1. Introduction1. Introduction

1. Introduction : What is the problem ?

2. Framework : Applicative formalism and Combinatory Logic with types

3. Four Theories of Quantification

I/ Fregean Theories

(A) Classical presentation : with bound variables

(B) with λ-calculus (Church, Montague)

(C) Illative quantifiers (Curry) without bound variables

II/ A non Fregean theory :

(D) Star Theory without bound variables

III/ Reduction of Illative Theory to Star Theory

IV/ Logical cube

PROBLEM :PROBLEM :

• Classical Logic (or fregean logic) is not fully adequate toa deep logical analysis of natural languages.

• In classical logic, there are different operations for buildingpropositions :

- connection between propositions ;- negation, modalities ;- predication (or application of a predicate onto terms) ;- quantification (or application of a quantifier ontopredicate to build a proposition or another predictate).

• In natural languages, these operations are used for buildingstatements but there are other fundamental operations :

- topicalization, voices (diatheses) ;- enunciatice operation (uttering or speech acts) ;- determination.

• A determination is used to bring information about anexpression (terms, predicates); different synytacticalcategories (adjectives, relatives, articles, adverbs …)express a determination operation.

• In classical logic, after Frege, determination is not takeninto account :

adjectives, common names, intransitive verbs are analysedby a predicative operation.

John is reading an excellent book

(∃ x) [ is-reading (John, x) & book (x) & excellent (x) ]

• With Anca Pascu, we are studying a new formalism to takeinto account the logical operations of determination innatural languages and their different consequences.

• This formalism is called Logic of Determination of Objects(LDO) (presented in other meetings).

Logic of Determination of Objects(LDO)

• We start with the fundamental distinction of G. Frege(1893) : concepts (predicates) are operators thatexpress an « unsaturation » (or « no completion ») andobjects express « saturation » (or completedexpression).

• In LDO, we study different kinds of objects :

- fully determinate objects ;- more and less determinate objects ;- typical objects ;- atypical objects ;- determination of more determinated objects from an

underterminate object …

• A concept is an operator such that it can be applied todifferent objects. If the object « falls under the concept », theresult of the application is « true ».

• To each concept ‘f’ is associated its « intension » and its« essence ».

• The « intension » ‘Int(f)’ is an ordered set of properties whosethe meanings are contained in the meaning of the concept ‘f’.

• The « essence », ‘Ess (f)’, is a part of the « intension », ‘Int(f)’.

• To each concept ‘f’ is canonically associated an uniqueabstract object ‘τ(f)’ and a determination operator ‘δ(f)’.

• The object ‘τ(f)’ represents, as an object, the concept ‘f’; it iscompletly undeterminate.

• The operator ‘δ(f)’, associated to the concept ‘f’, contributes todetermine different objects.

• The object ‘τ(f)’ is called « the typical object »; it representsthe concept ‘f’ as an completly undeterminate object.

• It is a generator of differnt more or less determinate objectswhich fall under the concept ‘f’.

• The object which falls under the concept (or, when the conceptis applied to it with the result « true ») is belongs to theexpanse of the concept ‘f’, noted ‘Exp(f)’.

• When this object is fully determinate, it belongs to theextension of f, noted ‘Ext(f)’, a part of the expanse of ‘f’.

To-be-dog

τ(to-be-dog)

to-be-animal

to-be-mortal-being

(black)(τ(dog))

u1 u2 u3(white)(τ(dog))

whiteblack

Int(to be dog)

Exp(to-be-dog)

concept

Typical object

v1 v2

to-be-mammal

determination by « black »

determination by « white »

τ(f)

f

δ(g1) (τ(f))

Int(f)

Expansion(f)

z

δ(g2) (τ(f)) u

Typical object

Ext (f)Extτ (f)Typical fully specified instances

does not belongto Expansion(f)

f

τ(f)

x

y = δ(g1)(x)

δ(g1)

y = δ(g2)(x)

y = δ(g3)(x) δ(g2) δ(g3)

τ

Typical instances

All instances

Ess(f)

h3 = Ng3

Int(f)h2 = Ng2

An atypical instance

An instance outthe category

specified instance of x

contradictiong1∉Int(f)g2∉Int(f) ∧(∃ h2∈Int(f) -Ess(f); h2= Ng2g3∉Int(f) ∧(∃ h3∈ Ess(f) ⊆ Int(f) ; h3 = Ng3

• An object is a typical in stance of a concept ‘f’ when this object

1°) falls under the concept ‘f’

2°) it inherits all properties of the intension of ‘f’.

