Logic and computer vol.4

Many-valued Logics
Karpenko A.
CONTENTS
1. Intuitive understanding of many-valued logics and its origin. Sources of manyvaluedness. Many-valued truth table for proving of independent of axioms. Logical principle twovaludness (bivalence). Aristotle’s fatalistic argument and its refutation. Forerunners of many-valued logic. Introducing in logic of the third truth-value by J. Lukasiewicz (pp. 7-16).
2. Three-valued logics. Literature three-valued logics. Lukasiewicz’s three-valued logic L₃. Distinction of L₃ from classical two-valued logica Ñ₂. Extension of L₃ to functionally complete logic. Three-valued modal logic of Lukasiewicz and its intuitive interpretation. Lukasiewicz’s three-valued implication. Kleene’s three-valued logics K₃: strong and weak logical connectives. Bochvars’s three-valued logics B₃: internal and external logical connectives. Two fragments of B₃ which are isomorphic to Ñ₂. Interrelationship between L₃, Ê₃ and Â₃. Heyting’s three-valued logic G₃. Sobocinski’s three-valued logic S₃. Paraconsistent three-valued logics: Rozonoer’s logic Pcont Sette’s logic Ð₁. A logic which is a counterpart of the Ð₁. Some applications of three-valued logics (pp. 17-35).
3. Logical matrices and lattices. A notion of logical matrix. Propositional calculus L. Hilbert calculi. Lindenbaum matrix. A model and an exact model for L: characteristic matrix for L. Operations over the matrices. An example of the product of the matrix for Ñ₂ with itself. Jaskowski’s matrices. Finite approximility of L. A proper extension of L and pretabularity of L. A notion of lattice. Quasi-lattices. De Morgan algebras, Kleene algebras, and Boolean algebras. Stone’s representation theorem for Boolean algebras. Heyting algebras, double Heyting algebras, and symmetrical Heytings algebras. Algebraic characterization of L₃. A notion of algebraic semantics (pp. 36-50).
4. Finite-valued logics. Finite-valued logic of Lukasiewicz L_n. A propositional calculus L_n. J_i-operations. McNaughton’s criterion of definibility in L_n. An extension L_n to functionally complete logic. Finite-valued logic of G? del G_n. Definition operations of G_n by L_n. Finite-valued-logic of Bochvar Â_n. Matrix logic of Post Ð_n and its three-valued case Ð₃. Definition in Ð_nthe most impotant operations. Lukasiewicz-Moisil algebras. Intuitionistic implication. Algebras corresponding propositional calculus L_n. Lukasiewicz’s n-valued implication. Post algebras. Propositional calculus Ð_n(pp. 51-64).
5. Axiomatization of finite valued logics. Prehistory and histore of the question. Anshakow’s and Rychkows’s algorithm of axiomatization of finite-valued (predicate) logics. Axiomatization of any finite-valued logic: quasi-Hilbert’s axiomatization. Some critical remarks. axiomatization of L_n. Have an every finite-valued matrix finite axiomatization? An example of three-valued matrix which is nonfinitely axiomatizable one (pp. 65-76).
6. Infinite-valued logics. Infinite-valued matrix for classical logic. Natural generalization of classical (Boolean) operations: infinite-valued logic of Kleene Êw. Infinite-valued logic of Lukasiewicz L w. Axiomatization of L w. MV-algebras and W-algebras for L w. Some properties. Intuitionistic propositional calculus H. Results of G? del and Jaskowski. Linear logic LC and L-algebras. Logics which are dual logics to LC. Linear H-B-logic and P-algebras. Comparising of L w and Í. Proper extensions of L w, ? and LC: denumerability, continuun and pretabularity. L w with pseudo-complementation: definibility of linear H-B-logic. Symmetrical Heyting monoid. A matrix logic without fixed points LH: generalization logico-algebraic properties of L w and Í. Intensional lattices. Relevant logic R ant its extension RM. Unexpected properties of RM. Sugihara’s matrix. Idempotent de Morgan monoid. Pretabulariti of LC, RM and S5. Lewis modal logics S4 and S5. Implicational logics BCWK(= H®)and BCWI (=R®). Separability of implication. Implicational logics ?CI and ???. Extensions of logic BCK: commutative BCK and an implicational fragment Lw® of logic Lw. BCI- and BCK-algebras. Algebraic versions of logic Lw®. Bounded commutative ???-algebra is MV-algebra and it is dual of W-algebra. Importance of MV-algebras. Uniqness of Lukasiewicz‘s many-valued logics. Jan Lukasiewicz against Lukasiewicz‘s many-valued logics (pp. 77-94).
7. Many-valued logic as functional system. A notion of function of n-valued logic. Different methods of definition of logical functions. Examples of initial functions and their basic properties. Formulas model of many-valued logic. A notion of functional system. Closure operation. Algebra of functions. The problem of definibility of functions. Properties of funcional systems: functional completeness, closure operation and closed classes of functions. Functional precompleteness and precompleteness classes. Criteria of functional completeness. Precompleteness of L₃. Maximal non-Postian n-valued logic Ò_n. Bases and Sheffers’s stroke. Sheffers’s stroke for Ð_n and L
_n. Results of E. Post. The principle differences of many-valued logic from two-valued one. Functional properties of G₃ (the first matrix of Jaskowski). Some functional properties of . Limited and continuous logics (pp. 95-108).
8. Fuzzy sets theory and fuzzy-valued logic. Characteristic membership function of fuzzy subset. Operarions over fuzzy subset. Fuzzy logic, Kleene algebra and Kleene logic Ê₃as its model. Different fuzzy logics. Two studies of fuzification. Lnguistic variable «Truth» and its truth-values. A fuzzy membership function. Fuzzy subsets of type 2 as truth-values. Non-distributive quasi-lattice as an algebra for fuzzy set theory of type 2. Models for fuzzy algebras of type 2. Hierarchy of minimal logical matrices.
9. Interpretation of many-valued logics.Truth-values. Formal interpretation of many-valued logic. The problem of interpretation of truth-values. Truth-values in classical logic. Modalization of truth-values. Prior’s interpretation of modal logic S5. T-F-sequences (Boolean vectors) as truth-values. Truth-functional logics and the dificulties of their interpretation. Post’s interpretation P_nand Byrd’s interpretation L_n. Truth-values as subsets of set of T-F-sequences. Factor-semantics for L_n, Ê_n and Ò_n. Bounds of application of factor-semantics. Discrete model for infinite-valued logic of Lukasiewicz Lw and its factorization. Matrix logic LS (pp. 119-132).
10. Stcructuralization of truth-values. Lindenbaum matrices and fuzzy-valued logics. Truth-values as subsets of set {T, F}. Boolean-valued and Heyting-valued models. Significance of Stone’s representation theorem. Topoi. Factor-semantics: algebras as truth-values. Implication in LS. Two algebraic levels of logics. Logic as science about truth-values (pp.133-138).
11. Classification of non-classical logics. Comlexity of this problem. Classification of many-valued logics functional properties. Classification of type of truth-values. V. A.Smirnov’s ideas: structural rules and different formulations of deduction theorem. Combinators and implicational formulae: Curry-Howard isomorphism. Logical constructions for generating og logics. Decision of the problem about an extension of an implicational fragment (independent) I, B, C, W, K₁ of intuitionistic logic to an implicational fragment (independent) I, B, C, W, K₁ ? of classical logic. Classical versions of BCI, BCK and BCIW logics. Lattices of implicational logics. Maximal lattice. Some applications. Proper extensions of some implicational logics. Full propositional logics. Basic principles of generating of new classes of logics (pp. 139-165).
12. Addendum: Functional properties Lukasiewicz’s logic L_n+1andthe law of prime number generation. Reformulation L_n in L_n+1. V.K. Finn’s theorem about functionally precomplete classes in L_n+1. Matrix logic Ê_n+1:Ê_n+1 has a class of tautologies iff n is a prime number. Functional properties of Ê_n+1:Ê_n+1 = L_n+1 iff n is a prime number. Sheffer’s stroke for Ê_n+1. Lukasiewicz’s implication and classes of prime numbers. The law of prime number generation. Theorem about generation of all prime numbers. Structuralization of prime numbers (pp. 166-178).
LITERATURE (pp. 179-212).
AUTHOR INDEX