I am a professor at the University of Southern California. My main interests are in metaphysics, epistemology, the philosophy of language and philosophical logic. I have written a book on vagueness, and most recently a book on higher-order logic aimed at philosophers.
Much of my recent work focuses on what it means for a property, relation (or operation of some other type) to be fundamental and what it means for one property, relation, etc. to be built out of, or metaphysically definable from, some others. Underlying a lot of this work is an abstract analogy between languages and reality that is most naturally cashed out using category theory (spelled out in the paper Substitution Structures). My views on the metaphysical issues are outlined in the paper Logical Combinatorialism and The Broadest Necessity.
(Links to penultimate drafts)
Mathematical Modality. Journal of Philosophical Logic. [Expand abstract].
An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the "width" of the set theoretic universe, such as Cantor's continuum hypothesis. In the higher-order framework I show that contingency about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent.
A Theory of Structured Propositions. Philosophical Review. [Expand abstract].
This paper argues that the theory of structured propositions is not undermined by the Russell-Myhill paradox. I develop a theory of structured propositions in which the Russell-Myhill paradox doesn’t arise: the theory does not involve ramification or compromises to the underlying logic, but rather rejects common assumptions, encoded in the notation of the lambda-calculus, about what properties and relations can be built. I argue that the structuralist had independent reasons to reject these underlying assumptions. The theory is given both a diagrammatic representation, and a logical representation in a novel language. In the latter half of the paper I turn to some technical questions concerning the treatment of quantification, and demonstrate various equivalences between the diagrammatic and logical representations, and a fragment of the lambda-calculus.
A Case for Higher-order Metaphysics. For Higher-order Metaphysics. [Expand abstract].
Higher-order logic augments first-order logic with devices that let us generalize into grammatical positions other than that of a singular term. Some recent metaphysicians have advocated for using these devices to raise and answer questions that bear on many traditional issues in philosophy. In contrast to these `higher-order metaphysicians', traditional metaphysics has often focused on parallel, but importantly different, questions concerning special sorts of abstract objects: propositions, properties and relations. The answers to the higher-order and the property-theoretic questions may coincide sometimes but will often come apart. I argue that when they do, the higher-order questions are closer to the metaphysical action and so it would be better for these debates to proceed in higher-order terms.
Actual Value in Decision Theory. Analysis. [Expand abstract].
Decision theory is founded on the principle that we ought to take the action that has the maximum expected value from among actions we are in a position to take. But prior to the notion of expected value is the notion of the actual value of that action: roughly, a measure of the good outcomes you would in fact procure if you were to take it. Surprisingly many decision theories operate without an analysis of actual value. I offer a definition of actual value, and show that a form of decision theory due to Stalnaker can be reformulated so as to be in line with the edict to maximize expected value. By contrast, I show that there is no quantity --- given by my definition or otherwise --- that plays the role of actual value in Jeffrey's decision theory.
The Logic of Logical Necessity. with Kit Fine. For Saul Kripke on Modal Logic[Expand abstract].
Prior to Kripke's seminal work on the semantics of modal logic, McKinsey offered an alternative interpretation of the necessity operator, inspired by the Bolzano-Tarski notion of logical truth. According to this interpretation, 'it is necessary that A' is true just in case every sentence with the same logical form as A is true. In our paper, we investigate this interpretation of the modal operator, resolving some technical questions, and relating it to the logical interpretation of modality and some views in modal metaphysics. In particular, we present an hitherto unpublished solution to problems 41 and 42 from Friedman's 102 problems, which uses a different method of proof from the solution presented in the paper of Tadeusz Prucnal.
A Theory of Necessities. With Jin Zeng. Journal of Philosophical Logic. [Expand abstract].
We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of being a necessity, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about their logical behaviour. We also wield the framework to probe the conditions under which completely reductive theories of necessities are possible.
Against Disquotation. With Jeremy Goodman. Australasion Journal of Philosophy.[Expand abstract].