• An object is an atypical instance of ‘f’ when

1°) it falls under the concept ‘f’

2°) there is some property of the intension of ‘f’ such that this object does not inherit.

• All objects that fall under ‘f’

inherit all properties of the essence of ‘f’.

EXTENSION = class of fully determinate men

Typical menAn atypical man

Abraham LincolnSocrates

ConfuciusLao Tseu

General properties : « to have two eyes, two legs, two arms … » in Intension of « man »

Specific Property :« to have onlyone leg »

Cyclopewith only one eye

Different expressions of LDO are defined insidethe general framework of Combinatory Logicof Curry with functional types.

2. Combinatory Logic with types2. Combinatory Logic with types

Applicative ExpressionsApplicative Expressions • We use applicative expressions and functional types (or categories of functors) from given basic types (or basic categories). • The application is a fundamental operation; it builds all applicative expressions from more elementary expressions. • The applicative expression ‘XY’ is the result of the application ‘@’ of the operator X to the operand Y. The application is represented by means of the following applicative tree :

X Y----------------> X applies to Y and builds a result XY = X@Y

• There are different types of applicative expressions

Typed Applicative ExpressionsTyped Applicative Expressions

• Application rule:[A-Exp] If X and Y are applicative expressions then XY is an

applicative expression.• Church’s Functional types:

•• Basic types •• Functional types are recursively defined by using a two-placeconstructor, designated by ‘FF’, with two rules:

[Type 1] the basic types are functional types;[Type 2]if α and β are types then ‘FFαβ’ is a functional type.

Assignment of a type α to an applicative expression X is designated [α : X ]

• Inference rule for application wit types:Oαβ : X α : Y

[APPL] ---------------------------->β : XY

CurryCurry’’s Combinatory Logics Combinatory Logic

The Curry’s Combinatory Logic is an applicative formalism with Combinators. It is a logic of operators and composition of operators.

A combinator is an abstract operator which builds a “complex operator” from more elementary operators or transform an operator in another. EXAMPLE: The combinator BB builds a complex operator ‘BBXY’ from the given applicative expressions X and Y. When this complex operator is applied to the operand ‘Z’, then the result is the applicative expression ‘X(YZ)’ :

BB X------------------> BBX Y --------------------------->

BBXY Z-------------------------------->

BBXYZ-------- =X(YZ)

β-reduction: BBXYZ ->β X(YZ)

Elimination rule Introduction rule Meaning

If f--- [e-I] --- [i-I] identity

f If

Kfa f----- [e-K] ---- [i-K] scope extension

f Kfa

Wfa faa----- [e-W] ------- [i-W] duplication,

faa Wfa diagonalization

C*xf fx------- [e-C*] ------- [i-C*] transposition

fx C*xf

Elimination rule Introduction rule Meaning

Cfab fba------ [e-C] ------ [i-C] converse, permutation

fba Cfa

Bfga f(ga)------ [e-B] ------ [i-B] functional composition

f(ga) Bfga

Sfga fa(ga)------ [e-S] ------- [i-S] complex functional

fa(ga) Sfga composition

φfgha f(ga)(ha)------- [e-φ] ---------- [i-φ] intricationf(ga)(ha) φfgha

3. Four Theories of Quantification3. Four Theories of Quantification

I/ Fregean Theories with bound variables

• Theory A : Classical Theory in First Order Language

• Theory B : Montague’s Quantification expressedin Church’s λ-Calculus

II/ Fregean Theory without bound variables

• Theory C : Illative Theory expressedin Curry’s Combinatory Logic

III/ Non Fregean Theory without bound variables

• Theory D : Star Theory expressedin Curry’s Combinatory Logic

Fregean Approach Non Fregean Approach

With bound variables

Without bound variables

Theory A : classicalFirst Order Languages (LPO)

Theory B using Church’s λ-calculus(Montague, Dowty …)

Theory C using Curry’s combinatory LogicIllative Theory of Quantification

Theory D using Curry’s Combinatory Logic: Star Theory of QuantificationFor processing of natural languages

Theories of QuantificationTheories of Quantification

Questions about Fregean QuantificationQuestions about Fregean Quantification

(i) Are there variables in natural languages ?

