
52This paper seeks to question the position of ZF as the dominant system of set theory, and in particular to examine whether there is any philosophical justification for the axiom of foundation. After some historical observations regarding Poincare and Russell, and the notions of circularity and hierarchy, the iterative conception of set is argued to be a semiconstructvist hybrid without philosophical coherence. ZF cannot be justified as necessary to avoid paradoxes, as axiomatizing a coherent …Read more

66Patterns in the Philosophy of MathematicsPhilosophical Quarterly 52 (207): 247255. 2002.Mathematics as a Science of Patterns . By Michael D. Resnik. (Oxford: Clarendon Press, 1997. Pp. xiii + 285. Price £35.00.) Naturalism in Mathematics . By Penelope Maddy. (Oxford: Clarendon Press, 1998. Pp. viii + 254. Price £32.50.) Realistic Rationalism . By Jerrold J. Katz. ( MIT Press, 1998. Pp. xxxiv + 226. Price £22.50.) The Principles of Mathematics Revisited . By Jaakko Hintikka. ( Cambridge UP, 1996. Pp. xii + 288. Price £40.00.)

45Was Quine right about subjunctive conditionals?The Monist 100 (2): 180193. 2017.Given his hostility to intensional locutions, it is not surprising that Quine was suspicious of the subjunctive conditional. Although he admitted its usefulness as a heuristic device, in order to introduce dispositional terms, he held that it had no place in a finished scientific theory. In this paper I argue in support of something like Quine’s position. Many contemporary philosophers are unreflectively realist about subjunctives, regarding them as having objective truth values. I contest this.…Read more

29The Beautiful Art of Mathematics†Philosophia Mathematica 26 (2): 234250. 2018.ABSTRACT Mathematicians frequently use aesthetic vocabulary and sometimes even describe themselves as engaged in producing art. Yet aestheticians, in so far as they have discussed this at all, have often downplayed the ascriptions of aesthetic properties as metaphorical. In this paper I argue firstly that the aesthetic talk should be taken literally, and secondly that it is at least reasonable to classify some mathematics as art.

55Defending a simple theory of conditionalsAmerican Philosophical Quarterly 52 (3): 253260. 2015.This paper extends the defense of a simple theory of indicative conditionals previously proposed by the author, in which the truth conditions are material, and Gricestyle assertability conditions are given to explain the paradoxes of material implication. The paper discusses various apparent counterexamples to the material account in which conditionals are not asserted, and so the original theory cannot be applied; it is argued that, nevertheless, the material theory can be defended.

76Moore’s Paradox, Introspection and Doxastic LogicThought: A Journal of Philosophy 4 (4): 215227. 2015.An analysis of Moore's paradox is given in doxastic logic. Logics arising from formalizations of various introspective principles are compared; one logic, K5c, emerges as privileged in the sense that it is the weakest to avoid Moorean belief. Moreover it has other attractive properties, one of which is that it can be justified solely in terms of avoiding false belief. Introspection is therefore revealed as less relevant to the Moorean problem than first appears

74The liar, the strengthened liar, and bivalenceErkenntnis 54 (2): 195203. 2001.A view often expressed is that to classify the liar sentence as neither true nor false is satisfactory for the simple liar but not for the strengthened liar. I argue that in fact it is equally unsatisfactory for both liars. I go on to discuss whether, nevertheless, Kripke''s theory of truth represents an advance on that of Tarski.

73An argument for finslerAczel set theoryMind 109 (434): 241253. 2000.Recent interest in nonwellfounded set theories has been concentrated on Aczel's antifoundation axiom AFA. I compare this axiom with some others considered by Aczel, and argue that another axiom, FAFA, is superior in that it gives the richest possible universe of sets consistent with respecting the spirit of extensionality. I illustrate how using FAFA instead of AFA might result in an improvement to Barwise and Etchemendy's treatment of the liar paradox.

28Naturalism in Mathematics (review)Philosophical Review 112 (3): 425427. 2003.Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problemrealismis examined and rejected in favor of another approachnaturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will b…Read more

88Voting on voting systems, or the limits of democracyAnalysis 71 (4): 641642. 2011.It is natural to think that a society can be organized in a way consistent with the overarching principle that all decisions should be democratic. A regress is constructed to demonstrate that this is, in fact, impossible.

138Paradox without basic law V: A problem with frege’s ontologyAnalysis 62 (4): 327330. 2002.No abstract available.

112Conditionals are material: the positive argumentsSynthese 190 (15): 31613174. 2013.A number of papers have argued in favour of the material account of indicative conditionals, but typically they either concentrate on defending the account from the charge that it has counterintuitive consequences, or else focus on some particular positive argument in favour of the theory. In this paper, I survey the various positive arguments that can be given, presenting simple versions where possible and showing the connections between them. I conclude with some methodological considerations
Glasgow, Scotland, United Kingdom of Great Britain and Northern Ireland
Areas of Specialization
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Social Choice Theory 