Famously, Michael Dummett argues that considerations concerning the role of language in communication lead to the rejection of classical logic in favor of intuitionistic logic. Potentially, this results in massive revisions of established mathematics. Recently, Neil Tennant (“The law of excluded middle is synthetic a priori, if valid”, Philosophical Topics 24 (1996), 205-229) suggested that a Dummettian anti-realist can accept the law of excluded middle as a synthetic, a priori principle grounded on a metaphysical principle of determinacy. This article shows (...) that the for the anti-realist, the law of excluded middle entails that humans have wildly implausible abilities. The proposed synthesis between anti-realism and classical mathematics thus fails. (shrink)

The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.

European Journal of Philosophy, Volume 30, Issue 3, Page 1010-1027, September 2022.

We explore the problems that confront any attempt to explain or explicate exactly what a primitive logical rule of inferenceis, orconsists in. We arrive at a proposed solution that places a surprisingly heavy load on the prospect of being able to understand and deal with specifications of rules that are essentiallyself-referring. That is, any rule$\rho $is to be understood via a specification that involves, embedded within it, reference to rule$\rho $itself. Just how we arrive at this position is explained by (...) reference to familiar rules as well as less familiar ones with unusual features. An inquiry of this kind is surprisingly absent from the foundations of inferentialism—the view that meanings of expressions (especially logical ones) are to be characterized by the rules of inference that govern them. (shrink)

In “The Runabout Inference Ticket” AN Prior (1960) examines the idea that logical connectives can be given a meaning solely in virtue of the stipulation of a set of rules governing them, and thus that logical truth/consequence.

The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the notion of “same meaning” (...) is itself context-sensitive, depending on the purposes of a given conversation. (shrink)

I consider the claim that analytic statements are "true in virtue meaning", giving the claim a ground-theoretic interpretation.

We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which postulates that (...) various logics are constitutive for thought within particular practices, but none are constitutive for thought as such. (shrink)

Realists claim that, amongst the statements weunderstand, there are some which are true, yetwhose truth potentially transcends the limits ofwhat we can recognize. Dummett and othershave argued that this realist thesis is incompatiblewith an account of understanding in termsof recognitional capacities. But careful analysis revealsthat this contention is mistaken. Thealleged incompatibility presupposes the truth of ametaphysical principle which cannot bevindicated on the basis of an account of understanding.Realists have independent reason toreject this metaphysical principle, as it leads to a collapseof (...) more modest forms of realisminto more contentious ones. According to moderate realism,we neither know that there areactually recognition-transcendent truthsnor do we know taht there are none. Pending apositive case for the metaphysical principle, thisagnosticism is sufficient to warrant therealists' key thesis, viz. that there arepotentially recognition-transcendent truths. Thusrealism, in its more moderate guise, turns outto be the default position. (shrink)

We are logical pluralists who hold that the right logic is dependent on the domain of investigation; different logics for different mathematical theories. The purpose of this article is to explore the ramifications for our pluralism concerning normativity. Is there any normative role for logic, once we give up its universality? We discuss Florian Steingerger’s “Frege and Carnap on the Normativity of Logic” as a source for possible types of normativity, and then turn to our own proposal, which postulates that (...) various logics are constitutive for thought within particular practices, but none are constitutive for thought as such. (shrink)

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)

On the Dummettian understanding, anti-realism regarding a particular discourse amounts to (or at the very least, involves) a refusal to accept the determinacy of the subject matter of that discourse and a corresponding refusal to assert at least some instances of excluded middle (which can be understood as expressing this determinacy of subject matter). In short: one is an anti-realist about a discourse if and only if one accepts intuitionistic logic as correct for that discourse. On careful examination, the strongest (...) Dummettian arguments for anti-realism of this sort fail to secure intuitionistic logic as the single, correct logic for anti-realist discourses. Instead, antirealists are placed in a situation where they fail to be justified in asserting monism (that intuitionistic logic is the unique correct logic). Thus, antirealists seem forced either to accept pluralism (i.e. one or more intermediate logic is at least as `correct’ as intuitionistic logic–an option I take to be unattractive from the anti-realist perspective), or they must be anti-realists about the realism/anti-realism debate (and, in particular, must refuse to assert the instance of excluded middle equivalent to logical monism or logical pluralism). (shrink)

forthcoming in S. Lindström, E. Palmgren, K. Segerberg, and V. Stoltenberg-Hansen (eds) Logicism, Intuitionism, and Formalism — What has Become of Them?, Synthese Library, Springer. Pdf file.

I argue there are some objects which do not respect the Law of the Excluded Middle, i.e., which are such that, for some property F, the disjunction Fo v ~Fo fails to be true. I call such objects “odd objects” and present three examples—fictional objects, nonsort objects, and quantum objects. I argue that each of these objects is best understood as violating LEM. I, then, discuss Jessica Wilson’s account of metaphysical indeterminacy. I show how the indeterminacy which arises with odd (...) objects can be accounted for on Wilson’s account. I, then, argue that my Wilson-inspired, but non-LEM-respecting, account of metaphysical indeterminacy is superior to Wilson’s in terms of costs and benefits. (shrink)