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 −! f0; 1g is a characteristic function of S provided that, for each x 2 U, cS (x) = 1 if and only if x 2 S: We can introduce some set-theoretic operations using the Boolean operations on f0; 1g. 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 fx 2 U j cF (x) > 0g; F is normal if 9x 2 U such that µF (x) = 1, otherwise it is subnormal; the height of F is supfµF (x) j x 2 D(F )g. 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 ) ' and the structural rules of exchange, weakening, and contraction are as follows: Γ; '; ; ∆ ) χ Γ; ∆ ) χ Γ; '; '; ∆ ) χ (e) (w) (c) Γ; ; '; ∆ ) χ Γ; '; ∆ ) χ Γ; '; ∆ ) 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 2 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: ' ! ' Monoidal t-norm logic (MTL) is FLew plus semilinearity: (' ! ) _ ( ! ') BL is MTL plus divisibility: ' · (' ! ) ! (' ^ ) G¨odellogic G is INT plus semilinearity Lukasiewicz logicL is BL plus involutiveness: ::' ! ' Consistent logics extending FLew are weaker than, or equivalent to, classical logic. Adding the schema LEM: ' _:' 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 ' _:' for an arbitrary formula '. Then T is a theory over classical logic. Example: LEM from the axiom of regularity (aka foundation): 8x(x 6= ; ! 9y 2 x(y \ x = ;)) [Grayson 1979]; see also https://ncatlab.org/nlab/show/well-founded+relation Consider the set 1 = f;g. Let f; 'g stand for fx 2 1 j 'g (obtained by separation). Consider z = f; '; 1g. Then z is nonempty and has a 2-minimal element. If ; is minimal then ' holds; if 1 is minimal then ' 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; 1g; in addition, we define: :' as ' ! 0 and ' ⊕ as :' ! and ' ≡ as (' ! ) ^ ( ! ') In fact, one can define all the above function symbols from, e.g., f!; 0g. (This is a semantic statement on definable functions.) The following can be taken as a Hilbert axiomatization of L : (L1) ' ! ( ! ') (L2) ( ' ! ) ! (( ! χ) ! (' ! χ)) (L3) ( :' !: ) ! ( ! ') (L4) (( ' ! ) ! ) ! (( ! ') ! ') Deduction rule: modus ponens. T `L ' means ' 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 8 The following are axioms for quantifiers 8, 9 over a general MTL-chain. (81) 8x'(x) ! '(t)(t substitutable for x in ') (91) '(t) ! 9x'(x)(t substitutable for x in ') (82) 8x(χ ! ') ! (χ ! 8x')(x not free in χ) (92) 8x(' ! χ) ! (9x' ! χ)(x not free in χ) (83) 8x(' _ χ) ! (8x' _ χ)(x not free in χ) Rule of generalization: from ' derive 8x'. In fact, L 8 (with above axioms) proves 9x' ≡ :8x:', and the axioms (91), (92), and (83) 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 k'kM;v of a formula ' 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:'kM;v = :k'kM;v A A A k' ! kM;v = k'kM;v ! k kM;v A V A k8x'k = 0 k'k 0 (! need not exist) M;v v≡x v M;v A A structure M is safe provided k'kM;v is defined for each ' and v. A formula ' in L is valid in a safe A-structure M provided that ^ L A k'kM;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: 8x9z8u(u 2 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: 8x(x 2 P(;) ≡ x ⊆ ;) Which sets are subsets of ;? Let us try to identify them in [0; 1]L . We have x ⊆ ; iff 8z(z 2 x ! z 2 ;) iff 8z(:(z 2 x)).