Proof Theory

Patterns of resemblance of order 2

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.

Paraconsistency, paracompleteness, Gentzen systems, and trivalent semantics

The logics we present in this tutorial, Godel logics, can be characterized in a rough-and-ready way as follows: The language is standard, defined at different levels: propositional, quantified-propositional, first-order. The logics are many-valued, and the sets of truth values considered are (closed) subsets of [0, 1] which contain both 0 and 1. 1 is the ‘designated value,’ i.e., a formula is valid if it receives the value 1 in every interpretation. The truth functions of conjunction and disjunction are minimum and maximum, respectively, and in the first-order case quantifiers are defined by infimum and supremum over subsets of the set of truth values.

Gödel logics: a survey

This article investigates the proof theory of the Quantified Argument Calculus (Quarc) as developed and systematically studied by Hanoch Ben-Yami [3, 4]. Ben-Yami makes use of natural deduction (Suppes-Lemmon style), we, however, have chosen a sequent calculus presentation, which allows for the proofs of a multitude of significant meta-theoretic results with minor modifications to the Gentzen’s original framework, i.e., LK. As will be made clear in course of the article LK-Quarc will enjoy cut elimination and its corollaries (including subformula property and thus consistency).

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Proof-Theoretic Analysis of the Quantified Argument Calculus

We present a general method for providing Kripke semantics for the family of fully-structural multiple-conclusion propositional sequent systems. In particular, many well-known Kripke semantics for a variety of logics are easily obtained as special cases. This semantics is then used to obtain semantic characterizations of analytic sequent systems of this type, as well as of those admitting cut-admissibility. These characterizations serve as a uniform basis for semantic proofs of analyticity and cut-admissibility in such systems.

/pdf/kripke-semantics-for-basic-sequent-systems-1xjnvuhlaz.pdf

Proof Theory

Citations

Patterns of resemblance of order 2

Paraconsistency, paracompleteness, Gentzen systems, and trivalent semantics

Gödel logics: a survey

Proof-Theoretic Analysis of the Quantified Argument Calculus

Kripke semantics for basic sequent systems

Related Papers (5)

Untersuchungen über das logische Schließen. II

Introduction to Metamathematics

The collected papers of Gerhard Gentzen

Basic proof theory

Linear logic