Survey of generalizations of Urquhart semantics (Q1104918)

This paper reviews the extant semantic approaches and results on contractionless logics closely related to the relevant system \(RW\) (the relevant logic \(R\) without the contraction axiom: \((A\to.A\to B)\to.A\to B)\) and its positive fragment \(RW+\). The author declares these various approaches to be compromises between the intuitiveness of the original Urquhart semantics and the difficulties of obtaining completeness proofs. The approaches considered include \textit{A. Q. Abraham} [``Completeness of quantified classical relevant logic'', Stud. Logica (forthcoming)], \textit{K. Fine} [J. Philos. Logic 3, 347-372 (1974; Zbl 0296.02013)], \textit{R. Routley} and \textit{R. K. Meyer} [J. Philos. Logic 1, 192-208 (1972; Zbl 0317.02019)], \textit{R. K. Meyer} and \textit{R. Routley} [Stud. Logica 33, 183-194 (1974; Zbl 0316.02030)], \textit{H. Ono} and \textit{Y. Komori} [J. Symb. Logic 50, 169-201 (1985; Zbl 0583.03018)] and \textit{A. Urquhart} [J. Symb. Logic. 37, 159-169 (1972; Zbl 0245.02028)]. For other related results not included in the paper, see \textit{G. Charlwood} [J. Symb. Logic 46, 233-239 (1981; Zbl 0479.03012)], \textit{S. Giambrone}, \textit{R. K. Meyer} and \textit{A. Urquhart} [J. Symb. Logic 52, 526-529 (1987; Zbl 0624.03015)], \textit{S. Giambrone} and \textit{A. Urquhart} [Z. Math. Logik Grundlagen Math. 33, 433-439 (1987; Zbl 0611.03005)], \textit{R. K. Meyer}, \textit{S. Giambrone}, \textit{A. Urquhart} and \textit{E. P. Martin} [ibid. 34, 301-304 (1988; Zbl 0656.03011)] and \textit{J. Slaney} [Notre Dame J. Formal Logic 28, 395-407 (1987; Zbl 0637.03018)]. The paper also presents a new Gentzenization of RW without distribution using signed formulae, e.g., `TA', `FB', etc., as in the book of \textit{I. D. Zaslavskij} [Constructive symmetric logic (Russian) (Erevan, 1978)]. This approach is extended to a Gentzen system for all of RW by \textit{R. Brady} [``The Gentzenization and decidability of \(RW\)'', J. Philos. Logic 19, No.1, 35-73 (1990; Zbl 0697.03006)].