Problems of substitution and admissibility in the modal system Grz and in intuitionistic propositional calculus

AbstractQuestions connected with the admissibility of rules of inference and the solvability of the substitution problem for modal and intuitionistic logic are considered in an algebraic framework. The main result is the decidability of the universal theory of the free modal algebra Fω(Grz) extended in signature by adding constants for free generators. As corollaries we obtain: (a) there exists an algorithm for the recognition of admissibility of rules with parameters (hence also without them) in the modal system Grz, (b) the substitution problem for Grz and for the intuitionistic calculus H is decidable, (c) intuitionistic propositional calculus H is decidable with respect to admissibility (a positive solution of Friedman's problem). A semantical criterion for the admissibility of rules of inference in Grz is given

Multiple Conclusion Rules in Logics with the Disjunction Property

We prove that for the intermediate logics with the disjunction property any basis of admissible rules can be reduced to a basis of admissible m-rules (multiple-conclusion rules), and every basis of admissible m-rules can be reduced to a basis of admissible rules. These results can be generalized to a broad class of logics including positive logic and its extensions, Johansson logic, normal extensions of S4, n-transitive logics and intuitionistic modal logics

Decidability of admissibility:On a problem by friedman and its solution by rybakov

Rybakov (1984) proved that the admissible rules of IPC are decidable. We give a proof of the same theorem, using the same core idea, but couched in the many notions that have been developed in the mean time. In particular, we illustrate how the argument can be interpreted as using refinements of the notions of exactness and extendibility

Speaking about Transistive Frames in Propositional Languages

This paper is a comparative study of the propositional intuitionistic (non-modal) and classical modal languages interpreted in the standard way on transitive frames. It shows that, when talking about these frames rather than conventional quasi-orders, the intuitionistic language displays some unusual features: its expressive power becomes weaker than that of the modal language, the induced consequence relation does not have a deduction theorem and is not protoalgebraic. Nevertheless, the paper develops a manageable model theory for this consequence and its extensions which also reveals some unexpected phenomena. The balance between the intuitionistic and modal languages is restored by adding to the former one more implication

Through and beyond classicality: analyticity, embeddings, infinity

Structural proof theory deals with formal representation of proofs and with the investigation of their properties. This thesis provides an analysis of various non-classical logical systems using proof-theoretic methods. The approach consists in the formulation of analytic calculi for these logics which are then used in order to study their metalogical properties. A specific attention is devoted to studying the connections between classical and non-classical reasoning. In particular, the use of analytic sequent calculi allows one to regain desirable structural properties which are lost in non-classical contexts. In this sense, proof-theoretic versions of embeddings between non-classical logics - both finitary and infinitary - prove to be a useful tool insofar as they build a bridge between different logical regions

Admissible Bases Via Stable Canonical Rules

We establish the dichotomy property for stable canonical multi-conclusionrules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics