 and D5) are not equivalent sincedifferent EUL-relations hold on them: A @ C holds on D4 in place of A ◃▹ Cof D1 (resp. C @ A and C @ B hold on D5 in place of A ◃▹ C and C ◃▹ Bof D1).

Our equation of diagrams may be explained in terms of a kind of “contin-uous transformation (deformation)” of named circles, which does not change

28

A B•a

C

D1

A B•a

C

D2

A B•a

C

D3

A B•a

C

D4

A B•a

C

D5

Fig. 2.4 Equivalence of EUL-diagrams.

any of the EUL-relations in a diagram. The named circle C in D1 of Fig.2.4can be continuously transformed, without changing the EUL-relations withA, with B and with a in such a way that C covers the intersection regionof A and B as it does in D2. Similarly, C in D1 can be continuously trans-formed, without changing the EUL-relations with A, with B and with a insuch a way that C is disjoint from the intersection region of A and B as itis in D3.

In what follows, the diagrams which are syntactically equivalent are iden-tified, and they are referred by a single name.

We summarize our notations.

Named points: a, b, c, . . . (x, y, z, . . . for meta variables)Named circles: A,B,C, . . . (X, Y, Z, . . . for meta variables)Objects: s, t, u, . . .EUL-diagrams: D, E ,F , . . .Minimal diagrams: α, β, γ, . . .

When D is an EUL-diagram, we denote by pt(D) the set of named pointsof D; by cr(D) the set of named circles of D; by ob(D) the set of objects ofD; by rel(D) the set of EUL-relations holding on D.

2.3 Set-theoretical semantics of EUL

In this section, we give a formal semantics for EUL. Here, we adopt thestandard set-theoretical semantics. 6 Intuitively, each circle is interpretedas a set of elements of a given domain, and each point is interpreted as an

6For similar set-theoretical approaches to semantics of Euler dia-grams, see [Hammer 1995, Hammer-Shin 1998, Swoboda-Allwein 2004,Howse-Stapleton-Taylor 2005] etc. Our semantics is distinct from theirs in thatour diagrams are interpreted in terms of binary relations, and not every region in adiagram has a meaning.

29

element of the domain. However, observe that each point of EUL can beconsidered as a special circle which does not contain, nor cross, any otherobjects. This observation enables us to interpret the EUL-relations @ and ⊢⊣uniformly as the subset relation and the disjointness relation, respectively.

Definition 2.3.1 (Model) A model M is a pair (U, I), where U is a non-empty set (the domain of M), and I is an interpretation function whichassigns to each diagrammatic object s a non-empty subset of U such that

• I(x) is a singleton for any named point x;

• I(x) = I(y) for any points x, y of distinct names.

Definition 2.3.2 (Truth-condition) Let D be an EUL-diagram. M =(U, I) is a model of D, written as M |= D, if the following truth-conditions(1) and (2) hold: For all objects s, t of D,

(1) I(s) ⊆ I(t) if s @ t holds on D,

(2) I(s) ∩ I(t) = ∅ if s ⊢⊣ t holds on D.

Note that when s is a named point a, for some e ∈ U , I(a) = {e}, and theabove I(a) ⊆ I(t) of (1) is equivalent to e ∈ I(t). Similarly, I(a) ∩ I(t) = ∅of (2) is equivalent to e ∈ I(t).

Remark 2.3.3 (Semantic interpretation of ◃▹-relation) By Definition2.3.2, the EUL-relation ◃▹ does not contribute to the truth-condition of EUL-diagrams. Informally speaking, s ◃▹ t may be understood as I(s) ∩ I(t) = ∅or I(s) ∩ I(t) = ∅, which is true in any model.

The well-definedness of the truth-conditions in Definition 2.3.2 followsfrom Proposition 2.2.4, which ensures that the EUL-relations holding on agiven diagram D are uniquely determined.

Definition 2.3.4 (Validity) An EUL-diagram E is a semantically validconsequence of EUL-diagrams D1, . . . ,Dn, written as D1, . . . ,Dn |= E , whenthe following holds: For any model M , if M |= D1 and . . . and M |= Dn,then M |= E .

Let D be an EUL-diagram. Let β be a minimal diagram consisting oftwo objects s and t which is obtained from D by deleting all objects otherthan s and t. Then, by definition, we have D |= β. (See also Section 3.4.2.)

30

Chapter 3

Diagrammatic inferencesystem GDS

In this chapter, we introduce Generalized Diagrammatic Syllogistic inferencesystem GDS for the EUL-diagrams defined in Section 2.2. There are twoinference rules of GDS: unification and deletion. We first give an informalexplanation of our unification in Section 3.1, and we then formalize it inSection 3.2. We give an inductive definition of diagrammatic proofs of GDSas is usual in the study of symbolic logical systems. In Section 3.3 our GDS isshown to be sound and complete with respect to the set-theoretical semanticsgiven in Section 2.3. In Section 3.4, we discuss some consequences of thecompleteness theorem of GDS. In particular, we define a class of normaldiagrammatic proofs of GDS and we show a normal form theorem.

3.1 Introduction to unification

Before giving a formal description of our diagrammatic inference system,we motivate our inference rule unification. Let us consider the followingquestion: Given the following diagrams D1,D2 and D3, what diagrammaticinformation on A,B and c can be obtained? (In what follows, in order toavoid notational complexity in a diagram, we express each named point, say•c, simply by its name c.)

A

c

D1

B

c

D2

A

B

D3

31

A

c

D1

Bc

D2R

AB

D3

AcB

D1 + D2 R �

AB

c

(D1 + D2) + D3

Fig. 3.1

A

c

D1

Bc

D2

AB

D3z

AB

cc

D1 + D3W

AB

c

D2 + (D1 + D3)

Fig. 3.2

A

c

D1

Bc

D2

AB

D3s

AB

cc

D2 + D3U

AB

c

D1 + (D2 + D3)

Fig. 3.3

Figs.3.1, 3.2, and 3.3 represent the three ways of solving the question.In Fig.3.1, at the first step, two diagrams D1 and D2 are unified to obtainD1 +D2, where the point c in D1 and D2 are identified, and B is added toD1 so that c is inside of B and B overlaps with A without any implication ofa relationship between A and B. Then, D1 + D2 is combined with anotherdiagram D3 to obtain (D1 +D2)+D3. Note that the diagrams D1 +D2 andD3 share two circles A and B: A ◃▹ B holds on D1 +D2 and A @ B holds onD3. Since the semantic information of A @ B on D3 is more accurate thanthat of A ◃▹ B on D1 + D2, according to our semantics of EUL (recall thatA ◃▹ B means just “true” in our semantics), one keeps the relation A @ Bin the unified diagram (D1 + D2) + D3. Observe that the unified diagramrepresents the information of these diagrams D1,D2, and D3, that is, theirconjunction.

Figs.3.2 and 3.3, illustrate other procedures to solve the question. Atthe first step of unifying diagrams D1 and D3 in Fig.3.2 (and D2 and D3

in Fig.3.3), there are two possible positions of the point c. However, EUL-diagrams do not have syntactic devices to represent such disjunctive infor-mation about positions of a point. One solution to this problem is, as illus-trated in Figs.3.2 and 3.3, to introduce Peirce’s linking of points. However,following the conception we explained in Section 2.1, we keep our diagramsfree from such disjunctive ambiguity. For that purpose, we impose someconstraint on unification, called the constraint for determinacy: Any twodiagrams are not permitted to be unified when the relations between eachpoint and all circles of the two diagrams are not determined. Thus D1 andD3 of Fig.3.2 (respectively D2 and D3 of Fig.3.3) are not permitted to beunified.

We impose another constraint on unification called a constraint for con-sistency, in order to avoid complexity due to conflicting graphical informa-

32

tion represented in a single diagram. For example, it is not permitted tounify two diagrams D1 and D2 when, as is shown in Fig.3.4, they share twocircles C and B such that a @ C and a @ B hold on D1 and C ⊢⊣ B holdson D2. Note that these relations a @ C, a @ B, and C ⊢⊣ B are incompati-ble in the same diagram. The diagrams D3 and D4 in Fig.3.4 are also notpermitted to be unified in our system. Recall that each circle is interpretedby non-empty set in our semantics of Definition 2.3.1, and hence D3 and D4

are also incompatible.

Ca

B

D1

C B

D2

AB

D3

A B

D4

Fig. 3.4 Inconsistency

3.2 Generalized diagrammatic syllogistic inferencesystem GDS

In this section, we introduce unification and deletion of GDS. We formalizeour unification of two diagrams by restricting one of them to be a minimaldiagram, except for one rule called the Point Insertion-rule. Our completeness(Theorem 3.3.10) ensures that any diagrams D1, . . . ,Dn may be unified, un-der the constraints for determinacy and consistency, into one diagram whosesemantic information is equivalent to the conjunction of that of D1, . . . ,Dn.(We will return to this issue in Section 3.4.1.)

We give a formal description of inference rules in terms of EUL-relations:Given a diagram D and a minimal diagram α, the set of relations rel(D+α)for the unified diagram D + α is defined. It is easily checked that the setrel(D + α) satisfies the properties of Lemma 2.2.5, and hence there is aconcrete instance of the set. (See also Chapter 4, where we give a graph-ical representation of unification.) We also give a schematic diagrammaticrepresentation and a concrete example of each rule. In the schematic rep-resentation of diagrams, to indicate the occurrence of some objects in acontext on a diagram, we write the indicated objects explicitly and indicatethe context by “dots” as in the diagram to the right below. 1 For example,

1Note that the dots notation is used only for abbreviation of a given diagram. For aformal treatment of such “backgrounds” in a diagram, see, for example, [Meyer 2001].

33

when we need to indicate only A and c on the left hand diagram, we couldwrite it as shown on the right.

B F

A

ED

c

b

Ac

Definition 3.2.1 (Inference rules of GDS) Axiom, unification, and dele-tion of GDS are defined as follows.

Axiom:

A1: For any circles A and B, any minimal diagram where A ◃▹ B holds isan axiom.

A2: Any EUL-diagram which consists only of (at least two) points is anaxiom.

Unification: We denote by D+α the unified diagram of D with a minimaldiagram α. D + α is defined when D and α share one or two objects. Wedistinguish the following two cases: (I) When D and α share one object, theymay be unified to D+ α by rules U1–U8 according to the shared object andthe relation holding on α. Each rule of (I) has a constraint for determinacy.(II) When D and α share two circles, if the relation which holds on α alsoholds on D, D + α is D itself; otherwise, they may be unified to D + α byrules U9 or U10 according to the relation holding on α. Each rule of (II)has a constraint for consistency. Moreover, there is another unification rulecalled the Point Insertion-rule (III).(I) The case D and α share one object:

U1: If b @ A holds on α and pt(D) = {b}, then D and α may be unified toa diagram D + α such that the set rel(D + α) of relations holding onit is the following:

rel(D) ∪ {b @ A} ∪ {A ◃▹ X | X ∈ cr(D)}

U1 is applied as follows:

b

D R

A

b

αU1

Ab

D + α

C Bb

D1 R

A

b

D2U1

C

b

B

AD1 + D234

U2: If b ⊢⊣ A holds on α and pt(D) = {b}, then D and α may be unified toa diagram D + α such that the set rel(D + α) of relations holding onit is the following:

rel(D) ∪ {b ⊢⊣ A} ∪ {A ◃▹ X | X ∈ cr(D)}


b

D R

Ab

αU2

Ab

D + α

Bb

C

D1

Bb

CA

D1 + D2

Ab

R U2 D2

U3: If b @ A holds on α and A ∈ cr(D), and if A @ X or A ⊢⊣ X holds forall circle X in D, then D and α may be unified to a diagram D + αsuch that the set of relations rel(D + α) is the following:

rel(D) ∪ {b @ A} ∪ {b @ X | A @ X ∈ rel(D)}∪ {b ⊢⊣ X | A ⊢⊣ X ∈ rel(D)}∪ {b ⊢⊣ x | x ∈ pt(D)}


A

D R

A

b

αU3

Ab

D + α

A

BC

D1 R

A

b

U3 D2

Ab

BC

D1 + D2

U4: If b ⊢⊣ A holds on α and A ∈ cr(D), and if X @ A holds for all circle Xin D, then D and α may be unified to a diagram D + α such that theset of relations rel(D + α) is the following:

rel(D) ∪ {b ⊢⊣ A} ∪ {b ⊢⊣ X | X @ A ∈ rel(D)}∪ {b ⊢⊣ x | x ∈ pt(D)}


35

A

D R

Ab

αU4

Ab

D + α

B

A

D1 R

Ab

D2U4

B

Ab

D1 + D2

U5: If A @ B holds on α and B ∈ cr(D), and if x ⊢⊣ B holds for all x ∈ pt(D),then D and α may be unified to a diagram D+ α such that the set ofrelations rel(D + α) is the following:

rel(D) ∪ {A @ B} ∪ {A ◃▹ X | X @ B or X ◃▹ B ∈ rel(D)}∪ {A @ X | B @ X ∈ rel(D)}∪ {A ⊢⊣ X | X ⊢⊣ B ∈ rel(D)}∪ {x ⊢⊣ A | x ∈ pt(D)}


B

D R

AB

αU5

AB

D + α

C

B

EF

D1 R

A

B

U5 D2

A C

B

EF

D1 + D2

U6: If A @ B holds on α and A ∈ cr(D), and if x @ A holds for all x ∈ pt(D),then D and α may be unified to a diagram D+ α such that the set ofrelations rel(D + α) is the following:

rel(D) ∪ {A @ B} ∪ {X ◃▹ B | A @ X or A ⊢⊣ X or A ◃▹ X ∈ rel(D)}∪ {X @ B | X @ A ∈ rel(D)}∪ {x @ B | x ∈ pt(D)}


A

D R

AB

αU6

AB

D + α

CA

E

D1 R

A

B

U6 D2

EC

A

B

D1 + D236

U7: If A ⊢⊣ B holds on α and A ∈ cr(D), and if x @ A holds for all x ∈ pt(D),then D and α may be unified to a diagram D+ α such that the set ofrelations rel(D + α) is the following:

rel(D) ∪ {A ⊢⊣ B} ∪ {X ◃▹ B | A @ X or A ⊢⊣ X or A ◃▹ X ∈ rel(D)}∪ {X ⊢⊣ B | X @ A ∈ rel(D)}∪ {x ⊢⊣ B | x ∈ pt(D)}


A

D R

A B

αU7

A B

D + α

Aa

CE

D1

A B

R U7 D2

Aa

C

E

B

D1 + D2

U8: If A ◃▹ B holds on α and A ∈ cr(D), and if pt(D) = ∅, then D andα may be unified to a diagram D + α such that the set of relationsrel(D + α) is the following:

rel(D) ∪ {X ◃▹ B | X ∈ cr(D)}


A

D R

A B

αU8

A B

D + α

C

AE

D1 R

A B

U8 D2

CA

E

B

D1 + D2

(II) When D and α share two circles, they may be unified to D + α by thefollowing U9 and U10 rules.

U9: If A @ B holds on α and A ◃▹ B holds on D, and if there is no object ssuch that s @ A and s ⊢⊣ B hold on D, then D and α may be unifiedto a diagram D + α such that the set of relations rel(D + α) is the

37

following:(rel(D) \ {A ◃▹ B} \ {A ◃▹ X | B @ X ∈ rel(D)} \ {A ◃▹ X | B ⊢⊣ X ∈ rel(D)}

\ {X ◃▹ B | X @ A ∈ rel(D)} \ {Y ◃▹ X | Y @ A and B @ X ∈ rel(D)}

\ {X ◃▹ Y | X @ A and Y ⊢⊣ B ∈ rel(D)})

∪ {A @ B} ∪ {A @ X | B @ X ∈ rel(D)} ∪ {A ⊢⊣ X | B ⊢⊣ X ∈ rel(D)}∪ {X @ B | X @ A ∈ rel(D)} ∪ {Y @ X | Y @ A and B @ X ∈ rel(D)}∪ {X ⊢⊣ Y | X @ A and Y ⊢⊣ B ∈ rel(D)}


A B

RD

AB

U9α

AB

D + α

A BCE

D1 R

A

B

U9 D2

AB

CE

D1 + D2

U10: If A ⊢⊣ B holds on α and A ◃▹ B holds on D, and if there is no objects such that s @ A and s @ B hold on D, then D and α may be unifiedto a diagram D + α such that the set of relations rel(D + α) is thefollowing:(

rel(D) \ {A ◃▹ B} \ {X ◃▹ B | X @ A ∈ rel(D)} \ {X ◃▹ A | X @ B ∈ rel(D)}

\ {X ◃▹ Y | X @ A and Y @ B ∈ rel(D)})

∪ {A ⊢⊣ B} ∪ {X ⊢⊣ B | X @ A ∈ rel(D)} ∪ {X ⊢⊣ A | X @ B ∈ rel(D)}∪ {X ⊢⊣ Y | X @ A and Y @ B ∈ rel(D)}


A B

RD

A B

U10 α

A B

D + α

A B

C FE

U10D1 R

A B

D2

A

CE

B

F

D1 + D2

38

(III) Point Insertion: If, for any circles X,Y and for any � ∈ {@,A,⊢⊣, ◃▹},X�Y ∈ rel(D1) iff X�Y ∈ rel(D2) holds, and if pt(D2) is a singleton {b}such that b ∈ pt(D1), then D1 and D2 may be unified to a diagram D1 +D2

such that the set of relations rel(D1 +D2) is the following:

rel(D1) ∪ rel(D2) ∪ {b ⊢⊣ x | x ∈ pt(D1)}

Point Insertion is applied as follows:

Aa

c

C

B

A bC

B

D1 D2R

Aa

cb

CB

D1 + D2

Deletion: When t is an object of D, t may be deleted from D to obtain adiagram D − t under the constraint that D − t has at least two objects.

It is easily checked that unification preserves EUL-relations other thans ◃▹ t of unified diagrams:

Lemma 3.2.2 (Preservation of EUL-relations) Let D + α be an EUL-diagram obtained by an application of unification rules between D and α.For any relation s�t with � ∈ {@, A,⊢⊣}, if s�t ∈ rel(D) ∪ rel(α), thens�t ∈ rel(D + α).

Remark 3.2.3 (Inference rules of other systems) One of the distinc-tive features of our inference system GDS is that inference rules consist oftwo kinds of rules: unification and deletion. In the literature of region-based diagrammatic reasoning systems (Venn diagrammatic systems andEuler diagrammatic systems with shading), various inference rules are pro-posed. Among those rules, well-known rules for diagrams without shadingor linking of points are the following: (R1) Introduction of a circle; (R2)Erasure of a diagrammatic object; (R3) Combining of two diagrams; (R4)Rearrangement of circles (Weakening). See [Stapleton 2005] for a survey,and [Howse-Stapleton-Taylor 2005] for a formalization of these rules. Al-though they are defined in terms of regions, they can be simulated in GDSas follows. (R1) and (R2) are simulated by our U8 and Deletion rules, re-spectively. (R3) is a generalization of our Point Insertion rule in that more

39

than one point can be inserted, and it can be simulated by repeated appli-cations of Deletion and Point Insertion rules. (R4) is also simulated by ourunification and deletion, and we give a general procedure for it after ourproof of completeness in Section 3.3.

We give an inductive definition of diagrammatic proofs (d-proofs) ofGDS.

Definition 3.2.4 (Diagrammatic proofs of GDS) A diagrammatic proof(or d-proof) π of GDS is defined inductively as follows:

1. A diagram D is a d-proof from the premise D to the conclusion D.

2. Let π1 be a d-proof from D1, . . . ,Dn to F and π2 be a d-proof fromE1, . . . , Em to E , respectively. If D is obtained by an application ofunification of F and E , then the following (i) is a d-proof π fromD1, . . . ,Dn, E1, . . . , Em to D in GDS:

π1

FR

π2

ED

(i)

3. Let π1 be a d-proof from D1, . . . ,Dn to E . If D is obtained by anapplication of Deletion to E , then the following (ii) is a d-proof π fromD1, . . . ,Dn to D in GDS:

π1

E?D

(ii)

HereπD means a d-proof π with D as the conclusion. The length of a d-proof

is defined as the number of applications of inference rules.

We sometimes denote by−→D a sequence D1, . . . ,Dn of EUL-diagrams,

where by a sequence, we mean a set of diagrams {D1, . . . ,Dn}, i.e., morethan one Di may be identified for each 1 ≤ i ≤ n. For instance, sequencesD1,D1,D1,D2,D2,D3 and D1,D2,D3 may be identified.

Definition 3.2.5 (Provability) Let−→D be a sequence of EUL-diagrams.

An EUL-diagram E is provable from−→D , written as

−→D ⊢ E , if there is a

d-proof of E in GDS from a sequence D1, . . . ,Dm which is a subset of−→D .

40

The following Fig.3.5 is an example of a d-proof from D1,D2,D4,D6,D7

to D11. Fig.3.6 represents the underlying tree structure of the d-proof ofFig.3.5, where each edge is labeled by a diagram (we indicate only thepremises and the conclusion explicitly in Fig.3.6) and � nodes (resp. ⃝nodes) represent applications of unification (resp. deletion).

Cb

D1

Bb

D2R U1

Cb

B

D3

A B

D4R U7

A C Bb

D5

C

D

D6

DB

D7R U6/U5

C

D

B

D8?Deletion

C

B

D9�R U9

ACB

b

D10?Deletion

ACb

D11

Fig. 3.5 D-proof in GDS.

D1 D2

�Unification D4

�Unification Deletion

�Unification

D6 D7

�Unification

Deletion

D11

Fig. 3.6 The underlying tree struc-ture of the d-proof of Fig.3.5.

3.3 Soundness and completeness of GDS

One of the main goals of logical analysis of an inference system L is toestablish soundness and completeness with respect to a formal semantics ofL. In particular, the completeness theorem of L is formulated as follows:If a formula E is a semantically valid consequence of formulas D1, . . . ,Dn

then there is a proof of E from D1, . . . ,Dn in L. For the completeness ofthe diagrammatic system GDS, we assume the following condition calledsemantic consistency: There exists a model M for the premises D1, . . . ,Dn

(i.e., M |= Di for any 1 ≤ i ≤ n). Without this condition, any diagram,say E where A ⊢⊣ C holds, is a valid consequence of an inconsistent set ofpremise diagrams D1 and D2 where a @ B and a ⊢⊣ B hold, respectively,although there is no d-proof of E from D1 and D2 in GDS. 2

2 In place of our syntactic constraint, it is possible to allow unification of in-consistent diagrams by extending GDS with an inference rule corresponding to theabsurdity rule of Gentzen’s natural deduction system: We can infer any diagramfrom a pair of inconsistent diagrams. (For natural deduction systems, see, for ex-

41

Under the above assumption, we prove the completeness theorem of GDS(Theorem 3.3.10) with respect to our formal semantics of Section 2.3. Basedon the semantics of EUL, the validity of any EUL-diagram E is determined bythe set of relations holding on it. Hence, the premise D1, . . . ,Dn |= E of thecompleteness is equivalent to saying that D1, . . . ,Dn |= β for any minimaldiagram β which corresponds to the relation holding on E . Thus we firstshow atomic completeness (Proposition 3.3.9), which states that, for anyminimal diagram β, if D1, . . . ,Dn |= β then β is provable from D1, . . . ,Dn

in GDS under the assumption of semantic consistency of D1, . . . ,Dn. Thenusing such provable minimal diagrams, we give a canonical way to constructa d-proof of E .

In order to show the atomic completeness, we construct syntactic mod-els, whose domain consists of diagrammatic objects, i.e., named points andcircles. The interpretation function is defined based on the provability of di-agrams in GDS so that the validity of any minimal diagram in such a modelimplies the provability of the minimal diagram in GDS. Recall that eachnamed point can be regarded as a special circle which does not contain, norcross, any other object. Then the set of relations rel(D) of an EUL-diagram Dcan be seen as an algebraic structure with @-relations as ordering relations.(See Chapter 4 for a detailed description of the algebraic structure.) Thusour canonical models are constructed in a similar way to the usual syntacticmodel, the Lindenbaum algebra, in the literature of algebraic semantics forvarious propositional logics. Each interpretation of a named circle or a pointis, like a proposition, essentially defined as a downward closed set of objects(named points and circles) with respect to the provability of @-relation forsome fixed context α, i.e., I(A) = {s | α ⊢ s @ A} ∪ {A}. Unfortunately,such a model is too weak to show the provability of a minimal diagram ofthe form A ⊢⊣ B from the validity of it in the model, and we only knowA ⊢⊣ B or A ◃▹ B is provable. That is, such a model is too coarse to dis-tinguish between the relations ⊢⊣ and ◃▹. If we modify the interpretation ofA as I ′(A) = {s | α ⊢ s @ A} ∪ {A} ∪ {s | α ⊢ s�A} for � ∈ {@, A,⊢⊣},then, in turn, the validity of A @ B does not imply its provability. Thisis because we do not assume the negation ¬ to be a primitive operation inour syntax of EUL, nor the complementation operator in our semantics. Our

ample, [Gentzen 1969, Prawitz 1965].) Such a rule is introduced in, for example,[Howse-Stapleton-Taylor 2005] for spider diagrams; [Hammer-Danner 1993] for Venn di-agrams; [Swoboda-Allwein 2004, Swoboda-Allwein 2005] for Euler/Venn diagrams. How-ever, such a rule requires linguistic symbol, say ⊥, or some arbitrary convention to rep-resent inconsistency, and hence we prefer our syntactic constraint in our framework of adiagrammatic inference system.

42

solution is to construct a number of canonical models depending on the formof conclusion β of the completeness, i.e., we construct a canonical model fora minimal diagram β of the form A @ B by using the interpretation I(A)as above, and we construct another canonical model for a minimal diagramβ of the form A ⊢⊣ B by using I ′(A) as above. Then we obtain the atomiccompleteness.

In what follows, we sometimes refer to any minimal diagram, say α wheres @ t holds, by the EUL-relation holding on it, as s @ t. Note that thisconvention is harmless since there is a trivial one-to-one correspondence be-tween the minimal diagrams and the EUL-relations under the identificationof diagrams defined in Definition 2.2.6 of Section 2.2.

We first show the soundness theorem of GDS with respect to our formalsemantics of EUL:

Theorem 3.3.1 (Soundness of GDS) Let D1, . . . ,Dn, E be EUL-diagrams.

If D1, . . . ,Dn ⊢ E in GDS, then D1, . . . ,Dn |= E .

Proof. By induction on the length of the d-proof from D1, . . . ,Dn to E . It issufficient to show that each rule of U1–U10, Point Insertion, and Deletion issound in the sense that if each premise of the rule is true in a model, thenthe conclusion is also true in the model. We prove it only for U3 and U10rules since the other rules are proved similarly. Let M = (U, I) be a model.

U3 Let M |= D and M |= b @ A. We show M |= D + (b @ A). We dividethe following cases according to the relation holding on D + (b @ A):

1. When R ∈ rel(D) ∪ rel(b @ A) such that R = s @ t, we haveI(s) ⊆ I(t) by the induction hypothesis. The case that R is s ⊢⊣ tis similar.

2. When R ∈ {b @ X | A @ X ∈ rel(D)}, we have I(b) ⊆ I(X) bythe induction hypotheses I(b) ⊆ I(A) and I(A) ⊆ I(X).

3. The case R ∈ {b ⊢⊣ X | A ⊢⊣ X ∈ rel(D)} is similar.4. When R ∈ {b ⊢⊣ x | x ∈ pt(D)}, we have I(b) = I(x) for any

x ∈ pt(D) by definition.

U10 Let M |= D and M |= A ⊢⊣ B. We show M |= D+(A ⊢⊣ B). We dividethe following cases according to the relation holding on D + (A ⊢⊣ B).

1. When R ∈ rel(D) ∪ rel(A ⊢⊣ B) such that R = s ⊢⊣ t, we haveI(s) ∩ I(t) = ∅ by the induction hypothesis. The case that R iss @ t is similar.

43

2. When R ∈ {X ⊢⊣ B | X @ A ∈ rel(D)}, we have I(X) ∩ I(B) = ∅by the induction hypotheses I(A) ∩ I(B) = ∅ and I(X) ⊆ I(A).

3. The case R ∈ {X ⊢⊣ A | X @ B ∈ rel(D)} is similar.4. When R ∈ {X ⊢⊣ Y | X @ A, Y @ B ∈ rel(D)}, we have I(X) ∩

I(Y ) = ∅ by the induction hypotheses I(A) ∩ I(B) = ∅, I(X) ⊆I(A), and I(Y ) ⊆ I(B).

For completeness, let us begin by defining the notion of semantic consis-tency:

Definition 3.3.2 (Semantic consistency) A sequence of diagramsD1, . . . ,Dn

is semantically consistent if there is a model M such that M |= Di for any1 ≤ i ≤ n.

When−→D is a sequence D1, . . . ,Dn of diagrams, we sometimes write M |=

−→D for the formula ∀1≤i≤n(M |= Di).It is obvious that the soundness theorem (Theorem 3.3.1) also holds

under the assumption of the semantic consistency of the premise diagrams.The following is an important consequence of semantic consistency:

Lemma 3.3.3 (Semantic consistency) Let α be a sequence α1, . . . , αn

of minimal diagrams which is semantically consistent. Then none of thefollowing holds in GDS for any objects s and t:

1. α ⊢ s @ t and α ⊢ s ⊢⊣ t.

2. There is an object u such that α ⊢ s ⊢⊣ t and α ⊢ u @ s and α ⊢ u @ t.

Proof. For (1), assume to the contrary that both α ⊢ s ⊢⊣ t and α ⊢ s @ thold. Since α is semantically consistent, there is a model M = (U, I) suchthat M |= α. Then from the soundness theorem, we have M |= s ⊢⊣ t andM |= s @ t, i.e., I(s) ∩ I(t) = ∅ and I(s) ⊆ I(t) in M . Note that, sinceI(s) = ∅ by definition, I(s) ⊆ I(t) implies that I(s) ∩ I(t) = ∅, which leadsto a contradiction.

(2) is similar.

In order to show the completeness theorem of GDS, we construct twokinds of syntactic models, called canonical models. We first define the sim-pler one.

Definition 3.3.4 (Canonical model Mα) Let α be a sequence α1, . . . , αn

of minimal diagrams which is semantically consistent. A syntactic modelMα = (Mα, Iα), called a canonical model, for α is defined as follows:

44

• The domain Mα is the set of diagrammatic objects (named circles andpoints):

Mα = {s | s is a diagrammatic object}.

• Iα is an interpretation function such that, for any object t,

Iα(t) = {s | α ⊢ s @ t in GDS} ∪ {t}.

Observe that in the above definition of Iα, when the object t is a point,say a, its interpretation Iα(a) is the singleton {a} since α ⊢ s @ a for anyobject s by soundness (Theorem 3.3.1).

Lemma 3.3.5 (Canonical model 1) Let α be a sequence α1, . . . , αn ofminimal diagrams which is semantically consistent. Then Mα is a model ofα.

Proof. We show that Mα |= αi for each αi ∈ α (1 ≤ i ≤ n). The caseαi = s ◃▹ t is trivial. Otherwise, we divide the following cases according tothe form of αi:

1. When αi ∈ α is s @ t, we have α ⊢ s @ t in GDS. We show Mα |= s @ t,i.e., Iα(s) ⊆ Iα(t). Let u ∈ Iα(s).

(a) When u ≡ s, we immediately have s ∈ Iα(t) by the fact α ⊢ s @ t.

(b) Otherwise, by the definition of Iα(s), we have α ⊢ u @ s. Bycomposing it with α ⊢ s @ t as seen in the following d-proof, wehave α ⊢ u @ t in GDS, that is, u ∈ Iα(t).

When u is a point:

su

R

st

U6/U3

st

u

?tu

When u is a circle:

us

R

st

U6/U5

ust

?

ut

2. When αi ∈ α is s ⊢⊣ t, we have α ⊢ s ⊢⊣ t in GDS. We show Mα |= s ⊢⊣ t,i.e., Iα(s)∩Iα(t) = ∅. When both s and t are points, the claim is trivial.Otherwise, assume to the contrary that some u ∈ Iα(s) ∩ Iα(t).

45

(a) When u ≡ s, we have s ∈ Iα(t), i.e., α ⊢ s @ t. This, togetherwith α ⊢ s ⊢⊣ t, is a contradiction by Lemma 3.3.3(1).

(b) The same applies to the case u ≡ t.

(c) Otherwise, s ≡ u ≡ t, and we have α ⊢ u @ s and α ⊢ u @ t bythe definition of Iα(s) and Iα(t). They contradict α ⊢ s ⊢⊣ t byLemma 3.3.3(2).

