
628Wand/Set Theories: A realization of Conway's mathematicians' liberation movement, with an application to Church's set theory with a universal setJournal of Symbolic Logic. forthcoming.Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlierfound object with any magic wand (from a given stock of magic wands). By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive way of fleshing out this idea is synonymous with…Read more

417The iterative conception of function and the iterative conception of setIn Carolin Antos, Neil Barton & Giorgio Venturi (eds.), The Palgrave Companion to the Philosophy of Set Theory, Palgrave. 2023.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 wellknown 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

387We 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 notC, 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

200Naturalism, Realism, and Normativity, by Hilary Putnam, edited by Mario de Caro (review)Philosophy 92 (2): 30515. 2017.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…Read more

147Other minds and God: Russell and Stout on James and SchillerIn Sarin Marchetti & Maria Baghramian (eds.), Pragmatism and the European Traditions: Encounters with Analytic Philosophy and Phenomenology Before the Great Divide, Routledge. pp. 86109. 2017.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.

117Review of: Ontology After Carnap, by Stephan Blatti and Sandra Lapointe (eds.) (review)Notre Dame Philosophical Reviews. 2016.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 neoCarnapianism.

408Realistic structuralism's identity crisis: A hybrid solutionAnalysis 66 (3). 2006.Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a nontrivial automorphism, distinct indiscernible positions within the structure cannot be shown to be nonidentical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include 'irreflexive twoplace relations'. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positio…Read more

479A fictionalist theory of universalsIn Peter Fritz & Nicholas K. Jones (eds.), HigherOrder Metaphysics, Oxford University Press. 2024.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.

305Symmetric relations, symmetric theories, and PythagrapheanismPhilosophy and Phenomenological Research (3): 583612. 2022.It is a metaphysical orthodoxy that interesting nonsymmetric 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.

148An open source open access logic textbook

406Mathematical Internal RealismIn Sanjit Chakraborty & James Ferguson Conant (eds.), Engaging Putnam, De Gruyter. pp. 157182. 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 modeltheoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the SkolemGödel 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

498Against Cumulative Type TheoryReview of Symbolic Logic 15 (4): 90749. 2022.Standard Type Theory, STT, tells us that b^n(a^m) is wellformed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed typerestrictions: according to CTT, b^β(a^α) is wellformed 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 typerestrictions are unjustifiable, the typerestrictions i…Read more

390Level theory, part 1: Axiomatizing the bare idea of a cumulative hierarchy of setsBulletin of Symbolic Logic 27 (4): 436460. 2021.The following barebones 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 wellordered levels, and suffices for quasicategoricity. I show this by presenting Level Theory, a simplification of set theories due to Scott, M…Read more

250Level theory, part 2: Axiomatizing the bare idea of a potential hierarchyBulletin of Symbolic Logic 27 (4): 461484. 2021.Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following barbones 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 wellfoundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non…Read more

232Level Theory, Part 3: A Boolean Algebra of Sets Arranged in WellOrdered LevelsBulletin of Symbolic Logic 28 (1): 126. 2022.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 wellordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; a…Read more

81Forall x: Dortmund (2nd ed.). 2021.forall x: Dortmund is an adaptation and German translation of forall x: Calgary. As such, it is a fullfeatured textbook on formal logic. It covers key notions of logic such as consequence and validity, the syntax of truthfunctional (propositional) logic and truthtable semantics, the syntax of firstorder (predicate) logic with identity and firstorder interpretations, formalizing German in TFL and FOL, and Fitchstyle natural deduction proof systems for both TFL and FOL. It also deals with so…Read more

2172Para Todxs: Natal  uma introdução à lógica formalPPGFILUFRN. 2022.Livrotexto de introdução à lógica, com (mais do que) pitadas de filosofia da lógica, produzido como uma versão revista e ampliada do livro Forallx: Calgary. Tratase da versão de 13 de outubro de 2022. Comentários, críticas, correções e sugestões são muito bemvindos.

230Review of: Hilary Putnam on Logic and Mathematics, by Geoffrey Hellman and Roy T. Cook (eds.) (review)Mind 129 (516): 13271337. 2019.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 unrelate…Read more

562Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of MathematicsPhilosophia Mathematica 24 (3): 283307. 2016.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.

397The Philosophical Significance of Tennenbaum’s TheoremPhilosophia Mathematica 20 (1): 114121. 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 modeltheoretic 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

275Review of: Reading Putnam, by Maria Baghramian (ed.) (review)Mind 123 (490): 569575. 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

2394Forall x: Calgary. An Introduction to Formal Logic (4th ed.)Open Logic Project. 2023.forall x: Calgary is a fullfeatured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truthfunctional propositional logic TFL and truthtable semantics, the syntax of firstorder (predicate) logic FOL with identity (firstorder interpretations), symbolizing English in TFL and FOL, and Fitchstyle 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

367Wittgenstein on Solipsism in the 1930s: Private Pains, Private Languages, and Two Uses of ‘I’Royal Institute of Philosophy Supplement 82 205229. 2018.In the earlytomid 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

255Grades of Discrimination: Indiscernibility, Symmetry, and RelativityNotre Dame Journal of Formal Logic 58 (4): 527553. 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 firstorder formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discr…Read more

76Philosophy and Model TheoryOxford University Press. 2018.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 wideranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found on…Read more

429SAD computers and two versions of the Church–Turing thesisBritish Journal for the Philosophy of Science 60 (4): 765792. 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 nonTuring computability and nonEuclidean geometry, showing t…Read more

366Brains in vats and model theoryIn Sanford Goldberg (ed.), The Brain in a Vat, Cambridge University Press. pp. 131154. 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

671There's no time like the presentAnalysis 66 (2). 2006.Nofuturists ('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 BraddonMitchell (2004) have argued that this position is unmotivated, since the privilege of presentness comes apart from the indexicality of 'this moment'. I respond that nofuturists should treat 'x is realasof y' as a nonsymmetric relation. Then different moments are re…Read more

284Exclusion Problems and the Cardinality of Logical SpaceJournal of Philosophical Logic 46 (6): 611623. 2017.Wittgenstein’s atomist picture, as embodied in his Tractatus, is initially very appealing. However, it faces the famous colourexclusion 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

299Spotty Scope and Our Relation to FictionsNoûs 46 (2): 24358. 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
Areas of Specialization
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Metaphysics 
Metaphilosophy 
Philosophy of Language 
Areas of Interest
HigherOrder Logic 
Methodology in Metaphysics 
Set Theory as a Foundation 