Canonical Gentzen-Type Calculi with (n,k)-ary Quantifiers

Zamansky, Anna; Avron, Arnon

doi:10.1007/11814771_22

Cited by 4 publications

(11 citation statements)

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“…In the following two subsections, we briefly reproduce the relevant definitions from [16] of canonical systems with (n, k)-ary quantifiers and of the semantic framework of Nmatrices.…”

Section: The Set Of L-formulas (Closed L-formulas) Denote By T Rm L mentioning

confidence: 99%

“…We use a simplified representation language from [16] for a schematic representation of canonical rules. …”

Section: Canonical Systems With (N K)-ary Quantifiersmentioning

confidence: 99%

“…Proposition 9 (Decidability of coherence) ( [16]) The coherence of a canonical calculus G is decidable.…”

Section: Definition 7 For Two Sets Of Clausesmentioning

confidence: 99%

“…In [16] the notion of a canonical system is extended to languages with (n, k)-ary quantifiers. An (n, k)-ary quantifier (for n > 0, k ≥ 0) is a generalized logical connective, which binds k variables and connects n formulas.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

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

Avron

Zamansky

Logical Foundations of Computer Science

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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.

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“…In the following two subsections, we briefly reproduce the relevant definitions from [16] of canonical systems with (n, k)-ary quantifiers and of the semantic framework of Nmatrices.…”

Section: The Set Of L-formulas (Closed L-formulas) Denote By T Rm L mentioning

confidence: 99%

“…We use a simplified representation language from [16] for a schematic representation of canonical rules. …”

Section: Canonical Systems With (N K)-ary Quantifiersmentioning

confidence: 99%

“…Proposition 9 (Decidability of coherence) ( [16]) The coherence of a canonical calculus G is decidable.…”

Section: Definition 7 For Two Sets Of Clausesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Avron

Zamansky

Logical Foundations of Computer Science

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Modular Cut-Elimination: Finding Proofs or Counterexamples

Ciabattoni

Terui

2006

Logic for Programming, Artificial Intelligence, and Reasoning

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Non-Deterministic Semantics for Logical Systems

Avron

Zamansky

2010

Handbook of Philosophical Logic

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100

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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.

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Canonical Gentzen-Type Calculi with (n,k)-ary Quantifiers

Cited by 4 publications

References 13 publications

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

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

Modular Cut-Elimination: Finding Proofs or Counterexamples

Non-Deterministic Semantics for Logical Systems

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