I am a philosopher at MIT. My research is in the intersection of the philosophy of logic and the philosophy of language. I have done work on understanding the relationship between our language and the world it represents, on clarifying certain connections between logic and mathematics, and on investigating the limits of communicable thought.

77 Massachusetts Ave 32-d932 | Cambridge, MA 02139 | (617) 253 2559

arayo@mit.edu

arayo@mit.edu

## Monograph

### The Construction of Logical Space

*Oxford University Press*, 2013

Our conception of logical space is the set of distinctions we use to
navigate the world. In

*The Construction of Logical Space*I defend the idea that one's conception of logical space is shaped by one's acceptance or rejection of 'just is'-statements: statements like 'to be composed of water*just is*to be composed of H2O', or 'for the number of the dinosaurs to be zero just is for there to be no dinosaurs'. The resulting picture is used to articulate a conception of metaphysical possibility that does not depend on a reduction of the modal to the non-modal, and to develop a trivialist philosophy of mathematics, according to which the truths of pure mathematics have trivial truth-conditions.### [Reply to Critics]

*Inquiry*, forthcoming

I reply to Cameron, Eklund, Hofweber, Linnebo, Russell and Sider, as part of a special issue of

*Inquiry*on*The Construction of Logical Space*.### [Gaos Lectures: La construcción del espacio de posibilidades]

Institute for Philosohpical Research, UNAM, forthcoming

This is the (Spanish-language) text of my Gaos Lectures, a series of six talks I gave at Mexico's Institute for Philosophical Research. The content of the talks was drawn from

*The Construction of Logical Space*.[Draft]

## Epistemology

### An Anti-Reductionist's Guide to Evidential Support

I develop an account of evidential support. My account is

*anti-reductionist*: it does not presuppose that the notion of evidential support is reducible to 'more fundamental' notions. It is also*localist*: it does not presuppose that the notion of evidential support is well-defined independently of substantial background assumptions.[draft]

### A Puzzle about Ineffable Propositions

*Australasian Journal of Philosophy*, vol. 89, pp. 289-295, 2011.

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

The

*Australasian Journal of Philosophy*is available online at http://www.tandf.co.uk/journals/, and the final version of the article is available here.## Truth and Vagueness

### A Plea for Semantic Localism

*Noûs*, forthcoming

I defend a conception of language according to which sentences don't usually have stable meanings, and use it to address the Sorites and Liar paradoxes.

### Vague Representation

*Mind*vol. 117 pp. 329-373, 2008. Also in The Philosopher's Annual, vol 28.

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 rule-based -- 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.

### A Metasemantic Account of Vagueness

In Dietz and Moruzzi (eds.), Cuts and Clouds: Vagueness, its Nature and its Logic

I argue that the root of vagueness lies not in the type of semantic-value that is best associated with an expression, but in the type of linguistic practice that renders the expression meaningful. More specifically, I suggest that whereas conventions about how to use sentences involving only attributions of vague predicates to clear cases are often prevalent to a very high degree, conventions about how to use sentences involving attributions of vague predicates to borderline cases are prevalent, if at all, to lesser degrees. (I wrote this paper a long time ago, and have since have grown dissatisfied with its content. See footnote 1 of the paper for further explanation, and 'Vague Representation' for my current position of vagueness.)

### The Unexplained Supervenience Objection

In Moruzzi, S. and Sereni, A. (ed.s),

*Issues on Vagueness: 2nd Workshop on Vagueness*, Il Poligrafo, Padova, 2005.
I offer a response to the Unexplained Supervenience Objection: the complaint that postulating unexplained supervenience in accounting for the phenomenon of vagueness is methodologically perverse in the same general kind of way that setting forth a theory sufficiently lacking in simplicity
is methodologically perverse.

### Field on Revenge (with P.D. Welch)

In The Revenge of the Liar, ed. JC Beall, Oxford University Press, 2007.

In a series of recent papers, Hartry Field has proposed a novel class of solutions to the semantic paradoxes, and argued that the new solutions are 'revenge-immune'. He has argued, in particular, that by building on a sufficiently expressive language one can get a language which is able to express its own semantic theory, including its own truth predicates and any intelligible determinacy predicates. The purpose of this note is to argue that the plausibility of Field's revenge-immunity claim depends crucially on the status of higher-order languages. We show that by availing oneself of higher-order resources one can give an explicit characterization of the key semantic notion underlying Field's proposal, and note that inconsistency would ensue if the languages under discussion were expressive enough to capture this notion.