We argue against the disquotational meaning schema – ‘φ’ means that φ – and suggest that rejecting it is the key to resolving further intensional paradoxes about the rimits of thought.
Vagueness at every order. Proceedings of the Aristotelian Society. [Expand abstract].
There are some properties, like being bald, for which it is vague where the boundary between the things that have it, and the things that do not, lies. A number of argument threaten to show that such properties can still be associated with determinate and knowable boundaries: not between the things that have it and those that don't, but between the things such that it is borderline at some order whether they have it, and the things for which it is not.
I argue that these arguments, if successful, turn on a contentious principle in the logic of determinacy: Brouwer's Principle, that every truth is determinately not determinately false. Other paradoxes which do not appear to turn on this principle often tacitly make assumptions about assertion, knowledge and higher order vagueness. In this paper I'll show how one can avoid sharp higher-order boundaries by rejecting these assumptions.
Logical Combinatorialism. Philosophical Review. [Expand abstract].
In explaining the notion of a fundamental property or relation, metaphysicians will often draw an analogy with languages. The fundamental properties and relations stand to reality as the primitive predicates and relations stand to a language: the smallest set of vocabulary God would need in order to write the `book of the world'. In this paper I attempt to make good on this metaphor. In order to do this I introduce a modality that, put informally, stands to propositions as logical truth stands to sentences. The resulting theory, formulated in higher-order logic, also vindicates the Humean idea that fundamental properties and relations are freely recombinable and a variant of the structural idea that propositions can be decomposed into their fundamental constituents via logical operations. Indeed, it is seen that, although these ideas are seemingly distinct, they are not independent, and fall out of a natural and general theory about the granularity of reality.
Opacity and Paradox. For Modes of Truth: The unified approach to truth, modality, and paradox. [Expand abstract].
In 1963 Prior proved a theorem that places surprising constraints on the logic of intentional attitudes, like "thinks that", "hopes that", "says that" and "fears that". Paraphrasing it in English, and applying it to "thinks", it states: If, at t, I thought that I didn't think a truth at t, then there is both a truth and a falsehood I thought at t.
In this paper I explore a response to this paradox that exploits the opacity of attitude verbs, exemplified in this case by the operator "I thought at t that", to block Prior's derivation. According to this picture, both Leibniz's law and existential generalization fail in opaque contexts. In particular, one cannot infer from the fact that I'm thinking at t that I'm not thinking a truth at t, that there is a particular proposition such that I am thinking it at t. Moreover, unlike some approaches to this paradox the failure of existential generalization is not motivated by the idea that certain paradoxical propositions do not exist, for this view maintains that there is a proposition that I'm not thinking a truth at t. Several advantages of this approach over the non-existence approach are discussed, and models demonstrating the consistency of this theory are provided. Finally, the resulting considerations are applied to the liar paradox, and are used to provide a non-standard justification of a classical gap theory of truth. One of the main challenges for this sort of theory --- to explain the point of assertion, if not to assert truths --- can be met within this framework.
Counterfactuals, Infinity and Paradox. For Outstanding Contributions to Logic: Kit Fine. [Expand abstract].
In this paper two paradoxes of infinity are considered through the lense of counterfactual logic, drawing heavily on a result of Kit Fine. I will argue that a satisfactory resolution of these paradoxes will have wide ranging implications for the logic of counterfactuals. I then situate these puzzles in the context of the wider role of counterfactuals, connecting them to indicative conditionals, probabilities, rationality and the direction of causation, and compare my own resolution of the paradoxes to alternatives inspired by the theories of Lewis and Fine.
Substitution Structures. Journal of Philosophical Logic.[Expand abstract].
An increasing amount of twenty-first century metaphysics is couched in explicitly hyperintensional terms. A prerequisite of hyperintensional metaphysics is that reality itself be hyperintensional: at the metaphysical level, propositions, properties, operators, and other elements of the type hierarchy, must be more fine-grained than functions from possible worlds to extensions. In this paper I develop, in the setting of type theory, a general framework for reasoning about the granularity of propositions and properties. The theory takes as primitive the notion of a substitution on a proposition (property, etc.) and, among other things, uses this idea to elucidate a number of theoretically important distinctions. A class of structures are identified which can be used to model a wide range of positions about the granularity of reality; certain of these structures are seen to receive a natural treatment in the category of M-sets.
