Logic Texts

A Second Course in Logic. (Version of December 2013.) This is a free book, 165 pages. It is for anyone who has had a solid introductory logic course and wants more. Topics covered include soundness and completeness for first-order logic, Tarski’s theorem on the undefinability of truth, Gödel’s incompleteness theorems, the undecidability of first-order logic, a smattering of second-order logic, and modal logic (both propositional and quantificational). I wrote it for use in my own course, because I thought I could present the most important results and concepts more clearly than the available textbooks.

Kripke’s Theory of Truth. This is not a research paper. It is just a handout that I prepared for a course some years ago. It is a presentation of Kripke’s theory of truth that I intend to be understandable even to people who have had only a first course in logic. Although elementary, it is completely precise. All the terms are defined and all the proofs (except one trivial induction) are given in detail. I am putting this on the web because I think there are probably a lot of people who want to think about truth and who recognize that they need to know something about Kripke’s theory but who are not sure whether they have the necessary background to follow the precise presentations that have been published.

A Completeness Theorem for a 3-Valued Semantics for a First-order Language. (Version of July 2015.) This document presents a Gentzen-style deductive calculus and proves that it is complete with respect to a 3-valued semantics for a language with quantifiers. The semantics resembles the strong Kleene semantics with respect to conjunction, disjunction and negation. The completeness proof for the sentential fragment fills in the details of a proof sketched in Arnon Avron (2003). The extension to quantifiers is original but uses standard techniques.

Hard to Get Papers

These are papers that are not readily available in publications currently in print.

Open Texture and Schematicity as Arguments for Non-referential Semantics. (Prepublication version. Published in Meaning, Context and Methodology, Sarah-Jane Conrad and Klaus Petrus, eds., de Gruyter Mouton, 2017.) Many of the terms of our language, such as “jar”, are open-textured in the sense that their applicability to novel objects is not entirely determined by their past usage.  Many others, such as the verbs “use” and “have”, are schematic in the sense that they have only a very general meaning although on any particular occasion of use they denote some more particular relation.  The phenomena of open texture and schematicity constitute a sharp challenge to referential semantics, which assumes that every non-logical term has a definite extension.  A different, non-referential approach to formal semantics defines truth as relative to a context and defines contexts as built up from exclusively linguistic entities.  For any given utterance of a sentence, there will be one of these contexts that pertains to it.  In this framework, open texture and schematicity can be understood as consequences of the complex nature of the pertaining relation between contexts and utterances.

Logical Nihilism in Contemporary French Philosophy (published in Teorema 32 (2013): 65-79, with reply by Recanati): Recanati takes for granted the conveyance conception of linguistic communication, although it is not very clear exactly where he lies on the spectrum of possible variations. Even if we disavow all such conceptions of linguistic communication, there will be a place for semantic theory in articulating normative concepts such as logical consistency and logical validity. An approach to semantics focused on such normative concepts is illustrated using the example of “It’s raining”. It is argued that Recanati’s conception of semantics as involving the pragmatics of saturation and modulation cannot account for the logical properties of “It’s raining”.

Semantics and Pragmatics. (Prepublication version. Published in Delia Graff Fara and Gillian Russell, eds., Routledge Companion to the Philosophy of Language, 2012) Semantics deals with the literal meaning of sentences. Pragmatics deals with what speakers mean by their utterances of sentences over and above what those sentences literally mean. However, it is not always clear where to draw the line. Natural languages contain many expressions that may be thought of both as contributing to literal meaning and as devices by which speakers signal what they mean. After characterizing the aims of semantics and pragmatics, this chapter will set out the issues concerning such devices and will propose a way of dividing the labor between semantics and pragmatics. To semantics belongs the job of defining the conditions under which a sentence is true relative to a context. To pragmatics belongs the job of explicating the conditions under which a given context pertains to a given conversation.

Indirect Discourse, Relativism and Contexts that Point to Other Contexts. (published in François Recanati, Isidora Stojanovic, Neftali Villanueva, eds., Context-dependence, Perspective and Relativity, de Gruyter Mouton, 2010, pp. 283–315.) Some expressions, such as “all” and “might”, must be interpreted differently, relative to a single context, when embedded under “says that” than when unembedded. Egan, Hawthorne and Weatherson have appealed to that fact to argue that utterance-truth is relative to point of evaluation. This paper shows that the phenomena do not warrant this relativistic response. Instead, contexts may be defined as entities that assign other contexts to contextually relevant people, and context-relative truth conditions for indirect discourse sentences can be satisfacto- rily formulated in terms of such contexts.