## Philosophy of Mathematics

### Nominalism, Trivialism, Logicism

This paper is an effort to extract 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 the language of arithmetic. I then suggest a way around the problem by developing a method for specifying nominalistic*contents*without corresponding nominalistic*paraphrases*.[Draft]

### Neo-Fregeanism Reconsidered

To appear in Ebert and Rossberg (eds.),

*Abstractionism in Mathematics - Status Belli*
I characterize a variety of mathematical Platonism according to which, e.g. for the number of the planets to be eight just is for there to be eight planets, and argue that Neo-Fregeans should be seen as defending such a view.

### Towards a Trivialist Account of Mathematics

In Bueno and Linnebo (eds.), New Waves in Philosophy of Mathematics

I defend mathematical trivialism: the view that the truths of pure mathematics have trivial truth-conditions and the falsities of pure mathematics have trivial falsity-conditions.

### On Specifying Truth-Conditions

*The Philosophical Review*, vol. 117 pp. 385-443, 2008.

I develop a technique for specifying truth-conditions, and use it to address a puzzle in the philosophy of mathematics.

### Logicism Reconsidered

In Shapiro (ed.) The Oxford Handbook for Logic and the Philosophy of Mathematics, OUP, 2005.

I show that the truth-values of various logicist theses can be conclusively established on minimal assumptions. In addition, I develop a notion of 'content-recarving' as a constraint on logicism, and offer a critique of 'Neo-Logicism'.

### Frege's Correlation

*Analysis*64(2), April 2004.

This is a philosophical companion to 'Frege's Unofficial Arithmetic'. I offer an explanation of why is it that mathematical knowledge can be relevant to knowledge about the natural world.

This is an electronic version of an article published in

*Analysis*, complete citation information for the final version of the paper, as published in the print edition of*Analysis*, is available on the Blackwell Synergy online delivery service, accessible via the journal's website at http://www.blackwellpublishing.com/anal or http://www.blackwell-synergy.com.### Success by Default

*Philosophia Mathematica*11(3), 2003, pp. 305-322.

I argue that Neo-Fregean accounts of arithmetical language and arithmetical knowledge tacitly rely on a thesis I call [Success by Default] -- the thesis that, in the absence of reasons to the contrary, we are justified in thinking that certain stipulations are successful. Since Neo-Fregeans have yet to supply an adequate defense of [Success by Default], I conclude that there is an important gap in Neo-Fregean accounts of arithmetical language and knowledge. I end the paper by offering a naturalistic remedy.

### Frege's Unofficial Arithmetic

*The Journal of Symbolic Logic*67(4), December 2002 pp. 1623-1638.

Also in Cook (ed),

*The Arche Papers on the Mathematics of Abstraction*. The Western Ontario Series in the Philosophy of Science, Springer, 2007.
This is a technical companion of to 'Frege's Correlation'. I show that n th-order arithmetic can be expressed within second-order logic in a way which preserves compositionality.

## Metaphysics and Absolute Generality

### Absolute Generality Reconsidered

To appear in

*Oxford Studies in Metaphysics*(edited by Karen Bennett and Dean Zimmerman).
Years ago, when I was young and reckless, I believed that there was such a thing as an all- inclusive domain. Now I have come to see the error of my ways.

### Ontological Commitment

*Philosophical Compass*vol. 2, 2007.

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

### A Completeness Theorem for Unrestricted First-Order Languages (with Tim Williamson)

In JC Beall (ed.) Liars and Heaps, OUP, 2003.

Reprinted in an abbreviated form as 'Formal Semantics and Unrestricted Quantification', in

*Logic, Ontology, and Linguistics/ Logica, Ontologia, Linguistica, Reti Saperi Linguaggi*, n. 1, 2004.
We identify a sound and complete axiomatization for first-order languages containing logically unrestricted quantifiers, that is, quantifiers that range, as a matter of logic, over everything there is.

### Introduction to *Absolute Generality* (with Gabriel Uzquiano)

In Rayo and Uzquiano (eds.),

*Absolute Generality*Oxford University Press, 2006.
We characterize the problem of absolute generality, and draw a map of the relevant philosophical terrain.

