Logics preserving degrees of truth and the hierarchies of abstract algebraic logic Josep Maria Font Department of Probability, Logic and Statistics Faculty of Mathematics University of Barcelona XV Simposio Latinoamericano de Lógica Matemática Bogotá, 4 8 June 2012 Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 1 / 24
Outline 1 Logics preserving degrees of truth In general In ordered sets 2 Abstract algebraic logic and its two hierarchies of logics 3 Logics preserving degrees of truth from residuated lattices (substructural logics and mathematical fuzzy logic) Presentation and basic facts Classification The Deduction Theorem Completeness Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 2 / 24
Preservation of truth in (sentential) many-valued logic ϕ 1,..., ϕ n ψ whenever all ϕ i are true, ψ is true. The traditional many-valued setting: A set V of truth values V > 2 (Łukasiewicz: V [0,1] IR) A subset D V of designated truth values (often D = {1}! ) A set Val of evaluations v: Language V ϕ 1,..., ϕ n ψ for all v Val, if v(ϕ i ) D for all i = 1,..., n, then v(ψ) D «truth comes in degrees» «truth of a fuzzy proposition is a matter of degree» Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 3 / 24
Preservation of truth when truth comes in degrees A degree of truth can be any D V A structure of degrees of truth on a set of truth values V is any family {D j : j J} with D j V ϕ 1,..., ϕ n ψ for all v Val, for all j J, if v(ϕ i ) D j for all i = 1,..., n, then v(ψ) D j No individual D j has a meaning concerning truth. Only the global structure has. No need to designate one among them. We call this the consequence preserving degrees of truth (with respect to this structure) Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 4 / 24
Preservation of truth in ordered algebras V is an algebra A = A,... of the type of F m (the algebra of formulas) There is an order on A Each truth value a A determines a degree of truth a := { b A : a b } A The logic that preserves this structure of degrees of truth is ϕ 1,..., ϕ n A ψ for all v Val, for all a A, if a v(ϕ i ) for all i = 1,..., n, then a v(ψ) When A has a lattice reduct determining, then ϕ 1,..., ϕ n A ψ for all v Val, v(ϕ 1 ) v(ϕ n ) v(ψ) A ψ for all v Val, v(ψ) = max A SCOTT, CLEAVE, PAVELA, NOWA, WÓJCICI, GIL, HÁJE, BAAZ DUNN, BELNAP, FITTING, AVRON, SHRAMO, WANSING Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 5 / 24
Abstract algebraic logic HENIN-MON-TARSI (1985) Cylindric algebras, Part II (Studies in Logic..., vol. 115) Section 5.6: Abstract algebraic logic and more algebraic logics «algebraic version of abstract model theory» General, abstract study of algebraic semantics of sentential logics BLO, PIGOZZI; CZELAOWSI; HERRMANN; FONT, JANSANA; RAFTERY; JÓNSSON; GALATOS, TSINAIS; GIL-FÉREZ; CINTULA, NOGUERA;... Workshop on abstract algebraic logic (Barcelona, 1997) Studia Logica special issues: 2000, 2003, 2012 (forthcoming) Mathematics Subject Classification (2010 revision) code 03G27 Two hierarchies of logics: The Leibniz hierarchy and the Frege hierarchy Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 6 / 24
The Leibniz hierarchy (main fragment) finitely equivalential finitely algebraizable equivalential implicative finitely regularly algebraizable algebraizable protoalgebraic regularly algebraizable weakly algebraizable regularly weakly algebraizable Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 7 / 24
The Leibniz hierarchy: some bridge theorems Beth s definability Epimorphisms are surjective: equivalential [BLO, HOOGLAND (2006)] logics Deduction Theorem EDPRC: finitary, finitely algebraizable logics Deduction Theorem Join-semilattice of compact theories is dually residuated: finitary, protoalgebraic logics [CZELAOWSI (1985)] Interpolation Amalgamation: equivalential logics CZELAOWSI, J. AND PIGOZZI, D. Amalgamation and interpolation in abstract algebraic logic. In: CAICEDO, X. AND MONTENEGRO, C. H. (eds.) Models, algebras and proofs. Selected papers of the Tenth Simposio Latinoamericano de Lógica Matemática (Bogotá, 1995). Marcel Dekker, New York, 1999. pp. 187 265. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 8 / 24
The Frege hierarchy fully Fregean fully selfextensional Fregean selfextensional Protoalgebraic + Fregean + has theorems = regularly algebraizable For finitary, weakly algebraizable logics, the Frege hierarchy reduces to selfextensional and Fregean. Hierarchies are orthogonal to one another: A logic can be in the highest level of one hierarchy and totally outside the other. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 9 / 24
