Interpolation in Normal Extensions of the Brouwer Logic

The Craig interpolation property and interpolation property for deducibility are considered for special kind of normal extensions of the Brouwer logic

Post Completeness in Congruential Modal Logics

Well-known results due to David Makinson show that there are exactly two Post complete normal modal logics, that in both of them, the modal operator is truth-functional, and that every consistent normal modal logic can be extended to at least one of them. Lloyd Humberstone has recently shown that a natural analog of this result in congruential modal logics fails, by showing that not every congruential modal logic can be extended to one in which the modal operator is truth-functional. As Humberstone notes, the issue of Post completeness in congruential modal logics is not well understood. The present article shows that in contrast to normal modal logics, the extent of the property of Post completeness among congruential modal logics depends on the background set of logics. Some basic results on the corresponding properties of Post completeness are established, in particular that although a congruential modal logic is Post complete among all modal logics if and only if its modality is truth-functional, there are continuum many modal logics Post complete among congruential modal logics

Proof Theory of Finite-valued Logics

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics

Propositional Logics of Dependence

In this paper, we study logics of dependence on the propositional level. We prove that several interesting propositional logics of dependence, including propositional dependence logic, propositional intuitionistic dependence logic as well as propositional inquisitive logic, are expressively complete and have disjunctive or conjunctive normal forms. We provide deduction systems and prove the completeness theorems for these logics

Metalogical properties, being logical and being formal

The predicate ‘being logical’ has at least four applications. We can apply it to concepts, propositions, sets of propositions (systems, theories) and methods. The concepts of quantifier or disjunction are logical but those of horse or water are not. Some propositions, for instance, the principle of excluded middle, are logical, others, for instance the law of gravity, are not. Propositional calculus is a logical theory (belongs to logic), but the theory of evolution is not. In a sense, the problem of logical propositions reduces itself to the question of logical systems, because we can say that A is logical if and only if it belongs to a logical systems (however, see below). Finally, deduction is a logical method of justification, but observation is not

Metalogical properties, being logical and being formal

The predicate 'being logical' has at least four applications. We can apply it to concepts, propositions, sets of propositions (systems, theories) and methods. The concepts of quantifier or disjunction are logical but those of horse or water are not. Some propositions, for instance, the principle of excluded middle, are logical, others, for instance the law of gravity, are not. Propositional calculus is a logical theory (belongs to logic), but the theory of evolution is not. In a sense, the problem of logical propositions reduces itself to the question of logical systems, because we can say that A is logical if and only if it belongs to a logical systems (however, see below). Finally, deduction is a logical method of justification, but observation is not

On two properties of structurally complete logics

Let S = (S, f1,..., fn) sentential language C, possibly with index, denotes a standard consequence. Sb is the consequence operation determined by the rule of substitution. By L(C) we denote the set of all Lindenbaum's extensions of the consequence C (L(C) = {X C S : C(X) = S and C(X U {a}) = S for every a e X}). End(S) is the set of all endomorphisms of S. Let U be the consequence operation induced by a matrix U . The symbol E(U ) stands for the set of all formulas which are valid in U . We also write U C- U, iff U is a submatrix of U1. In this paper we assume that in every functionally complete matrix U(A, D) the set D is proper non-empty subset of the domain A of A. By a rule of inference we mean a non-empty subset of 2S x S. A rule r is finitary iff for every X C S and every a e S : if (X, a) e r, then X is finite. A rule r is elementary iff r = {(h(X), h(a)) : h e End(S)} for some X and for some a. In turn, CA stands for the consequence operation determined by A (A C S) and by the set of the rules R. For simplicity the symbol CR will be used instead of CR. Two particular sets of rules will be used: MP - the set which contains only the modus ponens and the Godel's rule. CL stands for the set of all classical tautologies. I is the set of all theses of the intuitionistic logic and J denotes the set of all theses of the Johansson's minimal logic (Fragment tekstu)