### When Does 'Everything' Mean *Everything*

*Analysis*63(2), April 2003.

I offer an argument for the possibility of determinately quantifying over absolutely everything.

This is an electronic version of an article published in

*Analysis*, complete citation information for the final version of the paper, as published in the print edition of*Analysis*, is available on the Blackwell Synergy online delivery service, accessible via the journal's website at http://www.blackwellpublishing.com/anal or http://www.blackwell-synergy.com.## Higher-Order Resources

### A Guide to the Concept *Horse*

Frege famously set forth the claim that the concept

*horse*is not a concept. I develop a view of concepts and objects whereby Frege's claim is essentially correct.[Draft]

### Hierarchies Ontological and Ideological (with Øystein Linnebo)

*Mind*, forthcoming.

We assess the claim that there is no significant difference between the ontological hierarchy of sets and the ideological hierarchy of type theory, and discuss some technical issues concerning infinitary type theory.

### Response to Florio and Shapiro (with Øystein Linnebo)

*Mind*, forthcoming.

We respond to Florio and Shapiro's "Set Theory, Type Theory, and Absolute Generality", which is a response to our "Hierarchies Ontological and Ideological".

### Beyond Plurals

In Rayo and Uzquiano (eds.) Absolute Generality, OUP, 2006.

The paper has two main objectives. The first is to get a better understanding of what is at issue between friends and foes of higher-order quantification, and of what it would mean to extend a Boolos-style treatment of second-order quantification to third- and higher-order quantification. The second objective is to argue that in the presence of absolutely general quantification, proper semantic theorizing is essentially unstable: it is impossible to provide a suitably general semantics for a given language in a language of the same logical type. I claim that this leads to a trilemma: one must choose between giving up absolutely general quantification, settling for the view that adequate semantic theorizing about certain languages is essentially beyond our reach, and countenancing an open-ended hierarchy of languages of ever ascending logical type. I conclude by suggesting that the hierarchy may be the least unattractive of the options on the table.

### Plurals

*Philosophical Compass*vol. 2, 2007

I explain why plural quantifiers and predicates have been thought to be philosophically significant.

### Nota crítica sobre *La Paradoja de Orayen*

*Crítica: Revista Hispanoamericana de Filosofía*, 37(109), 2005, pp. 99-115.

Reprinted in García de la Sienra, A.

*Reflexiones sobre la paradoja de Orayen*, UNAM-IIF, Mexico City, Mexico.
This is a critical review of

*Orayen's Paradox*, a collection of essays devoted to the work of the late Argentine philosopher Raul Orayen.### Word and Objects

*Noûs*, 36(3), 2002, pp. 436-464.

I argue that, if we admit the possibility of quantifying over everything, languages of regimentation involving plural variables and predicates are better suited to meet our needs than their first-order counterparts.

### Nominalism through De-Nominalization (with Steve Yablo)

*Noûs*, 35(1), 2001, pp. 74-92.

We offer a translation method from second-order logic into English which adequately deals with the predicative nature of second-order variables and with non-monadic higher-order quantification.

### A Puzzle about *de rebus* Beliefs (with Vann McGee)

*Analysis*60(4), October 2000, pp. 297-299.

We show that an account of de rebus belief that parallels the standard account of de re belief leads to contradiction.

This is an electronic version of an article published in

*Analysis*, complete citation information for the final version of the paper, as published in the print edition of*Analysis*, is available on the Blackwell Synergy online delivery service, accessible via the journal's website at http://www.blackwellpublishing.com/anal or http://www.blackwell-synergy.com.### Toward a Theory of Second-Order Consequence (with Gabriel Uzquiano)

*The Notre Dame Journal of Formal Logic*, 40(3), 1999, pp. 315-325.

We offer a definition of truth in a model for second-order languages which is adequate when the objects in our domain of discourse are too many to form a set.

## Modality

### An Actualist's Guide to Quantifying-In

*Crítica: Revista Hispanoamericana de Filosofía*44, 2012, pp. 3-34.

I develop a device for simulating quantification over merely possible objects from the perspective of a modal actualist.

