On an Intuitionistic Modal Logic - Nuance Communications · On an Intuitionistic Modal Logic 7 4. A...

On an Intuitionistic Modal Logic

G.M. BiermanDepartment of Computer Science, University of Warwick. England.

V.C.V. de PaivaSchool of Computer Science, University of Birmingham. England.

Abstract. In this paper we consider an intuitionistic variant of the modal logicS4 (which wecall IS4). The novelty of this paper is that we place particular importance on the natural de-duction formulation ofIS4—our formulation has several important metatheoretic properties.In addition, we study models ofIS4, not in the framework of Kripke semantics, but in themore general framework of category theory. This allows not only a more abstract definition ofa wholeclassof models but also a means of modelling proofs as well as provability.

1. Introduction

Modal logics are traditionally extensions ofclassical logic with new oper-ators, or modalities, whose operation is intensional. Modal logics are mostcommonly justified by the provision of an intuitive semantics based upon‘possible worlds’, an idea originally due to Kripke. Kripkealso provided apossible worlds semantics for intuitionistic logic, and soit is natural to con-sider intuitionistic logic extended with intensional modalities. Indeed muchwork already exists on this topic (a good account is given by Simpson [37]).In fact there is an almost bewildering number of intuitionistic modal logics—most of which consist of extensions of intuitionistic logicwith some selectionof modal rules and axioms. Much of this work is justified by reference toa Kripke model—the number of choices arises from the large number ofcombinations of conditions one can impose on the accessibility relations.

In this paper we take a different approach. Rather than use Kripke models,we are motivated by the much more general language of category theory. Onereason is that, unlike the situation for Kripke semantics, we are interested inmodelling not just provability but also the proofs themselves. This approachis often termed categorical proof theory, or simply categorical logic [31]. Cat-egory theory provides a language for describing abstractlywhat is required ofa model or, more precisely, which structures are needed for an arbitrary cat-egory to model the logic. Checking that a candidate is a concrete model thensimplifies to checking that it satisfies the abstract definition. Thus soundness,for example, need only be checked once and for all, for the abstract defini-tion. All concrete models which satisfy the abstract definition are guaranteedto be sound. In this way categorical semantics provide a general and oftensimple formulation of what it is to be a model. This is of interest because it

c 1999Kluwer Academic Publishers. Printed in the Netherlands.

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2 Bierman and de Paiva

is often the case that more traditional models lack any generality or are quitecomplicated to describe (or both). In addition categoricalsemantics enableone to model some very powerful logics such as impredicativetype theoriesand various higher order logics.

Another important aspect of our approach is our interest in proof theory.Most work on (intuitionistic) modal logics simply gives thelogic using anaxiomatic formulation, the primary interest being provability. In contrast weare interested in (primarily) natural deduction and sequent calculus formu-lations, and their metatheoretic properties. (Interestingly, Satre [35] givesnatural deduction formulations of (classical) modal logics, but does not studytheir metatheoretic properties. This point will be addressed more fully inx4.)

This paper is not the place for a philosophical discussion ofthe signif-icance, or importance, of intuitionistic modal logics in general. However itis worth highlighting a number of important applications ofintuitionisticmodal logics in computer science. Stirling [38] uses an intuitionistic modallogic to capture a notion of bisimilarity of divergent processes. Fairtloughand Mendler [15] use an intuitionistic modal logic [4] to reason about thebehaviours of hardware circuits. Davies and Pfenning [13] have used partof the logic described in this paper to define a programming language withexplicit binding time constructors.

In this paper we study an intuitionistic version of the modallogic S4.As we shall see, given our interests in proof theory and categorical models,there is quite a lot to say about this logic. Extensions of ourapproach toother intuitionistic modal logics remains future work. Thepaper is organisedas follows. Inxx2–3 we give axiomatic and sequent calculus formulationsof IS4, respectively. Inx4 we consider the definition of a natural deductionformulation due to Satre [35] and show that it is not closed under substitution.We also consider Prawitz’s proposal [32], which we also reject. We give ourformulation and demonstrate a number of important properties. Inx4.1 weconsider the question of proof normalisation and consider the question of thesubformula property for our natural deduction formulation. We give details ofcommuting conversions which are sufficient to derive a subformula property.In x5 we sketch some details of an alternative natural deductionformulationwhich uses two assumption sets—one for modalised formulae,and one fornon-modalised formulae. Inx6 we define the�S4-calculus, which is given bythe Curry-Howard correspondence from our natural deduction formulation.In x7 we give in detail our categorical analysis of both the necessity andpossibility modalities. We give a sound definition of a categorical model forIS4. In x8 we give details of the serious proof and model theoretic problemsof Prawitz’s natural deduction formulation of the necessity modality. In x9we give some details of an alternative possibility modality.

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On an Intuitionistic Modal Logic 3

2. An Axiomatic Formulation of IS4

In this paper we shall only consider propositionalIS4. Consequently formu-lae are given by the grammarA ::= p j ? jA ^A jA _A jA � A j2A j3A;wherep is taken from some countable set of propositional constants.

In this section we give an axiomatic, or Hilbert-style, formulation ofIS4.As discussed in the introduction, this is the method favoured by most modallogicians. An axiomatic formulation uses a set of axioms anda handful ofrules. To tie up with subsequent sections on sequent calculus and natural de-duction formulations, we shall present deduction in an axiomatic formulationusing the notation� � A, where� is a set of formulae. The interpretationof � � A is that from the assumptions contained in� one can deduceA.(Clearly if � is empty, thenA is a theorem.) We first fix our collection ofaxioms, which consist of those for intuitionistic logic (see, for example [12])along with the following axioms for the modalities.2(A � B) � (2A � 2B)2A � A2A � 22A2(A � 3B) � (3A � 3B)A � 3ATwo deduction rules are immediate, i.e.Identity�; A � A and Axiom� � ATheIdentity rule is clear: if we assumeA, then we can deduceA. TheAxiomrule is only permitted whenA is (a substitution instance of) an axiom. We alsohave the familiar rule� � A � B � � A Modus Ponens:� � BThe rule for introducing a necessity modality is as follows.� A Box� � 2ANotice that we requireA to be a theorem. The complete axiomatisation ofIS4 is given in Figure 1.

