Set Theory in Fuzzy Logic

Zuzana Hanikov´a Institute of Computer Science, CAS

Logic and its applications, Faculty of Arts, CUNI : May 14, 2020

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic The notion of a fuzzy set

To accommodate fuzzy sets, we shall refer to (usual) sets as classical or crisp sets.

A crisp set S in a universe U, such as the set of prime numbers P ⊆ N, is fully determined by its characteristic function: cS : U −→ {0, 1} is a characteristic function of S provided that, for each x ∈ U,

cS (x) = 1 if and only if x ∈ S. We can introduce some set-theoretic operations using the Boolean operations on {0, 1}.

A fuzzy set F in a universe U is fully determined by its characteristic function µF : U −→ [0, 1], where

µF (x) is the degree to which x belongs to F . Now [0, 1] is the real unit interval, possibly with suitable operations.

Fuzzy sets were introduced in L. A. Zadeh: Fuzzy sets. Information and Control. 8(3): 338353, 1965. Investigation of the operations on [0, 1], “suitably” generalizing the Boolean ones, were historically an important path to fuzzy logic: see J. A. Goguen: The logic of inexact concepts, Synthese 19, 325–373, 1969 Fuzzy sets have many nice applications (and “applications”).

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Some notions on fuzzy sets

Let [0, 1]∗ be our algebra of truth degrees (in lieu of a Boolean one). Now we have a flat universe U (sets are not members of sets).

Figure: [Wikipedia]

Let F be a fuzzy set. The domain is U (or its subset); the support is {x ∈ U | cF (x) > 0}; F is normal if ∃x ∈ U such that µF (x) = 1, otherwise it is subnormal; the height of F is sup{µF (x) | x ∈ D(F )}.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Axiomatic theory of fuzzy sets

We work in classical ZFC. Noting that fuzzy logics are weaker than classical, we can experiment with set-theoretic axioms over a suitable fuzzy/substructural logic. Any such theory, with only ZFC (or weaker) axioms, is consistent relative to ZFC.

Our intended theory should: generate a cumulative universe of sets be provably distinct from the classical set theory be reasonably strong (possibly interpret ZF(C)) be consistent (relative to ZFC)

Which axioms should we adopt to capture the universe of sets? – Are they compatible with the logic? Which classical constructions can be reproduced? Which are the new and interesting concepts, not visible in a classical setting?

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Substructural logics

Some structural rules of (a Gentzen calculus) for INT may be absent. The building stones of the calculus are sequents Γ ⇒ ϕ and the structural rules of exchange, weakening, and contraction are as follows:

Γ, ϕ, ψ, ∆ ⇒ χ Γ, ∆ ⇒ χ Γ, ϕ, ϕ, ∆ ⇒ χ (e) (w) (c) Γ, ψ, ϕ, ∆ ⇒ χ Γ, ϕ, ∆ ⇒ χ Γ, ϕ, ∆ ⇒ χ

Removal of these rules may call for some changes: splitting of connectives changes to interpretation of a sequent

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Algebraic semantics for FLew

FLew stands for the Full Lambek calculus with exchange and weakening. It is just the logic of FLew-algebras.

AFL ew-algebra is an algebra A = hA, ·, →, ∧, ∨, 0, 1i such that: 1 hA, ∧, ∨, 0, 1i is a bounded lattice, 1 is the greatest and 0 the least element 2 hA, ·, 1i is a commutative monoid 3 for all x, y, z ∈ A, z ≤ (x → y) iff x · z ≤ y

FLew-algebras form a variety.

Subvarieties of this variety correspond to axiomatic extensions of FLew.

[Hiroakira Ono: Logics without the contraction rule and residuated lattices. Australian Logic Journal, 2010]

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic FLew and extensions

Take FLew as a ground logic.