## Spring 2014

24.118 - Paradox and Infinity

24.401 - Proseminar II (with Bob Stalnaker)

## Spring 2013

24.118 - Paradox and Infinity

24.401 - Proseminar II (with Alex Byrne)

## January 2013

Gaos Lectures (UNAM)

## Fall 2012

24.500 - Intentionality and Mental Representation (with Alex Byrne)

## Spring 2012

24.118 - Paradox and Infinity

## Fall 2011

24.400 - Proseminar I (with Richard Holton)

24.501 - Representation and Fragmentation (with Adam Elga)

## Spring 2011

24.400 - Proseminar II (with Bob Stalnaker)

## Fall 2010

24.502 - The Limits of Rationality (with Caspar Hare)

## Spring 2009

24.279 - Modeling Representation

## Fall 2008

24.118 - Paradox and Infinity

24.400 - Proseminar I (with Rae Langton)

## Spring 2008

24.260 - Topics in Philosophy: David Lewis

## Fall 2007

24.118 - Paradox and Infinity

24.711 - Possibility and Content

## Fall 2006

24.118 - Paradox and Infinity

24.400 - Proseminar I (with Rae Langton)

## Spring 2006

## Fall 2005

24.118 - Paradox and Infinity

24.729 - Vagueness

## Spring 2005

Phil 10 - Introduction to Logic

Phil 285 - A Vagueness Primer

## Winter 2005

Phil 134 - Philosophy of Language

## Fall 2004

Phil 120 - Symbolic Logic I

## Juegos Matemáticos

A bimonthly column in Investigación y Ciencia (which is the Spanish edition of Scientific American).[articles]

## 2012

Gotas y partículas cuánticas: Sistemas macroscópicos, fenómenos de interferencia y la interpretación de De Broglie-Bohm de la mecánica cuántica.

Colecciones no medibles: El conjunto de Vitali: una introducción a la teoría de la medida

Duques y marqueses: Por qué es ventajoso un título nobiliario que no comporta prerrogativas intrínsecas

La paradoja de los dos sobres: Un misterio probabilístico

Ordenadores y números naturales: Cómo demostrar el teorema de Godel a partir de la complejidad de Kolmogórov

Limones y hospitales: Un paseo por los mercados de información asimétrica

## 2011

La hiperesfera: La cuarta dimensión y la conjetura de Poincaré

Espejos y reflejos: Por qué los espejos invierten derecha e izquierda, pero no arriba y abajo

El teorema de la bola peluda: O sobre la imposibilidad de peinar un coco

¿Qué es la probabilidad?: O de cuánta información podemos extraer al cuantificar nuestra ignorancia

Teletransportadores y transplantes: El problema de la identidad personal

El programa de Woodin: Una respuesta a la paradoja de Russell (con Alejandro Pérez Carballo)

## 2010

El juego de la vida: Un clásico de John Conway popularizado por Martin Gardner

El juego del diablo: Cómo perderlo todo sin equivocarse nunca

Preferencias colectivas: La tragedia del teorema de Arrow

Computación cuántica: Un sueño que podría hacerse realidad

P = NP: Problema del milenio

Ladrillos, candados y progresiones: El fabuloso mundo de los números primos

## 2009

El teorema de Banach-Tarski: Cómo convertir una bola en dos

Viajes a través del tiempo: ¿Qué nos enseña la ciencia ficción acerca del determinismo y el libre albedrío?

Los prisioneros y María: Un problema matemático y uno filosófico

¿Qué son los números?: Reflexiones filosóficas sobre la naturaleza de los objetos matemáticos

Sombreros e infinitos: Cómo garantizar resultados en una situación aleatoria

Cartas, monedas y sombreros: Tres juegos matemáticos

## 2008

El infinito: Es posible razonar con precisión acerca del infinito, y cuando lo hacemos nos encontramos con un mundo de resultados sorprendentes

El problema de Newcomb: Cómo proceder en un mundo en el que se recompense a la irracionalidad

El duelo de los números grandes: ¿Cuál es el número más grande que puede escribirse en una pizarra?

## Big Number Duel

On January 26th 2007, Adam Elga and I got together to see who could come up with a bigger number.[more]

## Wi-Phi

Wi-Phi is a project to make philosophy accessible to non-philosophers.## Podcasts

Some interviews on philosophical topics.## Really? You're a Philosopher?

An explanation of what I do, for non-philosophers[Text]

.

I am the housemaster of MIT's oldest dorm, and a big fan of the American Association of Mexican Philosophers.