(ii) Are necessary variables for analysing phrases withquantifiers ?Can we conceive another logical analysis of naturallanguage,different of Frege’s and Montague’s analyses ?

(iii) Is there a formalism which can adequatly formalizethe quantifications expressed by natural languages ?

(iv) Is it possible to build logical representations“without (bound) variables” such that the structures ofthese representations are more adequate to thestructures (and meanings) of these sentences ?

I/ Fregean theoriesI/ Fregean theories

Theories (A), (B) and (C)Theories (A), (B) and (C)

Predicates and QuantifiersPredicates and Quantifiers

Predicates• [Otp : P1] means “ P1 is a one-place predicate”• [OtOtp : P2] means “ P2 is a two-place predicate”

Fregean Quantifiers• [OOtpp : Q ] means “Q is a fregean quantifier” : it builds a proposition from an one-place predicate.• [OOtpOOtpp : Q ] means “Q is a fregean restricted quantifier” : it builds a proposition from two one-place predicates. No Fregean quantifiers• [OOtpt : Q* ] means “Q* is an operator (star quantifier) which builds a term from a one-place predicate” :it builds a term from a predicate.

John is happy

John is happyN* S\N*---------> [T]S/(S\N*)----------------------------------------------->

S

0. John is happy----------------------------------------------

1. [N* : John] x [S/N* : is happy]2. [S/(S\N*): C*John] x [S/N* :is-happy]3. [S : (C*John) (is-happy)]

----------------------------------------------4. (p : (C*John’) (is-happy’))5. (p: (is-happy’)(John’))6. (is-happy) (John)

FREGEAN ANALYSIS of QUANTIFIERSin FIRST ORDER LANGUAGE (Theory A)

(1) Jane is pretty(1’) is-pretty’ (Jane’)

(2) Everybody is pretty(2’) (∀x) [is-pretty’(x)]

(3) Every girl is pretty(3’) (∀x) [girl’(x) => pretty’(x)]

(4) Someone is pretty(4’) (∃x) [is-pretty’(x)]

(5) Some girl is pretty(5’) (∃x) [girl’(x) & is-pretty’(x)]

LOGICAL REPRESENTATIONS OFLOGICAL REPRESENTATIONS OFQUANTIFIERS (Theory B)QUANTIFIERS (Theory B)

Using Church’s λ-calculus, logical interpretations of quantifiers everybody, someone, every and some (Montague, 1974, Partee, Dowty) become:

everybody’ =def λP.(∀x) [P(x)]

someone’ = def λP.(∃x) [P(x)]

every’ = def λP.λQ.(∀x) [P(x) => Q(x)]

some’ = def λP.λQ.(∀x) [P(x) & Q(x)]

Everybody is pretty(2”) (λP.((∀x) [P(x)]) (is-pretty’)(2’) (∀x) [is-pretty’(x)]

Every girl is pretty(3”) (λP.λQ.((∀x)[P(x) => Q(x)]) (girl’) (is-pretty’)(3’) (∀x) [girl’(x) => pretty’(x)]

Someone is pretty

(4”) (λP.((∃x)[P(x)]) (is-pretty’)(4’) (∃x) [is-pretty’(x)]

Some girl is pretty(5”) (λP.λQ.((∀x)[P(x) & Q(x)]) (girl’) (is-pretty’)(5’) (∃x) [girl’(x) & is-pretty’(x)]

PROBLEM : Are there bound variablesin natural languages ?

