•  3
    Mathematical Internal Realism
    In Sanjit Chakraborty & James Ferguson Conant (eds.), Engaging Putnam, De Gruyter. pp. 157-182. 2022.
  •  128
    Against Cumulative Type Theory
    Review of Symbolic Logic 1-43. forthcoming.
    Standard Type Theory, ${\textrm {STT}}$, tells us that $b^n$ is well-formed iff $n=m+1$. However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$, which has more relaxed type-restrictions: according to $\textrm {CTT}$, $b^\beta $ is well-formed iff $\beta>\alpha $. In this paper, we set ourselves against $\textrm {CTT}$. We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then a…Read more
  •  24
    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
  •  21
    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
  •  19
    On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural e…Read more
  •  43
    forall x: Dortmund is an adaptation and German translation of forall x: Calgary. As such, it is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity, the syntax of truth-functional (propositional) logic and truth-table semantics, the syntax of first-order (predicate) logic with identity and first-order interpretations, formalizing German in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with so…Read more
  •  539
    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
    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 05 maio de 2022. Comentários, críticas, correções e sugestões são muito bem-vindos.
  •  23
    Hilary Putnam on Logic and Mathematics, edited by HellmanGeoffrey and CookRoy T. Cham: Springer, 2018. Pp. x + 274.
  •  151
    Reading Putnam, edited by Maria Baghramian (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
  •  202
    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.
  •  128
    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
  •  362
    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), translating (formalizing) 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 truth-functional…Read more
  •  80
    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 §1, I unpack and vindicate his thinking.After discussing ‘my pains’, Wittgen…Read more
  •  64
    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
  •  18
    Philosophy and Model Theory
    with Sean Walsh
    Oxford University Press. 2018.
    Model theory is an important area of mathematical logic which has deep philosophical roots, many philosophical applications, and great philosophical interest in itself. The aim of this book is to introduce, organise, survey, and develop these connections between philosophy and model theory, for the benefit of philosophers and logicians alike.
  •  37
    In The American Pragmatists (2013), Cheryl Misak casts Peirce and Lewis as the heroes of American pragmatism. She establishes an impressive continuity between pragmatism and both logical empiricism and contemporary analytic philosophy. However, in casting James and Dewey as the villains of American pragmatism, she underplays the pragmatists' interest in action.
  •  215
    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
  •  193
    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
  •  194
    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
  •  157
    Brains in vats and model theory
    In Sanford Goldberg (ed.), The Brain in a Vat, Cambridge University Press. forthcoming.
    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
  •  316
    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
  •  70
    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
  •  121
    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
  •  196
    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
  •  127
    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
  •  86
    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
  •  26
    Hyperloops do not threaten the notion of an effective procedure
    Lecture Notes in Computer Science 5635 68-78. 2009.
    This paper develops my (BJPS 2009) criticisms of the philosophical significance of a certain sort of infinitary computational process, a hyperloop. I start by considering whether hyperloops suggest that "effectively computable" is vague (in some sense). I then consider and criticise two arguments by Hogarth, who maintains that hyperloops undermine the very idea of effective computability. I conclude that hyperloops, on their own, cannot threaten the notion of an effective procedure.
  •  1320
    The Limits of Realism
    Oxford University Press UK. 2013.
    Tim Button explores the relationship between minds, words, and world. He argues that the two main strands of scepticism are deeply related and can be overcome, but that there is a limit to how much we can show. We must position ourselves somewhere between internal realism and external realism, and we cannot hope to say exactly where.
  •  170
    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’
  •  424
    Truth by Analysis: Games, Names, and Philosophy By Colin McGinn (review)
    Analysis 73 (3): 577-580. 2013.
    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.