Our conception of logical space is the set of distinctions we use to navigate the world. Agustn Rayo argues that this is shaped by acceptance or rejection of 'just is'-statements: e.g. 'to be composed of water just is to be composed of H2O'. He offers a novel conception of metaphysical possibility, and a new trivialist philosophy of mathematics.

It would be good to have a Bayesian decision theory that assesses our decisions and thinking according to everyday standards of rationality — standards that do not require logical omniscience (Garber 1983, Hacking 1967). To that end we develop a “fragmented” decision theory in which a single state of mind is represented by a family of credence functions, each associated with a distinct choice condition (Lewis 1982, Stalnaker 1984). The theory imposes a local coherence assumption guaranteeing that as an agent's (...) attention shifts, successive batches of "obvious" logical information become available to her. A rule of expected utility maximization can then be applied to the decision of what to attend to next during a train of thought. On the resulting theory, rationality requires ordinary agents to be logically competent and to often engage in trains of thought that increase the unification of their states of mind. But rationality does not require ordinary agents to be logically omniscient. (shrink)

Gödel claimed that Zermelo-Fraenkel set theory is 'what becomes of the theory of types if certain superfluous restrictions are removed'. The aim of this paper is to develop a clearer understanding of Gödel's remark, and of the surrounding philosophical terrain. In connection with this, we discuss some technical issues concerning infinitary type theories and the programme of developing the semantics for higher-order languages in other higher-order languages.

I propose a way of thinking aboout content, and a related way of thinking about ontological commitment. (This is part of a series of four closely related papers. The other three are ‘On Specifying Truth-Conditions’, ‘An Actualist’s Guide to Quantifying In’ and ‘An Account of Possibility’.).

The aim of this essay is to show that the subject-matter of ontology is richer than one might have thought. Our route will be indirect. We will argue that there are circumstances under which standard first-order regimentation is unacceptable, and that more appropriate varieties of regimentation lead to unexpected kinds of ontological commitment.

Modal logicism is the view that a metaphysical possibility is just a non-absurd way for the world to be. I argue that modal logicists should see metaphysical possibility as "open ended'': any given possibilities can be used to characterize further possibilities. I then develop a formal framework for modal languages that is a good fit for the modal logicist and show that it delivers some attractive results.

The purpose of this paper is to defend a conception of language that does not rely on linguistic meanings, and use it to address the Sorites and Liar paradoxes.

In order to predict and explain behavior, one cannot specify the mental state of an agent merely by saying what information she possesses. Instead one must specify what information is available to an agent relative to various purposes. Specifying mental states in this way allows us to accommodate cases of imperfect recall, cognitive accomplishments involved in logical deduction, the mental states of confused or fragmented subjects, and the difference between propositional knowledge and know-how .

Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, treated as (...) variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals.1 Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables. (shrink)

This paper extracts some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase functions for mathematical languages, and suggest a way around the problem by developing a method for specifying nominalistic contents without corresponding nominalistic paraphrases. Although much of the material in this paper is drawn from the book — and from an earlier paper — I hope the present (...) discussion will earn its keep by motivating the ideas in a new way, and by suggesting further applications. (shrink)

I argue for a conception of essence that does not rely on distinctions of metaphysical fundamentality.

I propose a way of thinking about content, and a related way of thinking about ontological commitment.

Cameron, Eklund, Hofweber, Linnebo, Russell and Sider have written critical essays on my book, The Construction of Logical Space (Oxford: Oxford University Press, 2013). Here I offer some replies.

Forthcoming in Philosophical Compass. I explain why plural quantifiers and predicates have been thought to be philosophically significant.

Whether or not we achieve absolute generality in philosophical inquiry, most philosophers would agree that ordinary inquiry is rarely, if ever, absolutely general. Even if the quantifiers involved in an ordinary assertion are not explicitly restricted, we generally take the assertion’s domain of discourse to be implicitly restricted by context.1 Suppose someone asserts (2) while waiting for a plane to take off.

The goal of this paper is to develop a theory of content for vague language. My proposal is based on the following three theses: (1) language-mastery is not rulebased— it involves a certain kind of decision-making; (2) a theory of content is to be thought of instrumentally—it is a tool for making sense of our linguistic practice; and (3) linguistic contents are only locally defined—they are only defined relative to suitably constrained sets of possibilities. CiteULike Connotea Del.icio.us What's this?

I show that any sentence of nth-order (pure or applied) arithmetic can be expressed with no loss of compositionality as a second-order sentence containing no arithmetical vocabulary, and use this result to prove a completeness theorem for applied arithmetic. More specifically, I set forth an enriched second-order language L, a sentence A of L (which is true on the intended interpretation of L), and a compositionally recursive transformation Tr defined on formulas of L, and show that they have the following (...) two properties: (a) in a universe with at least [HEBREW LETTER BET] $_{n-2}$ objects, any formula of nth-order (pure or applied) arithmetic can be expressed as a formula of L, and (b) for any sentence $\ulcorner \phi \urcorner$ of L, $\ulcorner \phi^{Tr} \urcorner$ is a second-order sentence containing no arithmetical vocabulary, and nth $\mathcal{A} \vdash \ulcorner \phi \longleftrightarrow \phi^{Tr} \urcorner$. (shrink)

Modal contingentists face a dilemma: there are two attractive principles of which they can only accept one. In this paper I show that the most natural way of resolving the dilemma leads to expressive limitations. I then develop an alternative resolution. In addition to overcoming the expressive limitations, the alternative picture allows for an attractive account of arithmetic and for a style of semantic theorizing that can be helpful to contingentists.

