A comparative study of modal propositional semantics.
Martin Gerson · Summit (Simon Fraser University) · 1974
IWe define six semantics (systens of structures) for the modal language (the language obtained by adding to the language of classical propositional calculus the single unary connective o ) : the boolean semantics (derived from the algebraic structures of McKinsey aztd ~arski [ 9 ] ) , the neighbourhood (Scott-Montague) semantics, the relational (Kripke) semantics, the first-order semantics (Makinson's[lO] generalized "relational models"), and finally the connected and superconnected semantics.The dissertation studies the strengths of these semantics in various ways: first, with respect to containment of one within another and then with respect to width and depth (measures of the "span" and "density" of the set of logics which are complete with respect to the semantics in question).Previously known results in these areas are noted and some other results are proved.In particular, we note that the width of the relational semantics is that of normal logics and that any relational frame is in fact a neighbourhood frame, so that the "ground" common to both semantics is that of normal logics.Among the major results of the dissertation is the presentation of two normal logics, one between T and S4 and the other an extension of S4, which are incomplete with respect to the neighbourhood semantics, thus showing that, unlike the boolean and first-order semantics, (iii) Among all the people and institutions who have helped me in my work on this thesis, four particularly must be mentioned.The first is the National Research Council of Canada, whose assistance through a Postgraduate Scholarship freed me to spend more time on research than otherwise would have been possible.The second is Steven Thomason, whose interest and work in Modal Logic inspired mine.The third is Ronald Harrop, whose help at all stages of my work, from the initial development of my research to the proof reading of my amateur typing, has been constant and invaluable.His willingness to plunge with me into areas quite new to him and his continued ability to make helpful suggestions have been admired and appreciated.To him I am particularly grateful.The fourth is Carole, who has managed to stagger her periods of insanity with mine so well that we have both been able to stay on our feet.