•  242
    Necessitists about set theory think that the pure sets exists, and are the way they are, as a matter of necessity. They cannot explain why the sets (de rebus) are all the sets. This constitutes the Ur-Objection against necessitism; it is the primary motivation cited by potentialists about set theory. At least three families of potentialism draw motivation from the Ur-Objection. Contingentists think that any things could form a set even if they actually did not. Prioritists think that sets hyper…Read more
  •  934
    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 earlier-found 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
  •  709
    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
  •  666
    We offer two arguments against the halving repose to Sleeping Beauty. First, we show that halving violates the Epistemological Sure-Thing Principle, which we argue is a necessary constraint on any reasonable probability assignment. The constraint is that if hypothetically on C you assign to A the same probability you assign to A hypothetical on not-C, you must assign that probability to A simpliciter. Epistemically, it's a sure thing for you that A has this probability. Second, we show that halv…Read more
  •  363
    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
  •  292
    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.
  •  202
    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.
  •  560
    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
  •  787
    A fictionalist theory of universals
    In Peter Fritz & Nicholas K. Jones (eds.), Higher-Order 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.
  •  600
    Symmetric relations, symmetric theories, and Pythagrapheanism
    Philosophy and Phenomenological Research (3): 583-612. 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.
  •  684
    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
  •  722
    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
  •  613
    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
  •  418
    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
  •  437
    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 (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
  •  89
    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
  •  2575
    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.
  •  378
    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
  •  656
    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
  •  357
    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
  •  770
    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.
  •  3297
    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
  •  561
    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
  •  405
    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
  •  109
    Philosophy and Model Theory
    with Sean P. Walsh
    Oxford 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 wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found on…Read more
  •  440
    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
  •  764
    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
  •  488
    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