Operational semantics for positive R
type: Item
instance of: scholarly article
title: Operational semantics for positive R
zbMATH Open document ID: 0664.03017
author: I. L. Humberstone
published in: Notre Dame Journal of Formal Logic
publication date: 1988-01-01T00:00:00Z
review text: This paper presents a ``semilattice'' semantics for the positive fragment of the relevant logic R, i.e. \(R+\). Early investigations of the semantics for relevant logics [e.g. \textit{A. Urquhart}, J. Symb. Logic 37, 159-169 (1972; Zbl 0245.02028)] attempted to fit a straightforward semilattice semantics to \(R+\), but soon discovered that there was no such fit. Indeed, for the most part all hope of discovering a variation of that semantics was abandoned. (Although recent work such as that by the reviewer and \textit{A. Urquhart} [Z. Math. Logik Grundlagen Math. 33, 433- 439 (1987; Zbl 0611.03005)] revived some hope for doing so for contractionless relevant logics such as \(RW+.)\) The problem had always been to accomodate disjunction (not to mention negation, of course). Humberstone shows that with an appropriate enrichment of the structure of the underlying semilattice frames (model structures) it is possible to alter the truth conditions for disjunction in such a way as to yield the desired semantic completeness. In particular, a frame is to be an algebra \(<S,\cdot,+,1,0>\) of similarity type (2,2,0,0) in which \(<S,+,0>\) is a semilattice with zero (0), \(<S,\cdot,1>\) is a commutative semigroup with unit (1), and the following conditions are satisfied: (Pseudo-Idempotence) \(x(x+1)=x^ a+x;\) (Decomposition) If \(x\leq u+v,\) then there are u'\(\leq u\), v'\(\leq v\) with \(u'+v'=x;\) (Zeroing) \(0x=0\); (Ring Distribution) \(x(y+z)=xy+xz.\) Valuation functions (on atoms) must satisfy the conditions: \((C+)\) \(V(A,x+y)=T\) iff \(V(A,x)=V(A,y)=T\); (C0) \(V(A,0)=T\). They are extended to full interpretations via the usual recursive clauses for \& and \(\to\) along with the unusual: (*) \(I(A\vee B,x)=T\) iff there are y, z such that \(x=y+z\) and \(I(A,y)=I(B,z)=T.\) The completeness proof is then a straightforward adaptation of the usual, as by \textit{K. Fine} [J. Philos. Logic 3, 347-372 (1974; Zbl 0296.02013)] or \textit{R. Routley} and \textit{R. K. Meyer}…
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03A05
zbMATH DE Number: 4085619
zbMATH Keywords: relevance logics
zbMATH Keywords: R Mingle
zbMATH Keywords: sentential constants
zbMATH Keywords: quantified relevant logic
zbMATH Keywords: semantics for the positive fragment of the relevant logic R
zbMATH Keywords: semilattice semantics
zbMATH Keywords: disjunction
zbMATH Keywords: semantic completeness
zbMATH Keywords: fusion
reviewed by: Steve Giambrone
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093637771
OpenAlex ID: W2126017509
DOI: 10.1305/NDJFL/1093637771
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description: scientific article; zbMATH DE number 4085619