TY - JOUR

T1 - Quasi-canonical systems and their semantics

AU - Avron, Arnon

N1 - Publisher Copyright:
© 2018, Springer Nature B.V.

PY - 2021/10

Y1 - 2021/10

N2 - A canonical (propositional) Gentzen-type system is a system in which every rule has the subformula property, it introduces exactly one occurrence of a connective, and it imposes no restrictions on the contexts of its applications. A larger class of Gentzen-type systems which is also extensively in use is that of quasi-canonical systems. In such systems a special role is given to a unary connective ¬ of the language (usually, but not necessarily, interpreted as negation). Accordingly, each application of a logical rule in such systems introduces either a formula of the form ⋄ (ψ1, … , ψn) , or of the form ¬ ⋄ (ψ1, … , ψn) , and all the active formulas of its premises belong to the set { ψ1, … , ψn, ¬ ψ1, … , ¬ ψn}. In this paper we investigate the whole class of quasi-canonical systems. We provide a constructive coherence criterion for such systems, and show that a quasi-canonical system is coherent iff it is analytic iff it has a characteristic non-deterministic matrix (Nmatrix) which is based on some subset of the set of the four truth-values which are used in the famous Dunn–Belnap four-valued (deterministic) matrix for information processing. In addition, we determine when a quasi-canonical system admits (strong) cut-elimination.

AB - A canonical (propositional) Gentzen-type system is a system in which every rule has the subformula property, it introduces exactly one occurrence of a connective, and it imposes no restrictions on the contexts of its applications. A larger class of Gentzen-type systems which is also extensively in use is that of quasi-canonical systems. In such systems a special role is given to a unary connective ¬ of the language (usually, but not necessarily, interpreted as negation). Accordingly, each application of a logical rule in such systems introduces either a formula of the form ⋄ (ψ1, … , ψn) , or of the form ¬ ⋄ (ψ1, … , ψn) , and all the active formulas of its premises belong to the set { ψ1, … , ψn, ¬ ψ1, … , ¬ ψn}. In this paper we investigate the whole class of quasi-canonical systems. We provide a constructive coherence criterion for such systems, and show that a quasi-canonical system is coherent iff it is analytic iff it has a characteristic non-deterministic matrix (Nmatrix) which is based on some subset of the set of the four truth-values which are used in the famous Dunn–Belnap four-valued (deterministic) matrix for information processing. In addition, we determine when a quasi-canonical system admits (strong) cut-elimination.

KW - Coherence

KW - Gentzen-type systems

KW - Negation

KW - Non-deterministic matrices

KW - Quasi-canonical systems

UR - http://www.scopus.com/inward/record.url?scp=85058066246&partnerID=8YFLogxK

U2 - 10.1007/s11229-018-02045-0

DO - 10.1007/s11229-018-02045-0

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AN - SCOPUS:85058066246

SN - 0039-7857

VL - 198

SP - 5353

EP - 5371

JO - Synthese

JF - Synthese

ER -