•  1975
    The Limits of Realism
    Oxford University Press UK. 2013.
    Tim Button explores the relationship between words and world; between semantics and scepticism. A certain kind of philosopher – the external realist – worries that appearances might be radically deceptive. For example, she allows that we might all be brains in vats, stimulated by an infernal machine. But anyone who entertains the possibility of radical deception must also entertain a further worry: that all of our thoughts are totally contentless. That worry is just incoherent. We cannot, then, …Read more
  •  1744
    Para Todxs: Natal - uma introdução à lógica formal
    with P. D. Magnus, Robert Loftis, Robert Trueman, Aaron Thomas Bolduc, Richard Zach, Daniel Durante, Maria da Paz Nunes de Medeiros, Ricardo Gentil de Araújo Pereira, Tiago de Oliveira Magalhães, Hudson Benevides, Jordão Cardoso, Paulo Benício de Andrade Guimarães, and Valdeniz da Silva Cruz Junior
    PPGFIL-UFRN. 2022.
    Livro-texto de introdução à lógica, com (mais do que) pitadas de filosofia da lógica, produzido como uma versão revista e ampliada do livro Forallx: Calgary. Trata-se da versão de 13 de outubro de 2022. Comentários, críticas, correções e sugestões são muito bem-vindos.
  •  1030
    forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), symbolizing English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as modal logic, soundness, and fu…Read more
  •  512
    There's no time like the present
    Analysis 66 (2). 2006.
    No-futurists ('growing block theorists') hold that that the past and the present are real, but that the future is not. The present moment is therefore privileged: it is the last moment of time. Craig Bourne (2002) and David Braddon-Mitchell (2004) have argued that this position is unmotivated, since the privilege of presentness comes apart from the indexicality of 'this moment'. I respond that no-futurists should treat 'x is real-as-of y' as a nonsymmetric relation. Then different moments are re…Read more
  •  476
    In Truth by Analysis (2012), Colin McGinn aims to breathe new life into conceptual analysis. Sadly, he fails to defend conceptual analysis, either in principle or by example.
  •  420
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
  •  407
    Deflationary metaphysics and ordinary language
    Synthese 197 (1): 33-57. 2020.
    Amie Thomasson and Eli Hirsch have both attempted to deflate metaphysics, by combining Carnapian ideas with an appeal to ordinary language. My main aim in this paper is to critique such deflationary appeals to ordinary language. Focussing on Thomasson, I draw two very general conclusions. First: ordinary language is a wildly complicated phenomenon. Its implicit ontological commitments can only be tackled by invoking a context principle; but this will mean that ordinary language ontology is not a…Read more
  •  394
    Against Cumulative Type Theory
    Review of Symbolic Logic 15 (4): 907-49. 2022.
    Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, the type-restrictions i…Read more
  •  373
    Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001…Read more
  •  322
    SAD computers and two versions of the Church–Turing thesis
    British Journal for the Philosophy of Science 60 (4): 765-792. 2009.
    Recent work on hypercomputation has raised new objections against the Church–Turing Thesis. In this paper, I focus on the challenge posed by a particular kind of hypercomputer, namely, SAD computers. I first consider deterministic and probabilistic barriers to the physical possibility of SAD computation. These suggest several ways to defend a Physical version of the Church–Turing Thesis. I then argue against Hogarth's analogy between non-Turing computability and non-Euclidean geometry, showing t…Read more
  •  316
    Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a non-trivial automorphism, distinct indiscernible positions within the structure cannot be shown to be non-identical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include 'irreflexive two-place relations'. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positio…Read more
  •  315
    Dadaism: Restrictivism as Militant Quietism
    Proceedings of the Aristotelian Society 110 (3pt3): 387-398. 2010.
    Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some philosophers have claimed that we must be able to do so, since the doctrine that we cannot is self-stultifying. But this treats restrictivism as a positive doctrine. Restrictivism is much better viewed as a kind of militant quietism, which I call dadaism. Dadaists advance a hostile challenge, with the aim of silencing everyone who holds a positive position about ‘absolute generality’
  •  298
    The Philosophical Significance of Tennenbaum’s Theorem
    with P. Smith
    Philosophia Mathematica 20 (1): 114-121. 2012.
    Tennenbaum's Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, it offers us a way of responding to model-theoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standard model, then Tennenbaum's Theorem does not help. We show this by examining a parallel argument, from …Read more
  •  297
    Tallant (2007) has challenged my recent defence of no-futurism (Button 2006), but he does not discuss the key to that defence: that no-futurism's primitive relation 'x is real-as-of y' is not symmetric. I therefore answer Tallant's challenge in the same way as I originally defended no-futurism. I also clarify no-futurism by rejecting a common mis-characterisation of the growing-block theorist. By supplying a semantics for no-futurists, I demonstrate that no-futurism faces no sceptical challenges…Read more
  •  293
    A fictionalist theory of universals
    In Peter Fritz & Nicholas K. Jones (eds.), Higher-Order Metaphysics, Oxford University Press. forthcoming.
    Universals are putative objects like wisdom, morality, redness, etc. Although we believe in properties (which, we argue, are not a kind of object), we do not believe in universals. However, a number of ordinary, natural language constructions seem to commit us to their existence. In this paper, we provide a fictionalist theory of universals, which allows us to speak as if universals existed, whilst denying that any really do.
  •  285
    Brains in vats and model theory
    In Sanford Goldberg (ed.), The Brain in a Vat, Cambridge University Press. pp. 131-154. 2016.