Dan Greco and Jason Turner wrote two fantastic critiques of my book, The Construction of Logical Space. Greco’s critique suggests that the book can be given a Kuhnian interpretation, with a Carnapian twist. Here I embrace that interpretation. Turner criticizes one of the views I develop in the book. Here I identify an avenue of resistance.

Florio and Shapiro take issue with an argument in ‘Hierarchies Ontological and Ideological’ for the conclusion that the set-theoretic hierarchy is open-ended. Here we clarify and reinforce the argument in light of their concerns.

George Boolos (1984, 1985) has extensively investigated plural quantifi- cation, as found in such locutions as the Geach-Kaplan sentence There are critics who admire only one another, and he found that their logic cannot be adequately formalized within the first-order predicate calculus. If we try to formalize the sentence by a paraphrase using individual variables that range over critics, or over sets or collections or fusions of critics, we misrepresent its logical structure. To represent plural quantification adequately requires the logical (...) resources of the full second-order predicate calculus. (shrink)

This is a supplementary document to my "Why I am not an Absolutist (Or a First-Orderist)", which is forthcoming in a volume on higher-order logic edited by Peter Fritz and Nick Jones.

La Paradoja de Orayen es dos cosas en una. Primeramente, es un homenaje al filósofo argentino Raúl Orayen (1942–2003). Pocos filósofos hispanoamericanos han gozado de la solidez intelectual y agudeza filosófica de Orayen, y pocos han sido tan queridos. Se trata, pues, de un homenaje bien merecido y que mucho agradecemos los que tuvimos la fortuna de interactuar con Raúl y aprender de él. En segundo lugar, el libro es una contribución a la filosofía hispanoamericana. Alberto Moretti y Guillermo Hurtado (...) tuvieron el acierto de reconocer el valor de un proyecto que la prematura muerte de Orayen dejó inconcluso, y apreciar su potencial para generar discusión filosófica de alto nivel. El resultado es un volumen que recompensará la atención de sus lectores, y dará al trabajo de Orayen justa prominencia en el mundo hispanoamericano. (shrink)

I will argue for localism about credal assignments: the view that credal assignments are well-defined only relative to suitably constrained sets of possibilities. I will motivate the position by suggesting that it is the best way of addressing a puzzle devised by Roger White.

The seminar is intended as an introduction to vagueness. We'll survey some prominent accounts of vagueness, so that people get a sense of what `accounting for vagueness' is all about, and why it's hard.

I develop a device for simulating quantification over merely possible objects from the perspective of a modal actualist ---someone who thinks that everything that exists actually exists.

My thesis consists of three self-contained but interconnected papers. In the first one, 'Word and Objects', I assume that it is possible to quantify over absolutely everything, and show that certain English sentences containing collective predicates resist paraphrase in first-order languages and even in first-order languages enriched with plural quantifiers. To capture such sentences I develop a language containing plural predicates . ;The introduction of plural predicates leads to an extension of Quine's criterion of ontological commitment. I argue that theories (...) containing plural predicates can have plural ontological commitments in addition to singular ones. In this sense, I argue that the subject-matter of ontology is richer than one might have thought. ;Plural predicates turn out to be tremendously fruitful. For example, they provide us with natural formalizations for English plural definite descriptions and generalized quantifiers. They also allow us to state important set theoretic propositions, and give a formal semantics for second-order languages. Such a formal semantics is developed in the second paper, 'Toward a Theory of Second-Order Consequence', which is a collaboration with Gabriel Uzquiano. ;In the third paper, 'Frege's Unofficial Arithmetic', I consider an application of plural predicates to the philosophy of mathematics. By developing a suggestion of the later Frege, I show that any arithmetical predicate can be transformed into a plural predicate in such a way that the arithmetical predicate is true of the number of the Fs just in case the plural predicate is true of the Fs themselves. ;The transformation is important both because it can be put to use by nominalists about arithmetic and neo-Fregeans, and because it provides the foundations for an account of applied arithmetic. (shrink)

I develop an account of the sorts of considerations that should go into determining where the limits of possibility lie. (This is part of a series of four closely related papers. The other three are ‘On Specifying Truth-Conditions’, ‘Ontological Commitment’ and ‘An Actualist’s Guide to Quantifying-In’.).

Students in this class are expected to complete work on their own. Both problem sets and exams should consist entirely of the student's own work; they must not be copied from other students or any other source. Failure to comply constitutes plagiarism and is a serious violation of class and University policy. Cases of academic dishonesty will be pursued to the fullest extent possible.