2 INT is FLew plus contraction: ϕ → ϕ

Monoidal t-norm logic (MTL) is FLew plus semilinearity: (ϕ → ψ) ∨ (ψ → ϕ) BL is MTL plus divisibility: ϕ · (ϕ → ψ) → (ϕ ∧ ψ) G¨odellogic G is INT plus semilinearity Lukasiewicz logicL is BL plus involutiveness: ¬¬ϕ → ϕ

Consistent logics extending FLew are weaker than, or equivalent to, classical logic.

Adding the schema LEM: ϕ ∨ ¬ϕ to FLew yields classical logic; the same for any of its consistent extensions.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Example: strengthening the logic I

Assume a theory T , defined (ostensibly) over an extension of FLew, proves ϕ ∨ ¬ϕ for an arbitrary formula ϕ. Then T is a theory over classical logic.

Example: LEM from the axiom of regularity (aka foundation):

∀x(x 6= ∅ → ∃y ∈ x(y ∩ x = ∅))

[Grayson 1979]; see also https://ncatlab.org/nlab/show/well-founded+relation

Consider the set 1 = {∅}. Let {∅  ϕ} stand for {x ∈ 1 | ϕ} (obtained by separation). Consider z = {∅  ϕ, 1}. Then z is nonempty and has a ∈-minimal element. If ∅ is minimal then ϕ holds; if 1 is minimal then ϕ fails.

Thus regularity (in a weak setting with empty set, pairing, and separation) implies LEM for any sentence. One can similarly prove LEM for any formula with free variables/parameters.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Interpretations

How can we be sure that a theory T over a logic L is not classical ZF(C)?

Let us construct a “model” of T over L in ZFC. In such “models”, we can pinpoint sets that provide counterexamples to LEM.

More precisely, we shall define an interpretation of T (over L) in ZFC. The intp. is an analogue of a “Boolean-valued model”, over a complete, totally ordered FLew-algebra.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Lukasiewicz logic: L

Can be obtained as an extension of FLew (see above). Recall the language is {·, →, ∧, ∨, 0, 1}; in addition, we define: ¬ϕ as ϕ → 0 and ϕ ⊕ ψ as ¬ϕ → ψ and ϕ ≡ ψ as (ϕ → ψ) ∧ (ψ → ϕ) In fact, one can define all the above function symbols from, e.g., {→, 0}. (This is a semantic statement on definable functions.) The following can be taken as a Hilbert axiomatization of L : (L1) ϕ → (ψ → ϕ) (L2) ( ϕ → ψ) → ((ψ → χ) → (ϕ → χ)) (L3) ( ¬ϕ → ¬ψ) → (ψ → ϕ) (L4) (( ϕ → ψ) → ψ) → ((ψ → ϕ) → ϕ) Deduction rule: modus ponens.

T `L ϕ means ϕ is provable using elements of T and (substitution instances of) (L1) – (L4) as axioms.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Standard MV-algebra

MV-algebras (“many-valued”) provide algebraic semantics to L .

Consider the universe [0, 1] and the algebra [0, 1]L = h[0, 1], f¬, f→i, where

f¬(x) = 1 − x and

f→(x, y) = min(1, 1 − (x − y)). By induction, this provides a semantics to every L -formula.

Note that all the operations of [0, 1]L are continuous. (In particular, piecewise linear with integer coefficients; cf. McNaughton theorem). Hence, no two-valued operation is term-definable.

The algebra [0, 1]L “corresponds” to the (propositional!) logic L as follows:

theorems of L coincide with TAUT([0, 1]L ) provability from finite theories in L coincides with the finite consequence relation of [0, 1]L .

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Lukasiewicz logic, 1st order: L ∀

The following are axioms for quantifiers ∀, ∃ over a general MTL-chain. (∀1) ∀xϕ(x) → ϕ(t)(t substitutable for x in ϕ) (∃1) ϕ(t) → ∃xϕ(x)(t substitutable for x in ϕ) (∀2) ∀x(χ → ϕ) → (χ → ∀xϕ)(x not free in χ) (∃2) ∀x(ϕ → χ) → (∃xϕ → χ)(x not free in χ) (∀3) ∀x(ϕ ∨ χ) → (∀xϕ ∨ χ)(x not free in χ)

Rule of generalization: from ϕ derive ∀xϕ.