As an illustration of the canonical model, let us consider the followingexample.

Example 3.3.6 (Canonical model Mα) Let α be the following minimaldiagrams α1, α2, α3, α4:

A

a

α1

Ab

α2

A B

α3

B

c

α4

Observe that we have α ⊢ b @ B and α ⊢ b ⊢⊣ B. In such a case, we say thatthe point b is indeterminate with respect to the circle B. Let us constructa canonical model for the α. Following Definition 3.3.4, we define

Iα(A) = {A, a} and Iα(B) = {B, c}.

Note that the indeterminate point b w.r.t. B is not contained in the inter-pretation Iα(B) of B. With this interpretation, for any point x ∈ Iα(B),we have α ⊢ x @ B (i.e., for c ∈ Iα(B), α ⊢ c @ B). In general, validity of@-relation in the model Mα imply provability of @-relation.

However, x ∈ Iα(B), i.e., Iα(x) ∩ Iα(B) = ∅, does not necessarily implyα ⊢ x ⊢⊣ B; because we do not have α ⊢ b ⊢⊣ B, while b ∈ Iα(B) in the aboveexample. Thus, in the canonical model Mα of Definition 3.3.4, validity of⊢⊣-relation does not imply provability of ⊢⊣-relation, and hence the model isnot enough to establish completeness.

Let us try to modify the model Mα of Example 3.3.6 so that the in-determinate point b w.r.t. B is contained in the interpretation I ′α(B) ofB:

I ′α(A) = {A, a} and I ′α(B) = {B, c, b}.

This definition also provides a model of α, that is, all of α1, α2, α3, and α4

are true in the model. With this interpretation function I ′α, for any pointx ∈ I ′α(B), we have α ⊢ x ⊢⊣ B (i.e., for a ∈ I ′α(B), α ⊢ a ⊢⊣ B).

46

However, in this model, x ∈ I ′α(B) does not necessarily imply α ⊢ x @ B;because we do not have α ⊢ b @ B, while b ∈ I ′α(B).

Although the above two kinds of models alone are insufficient to establishcompleteness, we can obtain our completeness result in the following manner:we construct the model Mα of Definition 3.3.4 for validity of @-relation,which implies provability of @-relation, and the model Mα,B of the followingDefinition 3.3.7 for validity of ⊢⊣-relation, which implies provability of ⊢⊣-relation.

Let us construct the second syntactic model.

Definition 3.3.7 (Canonical model Mα,B) Let α be a sequence α1, . . . , αn

of minimal diagrams which is semantically consistent. Let B be a fixednamed circle. A canonical model Mα,B = (Mα,B, Iα,B) for α is defined asfollows:

• The domain Mα,B is the same set as Mα of Definition 3.3.4.

• Iα,B is an interpretation function such that

– for t ≡ B, or for any t such that α ⊢ B @ t in GDS,Iα,B(t) = Iα(t) ∪ {s | α ⊢ B @ s and α ⊢ s @ B and α ⊢ s ⊢⊣ B};

– for any t (≡ B) such that α ⊢ B @ t in GDS,Iα,B(t) = Iα(t).

As seen in Definition 3.3.4, observe that Iα,B(a) = {a} when a is a point.Note also that Iα,B(t) is equal to Iα(t) of Definition 3.3.4 when α ⊢ B @ t.

We sometimes writ α ⊢ s�t when none of α ⊢ s @ t, α ⊢ t @ s, and α ⊢s ⊢⊣ t holds. And in that case, we say that s (resp. t) is indeterminate withrespect to t (resp. s). Observe that our two kinds of models of Definition3.3.4 and 3.3.7 differ in their treatment of such indeterminate objects withrespect to a fixed circle.

Let us show that Mα,B is a model of α.

Lemma 3.3.8 (Canonical model 2) Let α be a sequence α1, . . . , αn ofminimal diagrams which is semantically consistent. Let B be a fixed namedcircle. Then Mα,B is a model of α.

Proof. We show that, for each αi ∈ α (1 ≤ i ≤ n), Mα,B |= αi. The caseαi = s ◃▹ t is trivial. Otherwise, we divide the following cases according tothe form of αi:

47

1. When αi ∈ α is s @ t, we have α ⊢ s @ t. We show Iα,B(s) ⊆ Iα,B(t).Let u ∈ Iα,B(s).

(a) When u ≡ s, by the fact α ⊢ s @ t, we immediately have s ∈Iα,B(t) by the definition of Iα,B(t).

(b) Otherwise (u ≡ s), we divide the following two cases accordingto s and B:(i) When s ≡ B or α ⊢ B @ s hold, by the definition of Iα,B(s),we have (i-1) α ⊢ u @ s or (i-2) α ⊢ u�B with � ∈ {@, A,⊢⊣}.(i-1) implies, together with α ⊢ s @ t, that α ⊢ u @ t, i.e.,u ∈ Iα,B(t) by the following d-proofs:

When u is a point:

su

R

st

U6/U3

st

u

?tu

When u is a circle:

us

R

st

U6/U5

ust

?

ut

For (i-2), α ⊢ B @ s and α ⊢ s @ t imply α ⊢ B @ t by thesame d-proof as above, where u is B. Hence, in conjunction withα ⊢ u�B, we have u ∈ Iα,B(t) by the definition of Iα,B(t).(ii) When α ⊢ B @ s, by the definition of Iα,B(s), we have α ⊢u @ s. Hence this case is the same as (i-1).

2. When αi ∈ α is B ⊢⊣ t, we have α ⊢ B ⊢⊣ t. Observe that it impliesα ⊢ B @ t since α is semantically consistent. Hence we have Iα,B(t) =Iα(t). In order to show Mα,B |= B ⊢⊣ t, assume to the contrary thatsome u ∈ Iα,B(B) ∩ Iα(t).

(a) The case u ≡ B is impossible since B ∈ Iα(t).

(b) When u ≡ t, we have t ∈ Iα,B(B). That is α ⊢ t @ B, whichcontradicts α ⊢ B ⊢⊣ t by Lemma 3.3.3(1).

(c) Otherwise (B ≡ u ≡ t), by the definitions of Iα,B(B) and Iα(t),there are the following two cases, each of which leads to a con-tradiction:

i. α ⊢ u @ B, and α ⊢ u @ t;

48

ii. α ⊢ u�B with � ∈ {@, A,⊢⊣}, and α ⊢ u @ t.

(i) contradicts α ⊢ B ⊢⊣ t by Lemma 3.3.3(2). For (ii), α ⊢ u @ tand α ⊢ B ⊢⊣ t imply, by the following d-proofs, that α ⊢ u ⊢⊣ B,which contradicts α ⊢ u�B:

When u is a point:

u

t

R

t B

U7/U3

t B

u

?B

u

When u is a circle:

ut

R

t B

U7/U5

t B

u

?u B

3. When αi ∈ α is s ⊢⊣ t with s ≡ B ≡ t, we have α ⊢ s ⊢⊣ t. We show thatIα,B(s)∩Iα,B(t) = ∅. When both s and t are points, the claim is trivial.Otherwise, assume to the contrary that some u ∈ Iα,B(s) ∩ Iα,B(t).

(a) When u ≡ s, we have s ∈ Iα,B(t). We divide the following twocases according to whether or not α ⊢ B @ t holds:(i) When α ⊢ B @ t holds, by the definition of Iα,B(t), we have(i-1) α ⊢ s @ t or (i-2) α ⊢ s�B with � ∈ {@, A,⊢⊣}. Case(i-1) contradicts α ⊢ s ⊢⊣ t by Lemma 3.3.3(1). For (i-2), fromα ⊢ s ⊢⊣ t and α ⊢ B @ t, we have, by the following d-proofs,α ⊢ s ⊢⊣ B, which contradicts α ⊢ s�B:

When s is a point:

ts

R

B

t

U5/U4

Bt

s

?B

s

When s is a circle:

t s

R

B

t

U5/U7

t sB

?B s

(ii) When α ⊢ B @ t, we have α ⊢ s @ t by the definition ofIα,B(t), which contradicts α ⊢ s ⊢⊣ t by Lemma 3.3.3(1).

(b) The same applies to the case u ≡ t.

(c) Otherwise (s ≡ u ≡ t), we divide the following cases:

49

i. α ⊢ B @ s and α ⊢ B @ t;ii. α ⊢ B @ s and α ⊢ B @ t;iii. α ⊢ B @ s and α ⊢ B @ t;iv. α ⊢ B @ s and α ⊢ B @ t.

(i) contradicts α ⊢ s ⊢⊣ t by Lemma 3.3.3(2). For (ii), by thedefinitions of Iα,B(s) and Iα,B(t), we have α ⊢ u @ s and α ⊢ u @t, which contradict α ⊢ s ⊢⊣ t by Lemma 3.3.3(2). For (iii), bythe definition of Iα,B(s), we have α ⊢ u @ s. By the definitionof Iα,B(t), we have (iii-1) α ⊢ u @ t or (iii-2) α ⊢ u�B with� ∈ {@, A,⊢⊣}. (iii-1), together with α ⊢ u @ s, contradictsα ⊢ s ⊢⊣ t. For (iii-2), α ⊢ u @ s, α ⊢ s ⊢⊣ t, and α ⊢ B @ t imply,by the following d-proofs, that α ⊢ u ⊢⊣ B, which contradictsα ⊢ u�B:

When u is a point:

u

s

R

s t

U7/U3

s tu

?t

u

q

B

t

)U5/U4

B

tu

?B

u

When u is a circle:

u

s

R

s t

U7/U5

s tu

?tu

q

B

t

)U5/U7

B

tu

?Bu

(iv) is similar to (iii).

Using the two kinds of canonical models introduced so far, we prove thefollowing atomic completeness, from which completeness (Theorem 3.3.10)of GDS is derived.

Proposition 3.3.9 (Atomic completeness) Let D1, . . . ,Dn be a sequenceof EUL-diagrams which is semantically consistent. Let β be a minimal dia-gram. We have:

If D1, . . . ,Dn |= β, then D1, . . . ,Dn ⊢ β in GDS.

50

Proof. We first consider the case where the premises D1, . . . ,Dn are re-stricted to minimal diagrams α1, . . . , αn. Then we extend to the generalcase. We denote by α the sequence α1, . . . , αn of given minimal diagrams.Assume α |= β. When β is s ◃▹ t, we immediately have α ⊢ s ◃▹ t since itis an axiom. Otherwise, we divide the following cases according to the formof β. We use the canonical model of Definition 3.3.4 for the case that β iss @ t, and we use the canonical model of Definition 3.3.7 for the case thatβ is s ⊢⊣ t.

1. When β is of the form s @ t, by the assumption α |= s @ t, we have,in particular for the canonical model of Definition 3.3.4, Mα |= α ⇒Mα |= s @ t. Then, since Mα is a model of α by Lemma 3.3.5, we haveMα |= s @ t, i.e., Iα(s) ⊆ Iα(t). Since s ∈ Iα(s) by Definition 3.3.4,we have s ∈ Iα(t), that is, α ⊢ s @ t in GDS.

2. When β is of the form s ⊢⊣ t, observe that if s and t are both points,then the assertion is trivial since β is an axiom in that case. Otherwise,we assume, without loss of generality, that t is a circle B. By theassumption α |= s ⊢⊣ B, we have, in particular for the canonical modelof Definition 3.3.7, Mα,B |= α⇒Mα,B |= s ⊢⊣ B. Then, since Mα,B isa model of α by Lemma 3.3.8, we have Mα,B |= s ⊢⊣ B, i.e., Iα,B(s) ∩Iα,B(B) = ∅. Hence we have s ∈ Iα,B(B) and B ∈ Iα,B(s). Thenby the definition of Iα,B(B) and Iα,B(s) of Definition 3.3.7, we haveα ⊢ s @ B, and α ⊢ B @ s and α ⊢ s�B for some � ∈ {@, A,⊢⊣}.This means that, for the objects s and B, s @ B or B @ s or s ⊢⊣ B isprovable from α in GDS, but it is neither s @ B nor B @ s. Therefore,we have α ⊢ s ⊢⊣ B in GDS.

Next, we extend the premises to general diagrams D1, . . . ,Dn insteadof minimal diagrams α. Let D1, . . . ,Dn |= β. Then, by the definition ofour semantics, it is equivalent to the fact that, for any model M , M |=α1 ∧ · · · ∧ M |= αn ⇒ M |= β, where αi is a sequence of all minimaldiagrams whose relations hold on Di. Thus there is a sequence α1, . . . , αk

of minimal diagrams such that each relation holding on αj (1 ≤ j ≤ k)holds on some Di (1 ≤ i ≤ n) and α1, . . . , αk |= β. Then there is a d-prooffrom α1, . . . , αk to β in GDS. Thus we have the following d-proof for β from

51

D1, . . . ,Dn, where ⇓ indicates some applications of the Deletion rule:

Di

⇓α1

Dj

⇓α2 . . .

Dl

⇓αk.. . .. . .. . .. . .. . ......

β

By extending the conclusion diagram β of atomic completeness to ageneral (not restricted to minimal) diagram E , we establish the completenessof GDS.

Theorem 3.3.10 (Completeness of GDS) Let D1, . . . ,Dn, E be EUL-diagrams.Let D1, . . . ,Dn be semantically consistent.

If D1, . . . ,Dn |= E , then D1, . . . ,Dn ⊢ E in GDS.

Proof. Using the atomic completeness theorem, we construct a d-proof of Efrom the given premise diagrams D1, . . . ,Dn in a canonical way, as follows(see also Example 3.3.11 given after this proof):

(I) From the premise diagrams D1, . . . ,Dn, by using atomic completenessand U1, U2-rules, we first construct EUL-diagrams, called Venn-likediagrams, each of which consists of a point and all circles of E , and ineach of which A ◃▹ B holds for any pair of circles of the diagram.

(II) Then, by unifying all Venn-like diagrams of (I) with the Point Insertionrule, we construct a Venn-like diagram consisting of all points andcircles of E .

(III) By using atomic completeness, we construct d-proofs for all point-freeminimal diagrams in each of which a relation A @ B or A ⊢⊣ B of Eholds.

(IV) We then construct a diagram F , by unifying the minimal diagrams of(III) and the Venn-like diagram of (II) with U9 and U10-rules.

(V) Finally, we check that the diagram F of (IV) coincides with the con-clusion E .

52

A diagrammatic proof is called a canonical diagrammatic proof when itis constructed in accordance with the above canonical construction.

We now formalize the above (I)–(V). We denote by−→D the sequence

D1, . . . ,Dn of the given premise diagrams. We first construct Venn-likediagrams each of which consists of a point of E and all circles of E , and ineach of which A ◃▹ B holds for any pair of circles of the diagram.(I) For each point a ∈ pt(E), let Pa be the set of relations holding on E eachof which consists of the point a and a circle of E: I.e., for � ∈ {@,⊢⊣},

Pa = {a�X | a�X ∈ rel(E)}.

Then the set Pa gives rise to an EUL-diagram Pa such that−→D ⊢ Pa in GDS.

Proof of (I). Let R1, R2, . . . , Rn be an enumeration of the elements of Pa,and β1, β2, . . . , βn be the sequence of corresponding minimal diagrams whereRi holds on βi for 1 ≤ i ≤ n. Note that all βi share the same point a andthey differ only in their circles. The assumption

−→D |= E of completeness

implies−→D |= βi since Ri ∈ rel(E). Hence we have

−→D ⊢ βi in GDS by atomic

completeness (Proposition 3.3.9). Then starting from β1, by successivelyapplying U1-rule (when βi is a @ Bi for 1 < i ≤ n) or U2-rule (when βi isa ⊢⊣ Bi for 1 < i ≤ n), we have a d-proof of Pa from

−→D in GDS as follows:−→D....β1

−→D....β2

R U1/U2

β1 + β2

−→D....β3

R U1/U2

(β1 + β2) + β3...Pa

�We next construct a d-proof of a Venn-like diagram which consists of all

points and circles of E .(II) Let {a1, . . . , am} = pt(E). Let P be the union of the relations of all Pai

(1 ≤ i ≤ m) of (I):P =

∪1≤i≤m Pai .

Then P gives rise to an EUL-diagram P such that−→D ⊢ P in GDS.

Proof of (II). Note that all diagrams Pai (1 ≤ i ≤ m) of (I) share the samecircles, where X ◃▹ Y holds for any X, Y ∈ cr(Pai). The diagrams Pai differ

53

only in their points. Hence by successively applying the Point Insertion rule,we have

−→D ⊢ P in GDS. �

Note that when E does not contain any point, the set∪

1≤i≤m Pai be-comes empty. In such a case, we construct a Venn-like diagram P (withoutany point) which consists of all circles of E . This is possible by successivelyapplying U8-rule to axioms of the form X ◃▹ Y for X, Y ∈ cr(E).

We construct d-proofs for all point-free minimal diagrams in each ofwhich a relation of E of the form A @ B or A ⊢⊣ B holds.(III) Let β be a minimal diagram such that A @ B or A ⊢⊣ B of rel(E) holds.Then we have

−→D ⊢ β in GDS.

Proof of (III). Since−→D |= E and rel(β) ⊆ rel(E), we have

−→D |= β. Then byatomic completeness (Proposition 3.3.9), we have

−→D ⊢ β in GDS. �

We finally construct a d-proof for the conclusion E by unifying minimaldiagrams of (III) and the Venn-like diagram P of (II).(IV) Let R1, . . . , Rl be all relations of the form A @ B or A ⊢⊣ B holding onE, and let β1, . . . , βl be the corresponding minimal diagrams, where Ri holdson βi for 1 ≤ i ≤ l. Let P be the set of relations of (II). Then the set

P ∪ {R1, . . . , Rl}

of relations gives rise to an EUL-diagram (· · · (P + β1) + · · · ) + βl such that−→D ⊢ (· · · (P + β1) + · · · ) + βl in GDS.Proof of (IV). By induction on l. Let P + Bl denote the diagram (· · · (P +β1) + · · · ) + βl. We show the induction step (l > 1) with the inductionhypothesis

−→D ⊢ P + Bl−1 in GDS, since the same applies to the base step(l = 1).(Induction step: l > 1) We divide the following two cases according to theform of βl: (1) A ⊢⊣ B holds on βl or (2) A @ B holds on βl.Case (1): Since cr(E) = cr(P+Bl−1) by the construction (II) of P and (III),we have A, B ∈ cr(P + Bl−1). We claim that A ◃▹ B or A ⊢⊣ B holds on thediagram P + Bl−1. Assume to the contrary that neither of them holds onP + Bl−1, that is, A @ B or B @ A holds by Proposition 2.2.4. If A @ B

holds on P + Bl−1, since−→D ⊢ P + Bl−1 by the induction hypothesis, we

have−→D |= P + Bl−1 by soundness of GDS (Theorem 3.3.1), which implies

that−→D |= A @ B. This contradicts the assumption that

−→D is semanticallyconsistent because we have

−→D |= A ⊢⊣ B from the assumptions−→D |= E and

54

Rl = A ⊢⊣ B ∈ rel(E). The same applies in case B @ A. Thus exactly one ofA ◃▹ B and A ⊢⊣ B holds on the diagram P + Bl−1.

Now we prove−→D ⊢ (P + Bl−1) + βl. When A ⊢⊣ B holds on P + Bl−1,

we obtain the assertion immediately by the induction hypothesis since (P +Bl−1) + βl is P + Bl−1 itself. When A ◃▹ B holds on P + Bl−1, by applyingU10-rule to βl and P + Bl−1, we have

−→D ⊢ (P + Bl−1) + βl in GDS. Theapplication of U10-rule is possible because there is no object s such thatboth s @ A and s @ B hold on P + Bl−1: If there were such an object s,since

−→D ⊢ P + Bl−1 by the induction hypothesis, we have−→D ⊢ s @ A and

−→D ⊢ s @ B by applying a series of Deletion. Then, using the soundnesstheorem, we would have

−→D |= s @ A and

−→D |= s @ B. This contradicts the

assumption that−→D is semantically consistent because we have

−→D |= A ⊢⊣ B.

Case (2) where A @ B holds on βl is similar. �

Thus constructed (· · · (P +β1)+ · · · )+βl is equivalent to the conclusionE .(V) For any EUL-relation R,

R ∈ rel((· · · (P + β1) + · · · ) + βl) iff R ∈ rel(E).

Proof of (V). We denote by P + Bl the diagram (· · · (P + β1) + · · · ) + βl.⇐) By the constructions (II) and (IV), all minimal diagrams of E are unifiedto obtain P + Bl. Hence by Lemma 3.2.2, we obtain rel(E) ⊆ rel(P + Bl).

⇒) Let R ∈ rel(P + Bl). We divide the following argument in two casesdepending on whether or not R is of the form s ◃▹ t:(1) When R = s ◃▹ t, assume to the contrary that s ◃▹ t ∈ rel(E). SinceE is a diagram, for some � ∈ {@,A,⊢⊣}, s�t ∈ rel(E) by Proposition 2.2.4.Then, by definition, for some j, βj is of the form s�t, which implies thats�t ∈ rel(P+Bl). This contradicts Proposition 2.2.4 for the diagram P+Bl,since s ◃▹ t ∈ rel(P + Bl) by the assumption.

(2) In case, R = s ◃▹ t, we show that R ∈ rel(P + Bl) ⇒ R ∈ rel(E) byinduction on l. We prove the induction step since the same applies to thebase step.(Induction step: l > 1) Assume to the contrary that R ∈ rel(E). Then,since rel(P) \ {X ◃▹ Y | X, Y ∈ cr(P)} ⊆ rel(E) by the construction (II),R should be a relation between circles (not points), and R = βi for anyi. Hence, there is some 1 ≤ i ≤ l such that R ∈ rel(P + Bi−1) but R ∈rel((P + Bi−1) + βi). We show the case (P + Bi−1) + βi is obtained by U10-rule. (The case of U9-rule is shown similarly.) Assume A ⊢⊣ B holds on βi.

55

By the definition of U10-rule, there are the following three cases accordingto the form of R: (i) R = X ⊢⊣ B such that X @ A ∈ rel(P + Bi−1); (ii)R = X ⊢⊣ A such that X @ B ∈ rel(P + Bi−1); (iii) R = X ⊢⊣ Y such thatX @ A, Y @ B ∈ rel(P+Bi−1). For case (i), by the induction hypothesis, wehave X @ A ∈ rel(E). Then, since A ⊢⊣ B ∈ rel(E), we have X ⊢⊣ B ∈ rel(E),contrary to the assumption R = X ⊢⊣ B ∈ rel(E). Similarly, cases (ii) and(iii) also lead to contradictions. Therefore, we have R ∈ rel(E). �

Example 3.3.11 (Canonical d-proof of GDS) As an illustration of thecanonical construction of d-proofs given in our proof of completeness (The-orem 3.3.10), let us consider the following diagrams D1,D2,D3, and E :

Aa

D1

,

Bab

D2

,

A B

D3

⊢A Ba

b

E

We have the following canonical d-proof of E from D1,D2,D3:

Ba

b

D2 ?B

b

A B

D3R B

ab

D2 ?

A B

b

?

Ba

b

D2 ?

Atomic

Completeness

Aa

D1

Ba

D4U2 R

A

b

D5

B

b

D6U1/U2�j

(I)A B

a

D7

A B

b

D8Point Insertionz 9

(II)A B

ab

D9

A B

D3U10z 9

(IV)A B

ab

E

We first derive, by using atomic completeness, all pointed minimal diagramsD1,D4,D5, and D6 each of which corresponds to an EUL-relation holding onthe conclusion E . Next, following the construction (I) with U1 and U2 rules,

56

we construct two Venn-like diagrams D7 and D8 each of which consists ofa point a (resp. b) and all circles A and B of E . Then, following theconstruction (II) with Point Insertion rule, we unify the Venn-like diagramsD7 and D8 to obtain a Venn-like diagram D9 consisting of all points a andb and all circles A and B of E . Finally, following the construction (IV) withU10 rule, we obtain the conclusion E .

3.4 Some consequences of completeness of GDS

In this section, we discuss some consequences of our completeness (Theorem3.3.10) of GDS.

3.4.1 Unification of any (two) diagrams

Let D1,D2 and E be EUL-diagrams such that for any model M , M |= E if andonly if M |= D1 and M |= D2, that is, E is semantically equivalent to theconjunction of D1 and D2. We may write such E as D1 +D2. Our complete-ness (Theorem 3.3.10) ensures thatD1,D2 ⊢ D1+D2 in GDS. This shows thatthe general notion of unification of two diagrams (cf. [Hammer-Shin 1998])is completely characterized by our formalization of unification of two dia-grams, where one of them is restricted to a minimal diagram.

3.4.2 Decomposition set of an EUL-diagram

Given an EUL-diagram D and two objects, say s and t, on D, a minimaldiagram obtained from D by deleting all objects other than s and t is calleda component minimal diagram of D. By Proposition 2.2.4, the set of compo-nent minimal diagrams of D is uniquely determined, and we call it the de-composition set (decomp(D)) of D. According to our semantics, decomp(D)and D are semantically equivalent in the following sense: for any model M ,for all α ∈ decomp(D), M |= α if and only if M |= D. Hence, by complete-ness (Theorem 3.3.10), decomp(D) and D are also provably equivalent.

In particular, the canonical construction of d-proofs of the completenessshows that any EUL-diagram D can be constructed from the componentminimal diagrams of D.

3.4.3 On normal diagrammatic proofs

In order to show a normal form theorem of GDS, we shall modify the seman-tic method introduced in our completeness proof, by applying a semantic

57

normal form proof for the linguistic proofs of e.g., [Okada 1999, Okada 2002].Let us define a class of normal diagrammatic proofs of GDS, called the

±-normal d-proofs:

Definition 3.4.1 (±-normal d-proofs) A d-proof π is in ±-normal formif a unification (+) and a deletion (−) appear alternately in π.

In Definition 3.3.4 and 3.3.7 of our canonical models, it is possible tomodify the interpretation of each object by restricting the provability witha ±-normal d-proof as follows:

• For Definition 3.3.4,

I ′α(t) = {s | α ⊢ s @ t with a ±-normal d-proof } ∪ {t}

• For Definition 3.3.7, for any t such that α ⊢ B @ t with a ±-normald-proof,

I ′α,B(t) = I ′α(t) ∪ {s | α ⊢ B @ s and α ⊢ s @ B and α ⊢ s ⊢⊣ B}

This slight modification of canonical models also enables us to prove theessential part of atomic completeness (Proposition 3.3.9) where the premisediagrams are restricted to minimal diagrams; because any d-proof appearingin our proof of Lemma 3.3.5 and 3.3.8 is in ±-normal form. Hence we obtainthe following version of atomic completeness:

Corollary 3.4.2 Let α be a sequence α1, . . . , αn of minimal diagrams whichis semantically consistent. Let β be a minimal diagram. Then we have:

If α |= β , then α ⊢ β in GDS with a ±-normal d-proof.

Then, together with soundness (Theorem 3.3.1) of GDS, we obtain thefollowing normal form theorem:

Theorem 3.4.3 (±-normal form for minimal diagrams) Let α be a se-quence α1, . . . , αn of minimal diagrams which is semantically consistent. Letβ be a minimal diagram. Then we have:

If α ⊢ β in GDS, then α ⊢ β in GDS with a ±-normal d-proof.

Proof. Let α ⊢ β in GDS. Then, by soundness (Theorem 3.3.1) of GDS, wehave α |= β, which implies that α ⊢ β in GDS with a ±-normal d-proof byCorollary 3.4.2.

58

This normal form plays an important role in characterizing chains ofAristotelian categorical syllogisms in GDS. See Chapter 6.

Although the above normal form theorem states only the existence ofnormal d-proofs, by defining a procedure to rewrite d-proofs, the theoremcan be extended to a normalization theorem of d-proofs: Any d-proof isrewritten into a ±-normal d-proof in a finite number of steps.

Observe that EUL-relations can be described linguistically by using usualfirst order formulas: A @ B is ∀x(A(x) → B(x)); a @ B is B(a); A ⊢⊣ Bis ∀x(A(x) → ¬B(x)); a ⊢⊣ B is ¬B(a). Then any EUL-diagram D canbe also described, through rel(D), as a conjunction of the above types offormulas. In that case, our diagrammatic inference rules are described interms of linguistic inference rules of a version of Gentzen’s natural deduc-tion: Deletion-rule corresponds to a parallel applications of ∧-eliminationrule; each Unification-rule essentially corresponds to a parallel applicationsof ∧-introduction rule among the relations described in Definition 3.2.1.(See, e.g., [Gentzen 1969, Prawitz 1965, Troelstra-Schwichtenberg 2000] fornatural deduction.) A natural deduction proof thus obtained is not in gen-eral in normal form, and it may contain some redexes (certain pairs of ∧-introduction and ∧-elimination rules). By normalizing such a natural deduc-tion proof with the well-known rewriting procedure, it is possible to obtaina normal proof. For such a normal natural deduction proof, we can define atranslation to a normal diagrammatic proof of GDS. (For a correspondencebetween normal natural deduction proofs and normal diagrammatic proofs,see also the discussion after Proposition 3.4.5.)

3.4.4 Structure of canonical diagrammatic proofs

In order further to investigate structure of canonical d-proofs of completeness(Theorem 3.3.10), we give a proposition, which is proved in a way similarto that of ±-normal form Theorem 3.4.3.

In our Definitions 3.3.4 and 3.3.7 of canonical models, it is possible tomodify the interpretation of each object by restricting the provability usingonly U3–U7 and Deletion rules as follows:

• For Definition 3.3.4,

I ′′α(t) = {s | α ⊢ s @ t with U3–U7 and Deletion rules} ∪ {t}

• For Definition 3.3.7, for any t such that α ⊢ B @ t with U3–U7 andDeletion rules,

59

I ′′α,B(t) = I ′′α(t) ∪ {s | α ⊢ B @ s and α ⊢ s @ B and α ⊢ s ⊢⊣ B}.

Recall that U3–U7 rules are unification where exactly one circle is sharedbetween the two premise diagrams.

These slight modification of canonical models also enables us, in a waysimilar to that in Corollary 3.4.2, to prove atomic completeness. Thus weobtain the following slightly stronger version of atomic completeness:

Corollary 3.4.4 Let−→D be a sequence D1, . . . ,Dn of EUL-diagrams which

is semantically consistent. Let β be a minimal diagram. Then we have:

If−→D |= β, then

−→D ⊢ β in GDS with U3–U7 and Deletion rules.

Thus soundness (Theorem 3.3.1) and Corollary 3.4.4 imply that anyminimal diagram is provable by using only U3–U7 and Deletion rules:

Proposition 3.4.5 (U3–U7 rules) Let−→D be a sequence D1, . . . ,Dn of EUL-

diagrams which is semantically consistent. Let β be a minimal diagram.

If−→D ⊢ β in GDS, then

−→D ⊢ β in GDS with U3–U7 and Deletion rules.

Completeness (Theorem 3.3.10), the ±-normal form theorem (Theorem3.4.3), and the above Proposition 3.4.5 give a more precise classification ofinference rules of GDS in terms of proof-construction as follows:

• U3–U7 and Deletion rules for derivation of a minimal diagram.

• U1, U2 (U8) rules for construction of a Venn-like diagram consistingof a single point (resp. no point).

• Point Insertion rule for construction of a Venn-like diagram consistingof multiple points.

• U9, U10 rules for construction of the conclusion.

See also the d-proof given in Example 3.3.11.The classification of inference rules and our canonical construction of

d-proofs show that the structure of each canonical d-proof essentially corre-sponds to the structure of a normal (linguistic) proof of Gentzen’s natural de-duction system. (See, e.g., [Gentzen 1969, Prawitz 1965, Troelstra-Schwichtenberg 2000]for natural deduction.) Observe that each pair of U3–U7 rules and a Deletionrule essentially corresponds to a combination of elimination rules of naturaldeduction. For example, the pair of U3 and Deletion rules corresponds to→-elimination accompanied by ∀-elimination as follows:

60

AB

R

Ac

U3

AB

c

?Deletion

Bc

∀x(A(x)→ B(x))A(c)→ B(c) ∀E

A(c)B(c) → E

Further note that U1,U2, Point Insertion, U9, U10 rules essentially correspondto ∧-introduction rule of natural deduction. For example, there is a corre-spondence between the following application of U1 rule and ∧-introductionrule:

Ac

Bc

R U1

Ac

B

A(c) B(c)A(c) ∧B(c) ∧I

In this way, in terms of natural deduction, it is shown that our canonicald-proof consists of, from top down, successive applications of eliminationrules and successive applications of introduction rules. See, for example,[Troelstra-Schwichtenberg 2000] for the structure of a normal natural de-duction proofs. A precise formulation of this discussion will appear in aforthcoming paper.