It is relatively straightforward to demonstrate three important propertiesof our formulation.11 These properties hold by a straightforward induction. To save space, in the rest of thispaper, facts which can be proven by a simple induction will bestated without proof.

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Axioms: 1 A � (B � A)2 (A � B � C) � ((A � B) � (A � C))3 A � (B � A ^B)4 A ^B � A5 A ^B � B6 (A � C) � ((B � C) � (A _B � C))7 A � A _B8 B � A _B9 ? � A10 2(A � B) � (2A � 2B)11 2A � A12 2A � 22A13 2(A � 3B) � (3A � 3B)14 A � 3ARules: Identity�; A � A Axiom� � A� � A � B � � A Modus Ponens� � B� A Box� � 2AFigure 1. Axiomatic formulation ofIS4.

THEOREM 1.

1. If � � B then�; A � B.

2. If �; A � B then� � A � B.

3. If 2� � A then2� � 2A.

The second of these properties is often called thededuction theorem. Whereit is not obvious by context, a deduction in the axiomatic formulation will beprefixed by an annotated turnstile,`A.

3. A Sequent Calculus Formulation of IS4

Gentzen’s sequent calculus is an important tool for formulating logics. How-ever, as Bull and Segerberg point out [9,x2], it has never really flourished inmodal logic (some important exceptions include [11] and [22]). Deductions

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On an Intuitionistic Modal Logic 5Identity�; A � A� � B B;� � C Cut�;� � C ?L�;? � A�; A � C �; B � C _ L�; A _B � C � � A _R� � A _B � � B _R� � A _B�; A � C ^ L�; A ^B � C �; B � C ^ L�; A ^B � C � � A � � B ^R� � A ^B� � A �; B � C � L�; A � B � C �; A � B � R� � A � B�; A � B 2 L�;2A � B 2� � A 2R2�;� � 2A2�; A � 3B 3 L�;2�;3A � 3B � � A 3R� � 3AFigure 2. Sequent calculus formulation ofIS4.

in the sequent calculus are also written� � A where� is a set of formulae.There are rules for introducing each connective on the left and the right of theturnstile. The complete formulation is given in Figure 2.

An important property of this formulation is that we have an admissible‘weakening’ rule (this explains the presence of the� in the Identity, 2 Rand3 L rules).

PROPOSITION 1. If � � B then�; A � B.

The sequent calculus formulation, where we use the symbol`S to rep-resent a sequent deduction, can be shown to be equivalent to the axiomaticpresentation given in the previous section.

THEOREM 2. `S � � A iff `A � � A.

An important property of sequent calculus formulations is the so-calledcut-elimination theorem—a concept due to Gentzen [18]. Instances of the

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6 Bierman and de PaivaCut rule are systematically replaced with instances on smallerproofs (thetechnical details are a little delicate; Gallier [17] givesa nice explanation).The two important new important cases forIS4 are as follows.2� � A 2R2� � 2A �; A � B 2 L�;2A � B Cut2�;� � Bwhich is rewritten to 2� � A �; A � B Cut2�;� � Band � � A 3R� � 3A 2�; A � 3B 3 L2�;3A � 3B Cut�;2� � 3Bwhich is rewritten to� � A 2�; A � 3B Cut:�;2� � 3BIt is possible to show that all occurrences of theCut rule can be systemati-cally removed from anIS4 sequent calculus deduction. This is a version ofGentzen’s so-called Hauptsatz.

THEOREM 3. Given a derivation� of � � A, a derivation�0 of � � Acan be found which contains no instances of theCut rule.

Proof.By a routine adaptation of Gentzen’s method [18].

An important consequence of this theorem is thesubformula property. Afact about our sequent calculus formulation is that every rule except theCutrule has the property that the premises are made up of subformulae of theconclusion.

PROPOSITION 2. In a cut-free proof of� � A, all the formulae whichoccur within the proof are contained in the set of subformulae of� andA.

A number of interesting facts can be derived from this property—thesehave been studied by Schwichtenberg [36]. As we shall see in the next section,this property is not so straightforward for a natural deduction formulation.

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4. A Natural Deduction Formulation of IS4

In a natural deduction system, originally due to Gentzen [18], but subse-quently expounded by Prawitz [32], a deduction is a derivation of a propo-sition from a finite set of assumptions using some predefined set of inferencerules. Assumptions are assumed to have been labelled by markers, which wewill write as Ax. Assumptions of the same form with the same labels forman assumptionclass. Within a deduction, we may ‘discharge’ any number ofassumption classes. The applications of these inference rules are annotatedwith the labels of those classes which they discharge.

The natural deduction formulation of intuitionistic logicis well-knownand uncontroversial. The difficulties arise in dealing withthe modalities. Forexample, take the paper of Satre [35]. There he defines the necessity rules ofS4 as follows [35, p468].��A 2 I2A and ��2A 2 EAwith the proviso that the introduction rule is only well-formed if the formulaecontained in� are all of the form2C, for some formulaC. Whilst adequatewith respect to provability, this rule suffers from a serious defect: it is notclosed under substitution of deductions. For example take the deduction2A1 � � �2Ak��B 2 I:2BConsider substituting for the open assumption2A1, the deductionC � 2A1 C � E :2A1We then get the following deductionC � 2A1 C � E2A1 � � �2Ak��B 2 I2Bwhich is clearlynot a valid deduction as the assumptions are not all boxed.

