1,721,373 research outputs found
Proof analysis beyond geometric theories: From rule systems to systems of rules
No full textA class of axiomatic theories with arbitrary quantifier alternations is identified and a conversion to normal form is provided in terms of generalized geometric implications. The class is also characterized in terms of Glivenko classes as those first-order formulas that do not contain implications or universal quantifiers in the negative part. It is shown how the methods of proof analysis can be extended to cover such axioms by means of conversion to systems of rules. The structural properties for the resulting extensions of sequent calculus are established and a generalization of the first-order Barr theorem is shown to follow as an immediate application. The method is also applied to obtain complete labelled proof systems for logics defined through their relational semantics. In particular, the method provides analytic proof systems for all the modal logics in the Sahlqvist fragment
Proofs and Countermodels in Non-Classical Logics
No full textProofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin's method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait-Schütte-Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope. © 2014 Springer Basel
PROOF ANALYSIS for LEWIS COUNTERFACTUALS
No full textA deductive system for Lewis counterfactuals is presented, based directly on the influential generalisation of relational semantics through ternary similarity relations introduced by Lewis. This deductive system builds on a method of enriching the syntax of sequent calculus by labels for possible worlds. The resulting labelled sequent calculus is shown to be equivalent to the axiomatic system VC of Lewis. It is further shown to have the structural properties that are needed for an analytic proof system that supports root-first proof search. Completeness of the calculus is proved in a direct way, such that for any given sequent either a formal derivation or a countermodel is provided; it is also shown how finite countermodels for unprovable sequents can be extracted from failed proof search, by which the completeness proof turns into a proof of decidability
Proof Theory for Extended Belnap–Dunn and Intuitionistic Logics
No full textFirst, G0-style sequent calculi for extended Belnap-Dunn and intuitionistic logics, including Nelson and Gurevich logics, are introduced. A theorem establishing the equivalence between G0- and G3-style sequent calculi for these logics is then presented, and the cut-elimination theorem for these G0-style calculi is obtained as a result. Next, natural deduction systems with general elimination rules are introduced for these logics, and a full normalization theorem for these natural deduction systems is proved. This proof is achieved using bi-directional translations between the proposed G0-style sequent calculi and the natural deduction systems with general elimination rules
Geometrisation of first-order logic
No full textThat every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms.Peer reviewe
A sequent calculus for preferential conditional logic based on neighbourhood semantics
No full textThe basic preferential conditional logic PCL, initially proposed by Burgess, finds an interest in the formalisation of both counterfactual and plausible reasoning, since it is at the same time more general than Lewis’ systems for counterfactuals and it contains as a fragment the KLM preferential logic P for default reasoning. This logic is characterised by Kripke models equipped with a ternary relational semantics that represents a comparative similarity/normality assessment between worlds, relativised to each world. It is first shown that its semantics can be equivalently specified in terms of neighbourhood models. On the basis of this alternative semantics, a new labelled calculus is given that makes use of both world and neighbourhood labels. It is shown that the calculus enjoys syntactic cut elimination and that, by adding suitable termination conditions, it provides a decision procedure
Non-Normal Modal Logics: A Challenge to Proof Theory
No full textA general procedure that follows the guidelines of inferentialism is presented for generating G3-style sequent calculi for non-normal modal logics on the basis of neighbourhood semantics
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