In fact, L ∀ (with above axioms) proves ∃xϕ ≡ ¬∀x¬ϕ, and the axioms (∃1), (∃2), and (∀3) become theorems.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Lukasiewicz logic, 1st order, semantics

Assume L is a first-order language.

Let A be the algebra [0, 1]L (or generally, an MV-chain).

An A-structure for L is an M = hM, (rP )P predicate symbol, (fF )F function symboli where domain M 6= ∅ n for each n-ary predicate symbol P of L, rP is an n-ary function rP : M → A n for each n-ary function symbol F of L, fF is a function fF : M −→ M

A The value kϕkM,v of a formula ϕ in an A-structure M and valuation v in M is: A kP(t1,..., tn)kM,v = rP (kt1kM,v ,..., ktnkM,v ) A A k¬ϕkM,v = ¬kϕkM,v A A A kϕ → ψkM,v = kϕkM,v → kψkM,v A V A k∀xϕk = 0 kϕk 0 (! need not exist) M,v v≡x v M,v A A structure M is safe provided kϕkM,v is defined for each ϕ and v. A formula ϕ in L is valid in a safe A-structure M provided that

^ L A kϕkM,v = 1 v an M−evaluation M is a model of a theory T iff all axioms of T are valid in M.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Example: power set axiom over [0, 1]L

Powerset axiom: ∀x∃z∀u(u ∈ z ≡ u ⊆ x); in ZFC, we define the operation P(x). Suppose we adopt powerset in a theory overLukasiewicz logic. Introducing P as a function, we get the following:

∀x(x ∈ P(∅) ≡ x ⊆ ∅)

Which sets are subsets of ∅? Let us try to identify them in [0, 1]L . We have x ⊆ ∅ iff ∀z(z ∈ x → z ∈ ∅) iff ∀z(¬(z ∈ x)). Note that ⊆ is fuzzy (provided membership is fuzzy). This means that kx ⊆ ∅kv(x)=a = infb∈U 1 − kz ∈ xkv(z)=b,v(x)=a. In particular: if a is normal in U, then it is indeed not a subset of ∅; if a is subnormal (and the height is not 1), then a is a subset of ∅ in a positive degree. Which subnormal sets are there in U? For each element y take a singleton {y}. Then separate from {y} using a (formula with) a value 0 < c < 1: we get {y | c}. Now for any y ∈ U and any 0 < c < 1, we have {y | c} in P(∅) in degree 1 − c. If U is not a set, the support of P(∅) is not a set.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic The ∆ operator

We shall introduce a weaker version of the powerset axiom. To that end, let us define a unary operator ∆ on [0, 1]: ∆(1) = 1, while ∆(x) = 0 for x < 1. Now the weak powerset axiom reads

∀x∃z∆∀u(u ∈ z ≡ ∆(u ⊆ x))

Some axioms: (∆1) ∆ϕ ∨ ¬∆ϕ (∆2) ∆(ϕ ∨ ψ) → (∆ϕ ∨ ∆ψ) (∆3) ∆ϕ → ϕ (∆4) ∆ϕ → ∆∆ϕ (∆5) ∆(ϕ → ψ) → (∆ϕ → ∆ψ) And the deduction rule of ∆-generalization: ϕ/∆ϕ.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Example: strengthening the logic II

Let L prove (p → p & p) → (p ∨ ¬p). Then, a set theory with separation (for open formulas), pairing, congruence: x = y & x ∈ z → y ∈ z proves ∀xy(x = y ∨ ¬(x = y)) over L . Proof: take x, y. Let z = {u ∈ {x} | u = x}, whence u ∈ z ≡ (u = x)2. Since (x = x)2, we have x ∈ z. If y = x then y ∈ z by congruence. Then (y = x)2. We proved y = x → (y = x)2, thus (by assumption on the logic) x = y ∨ ¬(x = y).