Linguists must explain how natural languages capture and encode by specificstructures the different quantification and determination operations.For analysing the constitution of sentences with quantification, the formalrepresentations given by the classical predicate calculus are not adequate sincenatural languages do not use bound variables in semiotic quantification strategy. For linguistics, bound variables are only useful artefacts. Is it possible to give logical representations “without (bound) variables” so that thestructures of these representations are more adequate to the syntactical structures ofsentences?

More precisely, the logical representations must explain how the analysed sentencesare build and also how they are related to a syntactic analysis.

ILLATIVE QUANTIFIERSILLATIVE QUANTIFIERSin illative Combinatory Logic Frameworkin illative Combinatory Logic Framework

(Theory C)(Theory C)

• Illative operators “represent” classical quantifiers insideCurry’s Combinatory Logic formalism. • Illative operators are adjoined to the “pure” applicativeformalism and their actions are defined, by means ofelimination and introduction rules in Gentzen’s NaturalDeduction style, without using bound variables.

« Classical » Categorial Analysis(Categorial Grammars) => Fregean Quantifiers

Every student admire some lecturer O. Every student admire some lecturerU*/N N (S\N)/N V*/N N---------------> ----------------------> U* V* 1. (every student) x admire x (some lecturer)

= S/(S\N) = (S\N) \((S\N)/N) -----------------------------------<

(S\N) 2. (every student) x ((some lecturer) admire)----------------------------------------------------->

S 3. (every student) ((some lecturer) admire)

-------------------------------------------------------------

4. [p : (every student) ((some lecturer) admire)]

5. (Π2 student) ((Σ222 lecturer) admire)

Illative Universal Quantifiers :Illative Universal Quantifiers :ΠΠ11 and and ΠΠ22€€

• Types and applicative quantified expressions building •• OOtpp : Π1 OOtpp : Π1 Otp : g

----------------------------------->p : Π1g

•• OOtpOOtpp : Π2 OOtpOOtpp : Π2 Otp : f--------------------------------------------->

Otp : Π2f Otp : g---------------------------------------------->

p : Π2fg

•• “Π1f” (“every is f”) and “Π2fg” (“every f is g”) are propositions

Elimination and introduction rules ofElimination and introduction rules ofUniversal Quantifiers Universal Quantifiers ΠΠ11 and and ΠΠ22

in Gentzenin Gentzen’’s styles style

Π1f Π2fg fx------ [e-Π1] ---------------------------> [e-Π2]fx gx

[t : x] [t : x]fx fx /- gx

-------- [i-Π1] ---------- [i-Π2]Π1f Π2fg

•• The elimination rule [e-Π2] generalizes modus ponens

•• λ-representations :[ Π1 =def λP. (∀x) [P(x)] ][ Π2 =def λP.λQ. (∀x) [P(x) => Q(x)] ]

The two quantifiers Π1 and Π2 are not independent since it is possible to defineΠ2 , inside Combinatory Logic, from Π1 by the following relation betweenoperators:

[ Π2 =def ((B(CB2)Φ) => Π1) ] This relation shows that the restricted illative quantifier Π2 is defined by means ofa Combinator B(CB2)Φ that combines the implication operator => with thequantifier Π1.

1. Π2fg hyp.

2. [ Π2 =def (B(CB2)Φ) => Π1 ] def. de Π2

3. ((B(CB2)Φ) => Π1) fg rempl. 2., 1.

4. (CB2)(Φ =>) Π1 fg [e-B]

5. B2 Π1 (Φ =>) fg [e-C]6. Π1 (Φ => fg) [e-B2]

The elimination rule [e-Π2] is deduced from [e-Π1]:

1. Π1 (Φ => fg) hyp.2. fx hyp.3. (Φ => fg) x [e-Π1],1.4. =>(fx)(gx) [e-Φ],3.5. gx modus ponens

Illative Existential QuantifiersIllative Existential QuantifiersΣΣ11 and and ΣΣ22

• Types and applicative building •• OOtpp : Σ1 OOtpp : Σ1 Otp : g

--------------------------------->p : Σ1g

•• OOtpOOtpp : Σ2 OOtpOOtpp : Σ2 Otp : f--------------------------------------------->