61

Chapter 4

EUL-structure

In this chapter, we describe the unification rules of Definition 3.2.1 of Section3.2 in terms of a graphical representation of EUL-diagrams, which may assistwith the understanding and motivation of our unification rules.

As seen in Section 2.2, given an EUL-diagram D, the set rel(D) of re-lations holding on it is uniquely determined by Proposition 2.2.4. rel(D)can be regarded as a kind of partially ordered structure (D, @,⊢⊣), called anEUL-structure, where D is the set of names of the objects of D and:

1. @ is an irreflexive transitive ordering relation on D;

2. ⊢⊣ is an irreflexive symmetric relation on D.

Furthermore, we have the following properties (cf. Lemma 2.2.5):

3. (⊢⊣-downward closedness) For any s, t, u ∈ D,

s ⊢⊣ t and t A u implies s ⊢⊣ u.

Hence the EUL-structure (D, @,⊢⊣) is an event structure of [Nielsen-Plotkin-Winskel 1980].

4. (Point minimality) For any s and point x of D,

not(s @ x).

5. (Point determinacy) For any s and any point x of D,

x @ s or x ⊢⊣ s.

62

Observe that above properties (1), (2), and (3) imply that, for any pairof elements of D, at most one of the relations @ and ⊢⊣ holds (cf. Proposition2.2.4); because if both of them hold, say s @ t and s ⊢⊣ t, the property (3)implies s ⊢⊣ s, which contradicts the irreflexivity of ⊢⊣-relation.

For example, rel(D1), rel(D4) and rel(D5) of Fig.2.4 in Section 2.2 areexpressed graphically as follows: Here the ordering relations @ are expressedby →-edges.

A C B

a

6

rel(D1)

A

C

B

a

6

6

rel(D4)

A

C

B

a

6Y >

rel(D5)

Observe that there is no edge for ◃▹-relation.

Remark Several authors introduce abstract (type) syntax, which is definedindependently of concrete (token) syntax. See, for example, [Molina 2001,Howse-Molina-Shin-Taylor 2002, Howse-Stapleton-Taylor 2005]. See also Sec-tion 5.1. For our EUL, it is possible to define EUL-structures, as abstractsyntax, independent of concrete plane diagrams of Definition 2.2.1 by defin-ing them using the above properties (1)–(5). We can show that any EUL-structure has an EUL-diagram as a concrete instance. However, in this thesis,we do not pursue the “abstract diagrams” separated in this way from theconcrete objects.

With the graphical representation of EUL-structures, our unification ofdiagrams can also be expressed by a simple unification of two graphs. Inorder to describe graphically the unification (U1–U8 rules) of EUL-diagramsD and α, we focus on the shared object of D and α, say A, and express theEUL-structure of rel(D) as follows:

X

A

6

Z/z

6Y/y W

rel(D)

→-edge denotes @-relation

⊢⊣-edge denotes ⊢⊣-relation

No edge for ◃▹-relation

“· · · ” denotes one of @, A,⊢⊣, ◃▹

The variables X, Y, Z,W (resp. y, z) are representative circles (resp. points)which are possibly related to A. When it makes no difference whether a

63

possibly related object is circle or point, we denote the object as Y/y (insteadof simply writing s). Each dotted line between objects expresses that theremay be one of the relations @, A,⊢⊣, ◃▹ between the objects. Note thatthere is no edge for each ◃▹-relation, as seen between A and W . We omitthe trivial transitive edge Z −→ X to avoid notational complexity. In thefollowing description of each unification rule for D and α, we give a graphicalrepresentation of the EUL-structures of rel(D) in the left-hand graph, andrel(D+ α) in the right-hand graph. We begin with U3-rule since U1 and U2rules are rather untypical cases:

U3 Under the constraint of U3-rule, there is no circle Z such that Z @ Aholds, and no circle W such that A ◃▹ W holds, which is expressedby × in the graph of rel(D). According to U3-rule of Definition 3.2.1,rel(D + (b @ A)) is represented by the graph on the right.

X

A

6

×Z/z

6Y/y ×W

rel(D)

X

A

6

z

6Y/y

b

I

K

rel(D + (b @ A))

It is easily seen that rel(D + (b @ A)) is an EUL-structure: I.e., theaugmented edges do not violate the properties of EUL-structure.

Note also that, without the constraint, i.e., if there is a circle Z orW as above, in order to preserve Point determinacy, we should fix arelation between b and Z (resp. W ) to @ or ⊢⊣. However, neither ofthem is sound with respect to our formal semantics of EUL. (See thesoundness theorem of GDS of Section 3.3.)

U4 Under the constraint of U4-rule, there is no circle X such that A @ Xholds, no circle Y such that A ⊢⊣ Y holds, and no circle W suchthat A ◃▹ W holds, which is expressed by × in the graph of rel(D).According to U4-rule of Definition 3.2.1, rel(D+(b ⊢⊣ A)) is representedby the right hand graph below.

×X

A

Z/z

6×Y/y ×W

rel(D)

A

Z/z

6y

b

rel(D + (b ⊢⊣ A))

64

It is easily seen that rel(D + (b ⊢⊣ A)) is an EUL-structure: I.e., theaugmented edges do not violate the properties of EUL-structure.

Without the constraint, i.e., if there is a circle X, Y or W as above, inorder to preserve Point determinacy, we should fix a relation betweenb and X (resp. Y, W ) to @ or ⊢⊣ in rel(D + (b ⊢⊣ A)). However, noneof them is sound with respect to our semantics of EUL.

U5 Under the constraint of U5-rule, there is no point z such that z @ Bholds. According to U5-rule of Definition 3.2.1, rel(D + (A @ B)) isrepresented by by the right hand graph below.

X

B

6

×Z/z

6Y/y W

rel(D)

X

B

6

Z

6Y/y W

A

K

I

rel(D + (A @ B))

Without the constraint, i.e., if there is a point z as above, in orderto preserve Point determinacy, we should fix a relation between z andA to @ or ⊢⊣. However, none of them is sound with respect to oursemantics of EUL.

U6 Under the constraint of U6-rule, there is no point y such that y ⊢⊣ Aholds. According to U6-rule of Definition 3.2.1, rel(D + (A @ B)) isrepresented by by the right hand graph below.

X

A

6

Z/z

6×Y/y W

rel(D)

X

A

6

Z/z

6Y W

B��

rel(D + (A @ B))

Without the constraint, i.e., if there is a point y as above, in orderto preserve Point determinacy, we should fix a relation between y andB to @ or ⊢⊣. However, none of them is sound with respect to oursemantics of EUL.

U7 Under the constraint of U7-rule, there is no point y such that y ⊢⊣ Aholds. According to U7-rule of Definition 3.2.1, rel(D + (A ⊢⊣ B)) isrepresented by by the right hand graph below.

65

X

A

6

Z/z

6×Y/y W

rel(D)

X

A

6

Z/z

6Y W

B

rel(D + (A ⊢⊣ B))

Without the constraint, i.e., if there is a point y as above, in orderto preserve Point determinacy, we should fix a relation between y andB to @ or ⊢⊣. However, none of them is sound with respect to oursemantics of EUL.

U8 Under the constraint of U8-rule, there is no point in D. According toU8-rule of Definition 3.2.1, rel(D + (A ◃▹ B)) is represented by by theright hand graph below.

X

A

6

×Z/z

6×Y/y W

rel(D)

X

A

6

Z

6Y W

B

rel(D + (A ◃▹ B))

Without the constraint, i.e., if there is a point y or z as above, inorder to preserve Point determinacy, we should fix a relation betweeny (resp. z) and B to @ or ⊢⊣. However, none of them is sound withrespect to our semantics of EUL.

U1 Under the constraint of U1-rule, there is no point y in D other than b.According to U1-rule of Definition 3.2.1, rel(D+(b @ A)) is representedby by the right hand graph below.

X

b

6

×Y/y

rel(D)

X

b

6Y

�A

rel(D + (b @ A))

Without the constraint, i.e., if there is a point y as above, in orderto preserve Point determinacy, we should fix a relation between y andA to @ or ⊢⊣. However, none of them is sound with respect to oursemantics of EUL.

66

U2 Under the constraint of U2-rule, there is no point y in D other than b.According to U2-rule of Definition 3.2.1, rel(D+(b ⊢⊣ A)) is representedby by the right hand graph below.

X

b

6

×Y/y

rel(D)

X

b

6Y

A

rel(D + (b ⊢⊣ A))

Without the constraint, i.e., if there is a point y as above, in orderto preserve Point determinacy, we should fix a relation between y andA to @ or ⊢⊣. However, none of them is sound with respect to oursemantics of EUL.

Observe that for the above U1–U8 rules, each constraint is introduced topreserve Point determinacy of EUL-structure after the unification.

In U9 and U10 rules of Definition 3.2.1, the unified diagrams D and αshare two circles, which makes the graphical description of rel(D) compli-cated. In order to avoid notational complexity, we omit irrelevant objectsand edges, which are retained after the application of U9 or U10 rule.

U9 Under the constraint of U9-rule, there is no object s such that s @ A ands ⊢⊣ B hold on D, i.e., in the following description of rel(D), the dottedline between Y/y and A should not be→, and the dotted line betweenZ/z and B should not be ⊢⊣. According to U9-rule of Definition 3.2.1,rel(D + (A @ B)) is represented by by the right hand graph below.

X

B

6

Z/z

6A Y/y

rel(D)

X�

B

6

�

Z/z

6

�A - Y/y

rel(D + (A @ B))

Observe that, after the unification, some of the dotted lines of rel(D)are fixed to → or ⊢⊣ in rel(D+ (A @ B)) according to Definition 3.2.1.We need to check that rel(D + (A @ B)) is an EUL-structure; for ex-ample, if the dotted line between A and X in rel(D) is A ⊢⊣ X (orA ← X), after the application of U9-rule, there are two incompati-ble edges ⊢⊣ (resp. ←) and → between A and X, which violates theirreflexivity of the ⊢⊣-relation of EUL-structure.

67

It is shown that, because of our constraint for U9-rule, the dottedline between A and X is ◃▹ (i.e., no edge) or →. Observe that, if wehave A ⊢⊣ X in rel(D), by the ⊢⊣-downward closedness of rel(D), wehave Z/z ⊢⊣ B in rel(D), which contradicts the constraint. If we haveA ← X in rel(D), by the transitivity of rel(D), we have A ← B inrel(D), which contradicts the presupposition of U9-rule, i.e., there isno edge between A and B in rel(D). Thus the dotted line between Aand X should be ◃▹ (i.e., no edge) or →, either of which is compatiblewith the edge A → X in rel(D + (A @ B)). Similarly, it is shownthat the other dotted lines of rel(D) are compatible with the edgesof rel(D + (A @ B)). Then it is easily checked that the properties(1)–(5) of EUL-structures hold in rel(D+ (A @ B)), and hence it is anEUL-structure.

U10 Under the constraint of U10-rule, there is no object s such that s @ Aand s @ B hold on D, i.e., in the following graph of rel(D), the dottedline between Z ′/z′ and A (and also between Z/z and B) should notbe →. According to U10-rule of Definition 3.2.1, rel(D + (A ⊢⊣ B)) isrepresented by by the right hand graph below.

B

Z ′/z′6

A

Z/z

6

rel(D)

B

Z ′/z′6

A

Z/z

6

rel(D + (A ⊢⊣ B))

We show that there are no incompatible edges in rel(D+(A ⊢⊣ B)). Forthe dotted line between Z/z and B, it is not → by the constraint forU10-rule. Furthermore, assume to the contrary that we have Z/z ← Bin rel(D). Then, by the transitivity of rel(D), we have A← B in rel(D),which contradicts the presupposition of U10-rule, i.e., there is no edgebetween A and B. Hence the dotted line between Z/z and B shouldbe ◃▹ (i.e., no edge) or ⊢⊣, either of which is compatible with the edgeZ/z ⊢⊣ B in rel(D+(A ⊢⊣ B)). Similarly, it is shown that the other twodotted lines of rel(D) are compatible with the edges of rel(D+(A ⊢⊣ B)).Then it is easily checked that the properties (1)–(5) of EUL-structureshold in rel(D + (A ⊢⊣ B)), and hence it is an EUL-structure.

68

Chapter 5

A relationship betweenEUL-diagrams and Venndiagrams

In this chapter, we study a relationship between our EUL-diagrams andVenn diagrams (or Euler diagrams with shading, which are region-basedEuler diagrams). We first review the definition of syntax and semanticsof Venn diagrams of [Howse-Stapleton-Taylor 2005, Fish-John-Taylor 2008,Howse-Molina-Shin-Taylor 2002, Shin 1994]. Then we give a transformationof each EUL-diagram to a Venn diagram which is semantically equivalent tothe original EUL-diagram.

5.1 Syntax and semantics of Venn diagrams

After Shin’s formalization of Venn-II system of [Shin 1994], the idea ofVenn diagram is extended, by regarding some empty regions as “miss-ing” regions, to Euler diagram with shading e.g., Euler/Venn diagrams of[Swoboda-Allwein 2004, Swoboda-Allwein 2005]; Spider diagrams ESD2 of[Molina 2001] and SD3 of [Howse-Stapleton-Taylor 2005]. Each Venn dia-gram is defined as a special Euler diagram with shading which contains allthe possible regions.

We briefly recall the syntax and semantics of Euler diagrams with shad-ing and Venn diagrams. See, for example, [Howse-Stapleton-Taylor 2005,Fish-John-Taylor 2008, Howse-Molina-Shin-Taylor 2002] for detailed descrip-tions.

69

In what follows, we do not consider named points for simplicity of dis-cussion. The following discussion can be easily extended to include namedpoints, since any named points of EUL can be regarded as special circles.

A concrete Euler diagram with shading consists of finite numbers ofnamed circles (simple closed curves) on a plane enclosed by a boundaryrectangle. A basic region is the bounded region of a plane enclosed by acircle or by the boundary rectangle. Regions are then defined recursively,based on the basic regions, by insisting that they be closed under union,intersection and difference. A zone, or minimal region, is a region havingno other region contained within it. For example, the following diagram Vconsists of two circles A and B, and four zones z1, z2, z3, and z4:

V

A Bz1 z2 z3

z4

Note that each zone is abstractly specified by the circles which en-close it and those which it lies outside: the zone z1 is inside A but out-side B; z2 is inside both A and B; z3 is outside A but inside B; and z4

is outside both A and B. Thus zones are identified by an ordered pairof names of circles that the zone lies inside and outside of, respectively:z1 = ({A}, {B}), z2 = ({A,B}, ∅), z3 = ({B}, {A}), and z4 = (∅, {A, B}).In this way, independently of concrete plane diagrams, abstract Euler dia-grams with shading are defined in terms of names of circles as follows: LetL be the finite set of the names of circles in a given diagram. An abstractzone z of an Euler diagram with shading is defined by using the names ofcircles of L,

z = (in(z), out(z)),

where in(z) and out(z) are finite subsets of L such that in(z) ∩ out(z) = ∅and in(z)∪out(z) = L. Note that for the fixed L for a diagram, each zone isidentified by the set of names in(z) (without indicating out(z)), and hence azone z is sometimes abbreviated by the set in(z). Each Euler diagram withshading is defined by specifying which zones are shaded, i.e., empty. Thus,for a fixed labels L of circles, an abstract Euler diagram with shadingV is defined by a pair of a finite set Z of zones and a finite subset Z• ⊆ Zof empty zones which are shaded:

V = (Z, Z•).

70

Note that the set Z of zones does not necessarily contain all the possiblezones, that is, {in(z) | z ∈ Z} is not necessarily equal to the power set P(L)of the labels L. Thus in an Euler diagram with shading, some regions maybe missing.

A Venn diagram is an Euler diagram with shading which contains allthe possible zones, i.e., {in(z) | z ∈ Z} = P(L).

A model of Venn diagrams is a pair M = (U, I) where U is a non-emptyset called the universe, and I is an interpretation function which assigns toeach circle a subset of U . Recall that each zone is specified by the circleswhich enclose it and those which it lies outside. Then the interpretationfunction I is extended to interpret zones and regions as follows: For anyzone z = (in(z), out(z)) the interpretation I(z) of a zone is defined by:

I(z) =∩

X∈in(z)

I(X) ∩∩

Y ∈out(z)

I(Y )

where I(Y ) is the complement of the set I(Y ). Any region r is interpreted asa union of zones contained in it: I(r) =

∪z∈r I(z). Observe that a model of

Venn diagrams is a natural extension of our semantics of EUL of Section 2.3,where the interpretation function is extended to interpret not only circlesbut also regions.

A pair M = (U, I) is a model of a Venn diagram V = (Z, Z•), denotedas M |= V, if each shaded zone represents the empty set:

For each z ∈ Z•, I(z) = ∅.

5.2 Transformation of EUL-diagrams to Venn dia-grams

We give a transformation of each EUL-diagram to a Venn diagram which issemantically equivalent to the original EUL-diagram. We define the transfor-mation by using EUL-structures of Chapter 4, since they contribute naturallyto an understanding of the transformation, and they also suggest a possibleextension of EUL (cf. Chapter 7).

In this section, for simplicity of discussion, we consider @-relations inEUL-structures to be reflexive, i.e., X @ X holds for any X, while they arenot in Chapter 4.

Let us begin with an example: Let D be an EUL-diagram such thatrel(D) = {A ◃▹ B, A ⊢⊣ C,C @ B}.

71

A B

C

D

A

B

C6

rel(D)

B

A ⊓ B6

3A C = B ⊓ C

6

venn(D)

A

C

B

Dv

Recall that each zone of a Venn diagram is specified by the intersectionof circles which enclose it. Based on this observation, we extend the EUL-structure rel(D) to venn(D) by introducing greatest lower bounds (glbs) (A⊓B and B ⊓ C), which correspond to intersections of basic regions, withpreservation of ⊢⊣-downward closedness. Note that in the above venn(D),the node A ⊓ B ⊓ C is not introduced since it destroys the ⊢⊣-downwardclosedness. From venn(D), we obtain a Venn diagram Dv by extracting theset of nodes of venn(D) as possibly non-empty zones:

{{A}, {B}, {A,B}, {B, C}} (∗)

where {A} (resp. {B}) corresponds to the node A (resp. B) of venn(D),and {A,B} (resp. {B, C}) corresponds to the node A ⊓ B (resp. B ⊓ C).Note that the node B ⊓ C = C corresponds to {B,C} instead of {C}. Theset of shaded zones of Dv is the complement of the above set (∗) with respectto the set of all possible zones, i.e.,

{{C}, {A,C}, {A,B, C}}. (∗∗)

The Venn diagram Dv is semantically equivalent to the original EUL-diagram D. In order to see this, let M = (U, I) be an EUL-model for D,that is, I(C) ⊆ I(B) and I(A) ∩ I(C) = ∅. Recall that M is a model of aVenn diagram if its interpretation function I assigns the empty set to eachshaded zone. It is easily checked that the following equations 1, 2, and 3hold in M :

1. I(C) ∩ I(A) ∩ I(B) = ∅

2. I(A) ∩ I(C) ∩ I(B) = ∅

3. I(A) ∩ I(B) ∩ I(C) = ∅

Thus we have M |= Dv. Conversely, note that the EUL-structure rel(D) is asubstructure of venn(D), that is, any EUL-relations which hold in the EUL-diagram D also hold in the Venn diagram Dv. Hence it is immediate thatany model M of the Venn diagram Dv is also a model of the EUL-diagramD.

72

We formalize the above discussion. We first extend EUL-structures byadding greatest lower bounds (glbs) with preservation of ⊢⊣-downward closed-ness as follows:

Definition 5.2.1 (EUL-structure with glbs) Let D be an EUL-diagram,and let rel(D) = (D, @,⊢⊣) be its EUL-structure. Let ⊓1≤i≤nXi be the great-est lower bound (glb) of a set of names {X1, . . . , Xn} ⊆ D. (In particularwhen n = 1, ⊓X1 = X1.) ⊓1≤i≤nXi is simply denoted by ⊓Xi when n is notimportant. An EUL-structure with glbs venn(D) = (Dv,@,⊢⊣) is defined asfollows:

⊓1≤i≤nXi ∈ Dv iff ¬∃1≤j,k≤n(Xj ⊢⊣ Xk ∈ rel(D))

⊓Xi ⊢⊣ ⊓Yj in venn(D) iff ∃Z, W ∈ D, Z ⊢⊣W ∈ rel(D), and⊓Xi @ Z, ⊓Yj @ W .

By Definition 5.2.1, for any EUL-diagramD, venn(D) is an EUL-structurewhich contains rel(D) as a substructure.

Lemma 5.2.2 Let D be an EUL-diagram. The venn(D) = (Dv,@,⊢⊣) sat-isfies the following properties (1)–(3):

1. @ is a reflexive transitive ordering relation on Dv;

2. ⊢⊣ is an irreflexive symmetric relation on Dv;

3. For any s, t, u ∈ Dv, s ⊢⊣ t and u @ t implies s ⊢⊣ u.

Thus venn(D) is an EUL-structure.

See Chapter 4 for EUL-structure.

From the EUL-structure with glbs venn(D), we obtain a Venn diagramDv which is semantically equivalent to the original EUL-diagram D.

In order to construct the Venn diagram Dv, we eliminate redundantnodes from venn(D). For example, we have C = B⊓C in the above example,and we eliminate the node C to construct the Venn diagram Dv. Roughlyspeaking, when several nodes are equivalent, we keep the node which hasthe longest name, and eliminate the other nodes as follows:

For any s ∈ Dv, let [s] be an equivalent class {t | s = t in venn(D)}. Wedefine an ordering relation 4 on [s] as follows: For any t, u ∈ [s], t 4 u if tis a segment of u (in particular, ⊓X1 4 ⊓X1). Then, u ∈ [s] is maximal ift 4 u for any t ∈ [s].

Let Dv′= Dv \ {t | t is not maximal with respect to 4 on some [s]}.

Thus Dv′consists only of maximal nodes with respect to the ordering 4.

73

Proposition 5.2.3 (Venn diagram) Let D be an EUL-diagram. Let ZDv

be the power set P(D) of the set of names of the circles of D. Let Z•Dv be

the following set:

ZDv \ {{X1, . . . , Xn} | ⊓1≤i≤nXi ∈ Dv′}.

Then the pair (ZDv , Z•Dv) gives rise to a Venn diagram Dv such that

M |= D iff M |= Dv

for any model M , that is, D and Dv are semantically equivalent.

74

Chapter 6

Normal form ofdiagrammatic proofs of GDSand syllogisms

In this chapter, we give a characterization of chains of Aristotelian categor-ical syllogisms in our diagrammatic inference system GDS. We first define aclass of syllogistic diagrams, which corresponds to the class of Aristoteliancategorical sentences. Then, we introduce a class of syllogistic normal di-agrammatic proofs, which corresponds to the class of chains of valid syllo-gisms.

6.1 Syllogistic diagrams

We introduce syllogistic diagrams.

Definition 6.1.1 (Syllogistic diagram) An EUL-diagram D is called asyllogistic diagram if it is in one of the following forms:

AB

A B Aa

B A

a

B

A-type E-type I-type O-type

We denote syllogistic diagrams by S,S1,S2, . . . .

Observe that among the four types, the first two (A-type and E-type) areminimal diagrams of EUL, while the other two (I-type and O-type) are not.

75

There is the following correspondence between syllogistic diagrams andcategorical sentences:

A-type: All A are B;E-type: No A are B, No B are A;I-type: Some A are B, Some B are A;O-type: Some A are not B.

The ±-normal form theorem for minimal diagrams (Theorem 3.4.3 ofSection 3.4.3) is slightly generalized for the syllogistic diagrams:

Theorem 6.1.2 (±-normal form for syllogistic diagrams) Let−→S ,S be

a sequence of syllogistic diagrams. Let−→S be semantically consistent. Then

we have:

If−→S ⊢ S in GDS, then

−→S ⊢ S in GDS with a ±-normal d-proof.

Proof. Observe that the canonical construction of d-proofs of completeness(Theorem 3.3.10) tells us the last rule to obtain the conclusion S accordingto the form of S:

• When the conclusion S is A-type or E-type, since any minimal diagramcannot be obtained by unification, S is obtained by deletion.

• When S is I-type or O-type, since they are Venn-like diagrams, bythe canonical construction, S is obtained by unification U1 or U2 rule,respectively.

Furthermore, the canonical construction also tells us what kind of rulesis applied to each premise according to its form:

• When a premise Si ∈−→S is A-type or E-type, since deletion cannot be

applied to any minimal diagram, unification is applied to Si.

• When a premise Si ∈−→S is I-type or O-type, by the canonical construc-

tion of d-proofs in atomic completeness (Proposition 3.3.9), deletion isapplied to Si.

The above observations show that there is a d-proof of S from−→S of the

following form:

76

S

?

... (−)

orS

?

... (−)

or

y

S

R (+)or

y

S

R (+)

y y

?

...(−) ?

...(−)

yy

?

...(−) ?

...(−)

...

...

...

...

±-normalby

Theorem 3.4.3

R

...(+)Si R

...(+)Sj

x

R

...(+)

x

R

...(+)

x

R

...(+)

x

Sk ?(−)

x

Sl ?(−)

x

Sm?(−)

Obviously, the above form of d-proof is in ±-normal form. Thus we havea ±-normal d-proof of S from

−→S .

6.2 Syllogistic normal diagrammatic proofs and chainsof Aristotelian categorical syllogisms

For a characterization of a chain of syllogisms, we introduce the followingnotion (cf. [Crabbe 2001]):

Definition 6.2.1 (Cycle) A sequence S1, . . . ,Sn,S of syllogistic diagramsis a cycle if

• cr(S1) = {A1, A2}, . . . , cr(Si) = {Ai, Ai+1}, . . . , cr(S) = {An+1, A1},and

• ∀x, y ∈ pt(S1, . . . ,Sn), x = y.

We introduce a further kind of normal form, called syllogistic normalform. The class of syllogistic normal d-proofs is a subclass of the ±-normald-proofs introduced in Definition 3.4.1 of Section 3.4.3.

Definition 6.2.2 (Syllogistic normal d-proof) A ±-normal d-proof πis in syllogistic normal form if it starts with a unification (+) and endswith a deletion (−) from top down.

77

Thus each syllogistic normal d-proof is of the form shown in Fig.6.1:

Ca

B

D1

A B

D2R U7

A C Ba

D3?A

aC

D4R )U5

DE

D5

EA

D6R U6/U5

DEA

D7?

D

A

D8

DA C

a

D9?D a

C

D10

Fig. 6.1 A syllogistic normal d-proof

D1 D2

�Unification

Deletion

�Unification

Deletion

�Unification

D5 D6

Deletion

D10

Fig. 6.2 The underlying treestructure of the syllogisticnormal d-proof of Fig.6.1

Fig.6.1 illustrates a syllogistic normal d-proof, where each pair of a unifica-tion rule and a deletion rule application corresponds to a valid pattern ofsyllogisms. For example, the subproof from D1 and D2 to D4 is a diagram-matic representation of syllogism of the form: Some C are B and No A areB; therefore Some C are not A. Indeed, each syllogistic normal d-proof canbe considered as a chain of valid patterns of Aristotelian categorical syllo-gisms. Note that, compared with the underlying proof tree of GDS in Fig.3.6in Section 3.2, the tree of the syllogistic normal d-proof has a canonical form,where a unification node, denoted by �, and a deletion node, denoted by ⃝,appear one after the other.

It is easily shown that for any (linguistic) chain of valid patterns ofAristotelian categorical syllogisms there is a corresponding syllogistic normald-proof in GDS.

Proposition 6.2.3 (Chain of syllogisms and a syllogistic normal d-proof)Let S1, . . . ,Sn,S be Aristotelian categorical sentences. Let SD1 , . . . ,SDn ,SDbe the corresponding syllogistic diagrams. Then S is a valid conclusion ofAristotelian categorical syllogisms from the premises S1, . . . ,Sn if and onlyif, there is a syllogistic normal d-proof of SD in GDS from the premisesSD1 , . . . ,SDn .

78

Moreover, it is shown that, in syllogistic fragment of GDS, if there is ad-proof (not necessarily in syllogistic normal form) of a syllogistic diagram,then there is a corresponding (linguistic) chain of syllogisms. Hence, thesyllogistic fragment of GDS completely characterize the chains of Aristoteliancategorical syllogisms. In order to show this, it is sufficient to prove thefollowing proposition:

Proposition 6.2.4 (Syllogistic normal form) Let−→S ,S be a sequence of

syllogistic diagrams which is a cycle. Let−→S be semantically consistent. Then

we have:

If−→S ⊢ S in GDS, then

−→S ⊢ S in GDS with a syllogistic normal d-proof.

Based on Theorem 6.1.2, we show that any ±-normal d-proof of S from−→S can be rewritten into a syllogistic normal d-proof of S from−→S . We

here illustrate a typical example of non-trivial cases which indicates howthe proposition is proved. Assume that the following syllogistic diagramsS1,S2,S3 and S are given:

Aa

B

S1

AC

S2

B D

S3

C

a

D

S

By following the canonical construction of d-proofs described in theproof of Theorem 6.1.2, we obtain the following ±-normal d-proof of S fromS1,S2,S3:

Ca

D

S

Ca

q

AC

a

?

Aa

R

Aa

B

?S1

AC

S2π1

Da

)

B Da

?

Ba

?

Aa

B

S1?B D

S3�

π2

79

As seen in the above d-proof, which starts with Deletion rules at the firststep, our canonical construction does not, in general, give syllogistic normald-proofs. However, it is possible to rewrite the ±-normal d-proof into asyllogistic normal d-proof as follows: We first rewrite the subproof π1 to thefollowing π′

1 by removing the first Deletion step and leaving the circle B forthe subsequent steps:

Ca

B

?

BAC

a

Aa

B

S1 R

AC

S2

π′1

Then, by connecting π2 under π′

1, with preserving the circle C of π′1, we

obtain the following syllogistic normal d-proof:

DaC

S

D C Ba

?

Ca

B

j

B D

S3�

BAC

a

?

Aa

B

S1 j

AC

S2�

It is easily seen that this syllogistic normal d-proof corresponds to thefollowing chain of syllogism, where each pair of unification and deletion rulescorresponds to a valid syllogism:

80

Some A are B All A are CSome C are B No B are D

Some C are not D

Remark 6.2.5 In [Mineshima-Okada-Sato-Takemura 2008], we introduceda subsystem DS of GDS, which characterizes only syllogistic normal d-proofs.The above Proposition 6.2.4 implies that GDS is a conservative extensionover its syllogistic fragment DS.

81

Chapter 7

Some extensions and futurework for Part I

In this chapter, we discuss some possible extensions of our diagrammaticsystem and we comment on our future work.

(1) We defined a formal Euler diagrammatic system in terms of the topo-logical relations, called EUL-relations, between two diagrammatic objects.Although, in this thesis, we considered only circles and points as diagram-matic objects and excluded any intersection region and linking of points,we can naturally extend our framework by introducing these diagrammaticobjects in the following way.

1. Our framework can be naturally extended by regarding intersectionregions of circles as diagrammatic objects. In Chapter 5, in order togive a translation of EUL-diagrams into Venn diagrams, we extended anEUL-structure of Chapter 4 by introducing greatest lower bounds whichpreserve the ⊢⊣-downward closedness of the EUL-structure. As seenthere, these greatest lower bounds correspond to intersection regionsof circles. With this extension, the diagrams D1,D2 and D3 of Fig.2.4in Section 2.2 can be distinguished.

2. We sometimes regarded a named point of EUL as a special named circlewhich does not contain, nor cross, any other object. Let us extend thenotion of point by allowing it to make crosses with circles as illustratedin the following diagram D.

82

A b

D

Ab b

D′

According to our semantic interpretation of ◃▹-relation (cf. Remark2.3.3 of Section 2.3), D can be considered to represent “the point bis inside A or outside A,” which is equivalently expressed by usingPeirce’s linking as D′. Although this linking does not fully correspondto Peirce’s linking (cf. [Peirce 1897, Shin 1994]), we can naturallyextend our framework to represent some disjunctive information onobjects in this way.

Our soundness, completeness, and normal form theorems are naturally re-tained with these extensions of EUL.