(Commutative and Integral) Residuated Lattices Definition A residuated lattice is an algebra A = A,,,,, 1, 0 such that 1 A,, is a lattice with maximum 1 (with order ), 2 A,, 1 is a commutative monoid with unit 1, 3 A,, is a residuated pair with respect to the order, i.e., c a b c a b a, b, c A. The class RL of these algebras is a variety. Also denoted in the literature by FL ei. [ Full LAMBE calculus... ] Some well-known subvarieties: FL ew, BL, MTL, MV, Π, G, HA, BA,... NOTATION: equations ϕ ψ ; order relations ϕ ψ := ϕ ψ ϕ Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 10 / 24
Logics preserving truth For RL: We turn truth-functional: Val = Hom(F m, A) The designated truth set in each A is {1} Definition is the finitary logic determined by: ϕ 1,..., ϕ n ψ = ϕ 1 1 &... & ϕ n 1 ψ 1 ψ = ψ 1 Informally, we call the logic of 1 (associated with ) GALATOS, N.; JIPSEN, P.; OWALSI, T.; ONO, H. Residuated lattices: an algebraic glimpse at substructural logics Studies in Logic, vol. 151. Elsevier, 2007. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 11 / 24
Logics preserving degrees of truth Definition For RL, is the finitary logic determined by: ϕ 1,..., ϕ n ψ = ϕ 1 ϕ n ψ Informally, we call ψ = ψ 1 the logic of order (associated with ) The logic depends only on the equations satisfied by. ϕ ψ = ϕ ψ For an arbitrary class RL, So will be a subvariety of RL from now on. = V() Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 12 / 24
BOU, F., Infinite-valued Łukasiewicz logic based on principal lattice filters, Journal of Multiple-Valued Logic and Soft Computing, (201x). To appear. BOU, ESTEVA, FONT, GIL, GODO, TORRENS, VERDÚ Logics preserving degrees of truth from varieties of residuated lattices Journal of Logic and Computation, 19 (2009), 1031 1069. FONT Taking degrees of truth seriously Studia Logica (Special issue on Truth Values, Part I), 91 (2009), 383 406. BOU, F., A first approach to the deduction-detachment theorem in logics preserving degrees of truth, in L. Magdalena, M. Ojeda-Aciego, and J. L. Verdegay, (eds.), Proceedings of IPMU 2008, Malaga, 2008, pp. 1061 1067. FONT, GIL, TORRENS, VERDÚ On the infinite-valued Łukasiewicz logic that preserves degrees of truth Archive for Mathematical Logic, 45 (2006), 839 868. FONT An abstract algebraic logic view of some multiple-valued logics In: M. FITTING and E. ORLOWSA (eds.), Beyond two: Theory and applications of multiple-valued logic, vol. 114 of Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 2003, pp. 25 58. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 13 / 24
A few basic facts The logics and have the same theorems or tautologies. ϕ 1,..., ϕ n ψ (ϕ 1 ϕ n ) ψ (ϕ 1 ϕ n ) ψ [ graded DT] ϕ ψ ϕ ψ [ weak DT: SCOTT (1973) for Ł n ] The logic is the inferential extension of The rule of Modus Ponens for : The rule of Adjunction for : The rule of -squaring : by any of: ϕ, ϕ ψ ψ If A then Fi (A) = { implicative filters of A}. If A then Fi (A) = { lattice filters of A}. ϕ, ψ ϕ ψ ϕ ϕ ϕ NOTATION: separates the two sides of rules or sequents ( no nor ) Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 14 / 24
Classification in the hierarchies For each, the logic is implicative, with equivalent algebraic semantics Alg ( ) =, equivalence formula x y := (x y) (y x), and defining equation x 1, but need not be even selfextensional. For each, the logic is fully selfextensional, with Alg ( ) ( = V ) = (the intrinsic variety), but need not be even protoalgebraic. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 15 / 24
Classification in the hierarchies (II) For each, the following conditions are equivalent: 1 The logic is Fregean. 2 The logic is weakly algebraizable. 3 The logic is selfextensional. 4 The two logics coincide: =. Equivalently, any of: 1 coincides with in. 2 is a variety of (generalized) Heyting algebras. 3 is an axiomatic extension of JOHANSSON s minimal logic. When this happens, the unique logic is both implicative and fully Fregean. Coincidence of and means we have left the properly substructural landscape. So it is perhaps more interesting to see what happens when belongs to lower levels of the Leibniz hierarchy. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 16 / 24
Classification in the hierarchies (III) For each, the following conditions are equivalent: 1 The logic is protoalgebraic. 2 The logic is equivalential. 3 The logic is finitely equivalential. Then (x y)n (y x) n, for some n ω, works as equivalence formula. 4 = x ( (x y) n (y x) n) y, for some n ω. 3 mutually excluding locations for in the Leibniz hierarchy: Non-protoalgebraic Finitely equivalential but not algebraizable Implicative (and then also: fully Fregean) Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 17 / 24