Otp : Σ2f Otp : g---------------------------------------------->

p : Σ2fg

•• “Σ1f” (“there is a f”) and “Π2fg” (“there is a f which is g”) are propositions

Elimination and introduction rules ofElimination and introduction rules ofExistential Quantifier Existential Quantifier ΣΣ11 and and ΣΣ22

in Gentzenin Gentzen’’s styles style

fx fx, gx------- [i-Σ1] --------- [i-Σ2]Σ1f Σ2fg

Σ1f /- B Σ2fg, fx /-B, gx /-B ---------- [e-Σ1] ------------------------- [e-Σ2]

B B • λ-representations :

[ Σ1 =def λP. (∃x) [P(x)] ][ Σ2 =def λP.λQ.(∃x) [P(x) & Q(x)] ]

• Expression of Σ2 in terms of & (conjunction) and Σ1 : [ Σ2 =def (B(CB2)Φ) & Σ1 ]

Illative Representations whitout bound variables(Theory C : Illative Combinatory Logic)

(1) Jane is pretty(1’) (Jane’*) (is-pretty’)(1’’) (C*Jane’)(is-pretty’)

(2) Everybody is-pretty(2’) (everybody’) (is-pretty’)(2’’) Π1 (is-pretty’)

(3) Every girl is pretty(3’) (every’(girl’)) (is-pretty’)(3’’) (Π2 (girl’)) (is-pretty’)

(4) Somebody runs

(4’) (somebody’) (runs’)

(4’’) Σ1 (runs’)

(5) Some girl is pretty

(5’) (some’ (girl’)) (is-pretty’)

(5’’) (Π2 (girl’)) (is-pretty’)

With two quantifiers ?

(6) Every boy love some girl

(6’) (some’(boy’)) ((some’(girl’)) loves’)

(6’’) (Π2 (boy’)) ((Σ 222 (girl’)) loves’)

« Classical » Categorial Analysis(Categorial Grammars) => Fregean Quantifiers

Every student admire some lecturer O. Every student admire some lecturerU*/N N (S\N)/N V*/N N---------------> ----------------------> U* V* 1. (every student) x admire x (some lecturer)

= S/(S\N) = (S\N) \((S\N)/N) -----------------------------------<

(S\N) 2. (every student) x ((some lecturer) admire)----------------------------------------------------->

S 3. (every student) ((some lecturer) admire)

-------------------------------------------------------------

4. [p : (every student) ((some lecturer) admire)]

5. (Π2 student) ((Σ222 lecturer) admire)

Arguments against theories (A), (B) and (C) (fregean theories)

• There are not bound variables in natural languages (against A). • Proper names and quantified phrases Subjects are not represented by the sameprocess (against A). • Logical decomposition in Subject and Predicate does not work (against A). • Common nouns, adjectives and intransitive verbs are represented by the sameprocess (against A, B, C). • Determination operations are not expressed (against A,B,C). • It is not easy to represent topicalization operations (against A,B,C). • The fregean logical represntations do not take in account the problems of typicalityand atypicality (from Rösch approach of categorization) (against A,B,C).

II/ Star Theory of QuantificationII/ Star Theory of Quantification

a non Fregean Approacha non Fregean Approachof Quantifiersof Quantifiers

in Natural Languagesin Natural Languages

Star Theory of Quantification :non Fregean Quantifiers Q*

Quantifiers considered as building terms operators and not as building propositionsfrom predicates

The expressions with a freagean restricted universal quantifier :

(∀x) (student’(x) => (∃ y) (lecturer’ (y) & admire’ (y)(x)))

(Π2 student’)((Σ 222 lecturer’) admire’) are logical representations of the sentence : Every student admires some lecturer

=> Fregean Quantifiers build propositions or predicates from predicates.

