Category Archives: Research

Abstraction Reconceived

Br J Philos Sci (2016) 67 (2): 579-615. doi: 10.1093/bjps/axu035

Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist’s account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma.

Semantic Pessimism and Absolute Generality

Book chapter in Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language (ed.  Alessandro Torza), Synthese Library 2015, pp 339-366

Abstract: Semantic pessimism has sometimes been used to argue in favour of absolutism about quantifiers, the view, to a first approximation, that quantifiers in natural or artificial languages sometimes range over a domain comprising absolutely everything.
Williamson argues that, by her lights, the relativist who opposes this view cannot state the semantics she wishes to attach to quantifiers in a suitable metalanguage. This chapter argues that this claim is sensitive to both the version of relativism in question and the sort of semantic theory in play. Restrictionist and expansionist variants of relativism should be distinguished. While restrictionists face the difficulties Williamson presses in stating the truth-conditions she wishes to ascribe to quantified sentences in the familiar quasi-homophonic style associated with Tarski and Davidson, the expansionist does not. In fact, not only does the expansionist fare no worse than the absolutist with respect to semantic optimism, for certain styles of semantic theory, she fares better. In the case of the extensional semantics of so called ‘generalised quantifiers’, famously applied to natural language by Barwise and Cooper, it is argued that expansionists enjoy optimism and absolutists face a significant measure of pessimism.

The Iterative Conception of Set

J Philos Logic (2013) 42: 697–725. doi:10.1007/s10992-012-9245-3

The final publication is available at www.springerlink.com

Abstract: The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously.
Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A modal stage theory, 𝖬𝖲𝖳, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that 𝖬𝖲𝖳 interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory.