Category Archives: Research

Infinite Types and the Principle of Union

Forthcoming in Nicolai and Stern, eds., Modes of Truth. The Unified Approach to Truth, Modalities, and Paradox 

Abstract: In addition to objects (i.e. entities of type 0) should we also believe in entities of other types? Are there also type 1 entities (e.g. first-level Fregean concepts, pluralities), and type 2 entities (e.g. second-level Fregean concepts, superpluralities), and so on? Øystein Linnebo and Agustín Rayo argue that ‘plausible’ assumptions lead to a surprising conclusion for those who accept that first-order quantifiers may range over absolutely every object: absolutists of this kind should countenance proper-class-many types. This chapter takes up their argument for this thesis—Infinite Types—and argues that one of its assumptions is rather less plausible than Linnebo and Rayo suggest. The problematic assumption is the Principle of Union which states that one should countenance any language which ‘pools together’ the expressive resources drawn from any set of languages already deemed legitimate.

Preprint [.pdf, new tab]

Hybrid relativism and revenge

 
Workshop: Semantic Paradox, Context, and Generality
June 26-28, 2019, University of Salzburg

Abstract: Absolutism about quantifiers is the view that quantifiers sometimes range over an absolutely comprehensive domain. Relativists about quantifiers often oppose this view on the grounds that concepts such as set are ‘indefinitely extensible. To assuage doubts about the coherence of this view, relativism and the argument from indefinite extensibility may be regimented with the help of ‘modalized quantifiers’. But what is the status of these putatively relativist-friendly generalizing devices? Some relativists may be tempted to combine their quantifier-relativism with a species of absolutism about modalized quantifiers. I argue that this hybrid view faces a revenge problem.

See also: Everything, More or Less, §7.5

Slides [.pdf, new tab]

Everything, more or less

Almost no systematic theorizing is generality-free. Scientists test general hypotheses; set theorists prove theorems about every set; metaphysicians espouse theses about all things regardless of their kind. But how general can we be and do we ever succeed in theorizing about absolutely everything? Not according to generality relativism.

In its most promising form, this kind of relativism maintains that what ‘everything’ and other quantifiers encompass is always open to expansion: no matter how broadly we may generalize, a more inclusive ‘everything’ is always available. The importance of the issue comes out, in part, in relation to the foundations of mathematics. Generality relativism opens the way to avoid Russell’s paradox without imposing ad hoc limitations on which pluralities of items may be encoded as a set. On the other hand, generality relativism faces numerous challenges: What are we to make of seemingly absolutely general theories? What prevents our achieving absolute generality simply by using ‘everything’ unrestrictedly? How are we to characterize relativism without making use of exactly the kind of generality this view foreswears?

This book offers a sustained defence of generality relativism that seeks to answer these challenges. Along the way, the contemporary absolute generality debate is traced through diverse issues in metaphysics, logic, and the philosophy of language; some of the key works that lie behind the debate are reassessed; an accessible introduction is given to the relevant mathematics; and a relativist-friendly motivation for Zermelo-Fraenkel set theory is developed.

Chapter 1 [pre-print]

This is a draft of a chapter/article that has been accepted for publication by Oxford University Press in the forthcoming book Everything, More or Less by James Studd due for publication in 2019.

Update: the book has now been published! OUP website

 

Generality, Extensibility, and Paradox

Proceedings of the Aristotelian Society (2017) 117 (1): 81–101 https://doi.org/10.1093/arisoc/aox003

Delivered to the Aristotelian Society, 28th November 2016

Slides [new tab, .pdf]

Abstract: Absolutism is the view that quantifiers like ‘everything’ sometimes range over an absolutely comprehensive domain. The debate between absolutists and the relativists opposing them comes down, in significant part, to a trade off between generality and collectability. But to reach this conclusion we must dispense with some long-standing concerns over the coherence of the argument from the paradoxes in favour of relativism and a heterodox absolutist response that seeks to reconcile the indefinite extensibility of set with the availability of an absolutely comprehensive domain.

 

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.