2004: Volume 2
Propositional Identity and Logical Necessity
David B. Martens
10 pages. Published March 12, 2004
In two early papers, Max Cresswell constructed two formal logics of propositional identity, PCR and FCR, which he observed to be respectively deductively equivalent to modal logics S4 and S5. Cresswell argued informally that these equivalences respectively “give … evidence” for the correctness of S4 and S5 as logics of broadly logical necessity. In this paper, I describe weaker propositional identity logics than PCR that accommodate core intuitions about identity and I argue that Cresswell’s informal arguments do not firmly and without epistemic circularity justify accepting S4 or S5. I also describe how to formulate standard modal logics (K, S2, and their extensions) with strict equivalence as the only modal primitive.
Possibility Semantics for Intuitionistic Logic
M. J. Cresswell
19 pages. Published April 30, 2004
The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.
The Classical and Maximin Versions of the Two-Envelope Paradox
14 pages. Published August 2, 2004
The Two-Envelope Paradox is classically presented as a problem in decision theory that turns on the use of probabilities in calculating expected utilities. I formulate a Maximin Version of the paradox, one that is decision-theoretic but omits considerations of probability. I investigate the source of the error in this new argument, and apply the insights thereby gained to the analysis of the classical version.
A Poor Concept Script
12 pages. Published August 2, 2004
The formal structure of Frege’s ‘concept script’ has been widely adopted in logic text books since his time, even though its rather elaborate symbols have been abandoned for more convenient ones. But there are major difficulties with its formalisation of pronouns, predicates, and propositions, which infect the whole of the tradition which has followed Frege. It is shown first in this paper that these difficulties are what has led to many of the most notable paradoxes associated with this tradition; the paper then goes on to indicate the lines on which formal logic—and also the lambda calculus and set theory—needs to be restructured, to remove the difficulties.
Throughout the study of what have come to be known as first-, second-, and higher-order languages, what has been primarily overlooked is that these languages are abstractions. Many well known paradoxes, we shall see, arose because of the elementary level of simplification which has been involved in the abstract languages studied. Straightforward resolutions of the paradoxes immediately appear merely through attention to languages of greater sophistication, notably natural language, of course. The basic problem has been exclusive attention to a theory in place of what it is a theory of, leading to a focus on mathematical manipulation, which ‘brackets off’ any natural language reading.
14 pages. Published August 2, 2004
Despite the wide acceptance of standard modal logic, there has always been a temptation to think that ordinary modal discourse may be correctly analyzed and adequately represented in terms of predicates rather than in terms of operators. The aim of the formal model outlined in this paper is to capture what I take to be the only plausible sense in which ‘possible’ and ‘necessary’ can be treated as predicates. The model is built by enriching the language of standard modal logic with a quantificational apparatus that is “substitutional” rather than “objectual”, and by obtaining from the language so enriched another language in which constants for such predicates apply to singular terms that stand for propositions.
Limiting Cases for Spectrum Closure Results
21 pages. Published October 26, 2004
The spectrum of a first-order sentence is the set of cardinalities of its finite models. Given a spectrum S and a function f, it is not always clear whether or not the image of S under f is also a spectrum. In this paper, we consider questions of this form for functions that increase very quickly and for functions that increase very slowly. Roughly speaking, we prove that the class of all spectra is closed under functions that increase arbitrarily quickly, but it is not closed under some natural slowly increasing functions.
Copyright © 2004, Philosophy Department, University of Melbourne.
Individual papers are copyright their authors.