From Fuzzy Type Theory to Fuzzy Intensional Logic

From Fuzzy Type Theory to Fuzzy Intensional Logic Vil´ em Nov´ ak University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic e-mail: [email protected]

Abstract In this paper, the concept of fuzzy intensional logic is elaborated based on fuzzy type theory recently presented by the author. Keywords: Fuzzy type theory, intensional logic, algebra of truth values. In the accepted linguistic theory, an important role is played by the, so called intensional logic (cf. [5]), which is based on a simple type theory. Thus, a challenge is raised whether the type theory as developed especially by A. Church [3] and L. Henkin [7, 8] can be generalized to fuzzy one. We will formulate fuzzy type theory (FTT) which is the basis for the intensional logic.

1

Fuzzy type theory

The fuzzy type theory can be developed along the way how fuzzy predicate logic has been developed. Namely, we first choose a structure of truth values and then introduce basic definitions of formulas, axioms and interpretation. Since as mentioned, our goal is to follow the way of development of classical type theory as initiated by A. Church and L. Henkin, we have to start with equivalence/equality symbol (this is the same symbol in syntax but interpreted differently depending on the given type). Let us remark that the Henkin formulation is very elegant and so, our goal is to preserve such elegancy also in the fuzzy type theory. In fuzzy logic, the equivalence/equality symbol is usually interpreted as fuzzy equality which ful-

fils the transitivity property with respect to some t-norm, which corresponds to the chosen truthvalues structure. Since the semantics of the type theory works with functions, a natural problem arises that the functions must keep the fuzzy equality (we speak about extensionality of the functions). If the latter is taken as the interpretation of the elementary symbol then all the functions must be extensional. However, it turns out that in general, this is impossible. Two essential structures of truth values seem to fit best the outlined requirements: the IMTLalgebra and Lukasiewicz MV-algebra. Note that in both cases, we have involutive negation at disposal, which significantly simplifies the syntax. The extensionality of all the functions can be required only in the case that the corresponding fuzzy equality is crisp (i.e. identical with the classical equality). However, certain generalization is possible when considering the, so called Baaz delta operation ∆ which maps all truth values to 0 except for 1. This operation enables to weaken some axioms concerning extensionality and so, fuzzy equality can still be introduced. At any case, fuzzy equality cannot be introduced in full generality as interpretation of the basic equvalence/equality symbol. 1.1 1.1.1

Syntax and semantics of IMTL-fuzzy type theory Types

Let , o be distinct objects. The set of types is the smallest set Types satisfying: (i) , o ∈ Types,

(ii) If α, β ∈ Types then (αβ) ∈ Types. The type  represents elements and o truth values. 1.1.2

Primitive symbols

The following symbols form the basic syntactical elements from which formulas of FTT are constructed.

We furthermore introduce the biresiduation operation a ↔ b = (a → b) ∧ (b → a) and the Baaz delta operation ( 1 if a = 1, ∆(a) = 0 otherwise. 1.2.2

(i) Variables xα , . . . where α ∈ Types. (ii) Special constants cα , . . . where α ∈ Types. We will consider the following concrete special constants: E(oα)α for every α ∈ Types, C(oo)o and Doo . (iii) Auxiliary symbols: λ, brackets. 1.1.3

Let D be a set of objects and L be a set of truth values. A basic frame based on D, L is a family of sets (Mα )α∈Types where (i) M = D is a set of objects, (iii) For each type γ = βα, the corresponding set Mγ ⊆ MβMα .

Formulas

(i) xα ∈ Formα and cα ∈ Formα , (ii) if B ∈ Formβα and A ∈ Formα then (BA) ∈ Formβ , (iii) if A ∈ Formβ then λxα A ∈ Formβα , If A ∈ Formα is a formula of the type α ∈ Types then we write Aα .

1.2.1

Basic frame

(ii) Mo = L is a set of truth values,

The set Formα is a set of formulas of type α ∈ Types, which is the smallest set satisfying:

1.2

Semantics Truth values

The structure of truth values is supposed to form a complete IMTL-algebra (see [4]), which is a complete residuated lattice L = hL, ∨, ∧, ⊗, →, 0, 1i

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1.2.3

Extensional functions

The concept of fuzzy equality has been introduced in [9, 14, 15] and elsewhere. We denote it by =α if considered in the set Mα of the type α. Let F : Mα −→ Mβ be a function. We say that it is extensional w.r.t fuzzy equalities =α , =β if there is a natural number q ≥ 1 such that [mα =α m0α ]q ≤ [F (mα ) =β F (m0α )] (3) holds for all mα , m0α ∈ Mα (the square brackets denote a truth value of the expression inside). If q = 1 then F is strongly extensional otherwise it is simply extensional; F is weakly extensional if [mα =α m0α ] = 1 implies that [F (mα ) =β F (m0α )] = 1.

fulfilling the prelinearity condition (a → b) ∨ (b → a) = 1,

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a, b ∈ L,

and, moreover, its negation function ¬a = a → 0 is involutive, i.e. ¬¬a = a holds for all a ∈ L. It is known that MTL-algebras are algebras (the negation needs not be involutive) of left-continuous t-norms. An example of left-continuous t-norm with involutive negation is nilpotent minimum.