We are going to define two new operators Π∗Π∗ and Σ∗Σ∗ (building terms) which aremore elementary than the quantifiers Π2 and Σ222 in the above representations:

[ Π2 =def ΒC*Π∗Π∗ ] [Σ222 = def ΒC*Σ∗Σ∗ ]

Applicative Tree with Star QuantifiersApplicative Tree with Star Quantifiers

Now, the building of the proposition is given by the applicative tree:

OOtpt : Σ∗Σ∗ Otp : lecturer’-------------------------------------------->

OtOtp : admire t : Σ∗Σ∗ lecturer’--------------------------------------------------------->Otp : admire (Σ∗Σ∗ lecturer’) OOtpt: Π∗Π∗ Otp:student’

--------------------------------------->t : Π∗Π∗ student’

--------------------------------------------------------------------------------------------------------------->p : (admire (Σ∗Σ∗ lecturer’)) (Π∗Π∗ student’)

Logical RepresentationsLogical Representationswith Star Quantifierswith Star Quantifiers

• In the applicative expression :

admire’ (Σ∗Σ∗ lecturer’) (Π∗Π∗ student’) the quantifiers Σ∗Σ∗ and Π∗Π∗ are operators whose operands are considered asproperties with the assigned type Otp.

It follows that the types of Σ∗Σ∗ and Π∗Π∗ must be O(Otp)t .

• These new operators Σ∗Σ∗ and Π∗Π∗ build terms from properties.

We have the following β-reduction : (Π2 student’) ((Σ222 lecturer’) admire’)

->β admire’ (Σ∗Σ∗ lecturer’) (Π∗Π∗ student’)

1. (Π2(student’)) ((Σ222 (lecturer’)) (admire’)) hyp.

2. [ Π2 =def ΒC*Π∗Π∗ ] def.3. [ Σ 222 =def ΒC*Σ∗Σ∗ ] def. 4. ΒC*Π∗Π∗(student’)(ΒC*Σ∗Σ∗(lecturer’)(admire’)) rempl.2,1.5. C*(Π∗Π∗ (student’)) (ΒC*Σ∗Σ∗ (lecturer’) (admire’)) [e-B],

4.6. (ΒC*Σ∗Σ∗ (lecturer’) (admire’)) (Π∗Π∗(student’)) [e-C*]1. C*(Σ∗Σ∗ (lecturer’)) (admire’) (Π∗Π∗ (student’)) [e-B], 6.

8. (admire’) (Σ∗Σ∗ (lecturer’)) (Π∗Π∗ (student’)) [e-C*],7

Every student admires some lecturerby means of Categorial Applicative Grammar with Type Raising and N*

Every student admire some lecturer O. Every student admires some lecturerN*/N N (S\N*)/N* N*/N N ---------------> -----------------> N* N* 1. [N*:every student] x [(S\N*)/N*:admires]

x [N*:some lecturer]------------->[T] ----------------->[T] S\(S/N*) S\(S/N*) 2. [S/(S\N*):C*(every student)]x[(S\N*)/N*:admire]

x [S\(S/N*): C*(some lecturer)]------------------------->[B]

S/N* 3. [S/N*: B(C*(every student)) admire)] x [S\(S/N*): C*(some lecturer)

--------------------------------------------------------<S 4. [S: (C*(some lecturer)) (B(C*(every student))

admire)]-------------------------------------------------------------------------------------5. (p:(C*(some’ lecturer’))(B(C*(every student’)) admire’)

6. (B(C*(every’ student’)) admire’) (some’ lecturer’) 7. (C*(every’ student’))(admire’ (some’ lecturer’)) 8. (admire’ (some’ lecturer’)) (every’ student’)

9. (admire’ (Σ∗Σ∗ lecturer’)) (Π∗Π∗ student’)

III /III / β β-Reduction-Reductionof Fregean Quantifiersof Fregean Quantifiers

totoNon Fregean QuantifiersNon Fregean Quantifiers

(Star Theory) (Star Theory)

Classical Universal Quantifier Π2reduces to the Star Quantifier ΠΠ**

[ Π2 = BC* Π* ] (law) Π2 is defined in terms of the quantifier ΠΠ** Reduction : (Π2f)g ->β g(ΠΠ**f)

1. (Π2 f)g hyp.