(2) As mentioned in Chapter 4, our EUL-structures can define an abstractsyntax of EUL-diagrams independently of concrete diagrams in a way similarto that of [Molina 2001, Howse-Molina-Shin-Taylor 2002, Howse-Stapleton-Taylor 2005].When we explore such an abstract formalization, the relationship betweenthe abstract (type) syntax and the concrete (token) syntax becomes cru-cial. In Chapter 4, we showed that any concrete diagram defines an EUL-structure. Conversely, it can be shown that any EUL-structure gives rise to a(plane) EUL-diagram. In order to show this, we should formalize a strategyfor drawing a concrete plane diagram from a given abstract EUL-structure.Recently, such drawability of concrete Euler diagrams from abstract typeshas been investigated by [Rodgers-Zhang-Fish 2008, Flower-Fish-Howse 2008]etc. Our formal strategy will appear in a forthcoming paper.

(3) In Chapter 5, we showed that any EUL-diagram can be transformedinto a semantically equivalent Venn diagram. The other direction is alsointeresting: Is there any EUL-diagram which is (semantically) equivalentto a given Venn diagram? However, it is known that there is a Venndiagram to which no Euler diagram without shading is equivalent. Cf.[Gil-Howse-Tulchinsky 2002]. Thus we will investigate the question: Whatkind of Venn diagram can be transformed into a (semantically) equivalentEUL-diagram? For this question, some conditions on shadings for Venn di-agrams are important, e.g., shaded regions should be connected, etc. Ourcharacterization of EUL-diagrams in terms of Venn diagrams will appear ina forthcoming paper.

83

(4) In Chapter 6, we illustrated that a syllogistic normal d-proof of GDScorresponds to a chain of Aristotelian categorical syllogisms. We will gen-eralize this correspondence to that between the ±-normal d-proofs of thefull fragment of GDS and the normal proofs of disjunction-free minimallogic. (See [Prawitz 1965, Troelstra-Schwichtenberg 2000] for minimal logic,which is essentially the ⊥-free fragment of intuitionistic logic.) In such acorrespondence, it will be shown that a pair of unification and deletion ina ±-normal d-proof of GDS corresponds to a combination of some inferencerules of Gentzen’s natural deduction formulation of minimal logic.

(5) In the context of the psychology of deductive reasoning, syllogistic rea-soning is one of the most intensively studied forms of reasoning. It wouldbe expected that the use of Euler diagrams might help subjects to solvesyllogistic reasoning tasks. In previous psychological researches, however, itis often reported that there is no evidence to show the advantage of Eulerdiagrams in solving syllogistic reasoning tasks (cf. [Rizzo-Palmonari 2005]etc.). Main reasons of this can be considered that Euler diagrams given inthe previous researches (i) cannot represent each premise of a syllogismin a single diagram; (ii) induce much disjunctive ambiguities in solvingsyllogisms. Because our EUL-diagrams are free from such problems, it isexpected that they improve syllogistic reasoning in general. In particularsome well-known errors in linguistic syllogistic reasoning could be blocked,mainly due to the fact that our diagrams make explicit the subject-predicaterelation of a categorical statement. We have designed cognitive psycholog-ical experiments to investigate the question of whether and how the useof Euler diagrams aid subjects to solve syllogistic reasoning tasks. See[Mineshima-Okada-Sato-Takemura 2008, Sato-Mineshima-Takemura-Okada 2009]for our ongoing cognitive psychological studies.

84

Part II

Focusing proofsin polarized linear logic

86

Chapter 8

Introduction to Part II

The notion of polarity was invented by Girard in his work on LC (LogiqueClassique, [Girard 1991]), which is an improvement of Gentzen’s standardformulation of classical logic LK. In particular, in LC, disjunction and ex-istential quantifier are divided in terms of positive/negative polarities, and,for hereditary positive formulas, intuitionistic disjunctive and existentialproperties hold in the classical logic framework. After the Girard’s intro-duction, polarity has turned out to be an important hidden parameter con-trolling linear proof-theory. In particular, it has been pointed out thatnegative/positive polarities correspond to dual proof-theoretical propertiesreversibility/focalization of linear logical connectives, which are clarified byAndreoli in his study on focusing proofs [Andreoli 1992]. There is a linkbetween polarity and focalization studied by [Danos-Joinet-Schellinx 1997]and clarified by [Laurent 2002]. (Cf. also Section 9.3 of this Part II.) Lau-rent’s formalization of polarized linear logic, in effect, provides a frameworkto construct focusing proofs in terms of the focalized sequent property. Thefocalized sequent property of a linear logical system, which is not necessarilya polarized system, means that if a sequent is provable with only polarizedformulas, especially in polarized linear logic, it contains at most one positiveformula, in which case we call the sequent “focalized.” Since the positive for-mula is always focused, each proof of polarized linear logic gives a focusingproof in Andreoli’s sense. Laurent [2002] shows a first order conservativ-ity theorem of linear logic LL over its polarized fragment LLpol. That is,if a (polarized) focalized sequent is provable in LL, then it is also in LLpol.Since all the proofs of LLpol are automatically focusing, it follows that anyfocalized sequent is provable with a focusing proof in LL. Combined withthe focalized sequent property of LL, the conservativity neatly captures a

88

main idea underlying polarity in linear logic: the “polarity” restriction onformulas leads naturally to focusing proofs. Moreover, seen from a logic pro-gramming viewpoint (cf. [Miller 2004]), the conservativity is also importantsince we have only to work with focusing proofs. In his proof of the firstorder conservativity, Laurent made essential use of the subformula propertyof LL, which ensures that if a focalized sequent is provable then it is provablewith only polarized formulas. When we try to extend the conservativity tothe second order linear logic LL2, we immediately encounter a difficulty withthe second order ∃-rule, which results in the loss of the subformula property.For this reason Laurent [2002] has left it open whether the conservativityresult can be extended to second order.

In order to give an answer to the above question, we introduce a phasesemantics for second order polarized linear logic. Phase semantics is in-troduced in [Girard 1987] for an algebraic semantics of LL, and a non-commutative version is also studied in [Abrusci 1991]. Phase semanticsis weaker than either game or categorical semantics (e.g. [Laurent 2002,Mellies 2003, Hamano-Scott 2007]) in the sense that only the notion of prov-ability (and not that of proof) is considered. However, phase semanticshas been quite useful to show various results for linear logic (e.g. finitemodel property [Lafont 1997, Okada-Terui 1999], uniform treatment of cut-elimination [Okada 2002, Terui 2007], relationship to denotational model[Bucciarelli-Ehrhard 2000, Ehrhard 2004]). In addition, compared with cat-egorical semantics, phase semantics with its “simpler” structure is morenaturally extendable to second order, as is seen in [Okada 1999]. In par-ticular, in the phase semantics for LL2 of [Okada 1999], a strong form ofcompleteness theorem is given to yield the second order cut-elimination forLL2, for which syntactical methods hardly work. The main feature of our po-larized phase semantics is its employment of a topological structure, whichaccommodates the two polarities as openness and closedness. This inter-pretation is an algebraic instance of the categorical construction developedin [Hamano-Scott 2007] and is based upon the adjunction between interiorand closure operators for the topology. A similar adjunction has recentlybeen discussed in [Mellies 2005] in order to solve the Blass problem, and todefine a game model of full linear logic (LL). See also [Mellies 2005b]. Ithas been also discussed in [Selinger 2001, Hofmann-Streicher 2002] in theircontinuation-passing-style models of λµ-calculus. Most recently motivatedby an observation of M. Hasegawa, Mellies-Tabareau [2007] advocate thatphase semantics should be seen as a model of tensorial logic, a more prim-itive logic than linear logic, where negation is not involutive anymore. Theusual phase space of linear logic is then recovered from the model of tensor

89

logic, see also [Tabareau 2008]. Moreover, general categorical principles im-ply that when a phase space model of tensorial logic is given, it induces aphase space model of polarized linear logic. On the other hand, few attemptshave so far been made to clarify the notion of polarity in terms of topol-ogy, although as regards usual linear logic, topolinear spaces are studied forexponential connectives (e.g., [Girard 1987, Sambin 1995]). In the follow-ing, we introduce two kinds of second order topological phase semantics−apolarized phase semantics for multiplicative additive polarized linear logicMALLP2, and an enriched polarized phase semantics for LLpol2−and provetheir strong completeness by revising Okada’s [1999] method, which impliessecond order cut-eliminations.

Then we first show by using a counter model construction that LL2 is notconservative over LLpol2 (Proposition 12.1.1), which is a rather unexpectedby-product of our polarized phase semantics. We next observe that LL2 doesnot have the focalized sequent property (Proposition 12.1.3). With these“negative” results, it appears that LL2 lacks the central idea of polarity in lin-ear logic mentioned above, and that it offers no bridge between polarity andfocusing. In order to remedy this shortcoming, we introduce an η-expandedfragment LLη2 of LL2, in which atoms are exponential forms (i.e., !X⊥ (resp.?X) for a positive (resp. negative) atom). Such a restriction, which was alsoadopted in [Laurent-Quatrini-Tortora 2005, Laurent 1999, Laurent 2005b],has a natural semantical counterpart in our polarized phase spaces; a topo-logical structure derived from the exponential connectives of linear logiccoincides with a topological structure for the polarity. Moreover, syntac-tically, under the restriction, the focalized sequent property is recovered.Accordingly our main goal in this paper is to establish the conservativity ofLLη2 over its polarized fragment LLη

pol2 (Theorem 12.1.16). The conservativ-ity follows from our main proposition (Proposition 12.1.15) which ensuresthat if a non polarized sequent is provable in LLη2, then it is canonicallydecorated with ! and ? so that the transformed polarized sequent is provablein LLη

pol2.

The rest of this Part II is organized as follows. In Chapter 9, we brieflyreview syntax of second order propositional linear logic LL2 of [Girard 1987],and its polarized fragment LLpol2 as well as its linear fragment MALLP2 of[Laurent 2002]. We also discuss a relationship between focalization andpolarization. In Chapter 10, we review phase spaces for linear logic as wellas related categorical constructions. In Chapter 11, we introduce two kindsof polarized phase semantics−one for MALLP and one for LLpol−and extendthem to second order so as to yield complete models of MALLP2 and LLpol2,

90

respectively. Chapter 12 is concerned with conservativity of the secondorder linear logic over its polarized fragment, and there the main theorem(Theorem 12.1.16) is established.

91

Chapter 9

Preliminary 1: Syntax oflinear logic and polarizedlinear logic

Polarized linear logic [Laurent 1999, Laurent 2002] provides a framework toconstruct focusing proofs ([Andreoli 1992]) in terms of the focalized sequentproperty. The focalized sequent property of a linear logical system, whichis not necessarily a polarized system, means that if a sequent is provablewith only polarized formulas, especially in polarized linear logic, it containsat most one positive formula, in which case we call the sequent “focalized.”Since the positive formula is always focused, each proof of polarized linearlogic gives a focusing proof in Andreoli’s sense.

In Section 9.1, we review second order linear logic LL2. Second order po-larized linear logic LLpol2 is obtained in Section 9.2.1 by imposing a naturalpolarity restriction on formulas of LL2. In Section 9.2.2, the linear fragmentMALLP2 of LLpol2 is obtained, where the exponential operators of LLpol2 arereplaced by more primitive polarity shifting operators. In Section 9.3, wediscuss a relationship between focalization and polarization.

In this thesis, we are concerned only with propositional fragments ofsecond order logics.

9.1 Second order linear logic LL2

We briefly recall the syntax of second order linear logic LL2. [Girard 1987]is the main reference for linear logic, but introductions may be found, for

92

example, in [Girard 1995, Okada 1999b], and for more comprehensive intro-ductions, see [Curien 2005, Troelstra 1992]. For a philosophical aspect oflinear logic, see [Okada 2004]. For an overview of linear logic programming,see [Miller 2004].

Formulas of LL2 are given by the following grammar:

A,B ::= X | A⊗B | A⊕B | 1 | 0 | !A | ∃X.AX⊥ | A

............................................................................................... B | A & B | ⊥ | ⊤ | ?A | ∀X.A

For any formula A, its linear negation A⊥ is defined by induction on Ausing the following de Morgan laws:

(X)⊥ := X⊥ (X⊥)⊥ := X

(A⊗B)⊥ := A⊥............................................................................................... B⊥ (A.................................................

.............................................. B)⊥ := A⊥ ⊗B⊥

(A⊕B)⊥ := A⊥ & B⊥ (A & B)⊥ := A⊥ ⊕B⊥

1⊥ := ⊥ ⊥⊥ := 1

0⊥ := ⊤ ⊤⊥ := 0

(!A)⊥ := ?A⊥ (?A)⊥ := !A⊥

(∃X.A)⊥ := ∀X.A⊥ (∀X.A)⊥ := ∃X.A⊥

Notation: Roman capitals A,B, . . . stand for formulas, and Greek capitalsΓ,∆, . . . ,Γ1, Γ2, . . . stand for arbitrary (finite) multisets of formulas, includ-ing the case of the empty sequence. When Γ is A1, A2, . . . , An, by !Γ, wemean !A1, !A2, . . . , !An and by ?Γ, we mean ?A1, ?A2, . . . , ?An. Sequents areexpressions of the form ⊢ Γ.

Under the concept of resource sensitivity (i.e., control of structural infer-ence rules of Gentzen’s sequent calculus), linear logical formulas are classifiedinto the following three groups:

Multiplicatives ⊗ (conjunction), ............................................................................................... (disjunction), 1 (true), ⊥ (false);

Additives & (conjunction), ⊕ (disjunction), ⊤ (true), 0 (false), ∀,∃ (quan-tifiers);

Exponentials !, ?.

93

We describe the one-sided sequent calculus formulation of LL2. Theexchange rule is left implicit in the following formulation, as Γ, ∆ denotemultisets of formulas.

Inference rules of LL2 are defined as follows:

Axiom and cut-rule

⊢ A,A⊥ ax ⊢ Γ, A ⊢ A⊥, ∆⊢ Γ, ∆ cut

Multiplicatives

⊢ Γ, A, B

⊢ Γ, A.................................................

.............................................. B

...............................................................................................

⊢ Γ, A ⊢ ∆, B

⊢ Γ, ∆, A⊗B⊗ ⊢ Γ

⊢ Γ,⊥ ⊥ ⊢ 1 1

Additives

⊢ Γ, A ⊢ Γ, B

⊢ Γ, A & B&

⊢ Γ, A

⊢ Γ, A⊕B⊕1

⊢ Γ, B

⊢ Γ, A⊕B⊕2 ⊢ Γ,⊤ ⊤

No rule for 0.

Exponentials

⊢ ?Γ, A

⊢ ?Γ, !A !⊢ Γ, A

⊢ Γ, ?A ? ⊢ Γ⊢ Γ, ?A ?w

⊢ Γ, ?A, ?A⊢ Γ, ?A ?c

Second order quantifiers

⊢ Γ, A

⊢ Γ,∀X.A∀

⊢ Γ, A[X := B]⊢ Γ,∃X.A

∃

In the ∀-rule, X does not appear free in the lower sequent.

The linear fragment, i.e. the exponential free fragment, of LL2 is calledsecond order multiplicative additive linear logic MALL2.

As well as the multiplicative/additive classification, the MALL2 connec-tives can be classified, from a proof-search viewpoint, into two groups ac-cording to whether or not they are reversible in the following sense:

94

Definition 9.1.1 (Reversible connective) A connective ⃝ is reversibleif for any proof π of a sequent ⊢ Γ, A, where ⃝ is the outermost connectiveof A, there is a cut-free proof π′ of ⊢ Γ, A whose last rule is ⃝-rule.A formula A is reversible if its outer most connective is reversible.

Remark 9.1.2 Reversibility of connectives is sometimes defined in termsof their introduction rule: A connective ⃝ is reversible if its premises arederived from the conclusion of ⃝-rule, see [Danos-Joinet-Schellinx 1997].With this definition, in addition to the above reversible connectives (exceptfor units), ! is also reversible.

Thus, in proof-search, the reversible formulas are decomposed immedi-ately upon their appearance in a sequent. The connectives .................................................

.............................................. , &,⊤,⊥,∀ are

reversible. (Cf. e.g., [Andreoli 1992, Danos-Joinet-Schellinx 1997]. See, forexample, [Laurent 2002] for a detailed proof.) Although the dual connec-tives ⊗,⊕,1,0,∃, called focalized connectives, are not reversible, Andreoliobserved that focalized connectives can be treated as a cluster in proof-search: Once a focalized formula is selected, i.e., focused, it is successivelydecomposed up to reversible subformulas.

Andreoli formalized this idea of focalization, in his Triadic sequent sys-tem, [Andreoli 1992]. Similar formalizations are found in [Girard 2001,Abramsky 2003, Curien 2005] among others. Andreoli’s system is shownequivalent to the usual linear logic, and the focusing proofs of his systemform a complete subset of proofs of linear logic. (See also Section 9.3.1 fora focalized sequent calculus of [Laurent 2005b].)

9.2 Second order polarized linear logic

In polarized linear logic, [Laurent 1999, Laurent 2002], the dual proof-theoreticalproperties of reversibility and focalization of linear logical connectives arecaptured simply by negative and positive polarities, respectively. One of theexcellent features of polarized linear logic is that the polarity restriction onlinear logical formulas provides a natural framework to construct Andreoli’sfocusing proofs inside the usual linear logic without modifying sequents orthe inference rules of LL2.

9.2.1 Second order polarized linear logic LLpol2

We review the syntax of the second order polarized linear logic, LLpol2, whichis the polarized fragment of LL2. LLpol2 has an important proof-theoretical

95

property called the focalized sequent property.

Polarized formulas, positive formula P and negative formula N , of LLpol2are given by the following grammar:

P ::= X | P ⊗ P | P ⊕ P | 1 | 0 | !N | ∃X.PN ::= X⊥ | N

............................................................................................... N | N & N | ⊥ | ⊤ | ?P | ∀X.N

Observe that the polarized formulas are naturally obtained by imposing thepolarity restriction on formulas of LL2.

Notation: P, Q (with or without subscript) (resp. N, L) denote positive(resp. negative) formulas; A denotes any (positive or negative) formula;Γ,∆ denote multisets of formulas; Q (resp. N ) denotes a sequent consistingonly of positive (resp. negative) formulas, called a positive sequent (resp.negative sequent). We write ?Q for a negative sequent obtained from apositive sequent Q by putting ? to each formula of Q.

Definition 9.2.1 (Focalized sequent) A sequent Γ of polarized formulasis called focalized when it contains at most one positive formula.

Inference rules of LLpol2 is the polarized fragment of LL2 where the ⊤-ruleis restricted to a focalized sequent.

Axiom and cut-rule

⊢ N, N⊥ ax ⊢ Γ, N ⊢ N⊥, ∆⊢ Γ, ∆ cut

Multiplicatives

⊢ Γ, N, L

⊢ Γ, N.................................................

.............................................. L

...............................................................................................

⊢ Γ, P ⊢ ∆, Q

⊢ Γ, ∆, P ⊗Q⊗ ⊢ Γ

⊢ Γ,⊥ ⊥ ⊢ 1 1

Additives

⊢ Γ, N ⊢ Γ, L

⊢ Γ, N & L&

⊢ Γ, P

⊢ Γ, P ⊕Q⊕1

⊢ Γ, Q

⊢ Γ, P ⊕Q⊕2 ⊢ Γ,⊤ ⊤

In the ⊤-rule, Γ is focalized, i.e., it contains at most one positiveformula.

96

Exponentials

⊢ ?Q, N

⊢ ?Q, !N !⊢ Γ, P

⊢ Γ, ?P ? ⊢ Γ⊢ Γ, ?P ?w

⊢ Γ, ?P, ?P⊢ Γ, ?P ?c


⊢ Γ, N

⊢ Γ,∀X.N∀

⊢ Γ, P [X := Q]⊢ Γ,∃X.P

∃

In the ∀-rule, X does not appear free in the lower sequent.In the ∃-rule, the formula Q substituted to X is restricted to be posi-tive.

The connectives for negative formulas except for ! are reversible:

Proposition 9.2.2 (Reversibility [Laurent 1999, 2002]) The connec-tives .................................................

.............................................. , &,∀,⊥,⊤ of LLpol2 are reversible.

As explained in [Laurent 2002], ⊗,⊕,1,∃, ?, ! are not reversible. In par-ticular, a counterexample for ! is given by the following provable sequent:⊢ !X⊥, ?X & ?X.

As for positive formulas, the following focalized sequent property 1holds,which explains why only focusing proofs are constructed because the positiveformula is always focused:

Definition 9.2.3 (Focalized sequent property (FSP)) A linear logicalsystem L, which is not necessarily a polarized system, has the focalized se-quent property (FSP) if the following holds: if ⊢ Γ is provable in L with onlypolarized formulas and by restricting the ⊤-rule to a focalized sequent, thenΓ is focalized.

Proposition 9.2.4 (Focalized sequent property [Laurent 1999, 2002])LLpol2 has FSP. That is, if ⊢ Γ is provable in LLpol2, then Γ contains at mostone positive formula.

Remark 9.2.5 LLP2 of [Laurent 2002] is obtained by adding the followingrules to LLpol2:

⊢ N , N

⊢ N , !N N ! ⊢ Γ⊢ Γ, N

Nw⊢ Γ, N,N

⊢ Γ, NNc.

1 This property is called the positive formula property in [Laurent 2002].

97

LLP2 is not a subsystem of LL2 because the above N !-, Nw- and Nc-rulesstrictly generalize the !-, ?w- and ?c-rules of LLpol2 respectively. These rulesof LLpol2 are restrictions of the above LLP2 rules in the sense that negativeformulas explicitly designated in the conclusions are of the particular form?P .

9.2.2 Second order multiplicative additive polarized linearlogic MALLP2

Laurent [2002] also introduced the linear fragment of LLpol2: second ordermultiplicative additive polarized linear logic MALLP2, which is the mainsyntax of Girard’s Ludics, [Girard 2001]. In MALLP2, the exponentials ! and? of LLpol2 are replaced by the more primitive polarity shifting operators ↓and ↑, respectively, which do not admit contraction or weakening.

Polarized formulas of MALLP2 are given by the following grammar:

P ::= X | P ⊗ P | P ⊕ P | 1 | 0 | ↓N | ∃X.PN ::= X⊥ | N

............................................................................................... N | N & N | ⊥ | ⊤ | ↑P | ∀X.N

↓ and ↑ are called polarity shifting operators.

Inference rules of MALLP2 are defined as follows:

Axiom and cut-rule

⊢ N, N⊥ ax ⊢ Γ, N ⊢ N⊥, ∆⊢ Γ, ∆ cut

Multiplicatives

⊢ Γ, N, L

⊢ Γ, N.................................................

.............................................. L

...............................................................................................

⊢ Γ, P ⊢ ∆, Q

⊢ Γ, ∆, P ⊗Q⊗ ⊢ Γ

⊢ Γ,⊥ ⊥ ⊢ 1 1

Additives⊢ Γ, N ⊢ Γ, L

⊢ Γ, N & L&

⊢ Γ, P

⊢ Γ, P ⊕Q⊕1

⊢ Γ, Q

⊢ Γ, P ⊕Q⊕2 ⊢ Γ,⊤ ⊤

In the ⊤-rule, Γ contains at most one positive formula.

Polarity shiftings⊢ N ,N⊢ N , ↓N

↓⊢ Γ, P

⊢ Γ, ↑P↑

In the ↓-rule, N consists only of negative formulas.

98


⊢ Γ, N

⊢ Γ,∀X.N∀

⊢ Γ, P [X := Q]⊢ Γ,∃X.P

∃

In the ∀-rule, X does not appear free in the lower sequent.In the ∃-rule, the formula Q substituted to X is restricted to be posi-tive.

Proposition 9.2.6 (Focalized sequent property [Laurent 2002])MALLP2 has FSP.

In contrast to the exponential ! of LLpol2, the more primitive polarityshifting operator ↓ of MALLP2 is shown to be reversible:

Proposition 9.2.7 (Reversibility [Laurent 2002]) The connectives.................................................

.............................................. , &,⊤,⊥,∀, ↓ of MALLP2 are reversible.

Thus, in MALLP2, all of the connectives for negative formulas are re-versible.

Remark 9.2.8 (Polarized logic and tensorial logic) MALLP is preciselythe monolateral presentation of the multiplicative additive tensorial logic of[Mellies-Tabareau 2007]. Tensorial logic is then extended with an expo-nential modality !exp, which does not change the polarity of formulas (theexponential !expP of a positive formula P is still a positive formula). This isa main difference with polarized logic LLpol. However LLpol may be easily en-coded in tensorial logic, the exponential modality ! of LLpol being recoveredby the formula:

!N := !exp↓N

where ↓ transforms the negative formula into a positive formula. This ex-plains why the exponential modality in LLpol changes the polarity of formu-las.

9.3 Focalization and polarization

In this section, we discuss a relationship between focalization and polariza-tion of LL without the second order quantifiers as depicted in the followingfigure:

99

LL

(1) focalization;

R

(2) polarization↑↓

LLfoc-� ≃

(3) LL↑↓pol

(1) In [Laurent 2005b], Laurent defined a focalized sequent calculus LLfoc,which formalizes the following main idea of Andreoli’s focalization: the de-compositions of two positive formulas are not interleaved. Then he defineda translation, we call focalization, of LL into LLfoc. (Cf. Proposition 9.3.1of Section 9.3.1.) Focusing proofs of LL are easily obtained through thetranslation as well as the cut-elimination theorem of LLfoc.

(2) Laurent also introduced a polarized system LL↑↓pol, which is obtained

from LLpol by adding polarity shifting operators ↑ and ↓ (see Appendix 2 of[Laurent 2005b]). We define a natural translation, we call polarization, ofLL into LL↑↓

pol: LL-sequents are naturally polarized by adding to each formulaexactly the required polarity shifting operators to get a polarized formula.LL-proofs are also polarized by inserting several polarity shifting rules andcut rules. (Cf. Proposition 9.3.2 of Section 9.3.2.)

(3) On the one hand, LLfoc is designed for focusing proofs of LL; on the otherhand, LL↑↓

pol is designed for the notion of polarity on LL-formulas. Althoughthey are formalized from different viewpoints, the two systems are shown tobe almost isomorphic. (Cf. Proposition 9.3.4 of Section 9.3.3.)

Furthermore, we show that the focalization procedure of LL into LLfoc

given in [Laurent 2005b] is exactly the same as our polarization procedureof LL into LL↑↓

pol given in Section 9.3.2. That is, focalization and polarizationcan be given by exactly the same procedure.

In this way, the notion of polarity neatly captures the notion of focaliza-tion.

9.3.1 Focalization of linear logic LL into focalized sequentcalculus LLfoc

Among several systems to formalize Andreoli’s focalization (e.g., [Andreoli 1992,Girard 2001, Abramsky 2003, Curien 2005]), we review a focalized sequentcalculus LLfoc of [Laurent 2005b], where the following main idea of focal-ization is formalized: The decompositions of two positive formulas are notinterleaved.

100

Formulas of LLfoc are those of LL, and a formula is called positive, de-noted by P or Q, if its outermost connective is one of ⊗,⊕, !,1,0; a formulais called negative, denoted by N or M , if its outermost connective is oneof .................................................

.............................................. , &, ?,⊥,⊤. In particular, an atomic formula X without negation is

positive, and X⊥ with negation is negative. 2 A sequent of LLfoc has theshape ⊢ Γ ; Π where Γ is a multiset of formulas and Π is either empty orcontains a unique positive formula.

The following inference rules of LLfoc are linear logic version of the rulesof Girard’s LC [Girard 1991].

⊢ X⊥ ; Xax ⊢ Γ ; P

⊢ Γ, P ;foc

⊢ Γ ; P ⊢ ∆, P⊥ ; Π⊢ Γ, ∆ ; Π

p-cut⊢ Γ, P ; Π ⊢ ∆, P⊥ ;

⊢ Γ,∆ ; Π n-cut

⊢ Γ, A, B ; Π⊢ Γ, A

............................................................................................... B ; Π

...............................................................................................

⊢ Γ ; P ⊢ ∆ ; Q

⊢ Γ, ∆ ; P ⊗Q⊗

⊢ Γ ; P ⊢ ∆,M ;⊢ Γ, ∆ ; P ⊗M

⊗ ⊢ Γ, N ; ⊢ ∆ ; Q

⊢ Γ, ∆ ; N ⊗Q⊗ ⊢ Γ, N ; ⊢ ∆,M ;

⊢ Γ,∆ ; N ⊗M⊗

⊢ Γ, A ; Π ⊢ Γ, B ; Π⊢ Γ, A & B ; Π &

⊢ Γ ; P

⊢ Γ ; P ⊕B⊕1

⊢ Γ, N ;⊢ Γ ; N ⊕B

⊕1⊢ Γ ; Q

⊢ Γ ; A⊕Q⊕2

⊢ Γ,M ;⊢ Γ ; A⊕M

⊕2

⊢ ; 1 1⊢ Γ ; Π⊢ Γ,⊥ ; Π ⊥ ⊢ Γ,⊤ ; Π ⊤

⊢ ?Γ, A ;⊢ ?Γ ; !A !

⊢ Γ ; P

⊢ Γ, ?P ; ?⊢ Γ, N ;⊢ Γ, ?N ; ?

2 Note that the formulas of LLfoc are not polarized formulas in general.

101

⊢ Γ ; Π⊢ Γ, ?A ; Π ?w

⊢ Γ, ?A, ?A ; Π⊢ Γ, ?A ; Π ?c

The focalized sequent calculus LLfoc enjoys cut-elimination ([Laurent 2005b]).

Laurent defined a translation ( )• of LL into LLfoc, which does not modifyformulas, translates a sequent ⊢ Γ to ⊢ Γ ; and acts on proofs by adding alot of cut rules. Then he obtained the following proposition:

Proposition 9.3.1 (Focalization) If ⊢ Γ is provable in LL, then ⊢ Γ ; isprovable in LLfoc.

Proof. See [Laurent 2005b]. See also Proposition 9.3.2.

From a given cut-free LLfoc-proof, simply by erasing all the “ ; ” in thesequents, we obtain a LL-proof which is focalized.

9.3.2 Polarization of linear logic LL into polarized linear logicwith shiftings LL↑↓

pol

Laurent introduced polarized linear logic with shiftings LL↑↓pol, which is ob-

tained from LLpol by adding the following polarity shifting rules:

⊢ Γ, P

⊢ Γ, ↑P↑

⊢ N ,M

⊢ N , ↓M↓

See Appendix 2 of [Laurent 2005b].

LL is naturally embedded into LL↑↓pol. In order to give a polarization

procedure of LL into LL↑↓pol, we first define a polarization σ on formulas

and sequents. Each LL-formula is polarized by adding exactly the requiredpolarity shifting operators to get a polarized formula:

σ(X⊥) = X⊥ σ(X) = Xσ(A.................................................

.............................................. B) = ↑σ(A).................................................

.............................................. ↑σ(B) σ(A⊗B) = ↓σ(A)⊗ ↓σ(B)

σ(?A) = ?↓σ(A) σ(!A) = !↑σ(A)σ(A & B) = ↑σ(A) & ↑σ(B) σ(A⊕B) = ↓σ(A)⊕ ↓σ(B)

102

We assume ↓P = P and ↑N = N for any positive formula P and for anynegative formula N .

LL-sequents are translated to LL↑↓pol-sequents as follows:

σ(⊢ Γ) = ⊢ ↑σ(Γ)

where ↑σ(Γ) is ↑σ(A1), . . . , ↑σ(An) when Γ is A1, . . . , An.

Then we have the following proposition:

Proposition 9.3.2 (Polarization) If ⊢ A1, . . . , An is provable in LL, then⊢ ↑σ(A1), . . . , ↑σ(An) is provable in LL↑↓

pol.

Proof. We give a translation of each inference rule of LL into a combinationof LL↑↓

pol rules. Note that the following translation is exactly the same as thetranslation, i.e., focalization, of LL into LLfoc given in [Laurent 2005b]. Weomit the cases for units:

1. ⊢ X⊥, Xax

is translated into

⊢ X⊥, Xax

⊢ X⊥, ↑X↑

2.⊢ Γ, A, B

⊢ Γ, A.................................................

.............................................. B

............................................................................................... is translated into

⊢ ↑σ(Γ), ↑σ(A), ↑σ(B)⊢ ↑σ(Γ), ↑σ(A).................................................

.............................................. ↑σ(B)

...............................................................................................

where ↑σ(A............................................................................................... B) = ↑σ(A).................................................

.............................................. ↑σ(B).

