Vann McGee

For accommodating defective utterances within compositional semantics, truth-value gaps (Kleene's 3-valued logic) and gluts (Priest's LP) are equally efficient, but in terms of classical logic gaps yield harmless incompleteness, whereas gluts precipitate collapse; _ex contradictione quodlibet_. The discrepancy is less deep than first appears, however. A dialetheist dual to van Fraassen's supervaluationism, implicit in Ramsey's ‘Theories,’ counts sentences true if they are true in at least one acceptable model. Divergent illucutionary norms compensate for differing semantics. Subvaluationists (so-called by Varzi) replace the supervaluationist maxim, ‘Do not assert what is untrue,’ by ‘Do not assert what is false.’ Both predict the same verbal behavior.

This chapter develops the thesis, loosely attributed to Gentzen, that the meanings of logical terms are given by the rules of inference. The author reverses the usual order of explanation, which has it that the inferences are permitted by the rules because they are truth-preserving. We want, instead, to start with deductive rules and use them to generate truth conditions. A theory of propositions is invoked to explain the role played by sentences in any possible language. This makes possible a robust conservativeness proof: on the 'realist' assumption that classical inferences are valid, the author shows how speaker usage can attach the classical meaning to the logical connectives in such a way that it will be preserved through any expansion of the language.

J. R. Lucas argues against mechanism that an ideal, immortal agent whose mental activities could be mimicked by a Turing machine would be able, absurdly, to prove the Gödel sentence for the set of arithmetical sentences she is able to prove. There are two main objections: “The agent cannot know her own program” and “The agent cannot be sure the things she can prove are consistent.” It is argued that accepting the first objection would hand the anti-mechanist a roundabout victory, since for an ordinary finite mechanical system, one can determine what its program is, but that one need not accept the first objection. The second objection can only be thwarted by adopting a conception of “proof” that treats proof as veridical. This reduces Lucas’s argument to Montague’s theorem on the undefinability of epistemic necessity, which is, it is argued, an obstacle to naturalized epistemology.

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.

If it is certain that performing an observation to determine whether P is true will in no way influence whether P is tme, then the proposition that the observation is performed ought to be probabilistically independent of P. Applying the notion of "observation" liberally, so that a wide variety of actions are treated as observations, this proposed new principle of belief revision yields the result that simple utihty maximization gives the correct solution to the Fisher smoking paradox and the two-box solution to Newcomb's paradox. Contrary intuitions are explained as arising from mistakenly treating subjective probability as a measure of the intensity of conscious assent, whereas it ought to be regarded as measuring dispositions to action.