2. [ Π2 = BC* ΠΠ** ] law

3. BC* ΠΠ** fg rempl.

4. C* (ΠΠ** f) g [B-e]

5. g( ΠΠ** f) [C*-e]

Ext(g) = { y ; build dams (y) }

Ext(f) = { x ; is-a-beaver (x) }

All f are g : All beavers build dams

Reference framework

Ext(f) ⊆ Ext(g)

Ext(g)

All f are g : A (typical) beaver builds damsIF all f are g THEN a typical f is (a fortiori) g

Reference framework

Ext(f)

Ext(f) ⊆ Ext(g)

Extτ(f)

A typical beaver builds dams

does not imply that

All beavers build dams

Example :

A (typical) Frenchman groans

does not imply that

All Frenchmen groan.

Extτ(f) ⊆ Ext(g)

Elimination and introduction rulesfor illative and star theories :

Universal quantifiers

[ Π2 = BC* Π* ]

(Π2f)g ->β g(Π*f)

Quantifier Π2 Quantifier Π* (Π2f)g, x ∈ Ext (f) g(Π*f), x ∈ Extτ (f)----------------------- [e-Π2] -------------------------- [e-Π*]

g(x) g(x) fx \- gx x ∈ Extτ (f), fx \- gx--------------- [i-Π2] ---------------------------- [i-Π*] (Π2f)g g(Π* f)

• In other words : Classical Universal Quantification Star Universal Quantification

Ext(f)€⊆ Ext(g), x ∈ Ext(f) Extτ(f) ⊆ Ext (g), x ∈ Extτ(f)------------------------------ --------------------------------- x ∈ Ext(g) x ∈ Ext(g) x ∈ Ext(f) => x ∈ Ext(g) x ∈ Extτ(f) => x ∈ Ext(g)-------------------------- -------------------------------- Ext(g)€ ⊆ Ext (f) Ext (g)€ ⊆ Extτ (f)

(Π2f)g <=> Ext(f) €⊆ Ext(g)

g(Π* f) <=> Extτ (f) €⊆ Ext (g)

We deduce :

Elimination and introduction rulesElimination and introduction rulesfor illative and star theories :for illative and star theories :

Existential QuantifiersExistential Quantifiers

[ Σ2 = BC* Σ* ]

(Σ2f)g ->β g(Σ*f)Quantifier Σ2 Quantifier Σ* x ∈ Extτ(f), fx \- gx x ∈ Ext(f), gx-------------------------- [i-Σ2] -------------------- [i-Σ*]

(Σ2f)g g(Σ* f) (Σ2f)g, fx /- B, gx /-B g(Σ* f),fx /- B,gx /-B-------------------------- [e-Σ2] ----------------------- [e-Σ*]

B B

We deduce : (Σ2f)g = True Extτ (f) ∩ Ext (g) ≠ ∅

g(Σ*f) = True Ext (f) ∩ Ext (g) ≠ ∅

The classical existantial Quantifier Σ2 reduces tothe existantial Star Quantifier ΣΣ**

[ Σ2 = BC* ΣΣ** ] (law) Reduction of Σ2 to Σ*

Reduction : (Σ2f)g ->β g(ΣΣ**f)

1. (Σ2f)g hyp.

2. [ Σ2 = BC* ΣΣ** ] law

3. BC* ΣΣ** fg rempl.

4. C* (ΣΣ** f) g [B-e]

5. g( ΣΣ** f) [C*-e]

Ext(g)

Ext(f)

Referential

Extτ(f) ∩ Ext(g) ≠ ∅

Some (typical) f is gSome (typical) beaver builds dams(Σ2f)g Extτ(f) ∩ Ext(g) ≠ ∅

Extτ(f)

IF there is a f which is g THEN a fortiorithere is an f which is gExample : Some typical beaver builds dams=> There is a beaver which builds dams

Ext(g)

Ext(f)

Referential

Ext(f) ∩ Ext(g) ≠ ∅

There is a beaver which builds damsg (Σ*f) Ext(f) ∩ Ext(g) ≠ ∅

Extτ(f)

IF there is an f which is gDoes not imply necessarly that the indetermined instance which is f and is g f et gIs a typical instance Example : There is a Frenchman who does not groan does not imply that this Frenchman is a typical instance of Frenchmeen

IV/ Logical IV/ Logical « « CubeCube » » of ofFregean and non FregeanFregean and non Fregean

Quantifiers.Quantifiers.A new Interpretation ofA new Interpretation of

Fregean and non FregeanFregean and non FregeanQuantifiersQuantifiers

Meaning of star quantifiers

• The terms Π∗(f) and Σ∗(f) denote objects ofExp (f) (Expanse of ‘f’).