Moreover, in a theory with extensionality, successors, and congruence, LEM for = implies LEM for ∈.

For that reason, we opt for including LEM for = in our theory, and modify extensionality: ∀xy(x = y ≡ (∆(x ⊆ y)&∆(y ⊆ x)))

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic The Universe V A

Work in classical ZFC. ∆ Let A denote the algebra [0, 1]L . (Note that the natural order on [0, 1] yields a complete lattice.)

Define the class V A by ordinal induction in ZFC. A+ = A \{0A}.

A V0 = {∅} A A + Vα+1 = {f : Fnc(f ) & Dom(f ) ⊆ Vα & Rng(f ) ⊆ A } for any ordinal α A S A Vλ = α<λ Vα for limit ordinals λ

A S A V = α∈Ord Vα

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic The Universe V A

We define two binary functions from V A into L, assigning to any u, v ∈ V A the values ku ∈ vk and ku = vk

ku ∈ vk = v(u) if u ∈ D(v), otherwise 0 ku = vk = 1 if u = v, otherwise 0

Using induction on the complexity of formulas, define for any formula ϕ(x1,..., xn) an A n n-ary function from (V ) into L, assigning to an n-tuple u1,..., un the value kϕ(u1,..., un)k. Say that ϕ is valid in V A iff kϕk = 1A is provable in ZFC.

If we verify that, on the other hand, axioms of a theory T are valid in V A, then we will have obtained an interpretation of T in ZFC. Moreover, T is distinct from ZF(C) unless A is a Boolean algebra.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic FST (“Fuzzy set theory”)

We define FST as a theory over the logic L ∀ with ∆ operator. The language is {∈, =}.

Equality axioms for set-theoretic language: reflexivity symmetry transitivity congruence ∀x, y, z(x = y & z ∈ x → z ∈ y) congruence ∀x, y, z(x = y & y ∈ z → x ∈ z) Moreover (for reasons stated), we postulate LEM for equality: ∀x, y(x = y ∨ ¬(x = y)) (we say that equality is crisp in our theory)

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Set-theoretic axioms of FST

(ext.) ∀xy(x = y ≡ (∆(x ⊆ y)&∆(y ⊆ x))) (empty) ∃x∆∀y¬(y ∈ x) (pair) ∀x∀y∃z∆∀u(u ∈ z ≡ (u = x ∨ u = y)) (union) ∀x∃z∆∀u(u ∈ z ≡ ∃y(u ∈ y & y ∈ x)) (weak power) ∀x∃z∆∀u(u ∈ z ≡ ∆(u ⊆ x)) (inf.) ∃z∆(∅ ∈ z & ∀x ∈ z(x ∪ {x} ∈ z)) (sep.) ∀x∃z∆∀u(u ∈ z ≡ (u ∈ x&ϕ(u, x))) for any ϕ not containing free z (coll.) ∀x∃z∆[∀u ∈ x∃v ϕ(u, v) → ∀u ∈ x∃v ∈ zϕ(u, v)] for any ϕ not containing free z (∈-ind.) ∆∀x(∆∀y(y ∈ x → ϕ(y)) → ϕ(x)) → ∆∀xϕ(x) for any ϕ (support) ∀x∃z(Crisp(z)&∆(x ⊆ z))) where Crisp(x) ≡ ∀u∆(u ∈ x ∨ ¬u ∈ x).

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Interpretation

Theorem Let ϕ be a closed formula provable in FST. Then ϕ is valid in V A, i. e., ZFC proves kϕk = 1.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic An Inner Model of ZF in FST

In FST, we can define a class H of hereditarily crisp sets. Theorem Let ϕ be a theorem of ZF. Then FST ` ϕH.