Some consequences is protoalgebraic = there is n ω such that = xn x n+1. (all algebras in are n-contractive) For most of best-known fuzzy logics (Ł, BL, MTL, FL ew, Π, etc.): the logic preserving degrees of truth is not protoalgebraic, the logic preserving truth is not selfextensional. If is a variety of MTL-algebras, then is protoalgebraic all chains in are ordinal sums of simple n-contractive MTL-chains, for some n ω. If is a variety of BL-algebras, then is protoalgebraic there is n ω such that = xn x n+1. The logic preserving degrees of truth with respect to any finite BL-chain is protoalgebraic, hence finitely equivalential. The Ł n are finitely equivalential but not algebraizable. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 18 / 24
Deduction Theorems for All the satisfy the Local Deduction Theorem for {x n y : n ω}: Γ, α β there is some n ω such that Γ α n β. [GALATOS, ONO (2006); POGORZELSI (1964) for Ł ] A logic satisfies the Deduction Theorem (DT) for some δ(x, y): Γ, α β Γ δ ( α, β ) if and only if all algebras in are n-contractive, for some n. Moreover δ(x, y) = x n y. [GALATOS (2003); POGORZELSI (1964), WÓJCICI (1973) for Ł n ] Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 19 / 24
Deduction Theorems for [BOU (2008)] For each, the following conditions are equivalent: 1 The logic satisfies the DT for δ(x, y) = x y. 2 The logic satisfies the DT for δ(x, y) = x y. 3 The logics and coincide. If is a variety of MTL-algebras, then satisfies the DT for some δ(x, y) is protoalgebraic. Moreover δ(x, y) = (x y) n y. [GIL, TORRENS, VERDÚ (1993) for subvarieties of MV] If is generated by a finite A, then satisfies the DT for some δ(x, y) the following hold: 1 The logic is protoalgebraic. 2 A is distributive as a lattice. 3 is residuated and every subalgebra of A is closed under the map a, b max{c A : a c b}. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 20 / 24
Axiomatizations: Gentzen-style, Tarski-style Sequents: Γ ϕ where Γ is a finite set of formulas All structural axioms and rules, and the logical axioms: 1 ϕ ϕ ϕ (ψ ξ) ψ (ϕ ξ) Logical rules: ϕ ξ ψ ξ ϕ ψ ξ ϕ ψ ξ ϕ ξ ϕ ψ ξ ψ ξ ϕ, ψ ψ ϕ ψ ϕ ϕ ψ ψ ϕ ψ ϕ ψ ϕ ψ ξ ϕ ψ ξ ϕ ψ ξ ϕ ψ ξ ϕ 1,..., ϕ n RL ψ ϕ 1,..., ϕ n ψ is derivable in this calculus Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 21 / 24
Axiomatizations: Hilbert-style 1 Axioms: all ϕ such that = ϕ 1 (i.e., all theorems of ) Rules: Adjunction for : ϕ, ψ ϕ ψ Restricted Modus Ponens: ϕ ψ when ϕ ψ is a theorem 2 Assume is axiomatized by Ax(), with MP as the only rule. Then is axiomatized by 1 as the only axiom, and the rules: -specific rule : Adjunction for : Modus Ponens for : Weakening for : α α ϕ for every ϕ Ax() ϕ, ψ ϕ ψ α ( ϕ (ϕ ψ) ) α ψ ϕ ψ ϕ Associativity for : (ϕ ψ) ξ ϕ (ψ ξ) Commutativity for : ϕ ψ ψ ϕ Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 22 / 24
Some conclusions There may be sound reasons to consider logics preserving degrees of truth in the framework of many-valued logic. Logics preserving degrees of truth from varieties of residuated lattices form an interesting and remarkably diverse family of logics, with a richer, less uniform and less predictable behaviour than logics preserving truth. Abstract algebraic logic provides a powerful and convenient framework to study both particular logics and large groups of logics. Locating a logic in the hierachies gives information on which algebraic properties to look for in order to prove certain metalogical theorems (and conversely). Having the hierachies is important because, along the history of mathematics, having good classifications has been an important driving force in the development of many areas. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 23 / 24
Some references on abstract algebraic logic CZELAOWSI, J. Protoalgebraic logics vol. 10 of Trends in Logic - Studia Logica Library. luwer Academic Publishers, Dordrecht, 2001. FONT, J. M., AND JANSANA, R. A general algebraic semantics for sentential logics. Second revised edition vol. 7 of Lecture Notes in Logic. Association for Symbolic Logic, 2009. Electronic version freely available through Project Euclid at http://projecteuclid.org/euclid.lnl/1235416965. FONT, J. M., JANSANA, R., AND PIGOZZI, D. A survey of abstract algebraic logic Studia Logica (Special issue on Abstract Algebraic Logic) 74 (2003), 13 97. With an update in vol. 91 (2009), 125 130. CINTULA, P., AND NOGUERA, C. A general framework for Mathematical Fuzzy Logic In Handbook of Mathematical Fuzzy Logic, vol. I, P. Cintula, P. Hájek, and C. Noguera, Eds., vol. 37 of Studies in Logic. College Publications, London, 2011. Josep Maria Font (Barcelona) Logics preserving degrees of truth and AAL XV SLALOM (Bogotá, 2012) 24 / 24