3.⊢ Γ, P ⊢ ∆, Q

⊢ Γ, ∆, P ⊗Q⊗ is translated into

⊢ ↑σ(∆), ↑σ(Q)

⊢ ↑σ(Γ), ↑σ(P )

⊢ σ(P )⊥, σ(P ) ⊢ σ(Q)⊥, σ(Q)

⊢ σ(P )⊥, σ(Q)⊥, σ(P )⊗ σ(Q)⊗

⊢ σ(P )⊥, σ(Q)⊥, ↑(σ(P )⊗ σ(Q))↑

⊢ ↓(σ(P )⊥), σ(Q)⊥, ↑(σ(P )⊗ σ(Q))↓

⊢ ↑σ(Γ), σ(Q)⊥, ↑(σ(P )⊗ σ(Q))cut

⊢ ↑σ(Γ), ↓(σ(Q)⊥), ↑(σ(P )⊗ σ(Q))↓

⊢ ↑σ(∆), ↑σ(Γ), ↑(σ(P )⊗ σ(Q))cut

where ↑σ(P ⊗Q) = ↑(σ(P )⊗ σ(Q)).

103

4.⊢ Γ, N ⊢ ∆, Q

⊢ Γ, ∆, N ⊗Q⊗ is translated into

⊢ ↑σ(Γ), ↑σ(N) = σ(N)⊢ ↑σ(Γ), ↓σ(N)

↓⊢ ↑σ(∆), ↑σ(Q)

⊢ (↓σ(N))⊥, ↓σ(N) ⊢ σ(Q)⊥, σ(Q)

⊢ (↓σ(N))⊥, σ(Q)⊥, ↓σ(N)⊗ σ(Q)⊗

⊢ (↓σ(N))⊥, σ(Q)⊥, ↑(↓σ(N)⊗ σ(Q))↑

⊢ (↓σ(N))⊥, ↓(σ(Q)⊥), ↑(↓σ(N)⊗ σ(Q))↓

⊢ ↑σ(∆), (↓σ(N))⊥, ↑(↓σ(N)⊗ σ(Q))cut

⊢ ↑σ(Γ), ↑σ(∆), ↑(↓σ(N)⊗ σ(Q))cut

where ↑σ(N ⊗Q) = ↑(↓σ(N)⊗ σ(Q)).

⊗-rule for P ⊗M is similar.

5.⊢ Γ, N ⊢ ∆,M

⊢ Γ, ∆, N ⊗M⊗ is translated into

⊢ ↑σ(Γ), ↑σ(N) = σ(N)⊢ ↑σ(Γ), ↓σ(N)

↓⊢ ↑σ(∆), ↑σ(M) = σ(M)⊢ ↑σ(∆), ↓σ(M)

↓

⊢ ↑σ(Γ), ↑σ(∆), ↓σ(N)⊗ ↓σ(M)⊗

⊢ ↑σ(Γ), ↑σ(∆), ↑(↓σ(N)⊗ ↓σ(M))↑

where ↑σ(N ⊗M) = ↑(↓σ(N)⊗ ↓σ(M)).

6.⊢ Γ, A ⊢ Γ, B

⊢ Γ, A & B& is translated into

⊢ ↑σ(Γ), ↑σ(A) ⊢ ↑σ(Γ), ↑σ(B)⊢ ↑σ(Γ), ↑σ(A) & ↑σ(B) &

where ↑σ(A & B) = ↑σ(A) & ↑σ(B).

7.⊢ Γ, P

⊢ Γ, P ⊕B⊕1 is translated into

⊢ ↑σ(Γ), ↑σ(P )

⊢ σ(P )⊥, σ(P )⊢ σ(P )⊥, σ(P )⊕ ↓σ(B)

⊕1

⊢ σ(P )⊥, ↑(σ(P )⊕ ↓σ(B))↑

⊢ ↓(σ(P )⊥), ↑(σ(P )⊕ ↓σ(B))↓

⊢ ↑σ(Γ), ↑(σ(P )⊕ ↓σ(B))cut

where ↑σ(P ⊕B) = ↑(σ(P )⊕ ↓σ(B)).

104

8.⊢ Γ, N

⊢ Γ, N ⊕B⊕1 is translated into

⊢ ↑σ(Γ), ↑σ(N) = σ(N)⊢ ↑σ(Γ), ↓σ(N)

↓

⊢ (↓σ(N))⊥, ↓σ(N)⊢ (↓σ(N))⊥, ↓σ(N)⊕ ↓σ(B)

⊕1

⊢ (↓σ(N))⊥, ↑(↓σ(N)⊕ ↓σ(B))↑

⊢ ↑σ(Γ), ↑(↓σ(N)⊕ ↓σ(B))cut

where ↑σ(N ⊕B) = ↑(↓σ(N)⊕ ↓σ(B)).

⊕2-rule is similar.

9.⊢ ?Γ, A

⊢ ?Γ, !A ! is translated into

⊢ ↑σ(?Γ), ↑σ(A)⊢ ↑σ(?Γ), !↑σ(A) !

⊢ ↑σ(?Γ), ↑!↑σ(A)↑

where ↑σ(?Γ) = ?↓σ(Γ), and ↑σ(!A) = ↑!↑σ(A).

10.⊢ Γ, P

⊢ Γ, ?P ? is translated into

⊢ ↑σ(Γ), ↑σ(P )

⊢ σ(P )⊥, σ(P )⊢ σ(P )⊥, ?σ(P )

?

⊢ ↓(σ(P )⊥), ?σ(P )↓

⊢ ↑σ(Γ), ?σ(P )cut

where ↑σ(?P ) = ?σ(P ).

11.⊢ Γ, N

⊢ Γ, ?N ? is translated into

⊢ ↑σ(Γ), ↑σ(N) = σ(N)⊢ ↑σ(Γ), ↓σ(N)

↓

⊢ ↑σ(Γ), ?↓σ(N) ?

where ↑σ(?N) = ?↓σ(N).

12. ⊢ Γ⊢ Γ, ?A ?w is translated into

⊢ ↑σ(Γ)⊢ ↑σ(Γ), ?↓σ(A) ?w

where ↑σ(?A) = ?↓σ(A).

105

13.⊢ Γ, ?A, ?A⊢ Γ, ?A ?c is translated into

⊢ ↑σ(Γ), ?↓σ(A), ?↓σ(A)⊢ ↑σ(Γ), ?↓σ(A) ?c

where ?↓σ(A) = ↑σ(?A).

14. ⊢ Γ, P ⊢ ∆, P⊥

⊢ Γ, ∆ cut is translated into

⊢ ↑σ(Γ), ↑σ(P )⊢ ↑σ(∆), ↑σ(P⊥) = σ(P )⊥

⊢ ↑σ(∆), ↓(σ(P )⊥)↓

⊢ ↑σ(Γ), ↑σ(∆)cut

In the above polarization, polarity shifting operators ↑, ↓ cannot be sim-ply replaced by ?, !, respectively, since the ↓-rule strictly generalizes the!-rule. However, when we consider η-expanded fragments LLη, LLη

foc, andLLη

pol, in which atoms are exponential forms (i.e., !X⊥ and ?X), since ↓coincides with ! in the fragment, it is possible to draw the same picture asbefore, where LL↑↓

pol is replaced by LLηpol (see Section 12.1.2 for second order

η-systems):

LLη

focalization;

R

polarization

?!

LLηfoc

-� ≃LLη

pol

From the polarization of LLη into LLηpol, we obtain a partial conservativity

result:

Proposition 9.3.3 (Conservativity on negative sequents) If a polar-ized negative sequent ⊢ N is provable in LLη, then it is provable in LLη

pol.

In order to establish the general conservativity result including polarizedsequents of the form ⊢ N , P , we need a more sophisticated polarizationprocedure as given in our proof of Proposition 12.1.15.

106

9.3.3 LLfoc and LL↑↓pol are almost isomorphic

On the one hand, LLfoc is designed for focusing proofs of LL; on the otherhand, LL↑↓

pol is designed for the notion of polarity on LL-formulas. Althoughthey are formalized from different viewpoints, the two systems are shown tobe almost isomorphic.

LLfoc-formulas (i.e., LL-formulas) are translated to LL↑↓pol-formulas by the

translation σ of Section 9.3.2. Then LLfoc-sequents are polarized to LL↑↓pol-

sequents as follows:

σ(⊢ Γ ; Π) = ⊢ ↑σ(Γ), σ(Π)

Each inference rule of LLfoc is translated as follows. We omit the casesfor units:

1. ⊢ X⊥ ; Xax

is translated into

⊢ X⊥, Xax

where X⊥ = ↑σ(X⊥), and X = σ(X).

2.⊢ Γ ; P

⊢ Γ, P ;foc is translated into

⊢ ↑σ(Γ), σ(P )⊢ ↑σ(Γ), ↑σ(P )

↑

3.⊢ Γ, A, B ; Π⊢ Γ, A

............................................................................................... B ; Π


⊢ ↑σ(Γ), ↑σ(A), ↑σ(B), σ(Π)⊢ ↑σ(Γ), ↑σ(A).................................................

.............................................. ↑σ(B), σ(Π)

...............................................................................................

4.⊢ Γ ; P ⊢ ∆ ; Q

⊢ Γ, ∆ ; P ⊗Q⊗ is translated into

⊢ ↑σ(Γ), σ(P ) ⊢ ↑σ(∆), σ(Q)⊢ ↑σ(Γ), ↑σ(∆), σ(P )⊗ σ(Q)

⊗

where σ(P ⊗Q) = σ(P )⊗ σ(Q).

107

5.⊢ Γ ; P ⊢ ∆,M ;⊢ Γ, ∆ ; P ⊗M

⊗ is translated into

⊢ ↑σ(Γ), σ(P )⊢ ↑σ(∆), ↑σ(M) = σ(M)⊢ ↑σ(∆), ↓σ(M)

↓

⊢ ↑σ(Γ), ↑σ(∆), σ(P )⊗ ↓σ(M)⊗

where σ(P ⊗M) = σ(P )⊗ ↓σ(M).

⊗-rule for N ⊗Q is similar.

6.⊢ Γ, N ; ⊢ ∆,M ;⊢ Γ, ∆ ; N ⊗M

⊗ is translated into

⊢ ↑σ(Γ), ↑σ(N) = σ(N)⊢ ↑σ(Γ), ↓σ(N)

↓⊢ ↑σ(∆), ↑σ(M) = σ(M)⊢ ↑σ(∆), ↓σ(M)

↓

⊢ ↑σ(Γ), ↑σ(∆), ↓σ(N)⊗ ↓σ(M)⊗

where σ(N ⊗M) = ↓σ(N)⊗ ↓σ(M).

7.⊢ Γ, A ; Π ⊢ Γ, B ; Π⊢ Γ, A & B ; Π & is translated into

⊢ ↑σ(Γ), ↑σ(A), σ(Π) ⊢ ↑σ(Γ), ↑σ(B), σ(Π)⊢ ↑σ(Γ), ↑σ(A) & ↑σ(B), σ(Π) &

where ↑σ(A & B) = ↑σ(A) & ↑σ(B).

8.⊢ Γ ; P

⊢ Γ ; P ⊕B⊕1 is translated into

⊢ ↑σ(Γ), σ(P )⊢ ↑σ(Γ), σ(P )⊕ ↓σ(B)

⊕1

where σ(P ⊕B) = σ(P )⊕ ↓σ(B).

9.⊢ Γ, N ;⊢ Γ ; N ⊕B

⊕1 is translated into

⊢ ↑σ(Γ), ↑σ(N) = σ(N)⊢ ↑σ(Γ), ↓σ(N)

↓

⊢ ↑σ(Γ), ↓σ(N)⊕ ↓σ(B)⊕1

where σ(N ⊕B) = ↓σ(N)⊕ ↓σ(B).

⊕2-rules is similar.

108

10.⊢ ?Γ, A ;⊢ ?Γ ; !A ! is translated into

⊢ ↑σ(?Γ), ↑σ(A)⊢ ↑σ(?Γ), !↑σ(A) !

where ↑σ(?Γ) = ?↓σ(Γ), and σ(!A) = !↑σ(A).

11.⊢ Γ ; P

⊢ Γ, ?P ; ? is translated into

⊢ ↑σ(Γ), σ(P )⊢ ↑σ(Γ), ?σ(P ) ?

where ↑σ(?P ) = ?σ(P ).

12.⊢ Γ, N ;⊢ Γ, ?N ; ? is translated into

⊢ ↑σ(Γ), ↑σ(N) = σ(N)⊢ ↑σ(Γ), ↓σ(N)

↓

⊢ ↑σ(Γ), ?↓σ(N) ?

where ↑σ(?N) = ?↓σ(N).

13.⊢ Γ ; Π⊢ Γ, ?A ; Π ?w is translated into

⊢ ↑σ(Γ), σ(Π)⊢ ↑σ(Γ), ?↓σ(A), σ(Π) ?w


14.⊢ Γ, ?A, ?A ; Π⊢ Γ, ?A ; Π ?c is translated into

⊢ ↑σ(Γ), ?↓σ(A), ?↓σ(A), σ(Π)⊢ ↑σ(Γ), ?↓σ(A), σ(Π) ?c


15. ⊢ Γ ; P ⊢ ∆, P⊥ ; Π⊢ Γ, ∆ ; Π

p-cut is translated into

⊢ ↑σ(Γ), σ(P ) ⊢ ↑σ(∆), ↑σ(P⊥), σ(Π)⊢ ↑σ(Γ), ↑σ(∆), σ(Π)

cut

where ↑σ(P⊥) = (σ(P ))⊥.

109

16. ⊢ Γ, P ; Π ⊢ ∆, P⊥ ;⊢ Γ,∆ ; Π n-cut is translated into

⊢ ↑σ(Γ), ↑σ(P ), σ(Π)⊢ ↑σ(∆), ↑σ(P⊥) = σ(P⊥)↑σ(∆), ↓σ(P⊥) = (↑σ(P ))⊥

↓

⊢ ↑σ(Γ), ↑σ(∆), σ(Π)cut

Conversely, a translation τ of LL↑↓pol to LLfoc is defined as follows: For any

LL↑↓pol formula A, a LLfoc formula τ(A) is obtained by simply erasing polarity

shifting operators ↑ and ↓ of A. Then, each sequent ⊢ N ,Π of LL↑↓pol, where

Π contains at most one positive formula, is translated as ⊢ τ(N ) ; τ(Π) ofLLfoc.

Each inference rule of LL↑↓pol is translated as follows. In what follows, for

notational simplicity, we write A for τ(A). We omit the cases for units:

1. ⊢ X⊥, Xax

is translated into

⊢ X⊥ ; Xax

2.⊢ N , P ⊢ M, Q

⊢ N ,M, P ⊗Q⊗ is translated into

⊢ N ; P ⊢ M ; Q

⊢ N ,M ; P ⊗Q⊗

3.⊢ N , N, M, Π⊢ N , N

............................................................................................... M, Π


⊢ N , N, M ; Π⊢ N , N

............................................................................................... M ; Π

...............................................................................................

4.⊢ N , N, Π ⊢ N ,M, Π⊢ N , N & M, Π & is translated into

⊢ N , N ; Π ⊢ N ,M ; Π⊢ N , N & M ; Π &

5.⊢ N , P

⊢ N , P ⊕Q⊕1 is translated into

⊢ N ; P

⊢ N ; P ⊕Q⊕1

⊕2-rule is similar.

110

6.⊢ ?Q, N

⊢ ?Q, !N ! is translated into

⊢ ?Q, N ;⊢ ?Q ; !N !

7.⊢ N , P

⊢ N , ?P ? is translated into

⊢ N ; P

⊢ N , ?P ; ?

8.⊢ N , Π⊢ N , ?P, Π ?w is translated into

⊢ N ; Π⊢ N , ?P ; Π ?w

9.⊢ N , ?P, ?P, Π⊢ N , ?P, Π ?c

⊢ N , ?P, ?P ; Π⊢ N , ?P ; Π ?c

10.⊢ N , P

⊢ N , ↑P↑, for P not of the form ↓M , is translated into

⊢ N ; P

⊢ N , P ;foc

11.⊢ N , ↓M⊢ N , ↑↓M

↑ is translated into

⊢ N ,M ;

12.⊢ N ,M

⊢ N , ↓M↓ is translated into

⊢ N ,M ;

13. ⊢ N , P ⊢ P⊥,M, Π⊢ N ,M, Π cut is translated into

⊢ N ; P ⊢ P⊥,M ; Π⊢ N ,M ; Π

p-cut

111

In order to translate any LL↑↓pol-proof to LLfoc-proof, we assume that

redundant applications of ↑, ↓ rules in LL↑↓pol-proofs are removed in advance

by using the following equations:

↑↓↑P◦−◦↑P and ↓↑↓N◦−◦↓N.

Based on the above two translations σ and τ , we obtain the followingproposition:

Proposition 9.3.4 (LLfoc and LL↑↓pol) There are translations σ : LLfoc −→

LL↑↓pol and τ : LL↑↓

pol −→ LLfoc such that:

• For any formula A of LLfoc, τ(σ(A)) = A; for any proof π of LLfoc,τ(σ(π)) = π.

• For any formula A of LL↑↓pol, σ(τ(A)) = A; for any proof π of LL↑↓

pol,σ(τ(π)) = π.

112

Chapter 10

Preliminary 2: Categories forphase semantics of linearlogic

Phase semantics was introduced in [Girard 1987] for a set-theoretical seman-tics of LL. It has been quite useful to show various results of linear logic, suchas the finite model property [Lafont 1997, Okada-Terui 1999], a semanticcharacterization of cut-elimination [Ciabattoni-Terui 2006, Terui 2007], andthe relationship of phase semantics to the denotational model [Bucciarelli-Ehrhard 2000,Ehrhard 2004]. In addition, phase semantics of LL is naturally extendable tosecond order, as is seen in [Okada 1999]. In particular, in the phase seman-tics for LL2 of [Okada 1999], a strong form of completeness is given, yield-ing the second order cut-elimination for LL2, for which syntactical methodshardly work.

We review phase spaces for multiplicative additive linear logic (MALL)in Section 10.1, and for linear logic (LL) in Section 10.3 as well as relatedcategorical constructions. Category theory may enable us to understandphase semantic constructions more abstractly and more structurally, andthis would help us see the full picture of the constructions. For that pur-pose, we also recall some definitions and properties related to categoricalmodels of linear logic sufficient for phase semantics. We refer mainly to[Asperti-Longo 1991, Amadio-Curien 1998, Mellies 2003, Mellies 2009, Blute-Scott 2004]for the following definitions.

113

10.1 ∗-autonomous category with products and phasespace for MALL

MALL is complete with respect to Girard’s phase space, [Girard 1987], whichdefines an algebraic instance of ∗-autonomous category with products. Inthis section, by recalling a construction of ∗-autonomous category, we reviewthe phase space for MALL. Let us begin with the following definition ofmonoidal category.

A monoidal category (C,⊗,1) consists of the following data:

• A category C.• A bifunctor ⊗ : C × C −→ C.• A unit object 1 ∈ C.• Natural isomorphisms: for any objects A, B,C ∈ C,

αA,B,C : A⊗ (B ⊗ C) −→ (A⊗B)⊗ C;λA : 1⊗A −→ A;ρA : A⊗ 1 −→ A;

such that the following diagrams commute:

(A⊗B)⊗ (C ⊗D) (A⊗B)⊗ (C ⊗D)

α

x α

yA⊗ (B ⊗ (C ⊗D)) ((A⊗B)⊗ C)⊗D

idA⊗α

y xα⊗idD

A⊗ ((B ⊗ C)⊗D) α−−−−→ (A⊗ (B ⊗ C))⊗D

A⊗ (1⊗ C) α−−−−→ (A⊗ 1)⊗ C

idA⊗λ

y yρ⊗idC

A⊗ C A⊗ C

As seen in the above diagrams, we sometimes omit, for notational con-venience, the indices on the maps α, λ, ρ in our diagrams.

A symmetric monoidal category (C,⊗,1) is a monoidal categorysuch that for all objects A,B there is a natural isomorphism γA,B : A⊗B −→B ⊗A such that the following diagrams commute:

114

A⊗BγA,B−−−−−−−→ B ⊗A

idA⊗B

y yγB,A

A⊗B A⊗B

B ⊗ 1γB,1−−−−−−→ 1⊗B

ρB

y yλB

B B

A⊗ (B ⊗ C) α−−−−→ (A⊗B)⊗ Cγ−−−−→ C ⊗ (A⊗B)

idA⊗γ

y yα

A⊗ (C ⊗B) α−−−−→ (A⊗ C)⊗Bγ⊗idB−−−−−−−−→ (C ⊗A)⊗B

A symmetric monoidal closed category (C,⊗,1,−◦) consists of thefollowing data:

• A symmetric monoidal category (C,⊗,1);

• an object A−◦B;

• a morphism evalA,B : A ⊗ (A −◦ B) −→ B for any objects A,B ∈ C,which satisfies the following universal property: For any morphismf : A⊗ C −→ B, there exists a unique morphism

h : C −→ A−◦B

making the following diagram commute:

A⊗ Cf−−−−→ B

idA⊗h

yA⊗ (A−◦B)

>evalA,B

We have the following simple algebraic example of a symmetric monoidalclosed category called naive phase space. The phase space is the multiplica-tive fragment of the intuitionistic phase space of [Okada-Terui 1999] whoseclosure operator ( )C is the identity.

Definition 10.1.1 (Naive phase space) Let M = (M, ·, ε) be a commu-tative monoid. We define operations on the power set P(M) of M as follows:For any α, β ∈ P(M),

α · β = {x · y | x ∈ α, y ∈ β};

115

α−◦ β = {x | x · y ∈ β for all y ∈ α};1 = {ε}.

ThenDM = (P(M), ·,1,−◦,⊆) is an algebraic example of symmetric monoidalclosed category, whose objects are subsets of M , and whose morphisms areset-inclusions among them.

Let (C,⊗,1,−◦) be a symmetric monoidal closed category. Let ⊥ bea distinguished object of C. For any object A ∈ C, we have the followingmorphism:

(A−◦ ⊥)⊗AγA−◦⊥,A−−−−−→ A⊗ (A−◦ ⊥)

evalA,⊥−−−−−→ ⊥

Then by the definition of monoidal closure, there is the following uniquemorphism ∂A called the canonical morphism:

∂A : A −→ (A−◦ ⊥)−◦ ⊥

A ∗-autonomous category (C,⊗,1,−◦,⊥) is a symmetric monoidalclosed category with a distinguished dualizing object ⊥, such that the canon-ical morphism ∂A : A −→ (A−◦ ⊥)−◦ ⊥ is an isomorphism for all A.

We write A⊥ for A−◦ ⊥, and A⊥⊥ for (A⊥)⊥.A ∗-autonomous category C is closed under duality of categorical con-

structions, and we may define A.................................................

.............................................. B = (A⊥ ⊗B⊥)⊥.

Definition 10.1.2 (Phase space for MALL) Let M = (M, ·, ε) be a com-mutative monoid, and let ⊥ be a fixed subset of M . We define operationson P(M) as follows: For any α, β ∈ P(M),

α⊗ β = (α · β)⊥⊥ α.................................................

.............................................. β = (α⊥ ⊗ β⊥)⊥ = (α⊥ · β⊥)⊥

α⊕ β = (α ∪ β)⊥⊥ α & β = α ∩ β

1 = ⊥⊥ = {ε}⊥⊥ α−◦ β = α⊥............................................................................................... β = (α · β⊥)⊥

0 = ⊤⊥ ⊤ = M

See [Girard 1987] for several properties which hold in phase spaces.Let DM⊥⊥ ⊆ P(M) be the set of facts, defined as {α ∈ P(M) | α =

α⊥⊥}. We often denote the set of facts DM⊥⊥ simply by D⊥⊥ when M isclear from the context. Then DM⊥⊥ = (DM⊥⊥,⊗,1,−◦,⊥, &,⊤,⊆) is analgebraic example of ∗-autonomous category with products, whose objectsare facts, and whose morphisms are set-inclusions among facts.

116

10.2 Monad and comonad

In the literature of categorical models of linear logic, monads and comon-ads are important notions to the study of the structure of the exponentials! and ? of linear logic. See, for example, [Seely 1989, Blute-Scott 2004,Asperti-Longo 1991].

A monad over a category C is a triple (T, µ, η), where T : C −→ C is afunctor, and µ : T 2 −→ T and η : IdC −→ T are natural transformations,such that the following diagrams commute for any object C ∈ C:

T 3CµT C−−−−−−→ T 2C

T (µC)

y yµC

T 2C −−−−−→µC

TC

TCηT C−−−−−−→ T 2C

T (ηC)←−−−−−−− TC

idT C

y yµC

yidT C

TC TC TC

µ is called the multiplication and η is called the unit of the monad.

The dual of monad is comonads, defined as follows:

A comonad over a category C is a triple (T, δ, ϵ), where T : C −→ C isa functor, and δ : T −→ T 2 and ϵ : T −→ IdC are natural transformations,such that the following diagrams commute for any object C ∈ C:

T 3CδT C←−−−−−− T 2C

T (δC)

x xδC

T 2C ←−−−−−δC

TC

TCϵT C←−−−−−− T 2C

T (ϵC)−−−−−−−→ TC

idT C

x xδC

xidT C

TC TC TC

δ is called the comultiplication and ϵ is called the counit of the comonad.

A preorder P = (P,≤) is a category, whose objects are elements of P ,and whose morphisms are the order relations among them. Let T = (T, µ, η)be a monad over a preorder P. The functor T : P −→ P gives an operationon P which is monotonic: A ≤ B implies TA ≤ TB. The multiplication µof the monad gives TTA ≤ TA (idempotency), and the unit η gives A ≤ TA(extensivity). Thus the monad T is a closure operator on P .

Dually, a comonad over the category P is an interior operator on P .

A monoidal structure is added to monads and comonads to obtain monoidalmonads and comonads. We first recall the definitions of monoidal functor

117

and monoidal natural transformation following [Blute-Scott 2004, Mellies 2003,Schalk 2004]. In what follows, we do not distinguish by our notation betweenthe tensor products, or the units, in different monoidal categories.

A symmetric monoidal functor between monoidal categories C andD is a triple (F, m1,m) where F : C −→ D is a functor, together with twonatural transformations m1 : 1 −→ F1 and mA,B : FA⊗FB −→ F (A⊗B)such that the following diagrams commute:

FA⊗ (FB ⊗ FC) α−−−−→ (FA⊗ FB)⊗ FC

idFA⊗mB,C

y ymA,B⊗idFC

FA⊗ F (B ⊗ C) F (A⊗B)⊗ FC

mA,B⊗C

y ymA⊗B,C

F (A⊗ (B ⊗ C))F (α)−−−−−−−→ F ((A⊗B)⊗ C)

FA⊗ 1ρ−−−−→ FA

idFA⊗m1

y xF (ρ)

FA⊗ F1mA,1−−−−−−−→ F (A⊗ 1)

1⊗ FBλ−−−−→ FB

m1⊗idFB

y xF (λ)

F1⊗ FBm1,B−−−−−−−→ F (1⊗B)

FA⊗ FBγ−−−−→ FB ⊗ FA

mA,B

y ymB,A

F (A⊗B)F (γ)−−−−−−→ F (B ⊗A)

A monoidal natural transformation θ : (F, m1,m) −→ (G,n1, n) be-tween symmetric monoidal functors is a natural transformation between theunderlying functors θ : F −→ G making the following diagrams commute:

1 m1−−−−−→ F1∥∥∥ yθ1

1 n1−−−−−→ G1

FA⊗ FBmA,B−−−−−−−→ F (A⊗B)

θA⊗θB

y yθA⊗B

GA⊗GBnA,B−−−−−−−→ G(A⊗B)

118

A monad (T, µ, η) (resp. comonad (T, ϵ, δ)) is symmetric monoidal if(T, m1, m) is a symmetric monoidal functor and µ and η (resp. ϵ and δ) aremonoidal natural transformations.

In Chapter 11, we extend the usual interior operator on a preorderedstructure to that over a monoidal preordered structure: the power set P(M)of a commutative monoid M , which is a posetal symmetric monoidal cat-egory. The interior operator on the structure is an algebraic example ofthe symmetric monoidal comonad. See Definition 11.1.1 and Proposition11.1.13 of Chapter 11.

10.3 Seely category and enriched phase space forLL

LL is complete with respect to enriched phase space introduced by [Lafont 1997],which is an algebraic instance of Seely category [Seely 1989]. After Seely’sintroduction of a categorical model of linear logic, several revisions wereproposed by [Benton-Bierman-Paiva-Hyland 1992, Bierman 1995] etc. See[Mellies 2003, Mellies 2009] for a review. Here we recall the definition ofSeely category [Seely 1989] following [Asperti-Longo 1991, Amadio-Curien 1998]since it is simple and sufficient to characterize enriched phase spaces.

A Seely category C = (C,⊗,1,−◦,⊥, &,⊤, !) consists of the followingdata:

• A ∗-autonomous category with products, C = (C,⊗,1,−◦,⊥, &,⊤).

• A comonad ! = (!, δ, ϵ) over C such that there exists the followingnatural isomorphisms:

I : !(A & B) ∼= !A⊗ !BJ : !⊤ ∼= 1

where ⊤ and 1 are the units for & and ⊗, the Cartesian and tensorproducts in C, respectively.

Important consequences of the definition are the following. If C is a Seelycategory, then:

• For each object A in C, there exist maps dA : !A −→ !A ⊗ !A andeA : !A −→ 1 such that (!A, dA, eA) is a comonoid, i.e., the followingdiagrams commute:

119

!A⊗!A dA←−−−−− !A dA−−−−−→ !A⊗!A

id!A⊗dA

y ydA⊗id!A

!A⊗ (!A⊗!A) (!A⊗!A)⊗!A-α

!A !A !A

λ−1!A

y dA

y yρ−1!A

1⊗!A eA⊗id!A←−−−−−−−−− !A⊗!A id!A⊗eA−−−−−−−−−→ !A⊗ 1

(Cf. Proposition 10.3.4.)

• The coKleisli category K associated with the comonad ! = (!, δ, ϵ) isCartesian closed.

Enriched phase space is obtained by augmenting the exponential ! tothe phase space for MALL. A key ingredient of the following formulationis the algebraic instance of Seely’s axiomatization of the exponential in thecategorical model (cf. [Seely 1989] as well as [Bierman 1995, Mellies 2003]).Similar axiomatizations are found in the literature of algebraic semantics[Troelstra 1992, Ono 1993, Abrusci 1990].

Definition 10.3.1 (Enriched phase space) An enriched phase space M =(M,⊥, !) is a phase space (M,⊥) together with an exponential operator !from D⊥⊥ to D⊥⊥ which satisfies the following conditions: For any α, β ∈D⊥⊥,

(Intensivity) !α ⊆ α

(Idempotency) !α ⊆ !!α

(Monotonicity) If α ⊆ β, then !α ⊆ !β

(Seely axiom 1) !α⊗ !β = !(α & β)

(Seely axiom 2) 1 = !⊤, where ⊤ = M

The dual operator ? of ! is defined as ?α = (!(α⊥))⊥.

Remark 10.3.2 (Seely axioms) Under the conditions Intensivity, Idem-potency, and Monotonicity of Definition 10.3.1, the Seely axioms 1 and 2are equivalent to the following conditions (Cf. [Ono 1993]):

120

(Contraction) !α ⊆ !α⊗ !α

(Weakening) !α ⊆ 1

(Openness of ⊗) !α⊗ !β ⊆ !(α⊗ β)

(Openness of 1) 1 ⊆ !1

Example 10.3.3 Let M be a phase space. Let J = {x ∈ 1 | x · x = x}, asubmonoid of the domain M . If we define !α = (α ∩ J)⊥⊥, then we have anenriched phase space. This definition of ! is due to [Lafont 1997].

Proposition 10.3.4 (Enriched phase space) LetD⊥⊥ be the ∗-autonomouscategory of facts of an enriched phase space M = (M,⊥, !). The exponentialoperator ! gives rise to a comonad over D⊥⊥ so that (D⊥⊥, !) forms a Seelycategory, and vice versa. In (D⊥⊥, !), for each α ∈ D⊥⊥, !α is endowed witha comonoid structure, and moreover, the comonad ! is monoidal.

Proof. By Monotonicity, the exponential gives rise to a functor ! from D⊥⊥to D⊥⊥. Then natural transformations ϵ : ! −→ IdD⊥⊥ and δ : ! −→ ! ◦ !are given by Intensivity and Idempotency, respectively. Thus we obtain acomonad ! = (!, δ, ϵ) over D⊥⊥. Seely axioms 1 and 2 correspond to theSeely’s natural isomorphisms.