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This defect was anticipated by Prawitz [32, Chapter VI] (surprisingly,Prawitz was not cited by Satre [35]). He highlights the problem as a lack ofclosure under proof normalisation, but this problem actually arises because ofthe lack of closure under substitution. Prawitz provides a solution using thenotion of “essentially modal formulae”. What this amounts to is a relaxationof the restriction on the introduction rule that all undischarged formulae mustbe modal. It is replaced with the restriction that for each formulaC in �, suchthatA depends onC, there is a modal formula2B in the path fromA to C.Diagrammatically this can be given as�1��2B1 � � � �k��2Bk��A 2 I:2AIt is important to notice that there may be several modal formulae in any givenpath. One of the contributions of this paper is to show that this fact has seriousproof and model theoretic consequences—these are detailedin x8.

Consequently, we shall use a different formulation which isboth closedunder substitution of deductions and has pleasant proof andmodel theoreticproperties. Our introduction rule is as follows.��2A1 : : : ��2Ak [[2Ax11 � � �2Axkk ℄℄��B 2 Ix1;:::;xk2BThus we insist not only that all the undischarged assumptions of the deductionof B are modal, but we then discharge them all (this is emphasisedby the useof semantic braces[[� � �℄℄) and reintroduce them. (As mentioned earlier, weannotate the inference rules with the labels of the assumption classes whichhave been discharged.)

REMARK 1. Curiously, our introduction rule actually appears in both [35]and [27, Page 242] (although we were unaware of this at the time). How-ever neither author seems aware of its proof theoretic significance, or of theproblems with Satre’s formulation [35].

There are similar difficulties in formulating the elimination rule for the pos-sibility modality. Our formulation is as follows.

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On an Intuitionistic Modal Logic 9��3B ��2A1 : : : ��2Ak [[2Ax11 � � �2Axkk Bxk+1℄℄��3C 3 Ex1;:::;xk+13CThe complete set of natural deduction rules forIS4 is given in Figure 3.

It is possible to present natural deduction rules in a ‘sequent-style’, wheregiven a sequent� � A, then� represents all the undischarged assumptionsandA represents the conclusion of the deduction. This formulation shouldnot be confused with the sequent calculus formulation, which differs by hav-ing rules which act on the left and right of the turnstile, rather than rulesfor the introduction and elimination of the connectives. The ‘sequent-style’formulation of natural deduction forIS4 is given in Figure 4.

PROPOSITION 3.The following rules are admissible

1. If � � B then�; A � B.

2. If � � A and�; A � B then� � B.

Our natural deduction formulation (where we use the prefix`N ) is equiv-alent to the axiomatic formulation given inx2.

THEOREM 4. `N � � A iff `A � � A.

4.1. NORMALISATION

With a natural deduction formulation we can produce so-called detours in adeduction, which arise where we introduce a logical connective only to elim-inate it immediately afterwards. We can define a reduction relation, written;� and called�-reduction, by considering each case in turn. The treatmentof the intuitionistic connectives is entirely standard andthe reader is referredto other works e.g. [32]. There are two new cases for the modalities. The firstis where2 I is followed by2 E . The deductionD1��2A1 : : : Dk��2Ak [[2A1 : : :2Ak℄℄��B 2 I2B 2 EBis reduced to

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��? ? EA[Ax℄��B � IxA � B ��A � B ��A � EB��A ��B ^ IA ^B ��A ^B ^ EA ��A ^B ^ EB��A _ IA _B ��B _ IA _B ��A _B [Ax℄��C [By℄��C _ Ex;yC��2A1 : : : ��2Ak [[2Ax11 � � �2Axkk ℄℄��B 2 Ix1;:::;xk2B ��2B 2 EB��A 3 I3A��2A1 : : : ��2Ak ��3B [[2Ax11 � � �2Axkk Bxk+1℄℄��3C 3 Ex1;:::;xk+13CFigure 3. Natural deduction formulation ofIS4.

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On an Intuitionistic Modal Logic 11�; A � A� � ? ? E� � A�; A � B � I� � A � B � � A � B � � A � E� � B� � A � � B ^ I� � A ^B � � A ^B ^ E� � A � � A ^B ^ E� � B� � A _ I� � A _B � � B _ I� � A _B� � A _B �; A � C �; B � C _ E� � C� � 2A1 � � � � � 2Ak 2A1; : : : ;2Ak � B 2 I� � 2B� � 2A 2 E� � A� � A 3 I� � 3A� � 2A1 � � � � � 2Ak � � 3B 2A1; : : : ;2Ak; B � 3C 3 E� � 3CFigure 4. Natural deduction formulation ofIS4 in sequent-style.

D1��[[2A1 : : : Dk��2Ak℄℄��B:The second is where3 I is followed by3 E . The deduction

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12 Bierman and de PaivaD1��2A1 Dk��2Ak Dk+1��B 3 I3B [[2A1 � � �2AkB℄℄��3C 3 E3Cis reduced to D1��[[2A1 : : : Dk��2Ak Dk+1��B℄℄��3C:A proof containing no instances of a�-reduction is said to be in�-normalform. Our formulation ofIS4 has the following property.

PROPOSITION 4. If � � A in IS4 then there is a natural deduction ofAfrom� which is in�-normal form.

An interesting question is whether there is a subformula property for agiven natural deduction formulation. As discussed in the previous section,this is a simple fact for most sequent calculus formulations. A good discussionof this issue for natural deduction can be found in the book byGirard [21].For intuitionistic logic, as explained by Girard, it is not sufficient that a de-duction be in�-normal form for it to satisfy the subformula property. Weare forced to introduce a new reduction relation, written; (and called -reduction), which Girard calls acommuting conversion(they are calledpermutative reductionsby Prawitz [33]). These deal with what Girard callsthe “bad” elimination rules (_ E and? E). A deduction of the form��A _B [A℄��C [B℄��C _ EC ... r EDis reduced to ��A _B [A℄��C ... r ED [B℄��C ... r ED _ ED

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and a deduction of the form ��? ? EA ... r EBis reduced to ��? ? EBwherer E representsanyelimination rule. A proof containing no instances ofa -reduction is said to be in -normal form. A�-normal form which is alsoa -normal form is said to be a(�; )-normal form. For intuitionistic logic wecan shown the following subformula property.