A strongly extensional function is both extensional as well as weakly extensional, and an extensional function is weakly extensional. 1.2.4

Frame

A frame is M = h(Mα , =α )α∈Types , L∆ i

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where (Mα )α∈Types is a basic frame such that L∆ is a structure of truth values being a complete, linearly ordered IMTL∆ algebra and =α is a fuzzy equality on Mα and =α ∈ M(oα)α for every α ∈ Types. If α = γβ then each function F ∈ Mα is weakly extensional w.r.t =β and =γ . Furthermore, we define interpretation of formulas in the frame and general model which is an interpretation I M such that for every formula Aα , α ∈ Types and every assignment p ∈ Asg(M) of elements to variables, IpM (Aα ) ∈ Mα

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1.2.7

Further definitions

(a) Representation of truth > := (λxo xo ≡ λxo xo ) (b) Representation of falsity ⊥ := (λxo xo ≡ λxo >) (c) Negation: ¬ := λxo (⊥ ≡ xo )

holds true. This means that the value of each formula is always defined in the general model. 1.2.5

Basic definitions

(a) Equivalence ≡ := λxα (λyα E(oα)α yα )xα .

(d) Implication: ⇒ := λxo (λyo ((xo ∧ yo ) ≡ xo ))

(b) Conjunction ∧ := λxo (λyo C(oo)o yo )xo . 1.2.6

Axioms

(FT1) ∆ (xα ≡ yα ) ⇒ (fβα xα ≡ fβα yα ) (FT2) (∀xα )(fβα xα ≡ gβα xα ) ≡ (fβα ≡ gβα ) (FT3) (λxα Bβ )Aα ≡ Cβ where Cβ is obtained from Bβ by replacing all substitutable occurrences of xα in it by Aα (lambda conversion). (FT4) ((Ao ≡ Bo ) ∧ (Bo ≡ Co )) ≡ (Ao ≡ Co ) (FT5) (Ao ≡ >) ≡ Ao (FT6) ((Ao ⇒ Bo ) ⇒ Co ) ⇒ (((Bo ⇒ Ao ) ⇒ Co ) ⇒ Co ) (FT7) Ao ∧ Bo ≡ Bo ∧ Ao (FT8) Ao ∧ > ≡ Ao

(e) Special connectives: (6) ∨ := λxo (λyo (((xo ⇒ yo ) ⇒ yo ) ∧ ((yo ⇒ xo ) ⇒ xo ))), (disjunction) ¬(xo ⇒ ¬ yo ))), & := λxo (λyo (¬ (strong conjunction) ¬(¬ ¬Ao & ¬ Bo ))). ∇ := λxo (λyo (¬ (strong disjunction) (f) Quantifiers: Let Ao ∈ Formo and xα be a variable of type α. Then we put: (∀xα )Ao := (λxα Ao ≡ λxα >), (General quantifier ) ¬Ao . (∃xα )Ao := ¬ (∀xα )¬ (Existential quantifier )

(FT9) (Ao ∧ Bo ) ∧ Co ≡ Ao ∧ (Bo ∧ Co ) (FT10) (Ao ⇒ (Bo ⇒ Co )) ≡ (Bo ⇒ (Ao ⇒ Co )) ¬Bo ⇒ ¬ Ao ) ≡ (Ao ⇒ Bo ) (FT11) (¬ (FT12) ∆ (Ao ∧ Bo ) ≡ ∆ Ao ∧ ∆ Bo (FT13) ∆ Ao ⇒ ∆∆ Ao (FT14) ∆Ao ∨ ¬∆Ao (FT15) (∀xα )(Ao ⇒ Bo ) ⇒ (Ao ⇒ (∀xα )Bo ) where xα is not free in Ao

As a special case, if A ∈ Formo then we put An := |A & ·{z · · & A}, n−times

∇ ·{z nA := |A∇ · · ∇ A} . n−times

With respect to the above definition, ⊥, >, (∀xα )Ao , (∃xα )Ao ∈ Formo , ¬ ∈ Formoo and ⇒ , ∨ , & , ∇ ∈ Form(oo)o .

1.2.8

Inference rule and provability

The following are inference rules: Let Aα ≡ A0α ∈ Formo and Bo ∈ Formo be formulas. Then we infer a formula Bo0 from them, where Bo0 comes from Bo by replacing one occurrence of Aα , which is not preceded by λ, by A0α .

