The formalization of abductive reasoning has received increasing attention from logicians. However, few work is found beyond abduction in propositional logic, given that in a first order formalism, the undecidability problem naturally appears, and therefore an abductive problem cannot even be appropriately formulated. Still, many applications in artificial intelligence allow finite domains to work with, and this gives an opportunity to apply abduction in first order logic with restricted domains. In this paper, we present an approach to abductive reasoning in (...) C-structures, first order structures with a finite domain in which each of its elements has a corresponding constant representing its interpretation. By using semantic tableaux with bounded depth, that is, C-tableaux (a modification of N-tableaux, introduced by the authors in previous papers) and δ-resolution calculus, we build an effective procedure for the searching of minimal abductive solutions within the proposed semantics. (shrink)

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

In Schwichtenberg, Schwichtenberg fine-tuned Tait’s technique so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic. In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4. The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of the logical (...) complexity of their end-sequent. (shrink)

This is an extended version of the lectures given during the 12-thConference on Applications of Logic in Philosophy and in the Foundationsof Mathematics in Szklarska Poręba. It contains a surveyof modal hybrid logic, one of the branches of contemporary modal logic. Inthe first part a variety of hybrid languages and logics is presented with adiscussion of expressivity matters. The second part is devoted to thoroughexposition of proof methods for hybrid logics. The main point is to showthat application of hybrid logics (...) may remarkably improve the situation inmodal proof theory. (shrink)

In this paper we show that, in Gentzen systems, there is a close relation between two of the main characters in algebraic logic and proof theory respectively: protoalgebraicity and the cut rule. We give certain conditions under which a Gentzen system is protoalgebraic if and only if it possesses the cut rule. To obtain this equivalence, we limit our discussion to what we call regular sequent calculi, which are those comprising some of the structural rules and some logical rules, in (...) a sense we make precise. We note that this restricted set of rules includes all the usual rules in the literature. We also stress the difference between the case of two-sided sequents and the case of many-sided sequents, in which more conditions are needed. (shrink)

When considering m-sequents, it is always possible to obtain an m-sequent calculus VL for every m-valued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiple-valued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2-Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a (...) reduct that is a distributive lattice or a pseudocomplemented distributive lattice. We also show that the sentential logic naturally associated with the provable sequents of this algebraizable Gentzen relation is the logic that preserves degrees of truth with respect to the original algebra (in contrast with the more common logic that merely preserves truth). Finally, for some particular logics we obtain 2-sequent calculi that axiomatize the algebraizable Gentzen relations obtained so far. (shrink)

ABSTRACT In this paper we study consequence relations on the set of many sided sequents over a propositional language. We deal with the consequence relations axiomatized by the sequent calculi defined in [2] and associated with arbitrary finite algebras. These consequence relations are examples of what we call Gentzen systems. We define a semantics for these systems and prove a Strong Completeness Theorem, which is an extension of the Completeness Theorem for provable sequents stated in [2]. For the special case (...) of the finite linear MV-algebras, the Strong Completeness Theorem was proved in [10], as a consequence of McNaughton's Theorem. The main tool to prove this result for arbitrary algebras is the deduction-detachment theorem for Gentzen systems. (shrink)

A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form , or of the form , and all the active formulas of its premises belong to the set . In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive coherence criterion (...) for such systems, and show that a quasi-canonical system of the type we investigate is coherent iff it is strongly paraconsistent or strongly paracomplete (in a sense defined in the paper), iff it has a trivalent, non-deterministic semantics of a special type (also defined in the paper) for which it is sound and complete. Finally, we determine when a system of this sort admits cut-elimination, and provide a simple procedure for transforming one which does not into one which does. (shrink)

Resolution and cut-free LK are the most popular propositional systems used for logical automated reasoning. The question whether or not resolution and cut-free LK have the same efficiency on the system of CNF formulas has been asked and studied since 1960 425–467). It was shown in Cook and Reckhow, J. Symbolic Logic 44 36–50 that tree resolution has super-polynomial speed-up over cut-free LK. Naturally, the current issue is whether or not resolution and cut-free LK expressed as directed acyclic graphs have (...) the same efficiency. In this paper, we introduce a new algorithm to eliminate atomic cuts and show that cut-free LK polynomially simulates resolution when the input formula is expressed as a k-CNF formula. As a corollary, we show that regular resolution does not polynomially simulate cut-free LK . We also show that cut-free LK polynomially simulates regular resolution. (shrink)

In this paper we address the problem of combining a logic with nonmonotonic modal logic. In particular we study the intuitionistic case. We start from a formal analysis of the notion of intuitionistic consistency via the sequent calculus. The epistemic operator M is interpreted as the consistency operator of intuitionistic logic by introducing intuitionistic stable sets. On the basis of a bimodal structure we also provide a semantics for intuitionistic stable sets.

I consider two cognitive phenomena: inquiring and justifying, as complementary processes running in opposite directions. I explain on an example that the former process is driven by questions and the latter is a codification of the results of the first one. Traditionally, proof theory focuses on the latter process, and thus describes the former, at best, as an example of a backward proof search. I argue that this is not the best way to analyze cognitive processes driven by questions, and (...) that proof-theoretical analysis of questions can bring mutual benefits for both proof theory and erotetic logic. In the second part of my paper (sect. 6.3) I introduce the structural rules of weakening and contraction. Results concerning their admissibility are discussed. I also sketch an erotetic interpretation of the rule of weakening as reducing information noise, and that of contraction as securing information. (shrink)