THEOREM 5. (Prawitz).If the natural deduction derivation of� � A (in in-tuitionistic logic) is in(�; )-normal form, then all formulae in the deductionare contained in the set of subformulae of� andA.

As one may have expected, the form of the rules for the modalities meansthat we require further commuting conversions for a subformula property tohold. Consider the introduction rule for the necessity modality.��2A1 : : : ��2Ak [[2Ax11 � � �2Axkk ℄℄��B 2 I2BThe problem is that there is no reason why the2Ai should be subformulaof 2B or an undischarged assumption. For example, theAi could be theconclusion of a bad elimination rule. Consider the following deduction whichis in (�; )-normal form.2(C � A � B) _ ? 2C�� D12(A � B) 2A [2(A � B)℄ 2 EA � B [2A℄ 2 EA � EB 2 I2BwhereD1 is the deduction

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2(C � A � B) _ ? [2(C � A � B)℄2C [2(C � A � B)℄C � A � B [2C℄C � EA � B 2 I2(A � B) [?℄2(A � B) _ E2(A � B)Unfortunately the formula2(A � B) is not a subformula of either theconclusion (2B) or of the undischarged assumptions (f2(C � A � B) _?;2C;2Ag). Consequently we shall add the following commuting conver-sion. ��2A1 : : : ...

��2Ai b E2Ai ��2Ak [[2A1 � � �2Ak℄℄��B 2 I2Bwhich is reduced to

...

��2A1 : : : ��2Ai : : : ��2Ak [[2A1 � � �2Ak℄℄��B 2 I2B b E2Bwhereb E is any bad elimination rule (_ E or? E).

Unfortunately this is still not enough; anAi could be the result of the2 Irule. Consider the following deduction which is a(�; )-normal form.2(C � A � B)2C�� D22(A � B) 2A [2(A � B)℄ 2 EA � B 2A 2 EAB 2 I2BwhereD2 is the deduction2(C � A � B)2C [2(C � A � B)℄ 2 EC � A � B [2C℄ 2 EC � EA � B 2 I2(A � B)

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Again2(A � B) is neither a subformula of the conclusion (2B) nor of theundischarged assumptions (f2(C � A � B);2C;2Ag). Consequently weadd another commuting conversion. The deduction��2A1 : : : ��2C1 : : : ��2Cj [[2C1 � � �2Cj ℄℄��Ai 2 I2Ai : : : ��2Ak [[2A1 � � �2Ak℄℄��B 2 I2Bis reduced to

��2A1 : : : ��2C1 : : : ��2Cj : : : ��2Ak [[2A1 � � �℄℄ [[2C1 � � �2Cj ℄℄ [[2C1 � � �2Cj℄℄��Ai 2 I2Ai [[� � �2Ak℄℄��B 2 I2BSimilar rules exist for the rules describing the possibility modality.

THEOREM 6. If the natural deduction derivation of� � A (in IS4) is in(�; )-normal form, then all formulae in the deduction are contained in theset of subformulae of� andA.

Proof. By a relatively straightforward adaptation of Prawitz’s proof forintuitionistic logic [33,xII.3].

5. Multi-context Formulation

It is possible to give alternative natural deduction formulations ofIS4. In thissection we shall sketch some details of one such alternative—some othersare listed inx11. This alternative originates from work on intuitionistic linearlogic [1].

Recall that the difficulties in formulating the introduction rule for thenecessity modality was, in part, in ensuring that all the undischarged assump-tions (the context) were modal. An alternative would be to split the contextinto two, where one part contains only modal formulae and theother the non-modal formulae. The condition then becomes a check that the non-modalcontext is empty. More precisely, judgements are of the form

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16 Bierman and de Paiva�;� � Awhere� is the modal context and� the non-modal context. To reiterate: theidea is that this judgement corresponds to the judgement2�;� � Ain our formulation of the previous section. Thus the2 I rule is given by�;� � A 2 I�;� � 2ATo ensure closure under substitution, the elimination ruleis slightly morecomplicated. �;� � 2A �; A;� � B 2 E�;� � BThe complete formulation, which we call a multi-context formulation is givenin Figure 5.

PROPOSITION 5.The following rules are admissible.

1. If �;� � B then�; A;� � B.

2. If �;� � B then�;�; A � B.

3. If �;� � A and�;�; A � B then�;� � B.

4. If �;� � A and�; A;� � B then�;� � B.

THEOREM 7. The following rules are derivable.

1. If �; A;� � B then�;�;2A � B.

2. If �;�;2A � B then�; A;� � B.

3. If �; A;� � B then�;� � 2A � B.Proof.

1. �;2A;� � 2A �; A;� � B 2 E�;�;2A � Bstudia.tex; 17/12/1999; 11:09; p.16

On an Intuitionistic Modal Logic 17�; A;� � A �;�; A � A�;�; A � B � I�;� � A � B �;� � A � B �;� � A � E�;� � B�;� � A �;� � B ^ I�;� � A ^B �;� � A ^B ^ E�;� � A �;� � A ^B ^ E�;� � B�;� � A _ I�;� � A _B �;� � B _ I�;� � A _B�;� � A _B �;�; A � C �;�; B � C _ E�;� � C�;� � A 2 I�;� � 2A�;� � 2A �; A;� � B 2 E�;� � B�;� � A 3 I�;� � 3A�;� � 3A �;A � 3C 3 E�;� � 3CFigure 5. Multi-context formulation ofIS4.

2. �; A;� � A 2 I�; A;� � 2A �;�;2A � B�; A;� � BThe last rule used is a substitution, shown to be admissible in Proposi-tion 5.

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3. �;2A;� � 2A �; A;� � B 2 E�;�;2A � B � I�;� � 2A � B6. Term Assignment for IS4

The Curry-Howard correspondence [23] relates natural deduction formula-tions of logics with typed�-calculi. It essentially annotates each stage ofa deduction with a ‘term’, which is an encoding of the construction of thededuction so far. Consequently a logic can be viewed as a typesystem for aterm assignment system. The correspondence also links proof normalisationto term reduction. We shall use this term assignment in the next section todevise a categorical model ofIS4.