    Hilary Putnam’s BIV argument first occurred to him when ‘thinking about a theorem in modern logic, the “Skolem–Löwenheim Theorem”’ (Putnam 1981: 7). One of my aims in this paper is to explore the connection between the argument and the Theorem. But I also want to draw some further connections. In particular, I think that Putnam’s BIV argument provides us with an impressively versatile template for dealing with sceptical challenges. Indeed, this template allows us to unify some of Putnam’s most e…Read more
  •  279
    Mathematical Internal Realism
    In Sanjit Chakraborty & James Ferguson Conant (eds.), Engaging Putnam, De Gruyter. pp. 157-182. 2022.
    In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is und…Read more
  •  260
    We argue against the halving response to Sleeping Beauty. First, we outline an appealing constraint on probability assignments: the Principle of Irrelevant Information. Roughly, this says: if you don’t know whether C, but you would assign probability p to X regardless of whether C or not-C, then you should assign p to X. This Principle is deeply plausible, but we show that it contradicts halving. Second, we show that halving either violates solid statistical reasoning or draws absurd distinction…Read more
  •  257
    In the early-to-mid 1930s, Wittgenstein investigated solipsism via the philosophy of language. In this paper, I want to reopen Wittgenstein's ‘grammatical’ examination of solipsism.Wittgenstein begins by considering the thesis that only I can feel my pains. Whilst this thesis may tempt us towards solipsism, Wittgenstein points out that this temptation rests on a grammatical confusion concerning the phrase ‘my pains’. In Section 1, I unpack and vindicate his thinking. After discussing ‘my pains’,…Read more
  •  250
    The Weight of Truth: Lessons for Minimalists from Russell's Gray's Elegy Argument
    Proceedings of the Aristotelian Society 114 (3pt3): 261-289. 2014.
    Minimalists, such as Paul Horwich, claim that the notions of truth, reference and satisfaction are exhausted by some very simple schemes. Unfortunately, there are subtle difficulties with treating these as schemes, in the ordinary sense. So instead, minimalists regard them as illustrating one-place functions, into which we can input propositions (when considering truth) or propositional constituents (when considering reference and satisfaction). However, Bertrand Russell's Gray's Elegy argument …Read more
  •  243
    Level theory, part 1: Axiomatizing the bare idea of a cumulative hierarchy of sets
    Bulletin of Symbolic Logic 27 (4): 436-460. 2021.
    The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, M…Read more
  •  238
    Spotty Scope and Our Relation to Fictions
    Noûs 46 (2): 243-58. 2012.
    Whatever the attractions of Tolkein's world, irrealists about fictions do not believe literally that Bilbo Baggins is a hobbit. Instead, irrealists believe that, according to The Lord of the Rings {Bilbo is a hobbit}. But when irrealists want to say something like “I am taller than Bilbo”, there is nowhere good for them to insert the operator “according to The Lord of the Rings”. This is an instance of the operator problem. In this paper, I outline and criticise Sainsbury's (2006) spotty scope a…Read more
  •  230
    Review of: Reading Putnam, by Maria Baghramian (ed.) (review)
    Mind 123 (490): 569-575. 2014.
    Reading Putnam consists largely of papers from the fantastic ‘Putnam @80’ conference (organised by Maria Baghramian in 2007) together with replies from Hilary Putnam. Given the diversity of Putnam’s work, the papers in this collection cover many different topics. This makes the collection difficulty to read but, ultimately, extremely rewarding. In this review, I focus on the contributions from Michael Devitt, Charles Parsons, Richard Boyd, Ned Block, Charles Travis and John McDowell, together wi…Read more
  •  223
    Prior’s Tonk is a famously horrible connective. It is defined by its inference rules. My aim in this article is to compare Tonk with some hitherto unnoticed nasty connectives, which are defined in semantic terms. I first use many-valued truth-tables for classical sentential logic to define a nasty connective, Knot. I then argue that we should refuse to add Knot to our language. And I show that this reverses the standard dialectic surrounding Tonk, and yields a novel solution to the problem of ma…Read more
  •  202
    Exclusion Problems and the Cardinality of Logical Space
    Journal of Philosophical Logic 46 (6): 611-623. 2017.
    Wittgenstein’s atomist picture, as embodied in his Tractatus, is initially very appealing. However, it faces the famous colour-exclusion problem. In this paper, I shall explain when the atomist picture can be defended in the face of that problem; and, in the light of this, why the atomist picture should be rejected. I outline the atomist picture in Section 1. In Section 2, I present a very simple necessary and sufficient condition for the tenability of the atomist picture. The condition is: logi…Read more
  •  198
    The iterative conception of function and the iterative conception of set
    In Carolin Antos, Neil Barton & Giorgio Venturi (eds.), The Palgrave Companion to the Philosophy of Set Theory, Palgrave. forthcoming.
    Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we first need to provide a full-…Read more
  •  183
    Grades of Discrimination: Indiscernibility, Symmetry, and Relativity
    Notre Dame Journal of Formal Logic 58 (4): 527-553. 2017.
    There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discr…Read more
  •  183
    Here is a template for introducing mathematical objects: “Objects are found in stages. For every stage S: (1) for any things found before S, you find at S the bland set whose members are exactly those things; (2) for anything, x, which was found before S, you find at S the result of tapping x with any magic wand (provided that the result is not itself a bland set); you find nothing else at S.” This Template has rich applications, it realizes John Conway’s (1976) Mathematicians’ Liberation Moveme…Read more
  •  181
    Symmetric relations, symmetric theories, and Pythagrapheanism
    Philosophy and Phenomenological Research. 2022.
    It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast, unlabelled, undirected graph.
  •  179
    Level theory, part 2: Axiomatizing the bare idea of a potential hierarchy
    Bulletin of Symbolic Logic 27 (4): 461-484. 2021.
    Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-…Read more