(R)

Let Ao ∈ Formo be a formula. Then infer ∆Ao from Ao .

(N)

Theorem 2 Every consistent theory T can be extended to a maximal consistent theory T which is complete. Theorem 3 A theory T is consistent iff it has a general model I M. Theorem 4 For every theory T and a formula Ao T ` Ao

The concept of provability and proof are defined in the same way as in classical logic. A theory T over FTT is a set of formulas of type o, i.e. T ⊂ Formo . If A ∈ Formo and it is provable in T then we write T ` A, as usual. Corollary 1 (Soundness) The fuzzy type theory is sound, i.e. the following holds for every theory T : If T ` Ao then Ip (Ao ) = 1 holds for every assignment p ∈ Asg(M) and every frame model I. We will define that a formula Ao is true in the theory T in the degree a ∈ L and write T |=a A if a=

^ {b | IpM (Ao ) = b, p ∈ Asg(M),

I M is a general model of T } (7)

iff

T |= Ao .

proof: In the same way as in classical logic, the implication left-to-right is soundness theorem. The opposite implication can be proved analogously as the completeness theorem of BL-fuzzy logic (see [6]). Let T |= Ao . We will show that T 6` Ao implies that there is a general model I M and an assignment p such that IpM (Ao ) 6= 1. S

Let us consider the canonical model I M and let S IpM (Ao ) = 1 for some assignment p. This means S that IpM (Ao ) = h(|>|) and since h is the embedS ding, it follows from the construction of I M that T ` Ao ≡ >, i.e. T ` Ao . Hence, T 6` Ao means S that I M (Ao ) 6= 1. 2

If a = 1 then we will simply write T |= Ao .

2

Theories in FTT and completeness

3

Lukasiewicz fuzzy type theory

If T be a theory and A ∈ Formo a formula the by T ∪ {A} is a theory whose set of special axioms is extended by A.

Another possibility is to introduce Lukasiewicz fuzzy type theory (LFT), which is based on the structure of truth values forming the Lukasiewicz MV-algebra

Theorem 1 (Deduction theorem) Let T be a theory, Ao ∈ Formo a formula. Then

LL = h[0, 1], ⊗, ⊕, ¬, 0, 1i

T ∪ {Ao } ` Bo

iff

T ` ∆ Ao ⇒ Bo

holds for every formula Bo ∈ Formo . A theory T is contradictory if T ` ⊥. Otherwise it is consistent. A theory T is complete if for every two formulas Ao , Bo either T ` Ao ⇒ Bo or T ` Bo ⇒ Ao . A theory T is maximal consistent if each its extension T 0 , T 0 ⊃ T is inconsistent.

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where a⊗b = 0∨(a+b−1) is Lukasiewicz conjunction, a ⊕ b = 1 ∧ (a + b) is Lukasiewicz disjunction, a → b = ¬a ⊕ b = 1 ∧ (1 − a + b) is implication and ¬a = 1 − a is negation (a, b ∈ [0, 1]). The Baaz delta operation ∆ is also included. Axioms and some definitions are modified in a way analogous to Lukasiewicz predicate fuzzy logic. A theorem analogous to Theorem 3 can also be proved in this case.

3.0.9

References

Axioms

The axioms of LFT are (FT1)–(FT5), (FT7)– (FT15). The only difference is axiom (FT6), which is replaced by:

[1] Anderson, C. A., General intensional logic. In Gabbay, D. and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. II, D. Reidel: Dordrecht, 1984, 355–385.

(LFT6) (Ao ∨ Bo ) ≡ (Bo ∨ Ao )

[2] Andrews, P., An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Kluwer, Dordrecht 2002.

LFT has also modified definition of the disjunction: ∨ := λxo (λyo (xo ⇒ yo ) ⇒ yo ),

4

Fuzzy intensional logic

Our goal for studying fuzzy type theory is especially in providing a tool for a more realistic model of natural language semantics, which includes also the vagueness phenomenon. This means that we should extend the definition of types by additional type ω for possible worlds, and possibly also the type τ for time. The resulting logical theory is called fuzzy intensional logic since it provides means for development of the mathematical model of intension and extension of natural language expressions. The former, in general, is a formula of the form λωλτ Aoα , i.e. the interpretation of the intension is a function assigning to each possible world and each time moment a fuzzy set of elements of type α. In the fuzzy intensional logic, theorems analogous to Theorems 3 and 4 can be proved. The whole theory also provides frame for introduction of other special connectives such as modal ones, and also for introduction of generalized quantifiers. A very interesting area, where fuzzy intensional logic may bring significant contribution is a generalized formulation of Montague grammar. All these problems will be studied in the future.

Acknowledgment It has been partly supported by project MSM ˇ ˇ 179000002 of MSMT CR.

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