Thus H is an inner model of ZF in FST and ZF is consistent relative to FST. Moreover, this interpretation is faithful.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Powell’s negative interpretation

In INT, define a set theory ZFI (in a language ∈ ): ZF-axioms plus double complement (∼∼)

∀x∃z∀u(u ∈ z ≡ ¬¬u ∈ x) Let ∼∼ x = {y | ¬¬y ∈ x}. (. . . defines ordinals and analyzes the notion . . . ) Define x ∈ St iff x =∼∼ x, and [ Sα = P(∼∼ Sβ ) ∩ St β∈α S and finallyS= α∈Ord Sα. Consider ZF in a language ¬, ∧, ∀; for each such formula ϕ, let ϕS result from ϕ by replacing each subformula ∀xψ with a ∀x ∈ Sψ.

Theorem (Powell 1975) For each formula (as above) such that ZF ` ϕ, we have ZFI ` ϕS.

ZFI proves S to be an inner model of ZF, and ZF is consistent relative to ZFI.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Grayson’s Heyting-valued models

Work in ZFI0: ZFI with collection instead of replacement. Consider a complete Heyting algebra A. Construct an A-valued universe V A, as usual. Define the functions kx ∈ yk and kx = yk, extend to formulas (as usual).

Theorem (Grayson 1979) ZFI0 proves itself valid in V A, i.e., for each axiom (theorem) of ZFI0, one has ZFI0 ` kϕk = 1 in A.

Then, fixing A as a complete Boolean algebra, V A interprets ZF in ZFI’. One can conceive a c.H.a. as P(1) and a c.B.a. as P(1) ∩ S. Moreover, ZFI’ + (∼∼) proves there is an isomorphism F between S and V A such that ∀x¯ ∈ V A(kϕ(¯x)k = 1A ≡ ϕ(F (¯x))S.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic Cantor’s comprehension axiom inL (?)

The axiom of comprehension

∀x1,..., xn∃z∀u(u ∈ z ≡ ϕ(u, x1,..., xn))

is classically inconsistent. To wit, let y = {x | x 6∈ x}; then y ∈ y ≡ y 6∈ y. However,L is weaker than classical logic, and x ≡ ¬x is satisfiable (setting x = 1/2). Generally one can investigate fixed points of defined functions. There has been a number of works investigating the consistency of comprehension in predicateLukasiewicz logic, under various assumptions about ϕ. [Skolem 1957] shows that this principle is consistent for open formulas. Some generalizations can be found in the works of Chang and Fenstad. White (1979) gave a proof of consistency of the full comprehension inL. (White’s paper has many citations, but most people do not understand it.) Terui (2014) claims to have found an error in White’s paper. As a result, the consistency of full comprehension inL seems an open problem.

However, the schema is consistent in weaker logics (such as FLew).

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic References

For intuitionistic and constructive set theory, see https://plato.stanford.edu/entries/set-theory-constructive/. Some (early!) papers on set theory over (variants of)Lukasiewicz logic: T. Skolem: Bemerkungen zum Komprehensionsaxiom. Z. Math. Logik Grundlagen Math., 3:1–17, 1957. D. Klaua: Uber¨ einen zweiten Ansatz zur Mehrwertigen Mengenlehre, Monatsb. Deutsch. Akad. Wiss. Berlin, 8:782–802, 1966. For consistency of full comprehension, see e.g. the website of Terui at Kyoto Uni http://www.kurims.kyoto-u.ac.jp/ terui/, and his paper Kazushige Terui: Light Affine Set Theory: A Naive Set Theory of Polynomial Time, Studia Logica 77(1), 9–40, 2004.

The paper G. Takeuti, S. Titani: Fuzzy logic and fuzzy set theory. Arch. Math. Logic, 32:1–32, 1992. is about axiomatic set theory in a rich expansion of G¨odellogic. My own work on my website www.cs.cas.cz/zuzana, e.g. P. H´ajekand ZH: A Development of Set Theory in Fuzzy Logic. Beyond Two: Theory and Applications of Multiple-Valued Logic, 273–285. Physica-Verlag, 2003.

Zuzana Hanikov´aInstitute of Computer Science, CAS Set Theory in Fuzzy Logic