The following morphisms dα : !α −→ !α ⊗ !α and eα : !α −→ 1 arederivable for each α ∈ D⊥⊥, showing that (!α, dα, eα) is a comonoid in D⊥⊥:

(Contraction) dα : !α!⟨id,id⟩−−−−→ !(α & α) = !α⊗ !α

(Weakening) eα : !α !∗α−−→ !⊤ = 1,where ∗α : α −→ ⊤ is the unique map to the terminal object ⊤ of theposetal D⊥⊥.

In addition, the following morphisms are derivable, describing the monoidal-ness of the comonad !: For any α, β ∈ D⊥⊥,

mα,β : !α⊗ !β = !(α & β) δ−→ !!(α & β) = !(!α⊗ !β)!(ϵ⊗ϵ)−−−→ !(α⊗ β)

m1 : 1 = !⊤ δ−→ !!⊤ = !1

10.4 Polarized ∗-autonomous category

Hamano-Scott [2007] introduced the polarized ∗-autonomous category forMALLP, [Laurent 2002]. Their category is based on an adjunction between

121

reflective/coreflective full subcategories C−/C+ of an ambient ∗-autonomouscategory C with products.

A similar adjunction has recently been discussed in [Mellies 2005] inorder to solve the Blass problem, and to define a game model of full lin-ear logic (LL). See also [Mellies 2005b]. It has been also discussed in[Selinger 2001, Hofmann-Streicher 2002] in their continuation-passing-stylemodels of λµ-calculus. Most recently, motivated by an observation of M.Hasegawa, Mellies-Tabareau [2007] announced a categorization of phasesemantics by using the continuation monad in their categorical semanticsbased on the adjunction.

Hamano-Scott’s polarized ∗-autonomous category (with products)C+,− consists of the following data:

• A ∗-autonomous category C with products (and hence coproducts).

• A full subcategory C+ of C (called the positive subcategory) which isclosed under the positive operations ⊗ and ⊕, along with their respec-tive unit 1 and 0, along with the induced monoidal structure withrespect to both connectives.

• A full subcateogry C− of C (called the negative subcategory), whichis closed under the negative operations .................................................

.............................................. and &, along with their

respective unit ⊥ and ⊤, along with the induced monoidal structurewith respect to both connectives.

• The contravariant equivalence ( )⊥ on C induces a contravariant equiv-alence of the two subcategories:

(−)⊥ : (C+)op −→ C−.

• The subcategory C− (resp. C+) is a reflective (resp. coreflective) sub-category of C. That is, there are distinguished functors

↓ : C −→ C+ and ↑ : C −→ C−

satisfying: ↓ is right adjoint to the inclusion Inj+ : C+ ↪→ C, and ↑ isleft adjoint to the inclusion Inj− : C− ↪→ C, i.e.,

C[Inj+(P ), X] ∼= C+[P, ↓X] and C[X, Inj−(N)] ∼= C−[↑X, N ]

for all A ∈ C, P ∈ C+ and N ∈ C−.

122

• De Morgan duality for ↓ and ↑:

(↓A)⊥ ∼= ↑(A⊥) and (↑A)⊥ ∼= ↓(A⊥)

See [Hamano-Scott 2007] for a detailed account of the polarized ∗-autonomouscategory.

In the next Chapter 11, we introduce polarized phase space for MALLP,which is an algebraic instance of polarized ∗-autonomous category with prod-ucts. Cf. Definition 11.1.9 and Proposition 11.1.10 and 11.1.11.

Remark 10.4.1 Polarized ∗-autonomous categories define a particular caseof dialogue categories, which play the same role for tensor logic as ∗-autonomouscategories for linear logic. In particular, the negation functor of a dialoguecategory

¬ : C+ −→ C+can be defined in a polarized ∗-autonomous category as the functor

¬(A) = ↓ ◦ Inj ◦ op(A)

that is,

¬ : C+op−→ C−

Inj−−−→ C ↓−→ C+Note in particular that C+[P ⊗ Q,¬R] ∼= C+[P ⊗ Q ⊗ R,¬1] where ∼= is anatural bijection.

123

Chapter 11

A phase semantics forpolarized linear logic

In this chapter, we introduce a phase semantics for second order polarizedlinear logic. The main feature of our polarized phase semantics is its employ-ment of a topological structure, which accommodates the positive/negativepolarities as openness/closedness. This interpretation is an algebraic in-stance of the categorical construction developed in [Hamano-Scott 2007]and is based upon the adjunction between interior and closure operatorsfor the topology. Few attempts have so far been made to clarify the no-tion of polarity in terms of topology, although as regards usual linear logic,topolinear spaces are studied for exponential connectives (e.g., [Girard 1987,Sambin 1995]). Indeed, to the best of our knowledge, no formulation ofphase semantics which completely characterizes provability of polarized lin-ear logic has previously appeared in the literature.

In the following, we introduce two kinds of topological phase semantics:polarized phase semantics for multiplicative additive polarized linear logicMALLP in Section 11.1, and enriched polarized phase semantics for LLpol

in Section 11.3. They are extended to second order so as to yield completesemantics of MALLP2 and LLpol2, respectively. We prove their strong com-pleteness by revising Okada’s [1999] method, which implies second ordercut-eliminations. In Section 11.5, as an application of polarized phase se-mantics, we give a semantic proof of the first order conservativity of LL overLLpol of [Laurent 2002].

124

11.1 Polarized phase space for MALLP

In this subsection, we introduce a polarized phase semantics. Our motiva-tion for the semantics is a simple question: What is a semantical counterpartof the syntactical notion of polarity in phase spaces? Our ingredient for theanswer is a topological structure, which accommodates the notion of polar-ity so that positive and negative can be captured by the semantical notionof openness and closedness respectively in the topological structure. Thusthe polarity shifting operator ↓ (dually ↑) is interpreted as an interior (resp.closure) operator. This interpretation stems from the categorical construc-tion of [Hamano-Scott 2007] when applied to phase spaces in particular. Letus start by defining our interior operator.

Definition 11.1.1 (Interior operator) Let M = (M, ·, ε) be a commu-tative monoid. An interior ↓ is an operator from the power set P(M) of Mto P(M) which satisfies the following conditions. For any α, β ∈ P(M),

1. ↓α ⊆ α (Intensivity)

2. ↓α ⊆ ↓↓α (Idempotency)

3. If α ⊆ β, then ↓α ⊆ ↓β (Monotonicity)

4. ↓α · ↓β ⊆ ↓(α · β) (Openness of monoid operator)

5. {ε} ⊆ ↓{ε} (Openness of unit)

Remark 11.1.2 (Interior operator) The usual definition of interior op-erator is (1), (2), and ↓(α ∩ β) = ↓α ∩ ↓β, which is stronger than ourconditions (1), (2) and (3). Our condition (3) is enough to guarantee thatan infinite union of open sets is open. See Proposition 11.1.13 below for acategory-theoretical status of the operator.

Remark 11.1.3 (Openness of the monoid operator) A map f : M −→M is open if f(↓α) ⊆ ↓(f(α)) holds for every α ∈ P(M). Thus a binary mapf(x1, x2) is open if f(↓α, ↓β) ⊆ ↓(f(α, β)) for every α, β ∈ P(M). Hencethe above condition (4) means that the monoid operator · is an open map.The condition (5) means that the unit seen as a map ε : {ε} −→ M is anopen map.

The interior operator induces an interior operator on facts of a phasespace by defining as ↓⊥⊥α = (↓α)⊥⊥ for any fact α ∈ D⊥⊥. We denote theinterior operator also by ↓. (See [Mellies-Tabareau 2007, Okada-Terui 2005]for similar constructions of the exponential of linear logic.)

125

A closure operator is defined, in a phase space, as the de Morgan dualoperator of ↓.

Definition 11.1.4 (Closure operator) Let M = (M,⊥) be a phase space,and let ↓ be an interior operator. A closure ↑ : D⊥⊥ −→ D⊥⊥ is defined asan operator which satisfies the following conditions:

↑(α⊥) = (↓α)⊥ and (↑α)⊥ = ↓(α⊥).

We say that a set α is open (or ↓-invariant) if α = ↓α, and α is closed(or ↑-invariant) if α = ↑α.

Lemma 11.1.5 (Duality) For any fact α ∈ D⊥⊥,

↑α = (↓(α⊥))⊥.

Proof. (↓(α⊥))⊥ = ↑(α⊥⊥) = ↑α.

From Definition 11.1.4 and Lemma 11.1.5 we have the following duality.

Lemma 11.1.6 α is closed iff α⊥ is open, for any fact α ∈ D⊥⊥.

Since ↓ and ↑ are de Morgan dual with respect to ⊥ to each other, bothoperators are shown to preserve ⊥⊥-invariance.

Lemma 11.1.7 (↓ and ↑ preserve ⊥⊥-invariance) For any fact α ∈ D⊥⊥,

↓α = (↓α)⊥⊥ and ↑α = (↑α)⊥⊥.

Proof. (↓α)⊥⊥ = (↑α⊥)⊥ = ↓α⊥⊥ = ↓α. (↑α)⊥⊥ = (↓α⊥)⊥ = ↑α⊥⊥ = ↑α.

Remark 11.1.8 (Closure operator of intuitionistic phase spaces and ↑)In studies of phase spaces for intuitionistic linear logic, several authors[Abrusci 1990, Troelstra 1992, Ono 1993, Okada-Terui 1999] introduce clo-sure operators by generalizing the ( )⊥⊥-operator of classical linear logic.Although such closure operators satisfy the dual conditions for ↑ of (1), (2)and (3) of Definition 11.1.1, they do not satisfy that of (4) and (5).

A polarized phase space is a phase space augmented with an interioroperator.

126

Definition 11.1.9 (Polarized phase space) A polarized phase space isM = (M,⊥, ↓) such that

− (M,⊥) is a phase space;

− ↓ is an interior operator on DM⊥⊥.

A polarized phase space yields the domains DM+ (or simply D+) of theset of open facts of M , i.e., {α ∈ DM⊥⊥ | α = ↓α}, and DM− (or D−) ofthe set of closed facts i.e., {α ∈ DM⊥⊥ | α = ↑α}, where positive formulasand negative formulas are interpreted, respectively.

Proposition 11.1.10 (Adjunctions)

(Inj+ ⊣ ↓) For any α ∈ D+ and β ∈ D⊥⊥,

α ⊆ β iff α ⊆ ↓β

That is, ↓ : D⊥⊥ −→ D+ is right adjoint to Inj+ : D+ ↪→ D⊥⊥.

(↑ ⊣ Inj−) For any β ∈ D⊥⊥ and γ ∈ D−,

β ⊆ γ iff ↑β ⊆ γ

That is, ↑ : D⊥⊥ −→ D− is left adjoint to Inj− : D− ↪→ D⊥⊥.

Note that the counit of the first adjunction is given by Intensivity (1) ofDefinition 11.1.1. These adjunctions give rise to the following adjunction⇑ ⊣ ⇓ between D+ and D−, where ⇓= ↓ ◦ Inj−, and ⇑= ↑ ◦ Inj+:

D+ -⇑⊤

� ⇓D−

↓⊤

�Inj+

∪

D⊥⊥

I Inj−

∪

⊤

R↑

Alternatively, this diagram says that the full subcategory D− (resp. D+) ofclosed (resp. open) facts is a reflective (resp. coreflective) subcategory ofthe ∗-autonomous category of facts D⊥⊥ with a reflector ↑ (resp. coreflector↓).

In polarized phase spaces, we have a nice semantical characterization forthe polarity:

127

Proposition 11.1.11 (Positives are open; negatives are closed)α⊗ β, α⊕ β are both open for any open facts α, β; and α

............................................................................................... β, α & β are both

closed for any closed facts α, β. Moreover, 1 = ⊥⊥ and 0 = ⊤⊥ are open,and ⊥ and ⊤ = M are closed. That is, the coreflective (resp. reflective)subcategory D+ (resp. D−) of D⊥⊥ is closed under positive operations ⊗,⊕,along with their respective units 1 and 0 (resp. under negative operations.................................................

.............................................. , &, along with their respective units ⊥ and ⊤).

Proof. We prove for ⊗,⊕, and 1. (Dually for ............................................................................................... , &, and ⊥, and the other

units are immediate.)As for ⊗, by Openness of the monoid operator ·, we have (α · β)⊥⊥ =

(↓α · ↓β)⊥⊥ ⊆ (↓(α ·β))⊥⊥ = ↓((α ·β)⊥⊥). The other direction is immediatefrom Intensivity.

As for ⊕, using the fact α ⊆ (α∪ β)⊥⊥ and Monotonicity, we have ↓α ⊆↓((α∪β)⊥⊥). Since the same applies to β, we have ↓α∪↓β ⊆ ↓((α∪β)⊥⊥).Thus we have (α∪β)⊥⊥ = (↓α∪↓β)⊥⊥ ⊆ (↓((α∪β)⊥⊥))⊥⊥ = ↓((α∪β)⊥⊥).The other direction is immediate from Intensivity.

As for 1, we have {ε} ⊆ ↓{ε} ⊆ ↓({ε}⊥⊥), by (5) of Definition 11.1.1and by {ε} ⊆ {ε}⊥⊥ with Monotonicity. Hence we have 1 = {ε}⊥⊥ ⊆(↓({ε}⊥⊥))⊥⊥ = ↓({ε}⊥⊥) = ↓1. The other direction is immediate fromIntensivity.

Propositions 11.1.10 and 11.1.11 imply that the triple (D⊥⊥,D+,D−)forms an algebraic example of polarized ∗-autonomous category of [Hamano-Scott 2007].

Remark 11.1.12 (Polarized ∗-autonomous category) A polarized ∗-autonomous category C+,− consists of a ∗-autonomous category with prod-ucts C, and a reflective (and a coreflective) full subcategory C− (resp. C+)of C with a reflector ↑ (resp. coreflector ↓). That is, there are distinguishedfunctors ↑ : C −→ C− and ↓ : C −→ C+ satisfying: ↑ is left adjoint to the in-clusion Inj− : C− ↪→ C, and ↓ is right adjoint to the inclusion Inj+ : C+ ↪→ C.(Cf. Proposition 11.1.10.) The subcategory C− (and C+) is required to beclosed under negative (resp. positive) operations. (Cf. Proposition 11.1.11.)See [Hamano-Scott 2007] for the detailed account.

Furthermore, category theory provides a nice characterization of ourinterior and closure operators on facts.

Proposition 11.1.13 (Interior as monoidal comonad) Let D⊥⊥ be the∗-autonomous category of facts of a polarized phase space M = (M,⊥, ↓).

128

The interior (resp. closure) operator ↓ (resp. ↑) on D⊥⊥ gives rise to amonoidal comonad (resp. comonoidal monad) over D⊥⊥, and vice versa.

Proof. We prove for ↓ (dually for ↑). By the equivalence of two concepts ofadjunction and of comonad, the first adjunction (Inj+, ↓, η, ϵ) of Proposition11.1.10 gives rise to a comonad ↓ = (↓, δ = η↓, ϵ) over D⊥⊥, and vice versa.The conditions (1) and (2) of Definition 11.1.1 correspond to the counit ϵand the comultiplication δ of the comonad, respectively. The condition (3)describes the functoriality of the comonad. Moreover the conditions (4) and(5) correspond to the following functorial morphisms (4’) and (5’) whichdescribe monoidalness of the comonad:

4’. mα,β : ↓α⊗ ↓βη↓α⊗↓β−−−−→ ↓(↓α⊗ ↓β)

↓(ϵα⊗ϵβ)−−−−−→ ↓(α⊗ β)

5’. m1 = η1 : 1 −→ ↓1.

We give some examples of polarized phase space.

Example 11.1.14 (Polarized phase space)

1. Let (M,⊥) be a phase space. For a submonoid I of M , by defining

↓α = (α ∩ I)⊥⊥,

we have an interior operator. Note that this construction is the same asthat of the exponential in an enriched phase space without weakeningand contraction. (See Example 10.3.3 of Section 10.3.)

2. The multiplicative monoid Z3 = Z/Z3 of integers of modulo 3 yieldsa phase space (Z3, {1, 2}) where M = Z3 and ⊥ = {1, 2}. MoreoverZ3 yields a polarized phase space: For a submonoid I = {1, 2}, theinterior operator ↓ is defined as ↓α = (α ∩ I)⊥⊥. In this polarizedphase space, ∅, {1, 2} and {0, 1, 2} are all facts, where ∅ is open, {1, 2}is clopen and {0, 1, 2} is closed.

Discussion 11.1.15 (Polarized phase spaces and Mellies-Tabareau’scategorical model of resources) Recently, [Mellies-Tabareau 2007] intro-duced dialogue category, which is a symmetric monoidal category equippedwith a non-involutive negation called tensorial negation. By introducing re-source modality over the dialogue category based on a symmetric monoidaladjunction, they define a categorical model for their tensorial logic, whichis a more primitive logic than linear logic, and is a syntactic counter partof their game semantics. A model of LL is induced from the category for

129

tensorial logic by introducing commutative continuation monad. Their cate-gorical framework may provide a more abstract framework for our polarizedphase spaces: Our interior operator is a resource modality (without weak-ening and contraction), and our category of open facts corresponds to theKleisli category associated with their continuation monad.

11.2 Second order polarized phase model for MALLP2and completeness

In this section, we extend our polarized phase space of Section 11.1 intosecond order so as to yield a complete semantics for MALLP2. We give asemantical restriction for the usual interpretation of the second order ex-istential quantifier ∃ of [Okada 1999]. Our restriction corresponds to thesyntactical constraint for the ∃-rule in MALLP2, in which rule a formulasubstituted to the variable X of ∃X.P is restricted to be positive (ratherthan arbitrary as in the case of the second order linear logic LL2). We referto [Okada 1999] for the definition of second order phase semantics for LL2.

Notation: POS (resp. NEG) denotes the set of second order positive (resp.negative) formulas. A[X1, . . . , Xn] (or more briefly A[X]) means that thesecond order free variables of A are contained in the list X1, . . . , Xn. Wewrite A[X := B] for the substitution of B for the free variables X in A, andwe abbreviate A[X1 := B1, . . . , Xn := Bn] by A[X := B].

In order to interpret second order quantifiers, the following operationsare introduced in polarized phase spaces. Let M be a polarized phase space.Let D+ ⊆ DM+ and D− ⊆ DM−. For any ξ : D+ −→ DM⊥⊥,

∃X.ξ(X) = (∪

α∈D+ξ(α))⊥⊥ ∀X.ξ(X) =

∩α∈D+

ξ(α)

Proposition 11.1.11 for the first order connectives is extended to thesecond order.

Proposition 11.2.1 (Positives are open; negatives are closed)∃X.ξ(X) is open for any ξ : D+ → D+, and ∀X.ξ(X) is closed for any ξ : D+ → D−.

Proof. From ξ(α) ⊆∪

β∈D+ξ(β) for any α ∈ D+, the adjunction of Propo-

sition 11.1.10 applies to have ξ(α) ⊆ ↓∪

β∈D+ξ(β) for any α ∈ D+ because

ξ(α) ∈ D+. Thus we obtain∪

α∈D+ξ(α) ⊆ ↓

∪β∈D+

ξ(β), which means∃X.ξ(X) ⊆ ↓∃X.ξ(X). Dually for ∀.

130

Definition 11.2.2 (Second order phase model for MALLP2) A secondorder phase model M = (M,⊥, ↓, D+, D−, ∗) for MALLP2 consists of

− a polarized phase space (M,⊥, ↓);

− an interpretation function ∗ from the set of atoms of MALLP2 to theset DM+ of open facts, which is extended to arbitrary formulas in anatural way;

− a subset D+ of DM+ such that P ∗[X := α] ∈ D+ for any P ∈ POSand for any αi ∈ D+;

− a subset D− of DM− such that N∗[X := α] ∈ D− for any N ∈ NEGand for any αi ∈ D+.

Note that, from Propositions 11.1.11 and 11.2.1, each positive (resp.negative) formula is interpreted by an open (resp. closed) fact.

The truth relation of sequents is defined by means of the set-theoreticalinclusion relation between facts of the subdomain D+∪{⊥} of DM⊥⊥. Thissemantical restriction on the subdomain naturally forces the truth relationof sequents to be defined only for the focalized sequents of Definition 9.2.3.

Definition 11.2.3 (Truth value of sequents) A sequent Γ[X] is true,denoted |= Γ[X] (or simply |= Γ when there is no need to indicate secondorder variables X), if the following holds: For any βi ∈ D+,

− N∗⊥1 [X := β] · · · · ·N∗⊥

n [X := β] ⊆ ⊥ when Γ is N1, . . . , Nn;− N∗⊥

1 [X := β] · · · · ·N∗⊥n [X := β] ⊆ P ∗[X := β] when Γ is N1, . . . , Nn, P .

Note that this interpretation of sequents is a restriction of that in phasesemantics for MALL2 such that only focalized sequents are considered.

We have the soundness and the strong completeness theorems of MALLP2,which together imply the cut-elimination theorem in a way similar to thatof [Okada 1999].

Proposition 11.2.4 (Soundness of MALLP2) If ⊢ Γ is provable in MALLP2,then |= Γ in any second order phase model for MALLP2.

Proof. By induction on the length of the proof of ⊢ Γ. We consider only thepolarity shifting rules since the other rules are the same as [Okada 1999].

− ⊢ N , N

⊢ N , ↓N↓

Let N be M1, . . . , Mn. We have M∗⊥1 · · · · ·M∗⊥

n ⊆ N∗ by the induction

131

hypothesis. Since M∗⊥i are open, from the first adjunction of Proposition

11.1.10, we have M∗⊥1 · · · ·M∗⊥

n ⊆ ↓N∗.

− ⊢ Γ, P

⊢ Γ, ↑P↑

Note that, by the focalization property of MALLP, Γ is a negative sequent.Hence we have Γ∗⊥ ⊆ P ∗ by the induction hypothesis. Since P ∗ ⊆ ↑P ∗,from the de Morgan dual of Intensivity (1) of ↓, we have Γ∗⊥ ⊆ ↑P ∗, andhence we have Γ∗⊥ · (↑P )∗⊥ ⊆ ⊥.

In order to show the strong completeness theorem, we construct a syn-tactical model MS = (MS ,⊥, ↓, D+, D−, ∗) called a canonical model.

− MS is the monoid of the sequents of MALLP2 formulas, which arenot restricted to focalized. Its monoid operator is the concatenationoperator between sequents, and the unit is the empty sequent.

− We define the outer-value [[A]] for each formula A as

[[A]] = {Γ |⊢ Γ, A is cut-free provable }.

− ⊥ := [[⊥]] = {Γ |⊢ Γ,⊥ is cut-free provable } = {Γ |⊢ Γ is cut-free provable }.− Let I be the submonoid of MS consisting only of negative sequents.

Then we define an interior operator ↓ as in Example 11.1.14(1):

↓α = (α ∩ I)⊥⊥.

− X∗ = [[X]] for each atom X.

− For each positive formula P , let ⟨P ⟩ be the set of open facts α of MS

such that P⊥ ∈ α ⊆ [[P ]]; and for each negative formula N , let ⟨N⟩ bethe set of closed facts α such that N⊥ ∈ α ⊆ [[N ]]. Then we define thesecond order domains D+ by

∪P∈POS⟨P ⟩, and D− by

∪N∈NEG⟨N⟩.

By distinguishing the inner-value A∗, which is the interpretation of A inthe canonical model, and the outer-value [[A]] = {Γ |⊢ Γ, A is cut-free provable },Okada [1999] proved the strong completeness theorem of LL2, which impliesthe cut-elimination theorem. This method is applicable to MALLP2.

Lemma 11.2.5 (Main lemma for MALLP2) In the canonical model MS

for MALLP2, for any A[Y ], for any Q ∈ POS and for any β ∈ ⟨Q⟩, we have

A⊥[Y := Q] ∈ A∗[Y := β] ⊆ [[A[Y := Q]]].

132

Proof. The first half of the lemma is obtained from the second one in thesame way as [Okada 1999]. We show the second half of the lemma by in-duction on the complexity of A. We prove the cases A is ↓N and ↑P sincethe other cases are treated in a way similar to that of [Okada 1999].Case A ≡ ↓N :Since outer-values are shown to be facts, it is sufficient to show N∗ ∩ I ⊆[[↓N ]]. Let Γ ∈ N∗ ∩ I. Then, since Γ ∈ N∗, we have Γ ∈ [[N ]] by theinduction hypothesis. Since Γ ∈ I means that Γ is negative sequent, by thefollowing ↓-rule, we have Γ ∈ [[↓N ]]:

⊢ Γ, N

⊢ Γ, ↓N↓

Case A ≡ ↑P :We first show ↑P ∈ P ∗⊥∩I. By the induction hypothesis we have P ∈ P ∗⊥,that is, P · P ∗ ⊆ ⊥. Hence, for any Λ ∈ P ∗, the sequent ⊢ P, Λ is cut-freeprovable. Then, by the following ↑-rule, ⊢ ↑P, Λ is provable:

⊢ P, Λ⊢ ↑P, Λ

↑

Thus we have ↑P · P ∗ ⊆ ⊥, which means ↑P ∈ P ∗⊥. Since ↑P is negative,we have ↑P ∈ P ∗⊥ ∩ I.We next show ↑P ∗ ⊆ [[↑P ]]. Let Γ ∈ ↑P ∗ = (P ∗⊥∩I)⊥, that is, Γ·(P ∗⊥∩I) ⊆⊥. This means that ⊢ Γ,∆ is cut-free provable for any ∆ ∈ P ∗⊥ ∩ I. Sincewe have ↑P ∈ P ∗⊥∩I in particular, ⊢ Γ, ↑P is cut-free provable. This showsΓ ∈ [[↑P ]].

Using the main lemma for MALLP2, we have the following strong com-pleteness theorem.

Proposition 11.2.6 (Strong completeness of MALLP2) If |= Γ in anysecond order phase model for MALLP2, then ⊢ Γ is provable without thecut-rule in MALLP2.

Proof. Let Γ be of the form N1, . . . , Nn, A where A is positive formula orempty. When A is empty, A is taken to be ⊥.

Let |= Γ, namely N∗⊥1 · · · · · N∗⊥

n ⊆ A∗ in canonical model MS . SinceN1 ∈ N∗⊥

1 , . . . , Nn ∈ N∗⊥n in MS , we have N1 · · · · · Nn ∈ A∗. Hence

⊢ N1, . . . , Nn, A is cut-free provable in MALLP by Lemma 11.2.5.

Combining this strong completeness theorem and the soundness theorem,we obtain the cut-elimination theorem for MALLP2.

133

Corollary 11.2.7 (Cut-elimination for MALLP2) If ⊢ Γ is provable inMALLP2, then it is provable without the cut-rule.

11.3 Enriched polarized phase space for LLpol

In this section, we introduce enriched polarized phase spaces, with respectto which LLpol is complete.

Definition 11.3.1 (Enriched polarized phase space) An enriched po-larized phase space is M = (M,⊥, !, ↓) such that

− (M,⊥, !) is an enriched phase space;

− (M,⊥, ↓) is a polarized phase space;

− !α ⊆ ↓α for any fact α ∈ D⊥⊥.

Note that the topological structure considered in M is that derived from↓ (not from !).

Remark 11.3.2 (Two modalities ↓ and !) In enriched polarized phasespaces, the interior operator ↓ does not necessarily coincide with !. If itdoes, X∗ ⊆ !?X∗ holds for a positive atom X, which, however, happens tobe a valid interpretation of an unprovable sequent ⊢ X⊥, !?X in LLpol.

Example 11.3.3 (Enriched polarized phase space) Let I and J be thesubmonoids of Example 11.1.14(1) and Example 10.3.3, respectively. Let ↓be the operation defined as in Example 11.1.14(1). Then by defining

!α = (α ∩ J ∩ I)⊥⊥,

we have an enriched polarized phase space. This essentially corresponds tothe decomposition of the exponentials of [Girard 2001]:

!α = ↓♯α and ?α = ↑♭α.

Note that ♯α is defined as α∩ J , and then !α is defined as the interior of ♯αby Example 11.1.14(1). I.e., !α = ↓♯α = (α∩ J ∩ I)⊥⊥. J ∩ I ⊆ I yields thecondition !α ⊆ ↓α. Note that ♯α is not necessarily a fact, hence its meaningis only given under the connective ↓. (Cf. [Laurent 2005b].)

Proposition 11.1.11 for MALLP connectives can be extended to LLpol

connectives: i.e., we have the following Proposition 11.3.4 for ! and ?.

134

Proposition 11.3.4 (Positives are open; negatives are closed)!α is open (i.e., ↓-invariant) and ?α is closed (i.e., ↑-invariant) for any factα.

Proof. We show !α is open, i.e., !α = ↓!α. We have !α ⊆ !!α ⊆ ↓!α by Idem-potency of ! and the definition !α ⊆ ↓α. The other direction is immediatefrom Intensivity of ↓. Dually for ?.

This proposition says that the topology derived from ! is coarser than thatfrom ↓. I.e., the set of !-invariant facts is a subset of ↓-invariant facts in anyenriched polarized phase space (M,⊥, !, ↓).

11.4 Second order polarized phase model for LLpol2and completeness

In this section, we extend our enriched polarized phase space of Section 11.3into second order so as to yield a complete semantics for the second orderextension LLpol2 of LLpol.

Definition 11.4.1 (Second order phase model for LLpol2) A second or-der phase model M = (M,⊥, !, ↓, D+, D−, ∗) for LLpol2 consists of

− an enriched polarized phase space (M,⊥, !, ↓);

− an interpretation function ∗ from the set of atoms of LLpol2 to the setDM+ of open facts ;

− second order domains D+ and D− which are defined in the same wayas for MALLP2. (See Definition 11.2.2.)

We have the soundness and the strong completeness theorems of LLpol2,which together imply the cut-elimination theorem in a way similar to thatof [Okada 1999].

Proposition 11.4.2 (Soundness of LLpol2) If ⊢ Γ is provable in LLpol2,then |= Γ in any second order phase model for LLpol2.

In order to show the completeness theorem, we construct the canonicalmodel MS = (MS ,⊥, !, ↓, D+, D−, ∗) by adding the following definition of !to the canonical model for MALLP2 of Section 11.2.

135

− Let J be the submonoid of MS which consists only of negative sequentsof the form ?Q. Here we identify more than one ?Q’s with one ?Q inMS and J ; for instance we identify ?Q, ?Q with ?Q. Then we define!α = (α ∩ J)⊥⊥ for α ⊆MS .

Since ?Q is a negative sequent, we have J ⊆ I, which implies !α ⊆ ↓α.

The strong completeness theorem which implies the cut-elimination the-orem is shown by using the same form of the lemma (Lemma 11.2.5) as forMALLP2.

Lemma 11.4.3 (Main lemma for LLpol2) In the canonical model MS forLLpol2, for any A[Y ], for any Q ∈ POS and for any β ∈ ⟨Q⟩, we have

A⊥[Y := Q] ∈ A∗[Y := β] ⊆ [[A[Y := Q]]].

Proposition 11.4.4 (Strong completeness of LLpol2) If |= Γ in any sec-ond order phase model for LLpol2, then ⊢ Γ is provable without the cut-rulein LLpol2.

Corollary 11.4.5 (Cut-elimination for LLpol2) If ⊢ Γ is provable in LLpol2,then it is provable without the cut-rule.

Remark 11.4.6 Let us consider an operator !−, which satisfies Intensivity,Idempotency, Monotonicity of Definition 10.3.1 and !α⊗!β ⊆!(α⊗β), 1 ⊆!1of Proposition 10.3.4. Then we can consider a complete subsystem LL−

pol2 ofLLpol2 without the weakening and contraction rules. (Cf. Remark 12.1.17of Section 12.1.6.)

11.5 An application of polarized phase semantics:First order conservativity

As an application of our polarized phase semantics, we give a semantic proofof the conservativity of LL without the second order quantifies over its polar-ized fragment LLpol of [Laurent 2002]. Let us start by summarizing the fourkinds of phase spaces for first order logics (Figure 1). In the following figure,on the top of the most primitive phase spaces, three kinds of phase spacesare obtained, where a vertical (resp. horizontal) line designates the augmen-tation of the interior ↓ (resp. the exponential !). A symbol ⊂ designates asubsystem relation: I.e., LLpol and MALL are subsystems of LL.

136

MALLP LLpol

LLMALL

(M,⊥, ↓)polarized

phase space

-! (M,⊥, ↓, !)

enriched polarizedphase space

?

↓

(M,⊥, !)enriched

phase space

-!(M,⊥)phase space

?