The Circle of Deference Proves the Normativity of Semantics (Published in Rivista di Estetica (special issue: essays in honor of Diego Marconi) 34 (2007): 181-198). According to normativism about meaning, as I define it, a statement to the effect that a word has a certain meaning is in effect a proposal. It is a proposal to use a word in a certain way. If the proposal is accepted, then it carries normative force. This paper is a defense of normativism, so defined. The key premise of my argument is that for every group of users of a word, the members of that group regard themselves as responsible to the usage of the other members of the group.

Semantics for Deflationists. (Prepublication version. Published in Deflationism and Paradox, Oxford U.P, 2005: 148-176.): According to deflationists, [p] is true is in some sense equivalent to p. The problem that the semantic paradoxes pose for the deflationist is to explicate this equivalence without relying on a semantics grounded in the sort of real reference relations that a deflationist thinks do not exist. More generally, the deflationist is challenged to give an account of logical validity that does not force us to countenance such relations. A precise semantics compatible with the deflationist philosophy can be had as follows: First, we define a context as a certain sort of set constructed from a basis of literals (atomic sentences and negations of atomic sentences). This formal account of contexts has to be supplemented with an account of the conditions under which a structure satisfying the formal definition is the structure of that kind pertinent ot a given conversation. For each syntactic type of sentence, we define the conditions under which a sentence of that type is assertible relative to a context. In particular, we define the conditions under which sentences of the form ” [p] is true” are assertible in a context, and we define the conditions under which sentences of the form “[p] is assertible in context G” are assertible in a context. Finally, logical validity is defined as preservation of assertibility in a context. It is demonstrated that this approach to semantics resists the semantic paradoxes.

On the Evidence for Prelinguistic Concepts (Published in Theoria (Spain) 54 (2005): 287-297; special issue on the relation between thought and language). Language acquisition is often said to be a process of mapping words into pre-existing concepts. Some researchers regard this theory as an immediate corollary of the assumption that all problem-solving involves the application of concepts. But in light of basic philosophical objections to the theory of language acquisition, that kind of rationale cannot be very persuasive. To have a reason to accept the theory of language acquisition despite the philosophical objections, we ought to have experimental evidence for the existence of concepts in prelinguistic children. One of the few lines of research that attempts to provide such evidence is the work of Paul Quinn, who claims that looking-time results show that four-month old infants form “category representations”. This paper argues that Quinn’s results have an alternative explanation. A distinction is drawn between conceptual thought and the perception of comparative similarity relations, and it is argued that Quinn’s results can be explained in terms of the latter rather than the former.

The Belief-Desire Law (Published in Facta Philosophica 7, 2005: 121-144). Many philosophers hold that for various reasons there must be psychological laws governing beliefs and desires. One of the few serious examples that they offer is the belief-desire law, which states, roughly, that ceteris paribus people do what they believe will satisfy their desires. This paper argues that, in fact, there is no such law. In particular, decision theory does not support the contention that there is such a law. The problem of incomparable value scales suggests, moreover, that there will be no such law.

Social Externalism and Linguistic Communication (Published in Meaning, Basic Self-Knowledge and Mind: Essays on Tyler Burge, María-José Frápolli and Esther Romero, eds., CSLI Publications, 2003: 1-33. Originally written in 1997). According to the expressive theory of communication, the primary function of language is to enable speakers to convey the content of their thoughts to hearers. According to Tyler Burge’s social externalism, the content of a person’s thought is relative to the way words are used in his or her surrounding linguistic community. This paper argues that Burge’s social externalism refutes the expressive theory of communication.

Deflationism and Logic (Facta Philosophica 1, 1999: 167–195). Inference rule deflationism is the thesis that the nature of truth can be explained in terms of the inference rules governing the word “true”. This paper argues, first, that, in light of the semantic paradoxes, the inference rule deflationist must reject some of the classical rules of inference. It is argued, secondly, that inference rule deflationism is incompatible with model theoretic approaches to the definition of logical validity. Here the argument focuses on the question whether the number of primitive referring expressions in a natural language is denumerably infinite. Finally, it is argued that these conclusions pertain to T-schema deflationism and Horwich’s minimal theory as well.

A New Skeptical Solution. (I wrote this in 1994, but it is still relevant to contemporary discussions. Published in Acta Analytica, 1995.) Kripke’s puzzle about rule-following is a form of the traditional problem of the nature of linguistic meaning. A skeptical solution explains not what meaning is but the role that talk of meaning plays in the linguistic community. Contrary to what some have claimed, the skeptical approach is not self-refuting. However, Kripke’s own skeptical solution is inadequate. He has not adequately explained the conditions under which we are justified in attributing meanings or the utility of the practice of attributing meanings. An alternative skeptical solution may be founded on a nonepistemic conception of assertibility. Roughly, a sentence is assertible if it facilitates cooperation. The function of meaning-talk is to resolve certain sorts of conflicts in assertion. Attributions of meaning to persons outside the community may be a proper expression of a practice whose reason for being lies entirely within the community.