Vann McGee

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.

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.

Tarski showed how to give satisfactory theories of truth for a wide variety of languages, but he required that the theory of truth for a language be formulated in an essentially richer metalanguage. Since there is no human language essentially richer than a natural language and since we would like to develop consistent theories of truth for natural languages, we would like to learn how to formulate a theory of truth for a language within that very language. ;Toward this end, I consider a class of formalized languages called partially interpreted languages, derived from the work of Carnap, in which sentences are classified as definitely true, definitely false, and intermediate. I give a condition of adequacy, analogus to Tarski's Convension T, requiring that is true be definitely true iff is definitely true , and show that it is possible to give, effectively and explicity, a theory of truth that meets the condition. Theories of truth that meet the condition are shown to have various pleasant properties. The construction depends heavily upon the work of Saul Kripke. ;In addition to the work of Tarski and Kripke, the "naive semantics" of Gupta and Herzberger is discussed. Of the paradoxes other than the liar, only Montague's paradox about necessity is discussed in any detail. To solve this paradox, I recommend a provability interpretation of modal logic of the type studied by Solovay. Prospects for extending Solovay's results into quantified modal logic are discussed. ;The work is entirely concerned with formal languages, although it is hoped that the tools developed can be usefully applied to natural languages.

Mathematical realism is the doctrine that mathematical objects really exist, that mathematical statements are either determinately true or determinately false, and that the accepted mathematical axioms are predominantly true. A realist understanding of set theory has it that when the sentences of the language of set theory are understood in their standard meaning, each sentence has a determinate truth value, so that there is a fact of the matter whether the cardinality of the continuum is א2 or whether there are measurable cardinals, whether or not those facts are knowable by us.

There is no preferred reduction of number theory to set theory. Nonetheless, we confidently accept axioms obtained by substituting formulas from the language of set theory into the induction axiom schema. This is only possible, it is argued, because our acceptance of the induction axioms depends solely on the meanings of aritlunetical and logical terms, which is only possible if our 'intended models' of number theory are standard. Similarly, our acceptance of the second-order natural deduction rules depends solely on the meanings of the logical terms, which implies, it is argued, that our second-order quantifiers have to be standard.