↓

Figure 1

The conservativity theorem of LL over LLpol was first obtained by Lau-rent [2002 (p.50)] with a proof-theoretical argument. We have a simplesemantical proof of the theorem as a corollary of the completeness theoremof LLpol. The proof is direct by virtue of the canonical forgetting ↓ mapfrom the bottom to the top on the right vertical relation of Figure 1.

Corollary 11.5.1 (LL ≻ LLpol [Laurent 2002]) LL is conservative over LLpol:For any focalized sequent Γ, if ⊢ Γ is provable in LL, then it is provable inLLpol.

Proof. Let Γ be a focalized sequent which is provable in LL, and let M =(M,⊥, !, ↓) be an arbitrary enriched polarized phase space. Then from thesoundness theorem of LL, |= Γ in the phase model (M,⊥, !) for LL. SinceΓ is polarized, it is also true in the polarized phase model M for LLpol.Therefore, from the completeness theorem of LLpol, which is obtained asa corollary of the strong completeness and the cut-elimination theorems ofLLpol2, the sequent ⊢ Γ is provable in LLpol.

137

Chapter 12

Second order conservativityof linear logic over itspolarized fragment

Laurent [2002] shows a first order conservativity theorem of linear logic LLover its polarized fragment LLpol. When we try to extend the conservativityto the second order linear logic LL2, we immediately encounter a difficultywith the second order ∃-rule, which results in the loss of the subformulaproperty. For this reason Laurent [2002] has left it open whether the con-servativity result can be extended to second order.

In this chapter, we first show by using a counter model construction thatLL2 is not conservative over LLpol2 (Proposition 12.1.1). We next observethat LL2 does not have the focalized sequent property (Proposition 12.1.3).In order to remedy this shortcoming, we introduce an η-expanded fragmentLLη2 of LL2, in which atoms are exponential forms (i.e., !X⊥ (resp. ?X)for a positive (resp. negative) atom). Such a restriction, which was alsoadopted in [Laurent-Quatrini-Tortora 2005, Laurent 1999, Laurent 2005a],has a natural semantical counterpart in our polarized phase spaces; a topo-logical structure derived from the exponential connectives of LL2 coincideswith a topological structure for the polarity. Moreover, syntactically, underthe restriction, the focalized sequent property is recovered. Accordingly ourmain goal in this chapter is to establish the conservativity of LLη2 over itspolarized fragment LLη

pol2 (Theorem 12.1.16). The conservativity followsfrom our main proposition (Proposition 12.1.15) which ensures that if a nonpolarized sequent is provable in LLη2, then it is canonically decorated with! and ? so that the transformed polarized sequent is provable in LLη

pol2. In

138

Section 12.2, we remark that restricted forms of the additives are definable inthe second order multiplicative exponential polarised linear logic MELLpol2.

12.1 Second order conservativity

In Section 12.1.2, we introduce a fragment LLη2 of LL2, then we prove ourmain technical proposition (Proposition 12.1.15) to obtain the main theorem(Theorem 12.1.16): LLη2 is conservative over its polarized fragment LLη

pol2.

12.1.1 LL2 is not conservative over LLpol2

In this subsection, we show that Corollary 11.5.1 does not extend to LL2.The semantic argument of Corollary 11.5.1 does not work for the secondorder LL2, because |= ∃X.P in LL2 does not necessarily imply the same inLLpol2. Note that |= ∃X.P means 1 ⊆ P ∗[X := α] for some α ∈ D⊥⊥ inLL2, but LLpol2 has the constraint that α ranges only over a subset D+ ofD⊥⊥.

Proposition 12.1.1 (LL2 ≻ LLpol2) LL2 is not conservative over LLpol2.

Proof. We show that a focalized sequent ⊢ Y, ?∃X.X is provable in LL2,but not in LLpol2: The following is an LL2 poof of the sequent.

⊢ Y, Y ⊥ ax

⊢ Y, ∃X.X∃(X := Y ⊥)

⊢ Y, ?∃X.X?

This LL2 proof is not an LLpol2 proof, since, in the application of the ∃-rule,the negative formula Y ⊥ is substituted to X. Note that this ∃-rule makesthe LL2 provable sequent non focalized (⊢ Y, ∃X.X).

In order to show the sequent is not provable in LLpol2, we construct acounter model for ⊢ Y, ?∃X.X. Let Z3 be the enriched polarized phase spaceof Example 11.1.14(2) where ! = ↓; and the second order domain D+ is theset of open facts {∅, {1, 2}}. Let Y ∗ be the open fact ∅. Then Y ⊥∗ is theclosed fact {0, 1, 2}. On the other hand, because ∃X.X∗ = (

∪α∈D+

α)⊥⊥,we have ?∃X.X∗ = {1, 2}. Hence, we have {0, 1, 2} = Y ⊥∗ ⊆ ?∃X.X∗ ={1, 2} in this model. Therefore, we have |= Y, ?∃X.X, and hence from thesoundness theorem, we conclude that ⊢ Y, ?∃X.X is not provable in LLpol2.

139

Remark 12.1.2 A direct syntactical proof of the unprovability of the se-quent of Proposition 12.1.1 is also possible with the help of the second ordercut-elimination theorem of LLpol2 (Corollary 11.4.5), which follows from ourstrong completeness theorem.

As seen in the LL2 proof of ⊢ Y, ?∃X.X of Proposition 12.1.1, a nonfocalized sequent ⊢ Y, ∃X.X is provable in LL2 with only polarized formulas.Thus we also do not have the focalized sequent property (FSP) in LL2.

Proposition 12.1.3 The focalized sequent property fails in LL2.

12.1.2 Second order η-expanded system LLη2

In order to remedy the shortcoming of LL2 in the previous section, we in-troduce a fragment LLη2, where atoms are restricted to exponential forms(i.e., of the form !X⊥ (resp. ?X) for a positive (resp. negative) atom). Sucha restriction of atoms is also adopted in [Laurent-Quatrini-Tortora 2005,Laurent 1999, Laurent 2005a] for LLpol without the second order quantifiers.

Definition 12.1.4 (LLη2) The syntax of LLη2 is defined as follows.

Formulas of LLη2 are given by the following grammar.

A ::= !X⊥ | A⊗A | A⊕A | 1 | 0 | !A | ∃X.A?X | A

............................................................................................... A | A & A | ⊥ | ⊤ | ?A | ∀X.A

We refer to connectives {⊗,⊕, !,∃,1,0} (resp. {............................................................................................... , &, ?,∀,⊥,⊤}) as pos-itive (resp. negative) connectives.

The substitution of B for the free occurrences of X in A (denoted A[X :=B]) is defined as usual for any LLη2 formulas A and B. Note that a sub-stitution such as X[X := !Y ⊥] is not allowed in LLη2 because X is not anLLη2 formula.

Inference rules of LLη2 are obtained from those of LL2 by replacing theusual axiom of the form ⊢ X, X⊥ with the form ⊢ ?X, !X⊥.

The restriction will be crucial in proving Lemma 12.1.10 of the nextsubsection.

Example 12.1.5 (A proof in LLη2) The following is an LLη2 proof.

140

⊢ ?X, !X⊥ ax

⊢ ?X, ?!X⊥ ?

⊢ ?X ............................................................................................... ?!X⊥

...............................................................................................

⊢ !(?X ............................................................................................... ?!X⊥)

!

⊢ ∃Y.!Y ⊥ ∃(Y := (?X ............................................................................................... ?!X⊥)⊥)

Since an application of the ∃-rule as in the proof of Proposition 12.1.1 isnot allowed in LLη2, the focalized sequent property (FSP) for LLη2 followsnaturally.

Lemma 12.1.6 LLη2 has the focalized sequent property (FSP).

Proof. We prove this by induction on the given proof of LLη2. Other than∃-rule, the assertion is straightforward by the induction hypothesis. Let usconsider a sequent ⊢ ∃X.P , Γ which is derived from a focalized sequent⊢ P [X := A] ,Γ via ∃-rule. We show that the formula P [X := A] ispositive. This is because (i) if P is atomic, it is of the form !X⊥ in LLη2,hence !X⊥[X := A] is positive; (ii) otherwise it is clear because the sequent⊢ P [X := A] , Γ is focalized. Thus we conclude that Γ is negative, and hencethe ∃-rule preserves FSP.

LLηpol2 denotes the polarized fragment of LLη2. Then a phase model for

LLηpol2 is defined as follows.

Definition 12.1.7 (Second order phase model for LLηpol2) A second or-

der phase model M = (M,⊥, !, ↓, D+, D−, ∗) for LLηpol2 is a second order phase

model for LLpol2 where

− ↓ coincides with !;

− an interpretation function is defined from the set of LLηpol2-atoms of

the form ?X to the set of negative facts of the form ?α for α ∈ D⊥⊥.

Since two modalities ↓ and ! coincide, LLηpol2 is natural from our topo-

logical semantic viewpoint.We have completeness of LLη

pol2 in a way similar to that of LLpol2.

Proposition 12.1.8 (Strong completeness of LLηpol2) If |= Γ in any phase

model for LLηpol2, then ⊢ Γ is provable without the cut-rule in LLη

pol2.

Remark 12.1.9 (LLηpol2 is equivalent to LLPη2) LLP2 of [Laurent 2002]

is obtained by adding the following rules to LLpol2:

141

⊢ N , N

⊢ N , !N N ! ⊢ Γ⊢ Γ, N

Nw⊢ Γ, N,N

⊢ Γ, NNc.

LLP2 is not a subsystem of LL2 because the above N !-, Nw- and Nc-rulesstrictly generalize the !-, ?w- and ?c-rules of LLpol2 respectively. These rulesof LLpol2 are restrictions of the above LLP2 rules in the sense that negativeformulas explicitly designated in the conclusions are of particular forms ?P .However when we consider η-expanded systems LLPη2 and LLη

pol2, the aboverules are shown to be derivable in LLη

pol2, and hence LLPη2 is equivalent toLLη

pol2.

12.1.3 Main proposition for LLη2: Polarization

This subsection is devoted to proving Proposition 12.1.15, which is our maintechnical result in this second part of the thesis. It is introduced to overcomethe difficulty, remarked in [Laurent 2002] (p. 51), with the following formof ∃-rule of LLη2:

⊢LLη2 N , P [X := A]⊢LLη2 N , ∃X.P

∃(X := A).

Although the lower sequent is focalized, the upper sequent is no longera polarized sequent because the formula A is arbitrary. Our Proposition12.1.15 ensures that the arbitrary A is appropriately decorated with ! and ?to a positive formula Q so that ⊢ N , P [X := Q] is provable in LLη

pol2.Throughout the proof of Proposition 12.1.15, we use the proof net rep-

resentation of the proofs for the sake of simplicity. Since we do not use anyparticular properties of proof net in the following proof, we only indicatethe links for LLη2 proof net (without units) here. The details can be foundin [Laurent 1999, Laurent 2002, Laurent-Tortora 2004].

!X⊥ ?Xax

...............................................................................................· · · ·

A1.................................................

.............................................. . . .

............................................................................................... An

A1 An ⊗· · · ·

A1 ⊗ · · · ⊗ An

A1 An ∀∀X.A

A

∃

∃X.A

A[X := B]

!

!A

A

?A1· · · ?An

?

?A

A?w

?A

?c

?A

?A ?A&

C C

A & B A1· · · An

⊕i

A1 ⊕ A2

Ai i ∈ {1, 2}

In the &-box, C⃝’s are additive contraction nodes. We consider the ............................................................................................... and

⊗ connectives as n-ary connectives for an appropriate n.

142

For the sake of simplicity of the argument, we do not consider any units(1,0,⊥,⊤) in the following proof.

We prepare some lemmas by recalling the following notions on proofnets.

Splitting link: A terminal link l of a proof net π splits, if l is removableto obtain proof nets which are premises of the corresponding rule to l. Aformula A in a conclusion of π splits, if the corresponding terminal link ofthe outermost connective of A splits. See [Andreoli-Maieli 1999].

Andreoli’s focusing property: Andreoli’s focusing property says that asplitting conclusion A can be chosen in such a way that each premises of A,if its outermost connective is positive, is again a splitting conclusion for theresulting proof nets after the splitting of A. I.e., whenever we write A asϕ[B1, . . . , Bn] by means of a cluster ϕ of positive connectives ⊗,⊕,∃, !, thenthe corresponding links to the connectives successively split until reachingBi’s. Note that the crucial case is that outermost connectives of Bi’s are nolonger positive. See [Andreoli-Maieli 1999, Andreoli 1992, Laurent 2005b].

By virtue of Andreoli’s focusing property, we have the following lemmain LLη2.

Lemma 12.1.10 (Context lemma for a splitting ?-link) Let A be anLLη2 formula, and ?Q be an LLη

pol2 sequent. For any LLη2 proof net ofconclusions A, ?Q, if A does not split and a ?-link of some formula in ?Qsplits, then A is of the form ?B.

Proof. Assume ?Q =?P, ?Q1 so that, after the splitting of ?P , the positiveP splits. We consider the following cases according to the form of P :(i) If P ≡!B1, the assertion is direct since the !-link of !B1 splits by theassumption. This means that A lies on one of the auxiliary doors of the!-box as follows:

A

?P?!B1

!B1

?Q1

Hence A is of the form ?B.(ii) If P ≡ B1 ⊗ B2, B1 ⊕ B2 or ∃X.B1, then P splits to yield proof netsof conclusions A,Bi[X := C], ?Q′

i with ?Q′i ⊆?Q1. We observe that the

outermost connective of Bi[X := C] is either {⊗,⊕,∃, !}; If Bi[X] is non

143

atomic, then the observation is clear because it is a subformula of positiveP . If Bi[X] is atomic, then it is of the form !X⊥, hence the observationholds. From the observation, we consider the following two cases.(ii− a) If the outermost connective of Bi[X := C] is !, then, since the !-linksplits from Andreoli’s focusing property, the assertion holds from (i).(ii− b) If the outermost connective of Bi[X := C] is ⊗,⊕ or ∃, after succes-sively applying this case (ii) using Andreoli’s focusing property, we finallyobtain a proof net of conclusions A, !D, ?Q′′

i with ?Q′′i ⊆?Q1. Thus the

assertion follows from (i).

Remark 12.1.11 Lemma 12.1.10 does not hold in (non η-expanded) LL2.The LL2 sequent ⊢ Y, ?∃X.X in the proof of Proposition 12.1.1 gives acounterexample.

In what follows, we make the following syntactical convention.

Convention: !P = P and ?N = N for any positive formula P and for anynegative formula N .

The convention is semantically valid in every polarized phase model ofLLη

pol2. This convention makes it possible to define uniformly the followingcanonical decoration σ.

Definition 12.1.12 (Canonical decoration) A canonical decoration σ ofLLη2 formulas is defined inductively as follows, where σ(A) is an LLη

pol2 for-mula from the above convention.

σ(?X) = ?X σ(!X⊥) = !X⊥

σ(A............................................................................................... B) = ?σ(A).................................................

.............................................. ?σ(B) σ(A⊗B) = !σ(A)⊗ !σ(B)

σ(?A) = ?!σ(A) σ(!A) = !?σ(A)σ(∀X.A) = ∀X.?σ(A) σ(∃X.A) = ∃X.!σ(A)σ(A & B) = ?σ(A) & ?σ(B) σ(A⊕B) = !σ(A)⊕ !σ(B)

This decoration is the identity map on the class of polarized formulas inthat σ(A) = A for each polarized LLη

pol2 formula A.We have the following lemma on the canonical decoration σ.

Lemma 12.1.13 (Substitution) Let A and B be LLη2 formulas. In LLηpol2,

we haveσ(A[X := B]) ≡ σ(A)[X := !σ(B)].

144

Note that, in the above lemma, the substituted formula B is decorated byσ into a positive formula !σ(B), which serves to overcome the difficulty of thesecond order ∃-rule of LLη2 mentioned in the beginning of this subsection.

Next we present the following lemma in preparation for the proof ofProposition 12.1.15. 1

Lemma 12.1.14 For any LLη2 formula A, and for any positive formulasP and Q, the following sequents are provable in LLη

pol2:

1. !?σ(A) ⊢ ?!σ(A).

2. ?P [X := Q] ⊢ ?∃X.P .

3. ?Pi ⊢ ?(P1 ⊕ P2) for i = 1, 2.

Proof. (1) If σ(A) is positive, we have !?σ(A) ⊢ ?σ(A) = ?!σ(A). If σ(A) isnegative, we have !?σ(A) = !σ(A) ⊢ ?!σ(A). (2) and (3) are straightforward.

Now we are ready to prove the main technical proposition:

Proposition 12.1.15 (Polarization) Let A1, . . . , An be LLη2 formulas, andlet ?Q be an LLη

pol2 sequent of the form ?Q1, . . . , ?Qm with m ≥ 0 (whenm = 0, ?Q is empty). If ⊢ A1, . . . , An, ?Q is provable in LLη2, then, inLLη

pol2,

⊢ σ(A), ?Q is provable if n = 1; and⊢ ?σ(A1), . . . , ?σ(An), ?Q is provable if n > 1.

Proof. In the following proof, ⊢ Γ means that there is an LLηpol2 proof net

of the conclusion Γ (i.e., the sequent ⊢ Γ is provable in LLηpol2).

We prove the proposition by induction on the size of a cut-free proof net ofconclusions A1, . . . , An, ?Q. In order to make the proof easy to read, all thesteps for the additive rules and for the weakening and the contraction rulesare separately moved to the next sections (Section 12.1.4 and Section 12.1.5,respectively). As demonstrated there, to accommodate with these rules isstraightforward. Before the proof, we recall that any terminal {............................................................................................... ,∀, &}-linkalways splits to yield a proof net of smaller size.

Base step for n = 1:This step is an axiom of the form !X⊥ ?X

axso that A ≡ !X⊥ and Q ≡ X.

In this case the assertion is clear since σ(!X⊥) = !X⊥.1 The lemma is used in the induction step for n > 1 of Proposition 12.1.15: (1) for the

case of ?B, (2) for the case of ∃X.B, and (3) for the case of B1 ⊕ B2.

145

Base step for n > 1:This step is the axiom of the form !X⊥ ?X

axso that A1 ≡ !X⊥, A2 ≡ ?X

and ?Q is empty. We have ⊢ ?!X⊥, ?X where ?!X⊥ = ?σ(!X⊥), and ?X =?σ(?X).

Induction step for n = 1:We divide the following into two cases depending whether A splits or not.(Case 1) When A splits, this case is divided according to the outermostconnective of A.If A ≡ B1

............................................................................................... · · · ............................................................................................... Bn, after removing the .................................................

.............................................. -link, we have ⊢ ?σ(B1), · · · , ?σ(Bn), ?Q

by the induction hypothesis (I.H.). Then by the ............................................................................................... -rules we have the

assertion since ?σ(B1).................................................

.............................................. · · · ............................................................................................... ?σ(Bn) = σ(B1

............................................................................................... · · · ............................................................................................... Bn).

If A ≡ ∀X.B, after removing the ∀-link, we obtain ⊢ σ(B), ?Q by I.H. Thenwe have ⊢ ∀X.?σ(B), ?Q by the ?-rule followed by the ∀-rule. This isthe assertion since ∀X.?σ(B) = σ(∀X.B).

If A ≡ ?B, each case is considered depending on the (kind of) splitting linkof ?B.

If the splitting link of ?B is the ?-link, after removing the ?-link, we have⊢ σ(B), ?Q by I.H. Then we have ⊢ ?!σ(B), ?Q by the !-rule followedby the ?-rule. Since ?!σ(B) = σ(?B), we obtain ⊢ σ(?B), ?Q.

If A ≡ !B, after removing the !-link, we have ⊢ σ(B), ?Q by I.H. Thenwe obtain ⊢ !?σ(B), ?Q by the ?-rule followed by the !-rule. Since!?σ(B) = σ(!B), we obtain ⊢ σ(!B), ?Q.

If A ≡ ∃X.B, after removing the ∃-link, we obtain ⊢ σ(B[X := C]), ?Qby I.H. Using the substitution lemma (Lemma 12.1.13), we have ⊢σ(B)[X := !σ(C)], ?Q, and hence we obtain ⊢ ∃X.!σ(B), ?Q by the!-rule followed by the ∃-rule. Since ∃X.!σ(B) = σ(∃X.B), we obtain⊢ σ(∃X.B), ?Q.

If A ≡ B1⊗· · ·⊗Bn, after removing the ⊗-link, we have ⊢ σ(Bi), ?Qi for all1 ≤ i ≤ n by I.H. Then we have ⊢ !σ(Bi), ?Qi by the !-rule, and hencewe have ⊢ !σ(B1) ⊗ · · · ⊗ !σ(Bn), ?Q by the ⊗-rules. Since !σ(B1) ⊗· · · ⊗ !σ(Bn) = σ(B1 ⊗ · · · ⊗Bn), we obtain ⊢ σ(B1 ⊗ · · · ⊗Bn), ?Q.

(Case 2) When A does not split, then ?P of ?Q splits. This case is divideddepending on the (kind of) splitting link of ?P .

If the ?-link of ?P splits, 2 note first that, in this case, by Lemma 12.1.10,A is of the form ?B, and the conclusions of the given LLη2 proof net

2This case is the case where Lemma 12.1.10 is used.

146

are written by ?B, ?P, ?Q′. Then, after removing the ?-link, we have⊢ ?σ(?B), ?σ(P ), ?Q′ by I.H. for the case n > 1. Since ?σ(P ) is ?P ,and ?σ(?B) is σ(?B), we obtain ⊢ σ(?B), ?P, ?Q′.

Induction step for n > 1:We divide the following into two main cases depending on whether A1, . . . , An

contains a splitting formula or not.

(Case 1) When there is no splitting formula among A1, . . . , An, then ?Pin ?Q splits. This case is divided depending on the (kind of) splitting linkof ?P .If the ?-link of ?P splits, after removing the ?-link, we have ⊢ ?σ(A1), . . . , ?σ(An),

?σ(P ), ?Q by I.H. Since ?σ(P ) = ?P , we obtain ⊢ ?σ(A1), . . . , ?σ(An), ?P, ?Q.

(Case 2) When there is a splitting formula among A1, . . . , An, we furtherdivide the following into two cases depending whether or not there is aterminal {............................................................................................... ,∀, &}-link in A1, . . . , An. In the following proof, A denotes acertain Ai by omitting its subscript, and Γ denotes the sequence of otherAj ’s with i = j.

− When there is a terminal {............................................................................................... ,∀, &}-link in A1, . . . , An, note that such anegative terminal link always splits.If A ≡ B1

............................................................................................... · · · ............................................................................................... Bn, then after removing the .................................................

.............................................. -link, we have

⊢ ?σ(B1), . . . , ?σ(Bn), ?σ(Γ), ?Q by I.H. Since ?σ(B1).................................................

.............................................. · · · ............................................................................................... ?σ(Bn) =

?σ(B1.................................................

.............................................. · · · ............................................................................................... Bn), by the .................................................

.............................................. -rules we obtain ⊢ ?σ(B1

............................................................................................... · · · ............................................................................................... Bn), ?σ(Γ), ?Q.

If A ≡ ∀X.B, then after removing the ∀-link, we have ⊢ ?σ(B), ?σ(Γ), ?Qby I.H. Then by the ∀-rule, we have ⊢ ∀X.?σ(B), ?σ(Γ), ?Q. Since∀X.?σ(B) = ?σ(∀X.B), we obtain ⊢ ?σ(∀X.B), ?σ(Γ), ?Q.

− Otherwise we choose a splitting A of A1, . . . , An since such A exists bythe assumption. We consider each case according to the splitting link of A.

If A ≡ ?B splits, each case is considered depending on the (kind of) splittinglink of ?B.

If the ?-link of ?B splits, after removing the ?-link, we obtain ⊢ ?σ(B), ?σ(Γ), ?Qby I.H. Then we have ⊢ !?σ(B), ?σ(Γ), ?Q by the !-rule. By com-posing this with !?σ(B) ⊢ ?!σ(B) of Lemma 12.1.14(1) by the cut-rule, we have ⊢ ?!σ(B), ?σ(Γ), ?Q. Thus ⊢ ?σ(?B), ?σ(Γ), ?Q since?!σ(B) = ?σ(?B).

If A ≡ !B splits, after removing the !-link, we have ⊢ ?σ(B), ?σ(Γ), ?Q byI.H. Then we have ⊢ ?!?σ(B), ?σ(Γ), ?Q by the !-rule followed by the?-rule. Since ?!?σ(B) = ?σ(!B), we obtain ⊢ ?σ(!B), ?σ(Γ), ?Q.

147

If A ≡ ∃X.B splits, after removing the ∃-link, we obtain ⊢?σ(B[X :=C]), ?σ(Γ), ?Q by I.H. Then we further divide the following into twocases according to the polarity of σ(B[X := C]):

(i). If σ(B[X := C]) is negative, we have ⊢ σ(B[X := C]), ?σ(Γ), ?Q.Using the substitution lemma, we have ⊢ σ(B)[X :=!σ(C)], ?σ(Γ), ?Q,and hence ⊢!σ(B)[X :=!σ(C)], ?σ(Γ), ?Q by the !-rule. Then, by the∃-rule followed by the ?-rule, we have ⊢?∃X.!σ(B), ?σ(Γ), ?Q. Since?∃X.!σ(B) =?σ(∃X.B), we obtain ⊢?σ(∃X.B), ?σ(Γ), ?Q.

(ii). If σ(B[X := C]) is positive, using the substitution lemma, we have⊢?σ(B)[X :=!σ(C)], ?σ(Γ), ?Q. By composing this with ?σ(B)[X :=!σ(C)] ⊢?∃X.σ(B) of Lemma 12.1.14(2) by the cut-rule, we have⊢?∃X.σ(B), ?σ(Γ), ?Q. Since ?∃X.σ(B) =?σ(∃X.B), we obtain⊢?σ(∃X.B), ?σ(Γ), ?Q.

If A ≡ B1⊗· · ·⊗Bn splits, we divide the following into two cases accordingto the polarities of each σ(Bi) for 1 ≤ i ≤ n:

(i). If all σ(B1), . . . , σ(Bn) are negative, after removing the ⊗-link, wehave ⊢ σ(Bi), ?σ(Γi), ?Qi for each 1 ≤ i ≤ n by I.H. Then by the !-rule we have ⊢!σ(Bi), ?σ(Γi), ?Qi. By applying the ⊗-rules followedby the ?-rule, we have ⊢?(!σ(B1) ⊗ · · ·⊗!σ(Bn)), ?σ(Γ), ?Q. Since?(!σ(B1) ⊗ · · ·⊗!σ(Bn)) = ?σ(B1 ⊗ · · · ⊗ Bn), we obtain ⊢?σ(B1 ⊗· · · ⊗Bn), ?σ(Γ), ?Q.

(ii). If some σ(Bi) is positive, then Bi is either ∃X.C, !C or C1 ⊕ C2.We show only the case of ∃X.C, and merely note that the other casesare similar.

If Bi is ∃X.C, then from Andreoli’s focusing property of LLη2, this∃X.C splits to obtain the LLη2 proof net of conclusions C[X := D], Γi, ?Qi.Then by applying ⊗-rules to this proof net and to the other (n − 1)-proof nets containing Bj (j = i) having been obtained by the ⊗ split-ting of this case, we obtain the following proof net in LLη2 (i.e., theproof net is obtained from the original one by skipping the ∃-link):

⊗

B1C[X: =D] Bn

Γ ?Q

The size of this LLη2 proof net is smaller than that of the originalone, hence we have ⊢?σ(B1 ⊗ · · · ⊗ C[X := D]⊗ · · · ⊗ Bn), ?σ(Γ), ?Qby I.H. On the other hand, using the substitution lemma, we have

148

!σ(C[X := D]) =!σ(C)[X :=!σ(D)] ⊢ ∃X.!σ(C). Hence we have⊢?(!σ(B1)⊗ · · · ⊗ ∃X.!σ(C)⊗ · · ·⊗!σ(Bn)), ?σ(Γ), ?Q by the cut-rule.Since ∃X.!σ(C) =!σ(∃X.C), we finally obtain ⊢?σ(B1 ⊗ · · · ⊗ ∃X.C ⊗· · · ⊗Bn), ?σ(Γ), ?Q.

12.1.4 Additives

For the additive connectives & and ⊕, we add the following cases to theprevious proof of Proposition 12.1.15.

Induction step for n = 1:(Case 1)

If A ≡ B1 & B2, after removing the &-link, we obtain ⊢ σ(B1), ?Q and⊢ σ(B2), ?Q by I.H. Then by the ?-rule followed by the &-rule, wehave⊢ ?σ(B1) & ?σ(B2), ?Q. Thus we have the assertion since ?σ(B1) &?σ(B2) = σ(B1 & B2).

If A ≡ B1⊕B2, after removing the ⊕i-link, we obtain ⊢ σ(Bi), ?Q for i = 1or i = 2 by I.H. By the !-rule, we have ⊢ !σ(Bi), ?Q, and hence weobtain⊢ !σ(B1) ⊕ !σ(B2), ?Q. Thus we have the assertion since !σ(B1) ⊕!σ(B2) = σ(B1 ⊕B2).

Induction step for n > 1:(Case 2)

If A ≡ B1&B2, then the &-link is always splitting. Thus after removing the&-link, we obtain ⊢ ?σ(B1), ?σ(Γ), ?Q and ⊢ ?σ(B2), ?σ(Γ), ?Q by I.H.Hence we obtain ⊢ ?σ(B1)&?σ(B2), ?σ(Γ), ?Q by the &-rule, which isthe assertion since ?σ(B1) & ?σ(B2) = ?σ(B1 & B2).

If A ≡ B1 ⊕ B2 splits, after removing the ⊕i-link for i = 1 or 2, we obtain⊢ ?σ(Bi), ?σ(Γ), ?Q by I.H. We assume i = 1 without loss of general-ity. Then we further divide the following two cases according to thepolarity of σ(B1).

(i). If σ(B1) is negative, we have ⊢ σ(B1), ?σ(Γ), ?Q. By the !-rulefollowed by the ⊕1-rule, we have ⊢ !σ(B1) ⊕ !σ(B2), ?σ(Γ), ?Q. Thuswe obtain⊢ ?(!σ(B1)⊕ !σ(B2)), ?σ(Γ), ?Q, which is the assertion since ?(!σ(B1)⊕ !σ(B2))= ?σ(B1 ⊕B2).

149

(ii). If σ(B1) is positive, we obtain ⊢ ?σ(B1), ?σ(Γ), ?Q. By composingthis to ?σ(B1) ⊢ ?(σ(B1) ⊕ !σ(B2)) of Lemma 12.1.14(3) by the cut-rule, we obtain ⊢ ?(σ(B1) ⊕ !σ(B2)), ?σ(Γ), ?Q. Thus we have theassertion since ?(σ(B1)⊕ !σ(B2)) = ?σ(B1 ⊕B2) in this case.

12.1.5 Weakening and contraction

In order to deal with the weakening and the contraction rules, we add thefollowing cases to the previous proof of Proposition 12.1.15 in Section 12.1.3.

Induction step for n = 1:(Case 1)A ≡ ?B.

If the splitting link of ?B is the ?w-link, after removing the link, weobtain a proof net of conclusion ?Q in LLη2. Note that ?Q is notempty, because of the consistency of LLη2 which is a consequence ofthe cut-elimination of LLη2 (cf. [Okada 1999]). Thus by I.H. we have⊢ ?Q. Hence we have ⊢ ?!σ(B), ?Q where ?!σ(B) = σ(?B).

If the splitting link of ?B is the ?c-link, after removing the link, we obtain⊢ ?σ(?B), ?σ(?B), ?Q by I.H. for the case of n > 1. Thus we obtain⊢ ?σ(?B), ?Q, and hence we have ⊢ σ(?B), ?Q since ?σ(?B) = σ(?B).

(Case 2)If the ?w-link of ?P splits, after removing the link, we obtain ⊢ σ(A), ?Q

by I.H. Thus we obtain ⊢ σ(A), ?P, ?Q by the ?w-rule.

If the ?c-link of ?P splits, after removing the link, we obtain ⊢ σ(A), ?P, ?P, ?Qby I.H. Thus we obtain ⊢ σ(A), ?P, ?Q by the ?c-rule.

Induction step for n > 1:(Case 1)If the ?w-link of ?P splits, we have the assertion by the same way as the

induction step for n = 1.

If the ?c-link of ?P splits, we have the assertion by the same way as theinduction step for n = 1.

(Case 2)A ≡ ?B.

