Decidable properties of logical calculi and of varieties of algebras

Abstract

We give an overview of decidable and strongly decidable properties over the propositional modal logics K, GL, S4, S5 and Grz, and also over the intuitionistic logic Int and the positive logic Int+. We consider a number of important properties of logical calculi: consistency, tabularity, pretabularity, local tabularity, various forms of interpolation and of the Beth property. For instance, consistency is decidable over K and strongly decidable over S4 and Int; tabularity and pretabularity are decidable over S4, Int and Pos; interpolation is decidable over S4 and Int+ and strongly decidable over S5, Grz and Int; the projective Beth property is decidable over Int, Int+ and Grz, etc. Some complexity bounds are found. In addition, we state that tabularity and many variants of amalgamation and of surjectivity of epimorphisms are base-decidable in varieties of closure algebras, of Heyting algebras and of relatively pseudocomplemented lattices. A.V.Kuznetsov. No non-trivial property of logics is decidable under recursive axiomatization. S.Linial, E.Post 49. The property ”to be an axiomatization of Cl” is undecidable. A.V.Kuznetsov 63. For every s.i.logic L the property ”to be an axiomatization of L” is undecidable. In particular, consistency and many other properties are, in general, undecidable. A property P is said to be decidable over a calculus L if there is an algorithm which, for any finite set Ax of additional axiom schemes, decides if the calculus L+Ax possesses the property P; P is said to be strongly decidable over L if there is an algorithm which, for any finite set Rul of additional axiom schemes and rules of inference, decides if the calculus L+Rul has the property P. Properties We consider the families of superintuitionistic calculi and of normal modal systems. Each calculus defines a logic, i.e. the set of its theorems. A logic is called tabular if it can be characterized by finitely many finite models; and pretabular if it is maximal among non-tabular logics. A logic L is called locally tabular if for any finite set P of propositional variables there exist only finitely many formulas of P nonequivalent in L. A logic L is said to have Craig’s interpolation property (CIP), if for every formula (A → B) ∈ L there exists C such that (i) both A → C and C → B belong to L, (ii) any variable of C occurs in both A and B. A logic L is said to have interpolation property (IPD), if A `L B implies that exists a formula C such that (i) A `L C, C `L B, and (ii) any variable of C occurs in both A and B. Beth’s definability properties Theorem on Implicit Definability (Beth 53) Any predicate implicitly definable in a first order theory is explicitly definable. Analogs of Beth’s property for prop. logics Let x, q, q′ be disjoint lists of variables not containing y and z, A(x,q, y) a formula. PB1. If `L A(x,q, y)A (2) V(L) has SES; (3) FI(V(L() has SES and V(L) has RAP. RAP: For any finitely generated and finitely indecomposable A,B,C in V such that A is a subalgebra of both B and C and for any b, c ∈ A, b 6= c, there exist a subdirectly irreducible algebra D in V and homomorphisms g : B → D, h : C → D such that g(z) = h(z) for all z in A and g(b) 6= g(c).