Operational semantics for positive R
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} [``The semantics of entailment'', in: Truth, Syntax and Modality, Stud. Logic Found. Math. 68, 199-243 (1973; Zbl 0317.02017)]. The paper gives some discussion of the interpretation and plausibility of the given semantics, as well as results concerning the addition of sentential constants, fusion (along with a very interesting discussion) and quantifiers. The author notes that changing Pseudo-Idempotence to full Idempotence, making \(<S,\cdot,1>\) a semilattice also, yields a semantics for \(R+\) Mingle.
semantics for the positive fragment of the relevant logic R
scientific article; zbMATH DE number 4085619
scientific article