Inductive Knowledge. Nous. [Expand abstract].
This paper formulates some paradoxes of inductive knowledge. Two responses in particular are explored: According to the first sort of theory, one is able to know in advance that certain observations will not be made unless a law exists. According to the other, this sort of knowledge is not available until after the observations have been made. Certain natural assumptions, such as the idea that the observations are just as informative as each other, the idea that they are independent, and that they increase your knowledge monotonically (among others) are given precise formulations. Some surprising consequences of these assumptions are drawn, and their ramifications for the two theories examined. Finally, a simple model of inductive knowledge is offered, and independently derived from other principles concerning the interaction of knowledge and counterfactuals.
Some results on the limits of thought. (with Gabriel Uzquiano). Journal of Philosophical Logic. [Expand abstract].
Generalising on some arguments due to Arthur Prior and Dmitry Mirimanoff, we provide some further limitative results on what can be thought.
Is Reality Fundamentally Qualitative?. Philosophical Studies. [Expand abstract].
Individuals play a prominent role in many metaphysical theories. According to an individualistic metaphysics, reality is determined (at least in part) by the pattern of properties and relations that hold between individuals. A number of philosophers have recently brought to attention alternative views in which individuals do not play such a prominent role; in this paper I will investigate one of these alternatives.
The Broadest Necessity. Journal of Philosophical Logic. [Expand abstract].
In this paper I explore the logic of broad necessity. Definitions of what it means for one modality to be broader than another are formulated, and I prove, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. I show, moreover, that it is possible to give a reductive analysis of this necessity in extensional language (using truth functional connectives and quantifiers). This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. I formulate this conjecture precisely in higher-order logic, and examine concrete cases in which it fails. I end by investigating the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. I give some philosophical reasons to think that the logic of broad necessity does not include the S5 principle.
The Logic of Opacity. (with Jeff Russell) Philosophical and Phenomenological Research. [Expand abstract].
We explore the view that Frege's puzzle is a source of straightforward counterexamples to Leibniz's law. Taking this seriously requires us to revise the classical logic of quantifiers and identity; we work out the options, in the context of higher-order logic. The logics we arrive at provide the resources for a straightforward semantics of attitude reports that is consistent with the Millian thesis that the meaning of a name is just the thing it stands for. We provide models to show that some of these logics are non-degenerate.
Radical Anti-Disquotationalism. Philosophical Perspectives. [Expand abstract].
A number of `no-proposition' approaches to the liar paradox find themselves implicitly committed to a moderate disquotational principle: the principle that if an utterance of the sentence `$P$' says anything at all, it says that $P$ (with suitable restrictions).\footnote{Restrictions might exclude sentences which express different propositions in different contexts, such as sentences involving indexical expressions.} I show that this principle alone is responsible for the revenge paradoxes that plague this view. I instead propose a view in which there are several closely related language-world relations playing the `semantic expressing' role, none of which is more central to semantic theorizing than any other. I use this thesis about language and the negative result about disquotation to motivate the view that people do say things with utterances of paradoxical sentences, although they do not say the proposition you'd always expect, as articulated with a disquotational principle.
Relative Locations. Oxford Studies in Metaphysics. (Awarded the 2016 Marc Sanders Prize in Metaphysics.)
. [Expand abstract].
The fact that physical laws often admit certain kinds of space-time symmetries is often thought to be problematic for substantivalism -- the view that space-time is as real as the objects it contains. The most prominent alternative, relationism, avoids these problems but at the cost of giving abstract objects (rather than space-time points) a pivotal role in the fundamental metaphysics. This incurs related problems, I shall argue, concerning the relation of the physical to the mathematical. In this paper I will present a version of substantivalism that respects Leibnizian theses about space-time symmetries, and argue that it is superior to both relationism and the more orthodox form of substantivalism.
