2018. . Joint with Sean Walsh, with a historical appendix by Wilfrid Hodges. Oxford University Press.

doi 10.1093/oso/9780198790396.001.0001

List of errata

Philosophy and Model Theory

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions

Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more.

The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas.

The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right.

2013. . Oxford University Press.

doi 10.1093/acprof:oso/9780199672172.001.0001

Sample chapters

The Limits of Realism

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, be external realists, who worry about the possibility of radical deception. Equally, however, we cannot be internal realists, who reject all possibility of deception. We must position ourselves somewhere between internal realism and external realism, but we cannot hope to say exactly where. We must be realists, for what that is worth, and realists within limits.

In establishing these claims, The Limits of Realism critically explores and develops several themes from Hilary Putnam’s work: the model-theoretic arguments; the connection between truth and justification; the brain-in-vat argument; semantic externalism; and conceptual relativity. The book establishes the continued significance of these topics for all philosophers interested in mind, logic, language, or the possibility of metaphysics.

Reviews by:

Lieven Decock, NDPR: published

Nicholas K Jones, Analysis: published | draft

Drew Khlentzos, AJP: published

Rory Madden, EJP: published | draft

J.T.M. Miller, Philosophy in Review: published

Joshua Thorpe, Argumenta: published

Jan Westerhoff, Mind: published

Nathan Wildman, Zeitschrift für philosophische Forschung: draft

2018. : Private pains, private languages, and two uses of ‘I’. Royal Institute of Philosophy Supplement 82, 205–29.

doi 10.1017/s1358246118000061


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’, Wittgenstein makes his now famous suggestion that the word ‘I’ has two distinct uses: a subject-use and an object-use. The purpose of Wittgenstein’s suggestion has, however, been widely misunderstood. I unpack it in §2, explaining how the subject-use connects with a phenomenological language, and so again tempts us into solipsism. In §§3–4, I consider various stages of Wittgenstein’s engagement with this kind of solipsism, culminating in a rejection of solipsism (and of subject-uses of ‘I’) via reflections on private languages.

2017. . Journal of Philosophical Logic 46.6, 611–23.

doi 10.1007/s10992-016-9412-zopen access

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 present a very simple necessary and sufficient condition for the tenability (in principle) of the atomist picture. The condition is: logical space is a power of two. This cardinality-condition supplies a cheap response to exclusion problems. (Moreover, this cheap response amounts to a distillation of a proposal due to Sarah Moss, 2012.) However, the cheapness of this response vindicates Wittgenstein’s ultimate rejection of the atomist picture. We have no guarantee that there are any solutions to a given exclusion problem but, if there are any, then there are far too many.

2017. : Indiscernibility, symmetry, and relativity. Notre Dame Journal of Formal Logic 58.4, 527–53.

doi 10.1215/00294527-2017-0007


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 discrimination have been studied in some detail. However, this paper also introduces grades of relativity, defined in terms of relativeness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas, and draws connections with definability theory.

2016. . Synthese.

doi 10.1007/s11229-016-1212-zopen access

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 trivial enterprise. Second: a wide variety of existence questions cannot be deflated using ordinary language, trivially or otherwise, for ordinary language often points in different directions simultaneously.

2016. : Nasty connectives on many-valued truth-tables for classical sentential logic. Analysis 76.1: 7–19.

doi 10.1093/analys/anv106


Prior’s Tonk is a famously horrible connective. It is defined by its inference rules. My aim in this paper 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 many-valued truth-tables for classical sentential logic. I close by outlining the technicalities surrounding nasty connectives on many-valued truth-tables.

2016. : Determinacy of reference and truth-value in the philosophy of mathematics. Joint with Sean Walsh. Philosophia Mathematica 24.3: 283–307.

doi 10.1093/philmat/nkw007


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.

2014. : Lessons for minimalists from Russell’s Gray’s Elegy argument. Proceedings of the Aristotelian Society 114.3: 261–89.

doi 10.1111/j.1467-9264.2014.00373.x


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, the minimalist regards 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 teaches us some important lessons about propositions and propositional constituents; and, when applied to minimalism, they show us why we should abandon it.

2012. . Joint with P. Smith. Philosophia Mathematica, 20.1: 114–21.

doi 10.1093/philmat/nkr031


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 doesn’t help. We show this by examining a parallel argument, from a simpler model-theoretic result.

2012. . Noûs 46.2: 243–58.

doi 10.1111/j.1468-0068.2010.00785.x


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 approach to the operator problem. Sainsbury treats the problem as syntactic, but the problem is ultimately metaphysical.

2011. . Erkenntnis 74.3: 321–49.

doi 10.1007/s10670-011-9270-6


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, 2007) that Putnam is largely immune to metamathematical challenges.

Philip Scowcroft has written a very useful review of this paper, on MathSciNet, MR2785345 (2012e:03005).

2010. . Proceedings of the Aristotelian Society 110.3: 387–98.

doi 10.1111/j.1467-9264.2010.00293.x


Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some authors 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 claims to hold a positive position about ‘absolute generality’.

2009. . Lecture Notes in Computer Science 5635: 68–78.

doi 10.1007/978-3-642-03073-4_8


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.

2009. . British Journal for the Philosophy of Science 60.4: 765–92.

doi 10.1093/bjps/axp038


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 barriers 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 that it is a non-sequitur. I conclude that the Effective version of the Church–Turing Thesis is unaffected by SAD computation.

