Modular Cut-Elimination: Finding Proofs or Counterexamples

Logical Foundations of Computer Science

Abstract. An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical Gentzen-type systems with (n, k)-ary quantifiers are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of an (n, k)-ary quantifier is introduced. The semantics of such systems for the case of k ∈ {0, 1} are provided in [16] using two-valued non-deterministic matrices (2Nmatrices). A constructive syntactic coherence criterion for the existence of a 2Nmatrix for which a canonical system is strongly sound and complete, is formulated there. In this paper we extend these results from the case of k ∈ {0, 1} to the general case of k ≥ 0. We show that the interpretation of quantifiers in the framework of Nmatrices is not sufficient for the case of k > 1 and introduce generalized Nmatrices which allow for a more complex treatment of quantifiers. Then we show that (i) a canonical calculus G is coherent iff there is a 2GNmatrix, for which G is strongly sound and complete, and (ii) any coherent canonical calculus admits cut-elimination.

Section: Corollary 27 If G Is a Coherent Canonical Calculus Then It Amentioning

confidence: 76%

Section: Summary and Further Researchmentioning

confidence: 94%

Section: Corollary 27 If G Is a Coherent Canonical Calculus Then It Amentioning

confidence: 98%

Section: Corollary 27 If G Is a Coherent Canonical Calculus Then It Amentioning

confidence: 99%

Section: Corollary 27 If G Is a Coherent Canonical Calculus Then It Amentioning

confidence: 99%

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Generalized Non-deterministic Matrices and (n,k)-ary Quantifiers

Logical Foundations of Computer Science

A Triple Correspondence in Canonical Calculi: Strong Cut-Elimination, Coherence, and Non-deterministic Semantics

Computer Science – Theory and Applications

Abstract. An (n, k)-ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)-ary quantifiers form a natural class of Gentzen-type systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using two-valued non-deterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence to characterize strong cutelimination in such systems. We show that the following properties of a canonical system G with arbitrary (n, k)-ary quantifiers are equivalent: (i) G is coherent, (ii) G admits strong cut-elimination, and (iii) G has a strongly characteristic two-valued generalized non-deterministic matrix.

Non-Deterministic Semantics for Logical Systems

Handbook of Philosophical Logic

2010

100

The rest of this survey is divided into two parts. Part I describes the propositional framework of Nmatrices. We begin with some preliminaries and a review of many-valued matrices in Section 2. The basic definitions of the framework of Nmatrices are presented in Section 3. In Section 4 we introduce canonical signed calculi, a natural family of proof systems manipulating sets of signed formulae (Gentzen-type systems can be thought of as a specific instance of such calculi). The relation between Nmatrices and canonical calculi is then explored in two complementary directions. In Section 4.1 we provide a general proof theory for Nmatrices using canonical calculi. In Section 4.2 modular non-deterministic semantics is provided for every canonical calculus (satisfying a simple syntactic condition). We then proceed to describe further applications of Nmatrices. In Section 5 we extend the modular approach to two non-canonical families of Gentzentype calculi: those that are obtained from the positive fragments of classical logic and intuitionistic logic by adding various natural Gentzen-type rules for negation. In Section 6 Nmatrices are used for yet another family of nonclassical logics: paraconsistent logics, designed for reasoning in the presence of contradictions. In Part II we handle the extension of the framework of Nmatrices to the first-order level and beyond. In Section 7 we briefly review the two standard approaches to interpreting unary quantifiers in many-valued logics. In Section 8 we extend the propositional framework of Nmatrices to languages with such quantifiers and discuss the problems that this move reveals (and were not evident on the propositional level). Section 9 is devoted to the particular case of the usual first-order quantifiers. An application of this case is presented in Section 10, where we extend the results from Section 6, and provide semantics for a large family of first-order paraconsistent logics. Section 11 further generalizes the framework of Nmatrices to multi-ary quantifiers and extends the relation between Nmatrices and canonical signed calculi to languages with such quantifiers. Due to lack of space, we omit in what follows most of the proofs, providing instead pointers to the relevant papers.Those of the proofs we do include are intended to give the reader a better insight into the nature of Nmatrices, and a flavour of the (mostly new) methods that can be employed in handling and applying them.