Higher-Order Free Logic and the Prior-Kaplan Paradox (with John Hawthorne and Gabriel Uzquiano). In Williamson on Modality. [Expand abstract].
The principle of Universal Instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox or Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free-logical approach to the intensional paradoxes and note how the free-logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our assessment of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivation at face value.
Scharp on Replacing Truth. Inquiry.
[Expand abstract].
Kevin Scharp's `Replacing Truth' is an ambitious and far reaching account of the semantic paradoxes. In this critical discussion we examine one the books central claims: to have provided a theory of truth that avoids the revenge paradoxes. In the first part we assess this claim, and in the second part we investigate some features of Scharp's preferred theory of truth, ADT, and compare it with existing theories such as the Kripke-Feferman theory. In the appendix a simple model of Scharp's theory is presented, and some potential consistent ways to strengthen the theory are suggested.
Tense and Relativity. Nous. [Expand abstract].
Those inclined to positions in the philosophy of time that take tense seriously have typically assumed that not all regions of space-time are equal: one special region of space-time corresponds to what is presently happening. When combined with assumptions from modern physics this has the unsettling consequence that the shape of this favored region distinguishes people in certain places or people traveling at certain velocities. In this paper I shall attempt to avoid this result by developing a tensed picture of reality that is nonetheless consistent with `hypersurface egalitarianism' -- the view that all hypersurfaces are equal.
Can The Classical Logician Avoid The Revenge Paradoxes? Philosophical Review. [Expand abstract].
Most work on the semantic paradoxes within classical logic has centred around what I call `linguistic' accounts of the paradoxes: they attribute to sentences or utterances of sentences some property that is supposed to explain their paradoxical or non-paradoxical status. `No proposition' views are paradigmatic cases of linguistic theories. This paper shows that linguistic accounts of the paradoxes endorsing classical logic are subject to a particularly acute form of the revenge paradox: that there is no exhaustive classification of sentences into `good' and `bad' such that the T-schema holds when restricted to the `good' sentences unless it is also possible to prove some `bad' sentences. The foundations for an alternative classical non-linguistic approach is outlined which is not subject to the same kinds of problems. Although revenge paradoxes of different strengths can be formulated, they are found to be indeterminate at higher orders and not inconsistent.
Stalnaker's Thesis in Context. The Review of Symbolic Logic. [Expand abstract]. (Reprinted in the Philosopher's Annual)
In this paper I present a precise version of Stalnaker's thesis and show that it is both consistent and predicts our intuitive judgments about the probabilities of conditionals. The thesis states that someone whose total evidence is E should have the same credence in the proposition expressed by 'if A then B' in a context where E is salient as they have conditional credence in the proposition B expresses given the proposition A expresses in that context. The thesis is formalised rigorously and two models are provided that demonstrate that the new thesis is indeed tenable within a standard possible world semantics based on selection functions. Unlike the Stalnaker-Lewis semantics the selection functions cannot be understood in terms of similarity. A probabilistic account of selection is defended in its place.
I end the paper by suggesting that this approach overcomes some of the objections often leveled at accounts of indicatives based on the notion of similarity.
Paradoxes of Logical Equivalence and Identity. Topoi. [Expand abstract].
In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories. It is suggested that this is problematic for theorists who endorse the principle that 'P' and ''P' is true' are always intersubstitutable.
Giving Your Knowledge Half a Chance. Philosophical Studies. [Expand abstract].
1000 fair causally isolated coins will be independently flipped tomorrow morning and you know this fact. I argue that the probability, conditional on your knowledge, that any coin will land tails is almost 1 if that coin in fact lands tails, and almost 0 if it in fact lands heads. I also show that the coin flips are not probabilistically independent given your knowledge. These results are uncomfortable for those, like Timothy Williamson, who take these probabilities to play a central role in their theorizing.
Representing Counterparts. The Australasian Journal of Logic.[Expand abstract].