2007. . Analysis 67.4: 325–32.

doi 10.1093/analys/67.4.325


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. I conclude by considering the problem of how to interpret the relation ‘x is real-as-of y’.

NB: a correction to this article appears in Analysis 68.1, and is available here. The Preprint incorporates the change made in this correction.

2006. : A hybrid solution. Analysis 66.3: 216–22.

doi 10.1093/analys/66.3.216


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. positions that have all the same properties as each other and exactly the same relations to all objects (including themselves). I conclude that realistic structuralists must compromise and treat some structures eliminativistically.

2006. . Analysis 66.2: 130–35.

doi 10.1093/analys/66.2.130


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 real-as-of different times. This reunites privilege with indexicality, but entails that no-futurists must believe in ineliminably tensed facts.

201x. . Submitted to J. Conant and S. Chakraborty (eds.), Engaging Putnam. Cambridge MA: Harvard University Press.


In ‘Models and Reality’, Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. The sketch was tantalising, but it was only a sketch. Mathematics was not the focus of any of his later writings on internal realism, and Putnam ultimately abandoned internal realism itself. As such, I have often wondered: What might a developed mathematical internal realism have looked like? In this paper, I try to answer that question.

By combining Putnam’s model-theoretic arguments and 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 understood in terms of a computable system of proof, and hence is incomplete. Whilst this might initially seem strange, I show how internal categoricity results for arithmetic and set theory allow us to face up to this Antinomy. In so doing, we come to see why ‘Models are not lost noumenal waifs looking for someone to name them’, but ‘constructions within our theory itself’, with ‘names from birth.’

2018. : Russell and Stout on James and Schiller. In M. Baghramian and S. Marchetti (eds.), Pragmatism and the European Traditions: Encounters with analytic philosophy and phenomenology before the great divide. London: Routledge, 86–109.

doi 10.31235/osf.io/u2ftt


In 1907–8, Russell and Stout presented an objection against James and Schiller, to which both James and Schiller replied. In this paper, I shall revisit their transatlantic exchange. Doing so will yield a better understanding of Schiller’s relationship to a worryingly solipsistic brand of phenomenalism. It will also allow us to appreciate a crucial difference between Schiller and James; a difference which James explicitly downplayed.

2016. . In S. Goldberg (ed.), The Brain in a Vat, Cambridge University Press: 131–54.

doi 10.1017/cbo9781107706965.009


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 enduring contributions to the realism/antirealism debate: his discussions of brains-in-vats, of Skolem’s Paradox, and of permutations. In all three cases, we have an argument which does not merely defeat the sceptic; it also shows us that we must reject some prima facie plausible philosophical picture.

2019. , by Geoffrey Hellman and Roy T. Cook (eds.). Mind

doi 10.1093/mind/fzz067 Preprint

Putnam’s most famous contribution to mathematical logic was his role in investigating Hilbert’s Tenth Problem; Putnam is the ‘P’ in the MRDP Theorem. This volume, though, focusses mostly on Putnam’s work on the philosophy of logic and mathematics. It is a somewhat bumpy ride. Of the twelve papers, two scarcely mention Putnam. Three others focus primarily on Putnam’s ‘Mathematics without foundations’ (1967), but with no interplay between them. The remaining seven papers apparently tackle unrelated themes. Some of this disjointedness would doubtless have been addressed, if Putnam had been able to compose his replies to these papers; sadly, he died before this was possible. In this review, I do my best to tease out some connections between the paper; and there are some really interesting connections to be made.

2017. , by Hilary Putnam, edited by Mario de Caro. Philosophy 92.2, pp.305–15.

doi 10.1017/S003181911600067X


Hilary Putnam’s Realism with a Human Face began with a quotation from Rilke, exhorting us to ‘try to love the questions themselves like locked rooms and like books that are written in a very foreign tongue’. Putnam followed this advice throughout his life. His love for the questions permanently changed how we understand them. In Naturalism, Realism, and Normativity – published only a few weeks after his death – Putnam continued to explore central questions concerning realism and perception, from the perspective of ‘liberal naturalism’. The volume’s thirteen papers were written over the past fifteen years (only one paper is new), and they show a man who fully inhabited the questions he loved. And the main significance of this book is that it shows – implicitly, but very clearly – quite how much of Putnam’s contribution to his philosophy is continuous with his ‘The Meaning of “Meaning”’.

2016. , by Stephan Blatti and Sandra Lapointe (eds.). Notre Dame Philosophical Reviews.



Ontology after Carnap focusses on metaontology in the light of recent interest in Carnap’s ‘Empiricism, Semantics and Ontology’. That paper is at the centre of things, as it is where Carnap formulates his internal/external dichotomy. If you haven’t already encountered the dichotomy, then neither Ontology after Carnap, nor this review, is for you. My aim in this review is to try to tease out some of the book’s themes, thereby giving some sense of contemporary neo-Carnapianism.

2014. , by Maria Baghramian (ed.). Mind 123.490, pp.569–75.

doi 10.1093/mind/fzu080


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 with Putnam’s responses. My aim is to highlight some connections between Putnam’s (evolving) views on ontology, conceptual relativism, and perception.

2013. , by Cheryl Misak. The Times Literary Supplement 18 October 2013.


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.

2013. , by Colin McGinn. Analysis 73.3: 577–80.

doi 10.1093/analys/ant026


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.

This page contains all my publications, by category (see the links above). Clicking a publication title will show you an abstract; clicking the doi will take you to the published version; pre-prints of most things are also available.