The Curry-Howard correspondence can be applied to the natural deductionformulation to obtain the term assignment system given in Figure 6. (In placeswe have used the shorthand~M instead of the sequenceM1; : : : Mk.) Theresulting calculus we shall call the�S4-calculus.

The syntax has been chosen so that the non-modal fragment is similar toseveral functional programming languages, and that used byGallier in histutorial article [17].

An important property of our system is that substitution is well-defined inthe following sense.

THEOREM 8. If � . N :A and�; x:A .M :B then� . M [x := N ℄:B.

The reduction rules discussed inx4.1 can be given at the level of terms. InFigure 7 we give the�-reduction rules denoted by;�. It is also possible togive the reduction rules associated with the commuting conversions. We shallomit them for reasons of space.

An important property of these reduction rules is given by the so-calledsubject reduction theorem.

THEOREM 9. If � . M :A andM ;� N then� . N :A.

(Of course, this theorem also extends to the commuting conversions.) Itshould be noted that the Curry-Howard correspondence can beapplied to themulti-context formulation given inx5, to yield a very compact term calculus.

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On an Intuitionistic Modal Logic 19�; x:A . x:A� . M :? ? E� .rA(M):A�; x:A .M :B ! I� . �x:A:M :A! B � . M :A! B � . N :A ! E� . MN :B� . M :A � . N :B � I� . hM;Ni:A �B � . M :A�B � E� . fst(M):A � . M :A�B � E� . snd(M):B� . M :A + I� . inl(M):A+B � . M :B + I� . inr(M):A+B� . M :A+B �; x:A . N :C �; y:B . P :C + E� . aseM of inl(x) ! N k inr(y) ! P :C� . M1:2A1 � � � � . Mk:2Ak x1:2A1; : : : ; xk:2Ak . N :B 2 I� . boxN with ~M for ~x:2B� . M :2A 2 E� . unbox(M):A� . M :A 3 I� .3M :3A� . M1:2A1 � � �� . Mk:2Ak� . N :3B x1:2A1; : : : ; xk:2Ak; y:B . P :3C 3 E� . let3y( N in P with ~M for ~x:3CFigure 6. Term assignment forIS4

7. The Categorical Model

The fundamental idea of a categorical treatment of proof theory is that propo-sitions should be interpreted as the objects of a category and proofs should beinterpreted as morphisms. The proof rules correspond to natural transforma-tions between appropriate hom-functors. The proof theory gives a numberof reduction rules, which can be viewed as equalities between proofs. Inparticular these equalities should hold in the categoricalmodel.

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20 Bierman and de Paiva(�x:A:M)N ;� M [x := N ℄fst(hM;Ni) ;� Msnd(hM;Ni) ;� N ase inl(M) of inl(x) ! N k inr(y) ! P ;� N [x :=M ℄ ase inr(M) of inl(x) ! N k inr(y) ! P ;� P [y := M ℄unbox(boxN with ~M for ~x) ;� N [~x := ~M ℄let3y( 3N in P with ~M for ~x ;� P [~x := ~M; y := N ℄Figure 7. �-reduction rules.

Other categorical studies have been carried out, notably byFlagg [16];Meloni and Ghilardi [19] and Reyes and Zolfaghari [34]. However these havebeen mainly concerned with categoricalmodeltheory, rather than categoricalproof theory. In particular, they all assume an isomorphism,22A �= 2A,whereas we see no reason to assume such a strong condition. Inthis work wehave morphisms in both directions, but we donot collapse the model so thatthey are isomorphic.

Let us fix some notation. We shall follow the convention common in com-puter science and write composition of morphisms in a left-to-right fashion,e.g.f ; g, rather than the mathematical convention of right-to-left, e.g.g Æ f .

The interpretation of a proof is represented using semanticbraces,[[�℄℄,making the usual simplification of using the same letter to represent a propo-sition as its interpretation. Given a term� . M :A whereM ;� N , we shallwrite � . M = N :A.

DEFINITION 1. A category,C , is said to be a categorical model of a logic/termcalculus iff

1. For all proofs� . M :A there is a morphism[[M ℄℄: �! A in C ; and

2. For all proof equalities� . M = N :A it is the case that[[M ℄℄ =C [[N ℄℄(where=C represents equality of morphisms in the categoryC ).

Given this definition we simply analyse the introduction andelimination rulesfor each connective. Both this and consideration of the reduction rules shouldsuggest a particular categorical structure to model the connective. The case

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for intuitionistic logic is well known; the reader is referred to Lambek andScott’s book [24] for a good discussion. Essentially the categorical modelof intuitionistic logic (with disjunction) is a cartesian closed category (CCC)with coproducts. Hence all we need do here is consider the twomodalities,which we shall do in some detail. The less-categorically minded reader maywish simply to skip to Definition 5.

The introduction rule for the necessity modality is of the form� . M1:2A1 � � � � . Mk:2Ak x1:2A1; : : : ; xk:2Ak . N :B 2 I� . boxN with ~M for ~x:2BTo interpret this rule we need a natural transformation withcomponents��: C (�;2A1 )�� C (�;2Ak )� C (2A1 �� 2Ak; B)! C (�;2B)Given morphismsei: � ! 2Ai, : �0 ! � andd:2A1 � � � � � 2Ak ! B,naturality gives the equation ; ��(e1; : : : ; ek; d) = ��0(( ; e1); : : : ; ( ; ek); d):In particular if we have morphismsmi: �! 2Ai then we take = hm1; : : : ;mki,ei to be thei-th product projection, written�i, andd to be some morphismp:2A1 � � � � �2Ak ! B, then by naturality we havehm1; : : : ;mki; �2A1;:::;2Ak(�1; : : : ; �k; p) = �2A1;:::;2Ak(m1; : : : ;mk; p):Thus�(m1; : : : ;mk; p) can be expressed as the compositionhm1; : : : ;mki; (p),where is a transformation: C (2A1 � � � � �2Ak; B)! C (2A1 � � � � �2Ak;2B):For the moment, the effect of this transformation will be written as(�)� andso we can make the preliminary definition[[� . boxN withM1; : : : ;Mk for x1; : : : ; xk:2B℄℄ def=h([[� . M1:2A1℄℄); : : : ; ([[� . Mk:2Ak℄℄)i;([[x1:2A1; : : : ; xk:2Ak . N :B℄℄)�The elimination rule for the necessity modality is of the form� . M :2A 2 E� . unbox(M):ATo interpret this rule we need a natural transformation withcomponents��: C (�;2A) ! C (�; A):