- Π∗(f) denotes an abstract « whatever, any » object; anytypical object of Ext(f) is determined from Π∗(f)

- Σ∗(f) denotes an abstract « undeterminate » object. Atypical object which exists in Ext(f) can be gerneratedby successive determinations from Σ∗(f).

All Frenchmen groan A (typical) Frenchman groans

There is a (typical)Frenchmanwho groans

Some Frenchman groans

There is a Frenchman

Typical and classical quantifiersTypical and classical quantifiers

(Π2) fg

(Σ2) fg

g(Π*f)

g(Σ*f)

All f are g A typical f is g

There is a typical fwhich is g

Some f is g

Σf

β-reduction

β-reduction

β-reduction

β-reduction

There is a f

AristotleAristotle’’s Squares Square

(Π2 f)g

(Σ2 f)g

(Π2 f) (Ng)

(Σ2 f) (Ng)

There is a f

contraries

contradictories

subcontraries

Square of the reduction of Square of the reduction of « « classicalclassical » » (fregean) (fregean)quantifiers toquantifiers to

star (non fregean) quantifiersstar (non fregean) quantifiers

(Π2f) g

(Σ2f) g

Σ1 f

g (Π∗Π∗f)

g (ΣΣ**f)

ΣΣ* * (f)

β-reduction

β-reduction

Square of Star QuantifiersSquare of Star Quantifiers(non Fregean Quantifiers)(non Fregean Quantifiers)

g (ΠΠ**ff)

g (ΣΣ**f)

(Ng) (ΠΠ**ff)

(Ng) (ΣΣ**f)

There isan f

contradiction

Square of star quantifiersSquare of star quantifiers

g (ΠΠ**f)

g (ΣΣ**f)

(Ng) (ΠΠ**f)

(Ng) (ΣΣ**f)

ΣΣ* (* (f)

(Π(Π22f) g

(Σ(Σ22f) g

(ΠΠ22 f) (Ng)

(ΣΣ2 2 f )(Ng)

ΣΣf

g(ΠΠ∗f)

g(ΣΣ*f)

(Ng)(ΠΠ*f)

(Ng)(ΣΣ*f)

ΣΣ*(f)

opposite

disjunction

All f are g No f is g

There isa f

A typical f is g

Some fis g

A typical f is not g

There is a fwhich is not g

There isa typical f

« « CubeCube » » of quantifiers of quantifiers

There is a typical f which is not g

Some typical f is g

All beavers build dams No beaver build dams

There is a beaver

A typical beaverbuilds dams

Some beaverbuilds dams

A typical beaverdoes not build dams

There is a beaverwhich does not build dams

There is a typical beaver

« Cube » of quantifiers : examples

There is a typical beaver which does not build damsThere is a typical beaver

which builds dams

All f are g

There is a typical f which is g

No f is g

There i a typical f which is not g

There is a f

A typical f is g

Some f is g

A f typique n’est pas g

There is a f which is not g

There is A typical f

Fregean quantificationFregean quantification

No fregeanNo fregeanquantification =quantification =star quantification star quantification

All f are g No f is g

There is f

A typical f is g

Some fis g

No typical f is g

There is a f which is not g

There is a f

Typical quantification

Aristotle’s square

Square of star quantifiers

There is a typical fwhich is g

There is a typical f which is not g

• In classical Logic

Exp(f) = Ext(f)Ext(f) ⊆ Ext(g) Int(f) ⊇ Int(g)

All objects are typical objects, no atypicalobjects

Star qunatifiers are classical quantifiers

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