If the ?w-link of ?B splits, after removing the link, we obtain ⊢ ?σ(Γ), ?Qby I.H. Hence we obtain ⊢ ?!σ(B), ?σ(Γ), ?Q where ?!σ(B) = ?σ(?B).

If the ?c-link of ?B splits, after removing the link, we obtain⊢ ?σ(?B), ?σ(?B), ?σ(Γ), ?Q by I.H. Hence we obtain ⊢ ?σ(?B), ?σ(Γ), ?Q.

150

12.1.6 LLη2 is conservative over LLηpol2

Using Proposition 12.1.15, we show the following main theorem:

Theorem 12.1.16 (LLη2 ≻ LLηpol2) LLη2 is conservative over LLη

pol2. Thatis, for any focalized sequent Γ, if ⊢ Γ is provable in LLη2, then it is provablein LLη

pol2.

Proof. We prove the theorem by induction on the length of a given proofπ of ⊢ Γ. If the last rule of π is other than the ∃-rule, the assertion isstraightforward from the induction hypothesis because the premises of therules are focalized. So we consider the case where a focalized sequent⊢ ∃X.P , Γ is provable from ⊢ P [X := A] , Γ via the ∃-rule. Since ∃X.P ispositive, Γ is a negative sequent. Since negative connectives .................................................

.............................................. ,∀ and & are

reversible, we may assume without loss of generality that Γ is of the form?Q. Then using Proposition 12.1.15 (when n = 1), ⊢ σ(P [X := A]), Γ isprovable in LLη

pol2. Since σ(P [X := A]) = P [X := !σ(A)] by the substitutionlemma (Lemma 12.1.13), ⊢ ∃X.P , Γ is provable by the ∃-rule of LLη

pol2.

Remark 12.1.17 (Theorem 12.1.16 without weakening and contraction)A variation of Theorem 12.1.16 is also valid when the two systems are re-stricted to the fragments without weakening and contraction: I.e., let LLη−2be the fragment of LLη2 without the weakening and the contraction rules.Then the variation states that LLη−2 is conservative over its polarized frag-ment LLη−

pol2. This variation is obtained because Proposition 12.1.15 holdsfor the fragment LLη−2 by virtue of our proof method (see the first paragraphof the proof of the proposition).

12.1.7 Some syntactical properties derived from Theorem12.1.16

In this subsection, as a consequence of Theorem 12.1.16, LLη2 is shown tosatisfy a stronger property than FSP (cf. Definition 9.2.3), which propertywe call the focalized proof property (FPP). This property explains a contrastbetween LL2 and LLη2, which we have also seen in conservativities over theirpolarized fragments.

Definition 12.1.18 (Focalized proof property (FPP)) If a focalized se-quent ⊢ Γ is provable in a logical system L by restricting ⊤-rules to focalizedsequents, then there is a proof π of ⊢ Γ which consists only of focalizedsequents.

151

Remark 12.1.19 FPP is both FSP of Definition 9.2.3 and its reverse. Thereverse property states as follows: If a focalized sequent ⊢ Γ is provable inL by restricting ⊤-rules to focalized sequents, then there exists a proof π of⊢ Γ which consists only of polarized formulas.

Combining Theorem 12.1.16 and FSP of LLη2 (cf. Lemma 12.1.6), wehave:

Corollary 12.1.20 LLη2 has focalized proof property (FPP).

Proof. It suffices to show that LLη2 has a property of Remark 12.1.19.Let ⊢ Γ be a focalized sequent which is provable in LLη2. Then by theconservativity theorem (Theorem 12.1.16), ⊢ Γ is provable in LLη

pol2, whereproofs consist only of polarized formulas.

As shown in Proposition 12.1.3, LL2 does not have FSP, hence neitherFPP. Let us summarize the states of these properties by the following table:

LL(η)pol2 LL LL2 LLη2

FSP Yes Yes No Yes(Prop 12.1.3) (Lem 12.1.6)

FPP Yes Yes No Yes(Cor 12.1.20)

Conservativity − Yes No Yesover pol frag. (Cor 11.5.1) (Prop 12.1.1) (Thm 12.1.16)

Table 1

Note that the left-most column of the table for LL(η)pol2 is automatic. The

column for LL is essentially due to [Laurent 2002] and its FPP is obtainedsince the property of Remark 12.1.19 holds by virtue of the subformulaproperty of the first order LL.

12.2 Second order definability of restricted addi-tives in polarized linear logic

In the polarized fragment MELLpol2 of the second order multiplicative expo-nential linear logic, only restricted forms of additives are definable. Hencein Section 12.1 it is impossible to make additive connectives redundant.

152

By linearizing Prawitz’s definition for the second order classical logic([Prawitz 1965]), the additive connectives are known to be definable inMELL2 by the following and by its De Morgan dual:

A⊕B := ∀X.!(A−◦X⊥)−◦ !(B −◦X⊥)−◦X⊥

A & B := ∃X.!(A............................................................................................... X⊥)⊗ !(B.................................................

.............................................. X⊥)⊗X.

By canonically polarizing these formulas, we define

P ⊕Q := !∀X.?(P ⊗X)............................................................................................... ?(Q⊗X).................................................

.............................................. X⊥

N&M := ?∃X.!(N ............................................................................................... X⊥)⊗ !(M .................................................

.............................................. X⊥)⊗X.

These then simulate the following restricted form of additive rules in MELLpol2:

⊢ Γ, Pi

⊢ Γ, P1 ⊕ P2⊕i

i = 1, 2

⊢ N , N ⊢ N , L

⊢ N , N & L&

I.e., in terms of categorical logic & (resp. ⊕) is a restriction of the usual(weaker) product (resp. coproduct) in D+,− in the following sense: ForN,M ∈ D−, N&M is a triple (N&M, N&M

proj1−−−→ N, N&Mproj2−−−→ M)

such that the usual (weaker) universal mapping property for the productholds for an arbitrarily given triple (P, P −→ N, P −→ M) for P ∈ D+.Note that the given triples are restrictions of the usual ones since P rangesonly in D+ but not in D−.

It is impossible to extend the &-rule for a general context Γ (not onlyN but also) containing at most one positive formula. This is because, thesimulation of the &-rule violates the focalized proof property of Definition12.1.18, which MELLpol2 retains (cf. Table 1 of Section 12.1.7).

The definability of the restricted additives ⊕ and & also works in MELLηpol2

if X and X⊥ are replaced by !X⊥ and ?X, respectively.

153

Chapter 13

Future work for Part II

A promising future work is to give a truth-valued semantics for the in-dexed logical system MALLP(I) of [Hamano-Takemura 2008] by employingour polarized phase spaces. The syntactical system MALLP(I) is a polarizedextension of Bucciarelli-Ehrhard [2000]’s indexed system and arises frommulti-pointed relations, which form an web-based instance of the MALLPcategorical model of [Hamano-Scott 2007]. For this work an I-product po-larized phase spaces are important analogously to [Ehrhard 2004]. A prod-uct topology on the spaces is a key ingredient to accommodate the polaritiesto the indexed system.

As seen unexpectedly in Section 12.1, the second order syntaxes con-trast with the first order ones in terms of conservativity and focalized se-quent property. To understand this contrast, it is important to extend thecategorical semantics of [Hamano-Scott 2007] into second order MALLP2.A categorical model, which is stronger in modeling proofs, may explainthis contrast observed in the weaker standpoint of provability, which con-servativity as well as FSP concerns. Polarized dinatural transformation of[Hamano-Scott 2007] is appropriate to model second order variable. It isalso anticipated that incorporating second order in the categorical modelprovides a better understanding of focalization, whose semantical counterpart is not yet clear.

The logical system MALLP2, which is complete with respect to our po-larized phase spaces, is considered as the basic syntax of Girard’s theory ofludics (without weakening). We intend to study a semantical strong normal-ization theorem for the L-nets of [Curien-Faggian 2005] for ludics by usingour polarized phase spaces. For that purpose, the phase semantic method

154

introduced in [Okada 1999, Okada-Takemura 2007] may be applied.

The categorical framework of [Mellies-Tabareau 2007] may provide amore abstract framework for our polarized phase semantics. In partic-ular, a notion of phase semantics for polarized linear logic follows from[Mellies-Tabareau 2007]’s categorical semantics for their tensorial logic. Al-though their categorical construction is limited to propositional logic with-out the second order quantifiers so far, our phase semantic constructioncould be a clue to extend their categorical framework to the second order.

Okada-Terui [2005] also introduced a similar framework as [Mellies-Tabareau 2007].They introduced a phase semantics called simple phase semantics for theirSimple Logic, which is a more primitive logic than linear logic. Simple phasesemantics is simple in the sense that no closure condition is needed to definea value (fact) in the space. Simple phase semantics was shown to be a com-plete model for Simple Logic. Then, from the simple phase semantics, usualclassical phase semantics for LL is naturally derived by imposing a doublenegation closure operator. We will study a relationship among our polar-ized phase semantics for polarized linear logic, and [Mellies-Tabareau 2007]’sphase semantics for their tensorial logic, and [Okada-Terui 2005]’s simplephase semantics for their Simple Logic, in order to understand a generalabstract construction shared by them.

155

Bibliography for Part I

[Allwein-Barwise 1996] Gerard Allwein and Jon Barwise, eds., Logical rea-soning with diagrams, Oxford Studies In Logic And Computation Series,1996.

[Blackett 1983] Donald W. Blackett, Elementary Topology, Academic Press,1983.

[Crabbe 2001] Marcel Crabbe, The formal theory of syllogisms, The Reviewof Modern Logic, Volume 9, Number 1-2, 29-52, 2001.

[Euler 1768] Leonhard Euler, Lettres a une Princesse d’Allemagne surDivers Sujets de Physique et de Philosophie, Saint-Petersbourg: Del’Academie des Sciences, 1768. (English Translation by H. Hunter, 1997,Letters of Euler to a German Princess on Different Subjects in Physicsand Philosophy, Thoemmes Press, 1997.)

[Fish-Flower 2008] Andrew Fish and Jean Flower, Euler Diagram Decom-position, Fifth International Conference on the Theory and Applicationof Diagrams (Diagrams 2008), Springer, Lecture Notes in ComputerScience 5223, 28-44, 2008.

[Fish-John-Taylor 2008] Andrew Fish, Chris John, John Taylor, A NormalForm for Euler Diagrams with Shading, Fifth International Conferenceon the Theory and Application of Diagrams (Diagrams 2008), Springer,Lecture Notes in Computer Science 5223, 206-221, 2008.

[Flower-Fish-Howse 2008] Jean Flower, Andrew Fish, John Howse, Euler di-agram generation, Journal of Visual Languages and Computing, 19(6),675-694, 2008.

[Flower-Masthoff-Stapleton 2004] Jean Flower, Judith Masthoff, and GemStapleton, Generating Readable Proofs: A Heuristic Approach to Theo-rem Proving With Spider Diagrams, Diagrammatic Representation and

156

Inference, Third International Conference (Diagrams 2004), LectureNotes in Computer Science 2980, 166-181, 2004.

[Flower-Stapleton 2004] Jean Flower and Gem Stapleton, Automated The-orem Proving with Spider Diagrams, Proceedings of Computing: TheAustralasian Theory Symposium (CATS) 2004, Electronic Notes inTheoretical Computer Science, 91, 246-263, 2004.

[Gentzen 1969] Gerhard Gentzen, Investigations into logical deduction. InM. E. Szabo, ed., The collected Papers of Gerhard Gentzen, 1969. (Orig-inally published as Unter suchungen uber das logische Scliessen, Math-ematische Zetischrift 39 (1935): 176-210, 405-431.)

[Gil-Howse-Tulchinsky 2002] Joseph Gil, John Howse, Elena Tulchinsky,Positive Semantics of Projections in Venn-Euler Diagrams, Journal ofVisual Languages and Computing, 13(2), 197-227, 2002.

[Grialou-Okada 2005] Pierre Grialou and Mitsuhiro Okada, Questions ontwo cognitive models of deductive reasoning, in P. Grialou, G. Longo,and M. Okada, eds., Images and Reasoning, Keio University Press, 31-67, 2005.

[Hammer 1994] Eric Hammer, Reasoning with sentences and diagrams, TheNotre Dame Journal of Formal Logic, 35(1), 73-87, 1994.

[Hammer 1995] Eric Hammer, Logic and Visual Information, CSLI Publi-cations, 1995.

[Hammer-Danner 1993] Eric Hammer and Norman Danner, Towards amodel theory of diagrams, in G. Allwein and J. Barwise, eds., WorkingPapers on Diagrams and Logic, Bloomington: Indiana University, 1993.

[Hammer-Shin 1998] Eric Hammer and Sun-Joo Shin, Euler’s visual logic,History and Philosophy of Logic, 19, 1-29, 1998.

[Howse 2008] John Howse, Diagrammatic Reasoning Systems, in Peter W.Eklund, Ollivier Haemmerle, eds., Conceptual Structures: KnowledgeVisualization and Reasoning, 16th International Conference on Con-ceptual Structures (ICCS 2008), Lecture Notes in Computer Science5113, Springer, 1-20, 2008.

[Howse-Molina-Shin-Taylor 2002] John Howse, Fernando Molina, Sun-JooShin, John Taylor, On Diagram Tokens and Types, in Mary Hegarty,

157

Bernd Meyer, N. Hari Narayanan, eds., Diagrammatic Representationand Inference, Second International Conference (Diagrams 2002), 146-160, 2002.

[Howse-Molina-Taylor 2000] John Howse, Fernando Molina, John Taylor,SD2: A Sound and Complete Diagrammatic Reasoning System, 2000IEEE International Symposium on Visual Languages, 127-134, 2000.

[Howse-Molina-Taylor-Kent-Gil 2001] John Howse, Fernando Molina, JohnTaylor, Stuart Kent, and Joseph Gil, Spider diagrams, Journal of VisualLanguages and Computing, 12(3), 299-324, 2001.

[Howse-Stapleton-Taylor 2005] John Howse, Gem Stapleton, and John Tay-lor, Spider Diagrams, LMS Journal of Computation and Mathematics,Volume 8, 145-194, London Mathematical Society, 2005.

[Kent 1997] Stuart Kent, Constraint diagrams, in Proceedings of the 1997ACM SIGPLAN Conference on Object-Oriented Programming Systems,Languages & Applications (OOPSLA ’97), 327-341, 1997.

[Lemon-Shin 2001] Oliver Lemon and Sun-Joo Shin, Diagrams, inStanford Encyclopedia of Philosophy, ed. Edward N. Zalta,http://plato.stanford.edu/entries/diagrams/, First published Aug 28,2001; substantive revision Sep 3, 2008.

[Meyer 2001] Bernd Meyer, Diagrammatic evaluation of visual mathemat-ical notations, in Diagrammatic Representation and Reasoning, P.Olivier, M. Anderson, and B. Meyer (Eds.), Springer, 261-277, 2001.

[Mineshima-Okada-Sato-Takemura 2008] Koji Mineshima, MitsuhiroOkada, Yuri Sato, and Ryo Takemura, Diagrammatic Reasoning Sys-tem with Euler Circles: Theory and Experiment Design, in Proceedingsof the 5th international conference on Diagrammatic Representationand Inference (Diagrams 08), Lecture Notes In Artificial Intelligence,Vol. 5223, Springer-Verlag, 188-205, 2008.

[Molina 2001] Fernando Molina, Reasoning with extended Venn-Peirce dia-grammatic systems, PhD Thesis, University of Brighton, 2001.

[Nielsen-Plotkin-Winskel 1980] Mogens Nielsen, Gordon Plotkin, and GlynnWinskel, Petri Nets, Event Structures and Domains, Part I, TheoreticalComputer Science, Vol. 13, No. 1, pages 85-108, 1980.

158

[Okada 1999] Mitsuhiro Okada, Phase Semantic Cut-Elimination and Nor-malization Proofs of First- and Higher-Order Linear Logic, TheoreticalComputer Science, 227(1-2), 333-396, 1999.

[Okada 2002] Mitsuhiro Okada, A uniform semantic proof for cut-elimination and completeness of various first and higher order logics,Theoretical Computer Science, 281(1-2), 471-498, 2002.

[Peirce 1897] Charles Sanders Peirce, Collected Papers IV, Harvard Univer-sity Press, 1897/1933.

[Prawitz 1965] Dag Prawitz, Natural Deduction, Almqvist & Wiksell, 1965(Dover, 2006).

[Rizzo-Palmonari 2005] Antonio Rizzo and Marco Palmonari, The Mediat-ing Role of Artefacts in Deductive Reasoning, In B. Bara, L. Barsalou,M. Bucciarelli (Eds.) Proceedings of 27th Annual conference of the Cog-nitive Science Society (CogSci 2005), 1862-1867, 2005.

[Rodgers-Zhang-Fish 2008] Peter Rodgers, Leishi Zhang, and Andrew Fish,General Euler Diagram Generation, in in Proceedings of the 5th interna-tional conference on Diagrammatic Representation and Inference (Dia-grams 08), Lecture Notes In Artificial Intelligence, Vol. 5223, Springer-Verlag, 13-27, 2008.

[Rodgers-Zhang-Stapleton-Fish 2008] Peter Rodgers, Leishi Zhang, GemStapleton, and Andrew Fish, Embedding Wellformed Euler Diagrams,in Proceedings of Information Visualisation 2008, IEEE Computer So-ciety Press, London, UK, 585-593, 2008.

[Sato-Mineshima-Takemura-Okada 2009] Yuri Sato, Koji Mineshima, RyoTakemura, and Mitsuhiro Okada, How can Euler diagrams improvesyllogistic reasoning?, 2009, in preparation.

[Shimojima 1996] Atsushi Shimojima, On the Efficacy of Representation,Ph.D. thesis, Indiana University, 1996.

[Shin 1994] Sun-Joo Shin, The Logical Status of Diagrams, Cambridge Uni-versity Press, 1994.

[Shin 1996] Sun-Joo Shin, Situation-theoretic account of valid reasoningwith Venn diagrams, in G. Allwein and J. Barwise (Eds.), Logical Rea-soning with Diagrams, Oxford University Press, 81-108, 1996.

159

[Stapleton 2005] Gem Stapleton, A survey of reasoning systems based onEuler diagrams, Proceedings of the First International Workshop onEuler Diagrams (Euler 2004), Electronic Notes in Theoretical Com-puter Science, Volume 134, 1, 127-151, 2005.

[Stapleton-Fish-Rodgers 2008] Gem Stapleton, Andrew Fish, and PeterRodgers, Abstract Euler Diagram Isomorphism, in Proceedings of 14thInternational Conference on Distributed Multimedia Systems, VisualLanguages and Computing, Boston, USA, Knowledge Systems Insti-tute, 310-317, 2008.

[Stapleton-Masthoff-Flower-Fish-Southern 2007] Gem Stapleton, JudithMasthoff, Jean Flower, Andrew Fish, and Jane Southern, AutomatedTheorem Proving in Euler Diagram Systems, Journal of AutomatedReasoning, volume 39 , issue 4, 431-470, 2007.

[Stapleton-Rodges-Howse-Taylor 2007] Gem Stapleton, Peter Rodgers,John Howse, and John Taylor, Properties of Euler diagrams, Elec-tronic Communication of the European Association of Software Scienceand Technology, Volume 7: Layout of (Software) Engineering Diagrams2007, 2-16, 2007.

[Stenning 2002] Keith Stenning, Seeing Reason, Oxford University Press,2002.

[Stenning-Oberlander 1995] Keith Stenning and Jon Oberlander, A Cogni-tive Theory of Graphical and Linguistic Reasoning: Logic and Imple-mentation, Cognitive Science, 19(1). 97-140, 1995.

[Swoboda-Allwein 2004] Nik Swoboda and Gerard Allwein, Using DAGtransformations to verify Euler/Venn homogeneous and Euler/VennFOL heterogeneous rules of inference, Journal on Software and Sys-tem Modeling, 3(2), 136-149, 2004.

[Swoboda-Allwein 2005] Nik Swoboda and Gerard Allwein, Heterogeneousreasoning with Euler/Venn diagrams containing named constants andFOL, Electronic Notes in Theoretical Computer Science, Volume 134,Elsevier, 153-187, 2005.

[Takeuti 1987] Gaisi Takeuti, Proof Theory, 2nd ed. North-Holland/ElsevierScience, 1987.

160

[Troelstra-Schwichtenberg 2000] Anne S. Troelstra and Helmut Schwicht-enberg, Basic Proof Theory, 2nd edition, Cambridge University Press,2000.

[Venn 1881] John Venn, Symbolic Logic, Macmillan, 1881.

161

Bibliography for Part II

[Abramsky 2003] Samson Abramsky, Sequentiality vs. Concurrency InGames and Logic, Mathematical Structures in Computer Science, 13(4),531-565, 2003.

[Abrusci 1990] V. Michele Abrusci, Sequent calculus for intuitionistic lin-ear propositional logic, Mathematical Logic, Plenum, New York, 223-242,1990.

[Abrusci 1991] V. Michele Abrusci, Phase Semantics and Sequent Calculusfor Pure Noncommutative Classical Linear Propositional Logic, Journalof Symbolic Logic, Vol. 56, No. 4, 1403-1451, 1991.

[Amadio-Curien 1998] Roberto M. Amadio, Pierre-Louis Curien, Domainsand Lambda-calculi, Cambridge Tracts in Theoretical Computer Science,Cambridge University Press, 1998.

[Andreoli 1992] Jean-Marc Andreoli, Logic Programming with FocusingProofs in Linear Logic, Journal of Logic and Computation, 2(3), 297-347,1992.

[Andreoli-Maieli 1999] Jean-Marc Andreoli, Roberto Maieli, Focusing andProof-Nets in Linear and Non-commutative Logic, Logic Programmingand Automated Reasoning, 321-336, 1999.

[Asperti-Longo 1991] Andrea Asperti and Giuseppe Longo, Categories,Types, and Structures: An Introduction to Category Theory for the Work-ing Computer Scientist, Foundations of Computing Series, M.I.T. Press,1991.

[Benton-Bierman-Paiva-Hyland 1992] Nick Benton, Gavin M. Bierman, Va-leria de Paiva, and Martin Hyland, Term assignment for intuitionisticlinear logic, Technical Report 262, Computer Laboratory, University ofCambridge, 1992.

162

[Bierman 1995] Gavin M. Bierman, What is a Categorical Model of Intu-itionistic Linear Logic?, In M. Dezani, editor, Second International Con-ference on Typed Lambda Calculi and Applications (TLCA ’95), Springer-Verlag LNCS 902, 78-93, 1995.

[Blute-Cockett-Seely 1996] Richard Blute, J. Robin B. Cockett, R. A. G.Seely, Storage as Tensorial Strength, Mathematical Structures in Com-puter Science, 6, 313-351, 1996.

[Blute-Scott 2004] Richard Blute, Philip Scott, Category theory for linearlogicians, in Linear Logic in Computer Science, T. Ehrhard, J.-Y. Girard,P. Ruet and. Ph. Scott, London Mathematical Society Lecture Note Series316, Cambridge University Press, 3-64, 2004.

[Bucciarelli-Ehrhard 2000] Antonio Bucciarelli and Thomas Ehrhard, Onphase semantics and denotational semantics in multiplicative-additive lin-ear logic, Annals of Pure and Applied Logic, 102(3), 247-282, 2000.

[Ciabattoni-Terui 2006] Agata Ciabattoni and Kazushige Terui, Towards asemantic characterization of cut-elimination, Studia Logica, Vol. 82, No.1, 95-119, 2006.

[Curien 2005] Pierre-Louis Curien, Introduction to linear logic and ludics,part I, and part II, Advances in Mathematics (China) 34 (5), 513-544,2005, and 35 (1), 1-44, 2006.

[Curien-Faggian 2005] Pierre-Louis Curien and Claudia Faggian, L-nets,strategies and proof-nets, Computer Science Logic, 19th InternationalWorkshop (CSL 2005), Lecture Notes in Computer Science, 3634,Springer, 167-183, 2005.

[Danos-Joinet-Schellinx 1997] Vincent Danos, Jean-Baptiste Joinet, HaroldSchellinx, A New Deconstructive Logic: Linear Logic, Journal of SymbolicLogic, 62(3), 755-807, 1997.

[Ehrhard 2004] Thomas Ehrhard, A completeness theorem for symmetricproduct phase spaces, Journal of Symbolic Logic, Volume 69, Issue 2,340-370, 2004.

[Girard 1987] Jean-Yves Girard, Linear Logic, Theoretical Computer Sci-ence, 50, 1-102, 1987.

[Girard 1991] Jean-Yves Girard, A New Constructive Logic: ClassicalLogic, Mathematical Structures in Computer Science, 1 (3), 255-296, 1991.

163

[Girard 1995] Jean-Yves Girard, Linear Logic: its syntax and semantics,in Advances in Linear Logic, Girard, Lafont, Regnier, eds., CambridgeUniversity Press, 1995.

[Girard 2001] Jean-Yves Girard, Locus Solum: From the rules of logic tothe logic of rules, Mathematical Structures in Computer Science, 11(3),301-506, 2001.

[Hamano-Scott 2007] Masahiro Hamano and Philip Scott, A Categorical Se-mantics for Polarized MALL, Annals of Pure and Applied Logic, 145, 276-313, 2007.

[Hamano-Takemura 2008] Masahiro Hamano and Ryo Takemura, An In-dexed System for Multiplicative Additive Polarized Linear Logic, Com-puter Science Logic, 22nd International Workshop (CSL 2008), LectureNotes in Computer Science, 5213, Springer, 262-277, 2008.

[Hamano-Takemura 2009] Masahiro Hamano and Ryo Takemura, A PhaseSemantics for Polarized Linear Logic and Second Order Conservativity,accepted for publication in Journal of Symbolic Logic, 33 pages, 2009.

[Hofmann-Streicher 2002] Martin Hofmann and Thomas Streicher, Com-pleteness of Continuation Models for λµ-Calculus, Information and Com-putation, Volume 179(2), 332-355, 2002.

[Lafont 1997] Yves Lafont, The Finite Model Property for Various Frag-ments of Linear Logic, Journal of Symbolic Logic, 62(4), 1202-1208, 1997.

[Laurent 1999] Olivier Laurent, Polarized Proof-Nets: Proof-Nets for LC,Proceedings of the 4th International Conference on Typed Lambda Calculiand Applications (TLCA 1999), 213-227, 1999.

[Laurent 2002] Olivier Laurent, Etude de la polarisation en logique, Thesede Doctorat, Institut de Mathematiques de Luminy, Universite Aix-Marseille II, 2002.

[Laurent 2005a] Olivier Laurent, Syntax vs. semantics: A polarized ap-proach, Theoretical Computer Science, 343 (1-2), 177-206, 2005.

[Laurent 2005b] Olivier Laurent, A proof of the focalization property ofLinear Logic, draft, 2005.

[Laurent-Quatrini-Tortora 2005] Olivier Laurent, Myriam Quatrini, andLorenzo Tortora de Falco, Polarized and focalized linear and classical

164

proofs, Annals of Pure and Applied Logic, Volume 134(2-3), 217-264,2005.

[Laurent-Tortora 2004] Olivier Laurent and Lorenzo Tortora de Falco, Slic-ing polarized additive normalization, in Linear Logic in Computer Sci-ence, eds. T. Ehrhard, J.-Y. Girard, P. Ruet, and P. Scott, LMS LectureSeries 316, Cambridge University Press, 247-282, 2004.

[Mellies 2003] Paul-Andre Mellies, Categorical models of linear logic revis-ited, Prepublication de l’equipe PPS (September 2003, number 22), toappear in Theoretical Computer Science.

[Mellies 2005] Paul-Andre Mellies, Asynchronous games 3, An innocentmodel of linear logic, Electronic Notes in Theoretical Computer Science,122, 171-192, 2005.

[Mellies 2005b] Paul-Andre Mellies, Asynchronous Games 4: A Fully Com-plete Model of Propositional Linear Logic, 20th IEEE Symposium onLogic in Computer Science (LICS 2005), 386-395, 2005.

[Mellies 2009] Paul-Andre Mellies, Categorical Semantics of Linear Logic,to appear in Panoramas et Syntheeses, Societe Mathematique de France,2009.

[Mellies-Tabareau 2007] Paul-Andre Mellies and Nicolas Tabareau, Re-source modalities in game semantics, Proceedings of the 22nd AnnualIEEE Symposium on Logic in Computer Science (LICS 2007), 389-398,2007.

[Miller 2004] Dale Miller, An overview of linear logic programming, in Lin-ear Logic and Computer Science, eds. T. Ehrhard, J.-Y. Girard, P. Ruet,and P. Scott, LMS Lecture Series 316, Cambridge University Press, 119-150, 2004.

[Okada 1999] Mitsuhiro Okada, Phase Semantic Cut-Elimination and Nor-malization Proofs of First- and Higher-Order Linear Logic, TheoreticalComputer Science, 227(1-2), 333-396, 1999.

[Okada 1999b] Mitsuhiro Okada, An Introduction to Linear Logic: PhaseSemantics and Expressiveness, in Theories of Types and Proofs, eds. M.Dezani, M. Okada, and M. Takahashi, Memoirs of Mathematical Societyof Japan, vol.2 (1998), 255-295, second edition 1999.

165

[Okada 2002] Mitsuhiro Okada, A Uniform Proof for Higher Order Cut-Elimination and Normalization Theorem, Theoretical Computer Science,281, 471-498, 2002.

[Okada 2004] Mitsuhiro Okada, Linear Logic and Intuitionistic Logic, Larevue internationale de philosophie, No. 230, pp. 449-481, special issue“Intuitionism”, 2004.

[Okada-Takemura 2007] Mitsuhiro Okada and Ryo Takemura, Remarkson Semantic Completeness for Proof-Terms with Laird’s DualAffine/Intuitionistic λ-Calculus, in Rewriting, Computation and Proof:Essays Dedicated to Jean-Pierre Jouannaud on the Occasion of His 60thBirthday, Lecture Notes in Computer Science, Springer, Volume 4600,167-181, 2007.

[Okada-Terui 2005] Mitsuhiro Okada and Kazushige Terui, Note on simplephase models and a simple fragment of linear logic which induce the fulllinear logic, manuscript.

[Okada-Terui 1999] Mitsuhiro Okada and Kazushige Terui, The FiniteModel Property for Various Fragments of Intuitionistic Linear Logic,Journal of Symbolic Logic, 64, 790-802, 1999.

[Ono 1993] Hiroakira Ono, Semantics for substructural logics, in Substruc-tural Logics, K. Dosen and P. Schroeder-Heister eds., Oxford UniversityPress, 259-291, 1993.

[Prawitz 1965] Dag Prawitz, Natural Deduction - A proof theoretical study,Almquist and Wiksell, Stockholm, 1965.

[Sambin 1995] Giovanni Sambin, Pretopologies and Completeness Proofs,Journal of Symbolic Logic, Vol. 60, No. 3, 861-878, 1995.

[Seely 1989] R. A. G. Seely, Linear logic, ∗-autonomous categories and cofreecoalgebras, in J. Gray and A. Scedrov (eds.), Categories in ComputerScience and Logic, Contemporary Mathematics 92, 371-382, 1989.

[Selinger 2001] Peter Selinger, Control categories and duality: on the cate-gorical semantics of the lambda-mu calculus, Mathematical Structures inComputer Science, 11, 207-260, 2001.

[Schalk 2004] Andrea Schalk, What is a Categorical Model of Linear Logic?,Manuscript, 2004, http://www.cs.man.ac.uk/ schalk/notes/index.html.

166

[Tabareau 2008] Nicolas Tabareau, Modalites de Ressource et Controle enLogique Tensorielle, PhD thesis, Universite Paris Diderot (Paris 7), 2008.

[Takeuti 1987] Gaisi Takeuti, Proof theory, 2nd ed., North-Holland, Ams-terdam, 1987.

[Terui 2007] Kazushige Terui, Which Structural Rules Admit Cut Elimi-nation? —An Algebraic Criterion, Journal of Symbolic Logic, 72 (3),738-754, 2007.

[Troelstra 1992] Anne S. Troelstra, Lectures on Linear Logic, CSLI LectureNotes, vol. 29, Stanford, 1992.

[Troelstra-Schwichtenberg 2000] Anne S. Troelstra and Helmut Schwichten-berg, Basic Proof Theory, Cambridge University Press, Cambridge U.K.,Second edition 2000.

167

abelard.flet.mita.keio.ac.jpabelard.flet.mita.keio.ac.jp/person/takemura/paper/18... · 2009. 10....

Documents

Transcript of abelard.flet.mita.keio.ac.jpabelard.flet.mita.keio.ac.jp/person/takemura/paper/18... · 2009. 10....