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It follows from the Yoneda Lemma [25, Page 61] that there is the bijection[C op ;Sets℄(C (�;2A); C (�; A)) �= C (2A;A):By constructing this isomorphism one can see that the components of� areinduced by post-composition with a morphism":2A ! A. Thus we makethe definition [[� . unbox(M):A℄℄ def= [[� . M :2A℄℄; ":From Figure 7 we have the term equality� . M1:2A1 � � � � . Mk:2Ak x1:2A1; : : : ; xk:2Ak . N :B� . unbox(boxN with ~x for ~M) = N [~x := ~M ℄:BTaking morphismsmi: � ! 2Ai andp:2A1 � � � � � 2Ak ! B, say, thisterm equality amounts to the categorical equalityhm1; : : : ;mki; (p)�; " = hm1; : : : ;mki; p: (1)

We can certainly define an operation2: C (�; A) ! C (2�;2A);f 7! ("; f)�:We shall make the simplifying assumption that this operation is a functor.However, notice that if� is the objectA1 � � � � � Ak, then2� will berepresented by2(A1 � � � � � Ak), but clearly we mean2A1 � � � � � 2Ak.Thus we shall make the further simplifying assumption that2 is asymmetricmonoidal functor, (2;mA;B ;m1). This notion is originally due to Eilenbergand Kelly [14]. In essence this provides a natural transformationmA;B:2A�2B ! 2(A�B)and morphism m1: 1! 21which satisfy a number of conditions which are detailed below.

DEFINITION 2. A monoidal functor, (2;mA;B ;m1), on a CCCC satisfiesthe four following equations.

1. id2A �m1;mA;1;2�1 = �12. m1 � id2A;m1;A;2�2 = �23. �;mA;B � id2C ;mA�B;C = id2A �mB;C ;mA;B�C ;2(�)

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4. ;mB;A = mA;B;2( )where� def= hh�1; (�2;�1)i; (�2;�2)i:A� (B � C)! (A�B)� C def= h�2; �1i:A�B ! B �AEquation 1 gives ("A; f)�; "B = "A; ffor any morphismf :A! B; or, in other words the diagram2A

A2BB

-2f?" ?"-f

commutes. Given the assumption that2 is a symmetric monoidal functor, thisdiagram suggests that" is a monoidal natural transformation.

We have that from the identity morphismid2A:2A ! 2A, we can form

the canonical morphismÆA def= (id2A)�. Equation 1 givesÆA; "2A = id2A:The categorically-minded reader will recognise this equation as one of thethree for acomonad. We shall make the simplifying assumption that notonly does(2; "; Æ) form a comonad but thatÆ is also a monoidal naturaltransformation. Hence the comonad is actually amonoidalcomonad.

DEFINITION 3.

1. The triple(2; "; Æ) forms acomonadon a CCCC , if 2 is an endofunctoron C , and":2A ! A andÆ:2A ! 22A are natural transformationswhich satisfy the following three equations.

a) ÆA; "2A = id2Ab) ÆA;2("A) = id2Ac) ÆA; Æ2A = ÆA;2(ÆA)

2. A comonad(2; "; Æ) is in addition amonoidal comonadif (2;mA;B ;m1)is a monoidal functor and the following four equations hold.

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a) mA;B; "A�B = "A � "Bb) m1; "1 = id1c) mA;B; ÆA�B = ÆA � ÆB ;m2A;2B;2(mA;B)d) m1; Æ1 = m1;2m1

We can now finalise the interpretation of the introduction rule for the neces-sity modality.[[� . boxN with ~M for ~x:2B℄℄ def= h[[� . M1:2A1℄℄; : : : ; [[� . Mk:2Ak℄℄i;ÆA1 � � � � � ÆAk ;m2A1;:::;2Ak ;2[[x1:2A1; : : : ; xk:2Ak . N :B℄℄The introduction rule for the possibility modality is of theform� . M :A 3 I� .3M :3ATo interpret this rule we need a natural transformation�: C (�; A) ! C (�;3A)It follows from the Yoneda Lemma [25, Page 61] that there is the bijection[C op ;Sets℄(C (�; A); C (�;3A)) �= C (A;3A):By constructing this isomorphism one can see that the components of� areinduced by post-composition with a morphism�:A ! 3A. Thus we makethe definition [[� . M :A℄℄ def= [[� . M :2A℄℄; �:The elimination rule for the possibility modality is of the form� . M1:3A1 � � �� . Mk:3Ak� . N :3B x1:3A1; : : : ; xk:3Ak; y:B . P :3C 3 E� . let3y( N in P with ~M for ~x:3CTo interpret this rule we need a natural transformation withcomponents��: C (�;2A1 )� � � � � C (�;2Ak )�C (�;3B) � C (2A1 � � � � �2Ak �B;3C)! C (�;3C)Given morphismsei: �! 2Ai, : �0 ! �, d: �! 3B, andf :2A1 � � � � �2Ak �B ! 3C, naturality gives the equation