This paper presents a counterpart theoretic semantics for quantified modal logic based on a fleshed out account of Lewis's notion of a 'possibility'. According to the account a possibility consists of a world and some haecceitistic information about how each possible individual gets represented de re. A semantics for quantified modal logic based on evaluating formulae at possibilities is developed. It is shown that this framework naturally accommodates an actuality operator, addressing recent objections to counterpart theory, and is equivalent to the more familiar Kripke semantics for quantified modal logic with an actuality operator.
Quantificational Logic and Empty Names. Philosophers' Imprint. [Expand abstract]. (Awarded the APA Article prize for 2012 and 2013. Also reprinted in the Philosopher's Annual.)
The result of combining classical quantificational logic with modal logic proves necessitism -- the claim that necessarily everything is necessarily identical to something. This problem is reflected in the purely quantificational theory by theorems such as ∃xt=x; it is a theorem, for example, that something is identical to Timothy Williamson. The standard way to avoid these consequences is to weaken the theory of quantification to a certain kind of free logic. However, it has often been noted that in order to specify the truth conditions of certain sentences involving constants or variables that don't denote, one has to apparently quantify over things that are not identical to anything.
In this paper I defend a contingentist, non-Meinongian metaphysics within a positive free logic. I argue that although certain names and free variables do not actually refer to anything, in each case there might have been something they actually refer to, allowing one to interpret the contingentist claims without quantifying over mere possibilia.
A New Conditional for Naive Truth Theory. Notre Dame Journal of Formal Logic. [Expand abstract].
In this paper a logic suitable for reasoning disquotationally about truth, TJK^+, is presented and shown to have a standard model. This work improves on Hartry Field's recent results establishing consistency and omega-consistency of truth-theories with strong conditional logics. A novel method utilising the Banach fixed point theorem for contracting functions on complete metric spaces is invoked, and the resulting logic is shown to validate a number of principles which existing revision theoretic methods have so far failed to provide.
Curry's Paradox and Omega-inconsistency. Studia Logica. [Expand abstract].
In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naive truth theory. On this basis I identify two natural subsystems of Pukasiewicz logic which individually, but not jointly, lack the problematic feature.
Non-classical Metatheory for Non-classical Logics. Journal of Philosophical Logic. [Expand abstract].
A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical logics are usually characterised by a classical meta-theory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for non-classicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples.
The aim of this paper is to show that a large class of non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the meta-theory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete.
Non-wellfounded Mereology. (with Aaron J. Cotnoir) The Review of Symbolic Logic. [Expand abstract].
This paper is a systematic exploration of non-wellfounded mereology. Motivations and applications suggested in the literature are considered. Some are exotic like Borges’ Aleph, and the trinity; other examples are less so, like time traveling bricks, and even Geach’s Tibbles the Cat. The authors point out that the transitivity of non-wellfounded parthood is inconsistent with extensionality. A non-wellfounded mereology is developed with careful consideration paid to rival notions of supplementation and fusion. Two equivalent axiomatizations are given, and are compared to classical mereology. We provide a class of models with respect to which the non-wellfounded mereology is sound and complete.
A Paradox for Supertask Decision Makers. Philosophical Studies. [Expand abstract].
I consider two puzzles in which an agent undergoes a sequence of decision problems. In both cases it is possible to respond rationally to any given problem yet it is impossible to respond rationally to every problem in the sequence, even though the choices are independent. In particular, although it might be a requirement of rationality that one must respond in a certain way at each point in the sequence, it seems it cannot be a requirement to respond as such at every point for that would be to require the impossible.
Quantificational Logic and Empty Names. Handout. USC, March 2013
In Defence of a Naive Conditional Epistemology. Handout. UC Irvine, January 2013. University of Barcelona, June 2013
What A Revenge-Free Solution to the Semantic Paradoxes Can and Can't Look Like. Handout. USC, January 2012. Brown, February. Yale, February 2012. Munich, March 2012
Vagueness at Every Order. Handout. Barcelona, September 2010.