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On an Intuitionistic Modal Logic 25 ; ��(e1; : : : ; ek; d; f) = ��0(( ; e1); : : : ; ( ; ek); ( ; d); f):In particular if we have morphismsmi: � ! 2Ai andn: � ! 3B, then wetake = hm1; : : : ;mk; ni, ei to be thei-th product projection, written�i, andf to be some morphismp:2A1 � � � � �2Ak �B ! 3C, then by naturalitywe havehm1; : : : ;mk; ni; �(�1; : : : ; �k; �k+1; p) = �(m1; : : : ;mk; n; p):Thus�(m1; : : : ;mk; n; p) can be expressed as the compositionhm1; : : : ;mk; ni; (p),where is a transformation: C (2A1 � � � � �2Ak �B;3C)! C (2A1 � � � � �2Ak �3B;3C):For the moment, the effect of this transformation will be written as(�)� andso we can make the preliminary definition[[� . let3y( N in P with ~M for ~x:3C℄℄ def=h([[� . M1:2A1℄℄); : : : ; ([[� . Mk:2Ak℄℄)([[� . N :3B℄℄)i;([[x1:2A1; : : : ; xk:2Ak; y:B . N :3C℄℄)�From Figure 7 we have the term equality� . M1:2A1 � � �� . Mk:2Ak� . N :B x1:2A1; : : : ; xn:2Ak; y:B . P :3C� . let3y( 3N in P with ~M for ~x = P [~x := ~M; y := N ℄:3CTaking, for example, morphismsmi: � ! 2Ai, n: � ! B, andp:2A1 �� 2Ak�B ! 3C, the term equality amounts to the categorical equalityhm1; : : : ;mk; (n; �)i; (p)� = hm1; : : : ;mk; ni; p (2)

We can then define an operation3: C (�; A) ! C (3�;3A);f 7! (f ; �)�:Again we shall make the simplifying assumption that this operation is afunctor. Equation 2 gives �; (f ; �)� = (f ; �)for any morphismf :A! B, or, in other words, the diagram

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26 Bierman and de PaivaAB

3A3B

-�?f ?3f-�

commutes. Given our assumption that3 is a functor, this diagrams suggeststhat� is a natural transformation. Given the identity morphismid3C , we can

form the canonical morphism�C def= (id3C)�. Equation 2 gives�;� = �:Again the categorically-minded reader will recognise thisequation as oneof the three for amonad. Thus we shall expect that(3; �; �) forms somesort of monad. A first guess might be that it should be the dual of the struc-ture required for the necessity modality, i.e. acomonoidal monad. Howeverthis seems far too strong a requirement—for example, it requires a naturaltransformation3(A+B)! 3A+3B, which is not even a theorem ofIS4!

Recalling the form of the3 E rule, it seems clear that the meaning of thepossibility modality is actually related to the necessity modality. Fortunately,category theory provides a means to describe such a situation.

DEFINITION 4.

1. The triple(3; �; �) forms amonadon a CCCC if 3 is an endofunctoron C , and�:A ! 3A and�:33A! 3A are natural transformationswhich satisfy the following three equations.

a) �3A;�A = id3Ab) 3(�A);�A = id3Ac) �3A;�A = 3(�A);�A

2. Given a monoidal comonad(2; "; Æ;m1;mA;B) and a monad(3; �; �)on a CCCC , we say that3 is a2-strong monad, if there exists a naturaltransformation stA;B:2A�3B ! 3(2A�B)which satisfies the following four equations.

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a) "1 � id3A;�2 = st1;A;3("1 � idA);3(�2)b) �;mA;B�id3C ; stA�B;C = id2A�stB;C ; stA;2B�C ;3(�);3(mA;B�idC)c) id2A � �B ; stA;B = �2A�Bd) stA;2B;3(stA;B); " = stA;B

Clearly we can define n-ary versions of the strength. For example the 3-aryversion stA;B;C :2A�2B �3C ! 3(2A�2B � C)can be defined as id2A � stB;C ; stA;2B�C :We can now fix the interpretation of the3 E rule as follows.[[� . let3y( N in ~M with ~x for :3C℄℄ def=h[[� . M1:2A1℄℄; : : : ; [[� . Mk:2Ak℄℄; [[� . N :3B℄℄i;stA1;:::;Ak;B;3([[x1:2A1; xn:2Ak; y:B . P :3C℄℄);�CThus our definition of a categorical model forIS4 is as follows.

DEFINITION 5. A categorical model forIS4 consists of a cartesian closedcategory with coproducts,C , together with a monoidal comonad(2; "; Æ;mA;B ;m1)and a2-strong monad(3; �; �; stA;B).To reiterate, this notion of a categorical model satisfies the following proper-ties.

PROPOSITION 6.Given a category,C , which satisfies Definition 5.

1. For every deductionD of� � A in IS4 there is a morphism[[D℄℄: �! Ain C .

2. If a deductionD �-reduces toD0, then[[D℄℄ = [[D0℄℄.3. If a deductionD -reduces toD0, then[[D℄℄ = [[D0℄℄.

In other words, any concrete model which satisfies the conditions given inDefinition 5, is asoundmodel ofIS4.

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8. Prawitz’s Formulation and the Categorical Model

Although Prawitz’s formulation has the appearance of beingequivalent to thatpresented in this paper, it has rather unfortunate proof andmodel theoreticconsequences. Consider the following deduction in Prawitz’s formulation.(1) 222A 2 E(2) 22A 2 E(3) 2A 2 EA 2 I2AAs explained inx4, the application of the2 I rule is valid as there is a modalformula in the path from the conclusion,A, to the undischarged assumption,222A. In fact there are three candidates for this modal formula, which arenumbered in the deduction. In our formulation these alternatives representdistinct deductions. For example, alternatives(2) and (3) are given by thedeductions 222A 2 E22A [[22A℄℄ 2 E2A 2 EA 2 I2Aand 222A 2 E22A 2 E2A [[2A℄℄ 2 EA 2 I2AIt is important to note that both these deductions are in(�; )-normal form,and so aredistinct deductions. Prawitz’s formulation essentially collapsesthese two deductions into one. In other words his formulation forces a seem-ingly unnecessary identification of deductions. Let us consider the conse-quences of this identification with respect to the categorical model. The twoderivations above are modelled by the morphisms"22A; Æ2A;2("22A);2("2A):222A! Aand "22A; "2A; ÆA;2"A:222A! Arespectively. Insisting on these being equal amounts to theequality"22A;2"A = "22A; "2A:

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Precomposing this equality with the morphismÆ2A gives2"A = "2A:It is easy to see that this is sufficient to make the comonadidempotent, i.e.2A �= 22A. Unfortunately this is a very strong condition to ask of a comonad.Indeed nearly all models ofIS4 that we are aware of do not satisfy thisidempotency. It is worth reiterating that our natural deduction formulationdoesnot impose this identification of proofs and consequently does not forcean idempotency.

9. An Alternative Possibility Modality

In x7 we saw that the possibility modality is modelled by a2-strong monad.A natural weakening of this condition is to only insist that it be astrongmonad.

DEFINITION 6. Given a monad(3; �; �) on a CCCC , we say that it is astrong monadif there exists a natural transformationsA;B:A�3B ! 3(A�B)which satisfies the following four equations.

1. s1;A;3(�2) = �22. idA � �B ; sA;B = �A�B3. �; idA � sB;C ; sA;B�C = sA�B;C ;3(�)4. sA;3B;3(s);�A�B = idA � �B ; sA;B

This weaker possibility modality has the following naturaldeduction rules.� � A 3 I� � 3A � � 3A �; A � 3B 3 E� � 3BIt can also be axiomatised by the following two axioms.A � 3A3A � ((A � 3B) � 3B)Somewhat surprisingly, this rather odd possibility modality has been discov-ered before [10, 4]. Indeed, in terms of computer science, this modality hasmany important applications [30, 15].

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10. Additional Properties

The computationally-inclined reader may have expected some additional re-duction rules for our�S4-calculus. Two interesting additional reduction rulesare as follows.boxM with N1; : : : ; Ni�1; Ni; Ni+1; : : : ; Nkfor x1; : : : ; xi�1; xi; xi+1; : : : ; xk;�boxM with N1; : : : ; Ni�1; Ni+1; : : : ; Nkfor x1; : : : ; xi�1; xi+1; : : : ; xkwhere xi 62 FV(M) (G)boxM with N1; : : : ; Ni�1; P; P;Ni+2; : : : ; Nkfor x1; : : : ; xi�1; xi; xi+1; xi+2; : : : ; xk;�boxM [xi; xi+1 := y℄ with N1; : : : ; Ni�1; P;Ni+2; : : : ; Nkfor x1; : : : ; xi�1; y; xi+2; : : : ; xk (C)

The first of these rules,(G), is a sort of ‘garbage collection’ rule, thesecond,(C), a ‘coalescing’ of identical explicit substitutions.

These rules are also interesting from our categorical perspective in that wecan characterise them quite succinctly.2PROPOSITION 7. (Schalk).A categorical model forIS4 (as given in Defi-nition 5), which in addition satisfies the property that the maps!2A:2A! 1and�2A:2A! 2A�2A are (free) coalgebra morphisms, satisfies the tworeduction rules(G) and(C).

This extra condition on the terminal and diagonal morphismsamounts torequiring that the following two equations hold.!2A;m1 = ÆA;2(!2A)ÆA;2(�2A) = �2A; ÆA � ÆA;m2A;2A

This richer notion of a categorical model ofIS4 actually amounts to alinear category (in the sense of Bierman [5]) where the underlying sym-metric monoidal category is, in fact, a cartesian closed category. Furtherinvestigation of this categorical model remains future work.

2 This fact was proved by A. Schalk (personal communication).

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11. Conclusions

In this paper we have considered the propositional, intuitionistic modal logicIS4. We have given equivalent axiomatic, sequent calculus and natural de-duction formulations, the corresponding term assignment system and the def-inition of a general categorical model.

As mentioned in the introduction we place particular importance on a nat-ural deduction formulation and its metatheoretic properties. We have shownthat the formulation proposed by Satre [35] is not closed under substitution,and the formulation proposed by Prawitz [32] introduces seemingly unneces-sary identifications of proofs, which in the model forces an idempotency.

A number of authors have considered the question of providing naturaldeduction formulations of (intuitionistic) modal logics (some in response toour earlier work [6]). These include Benevides and Maibaum [2], Bull andSegerberg [9, pages 29–30], Davies and Pfenning [13], Martini and Masini [26],Mints [27, Pages 221–294] and Simpson [37]. However they alluse exten-sions of one form or another to the nature of natural deduction (for example,by indexing formulae with possible worlds information). Again we reiteratethe conceptual simplicity of our proposal—we use no new features of naturaldeduction.

The techniques used in this paper to formulate modalities, also apply toGirard’s linear logic [20]. Conversely there are a number ofpapers concerninglinear logic which could be usefully studied in the context of modal logics,e.g. [3, 29, 28]. We leave this to future work.

We also prefer the use of categorical models over traditional Kripke-stylepossible worlds models. Unlike other categorical work we have placed em-phasis on modelling the proof theory not just provability. Our resulting modelis considerably simpler than other proposals.

In the future we should like to consider other modal logics within ourframework. It is clear that not all of the hundreds of modal logics will fit intoour framework. However we do not view this as a weakness of ourwork.Rather we feel it is important to identify those modal logicswhich havesimple, but interesting proof theories and mathematicallyappealing classesof (categorical) models.

Acknowledgements

This paper has suffered a little at the hands of time. The material (concerningjust the necessity modality) was first presented at the Logicat Work con-ference in Amsterdam in December 1992 and appears in draft form in theinformal proceedings [6]. For the next three years it was to appear as a chapterof a collected works, before being dropped at the final minuteby the editors.

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It was then simply issued as [8]. The details of the possibility modality werepresented at the 1994 European meeting of the ASL—an abstract appearsas [7]. We are grateful to Professor Mints for inspiring us tofinally write upour material.

Bierman’s work was funded by EPSRC Grant GR/M04716 and Gonvilleand Caius College, Cambridge. Over the years we have benefited from dis-cussions of matters modal with Richard Crouch, Rajeev Gore, Martin Hyland,Grigori Mints, Luiz Carlos Pereira, Frank Pfenning and AlexSimpson.

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