Vann McGee

Truth, vagueness, and paradox : an essay on the logic of truth

Awarded the 1988 Johnsonian Prize in Philosophy. Published with the aid of a grant from the National Endowment for the Humanities.

The Degree of the Set of Sentences of Predicate Provability Logic that are True Under Every Interpretation

The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability” indicates that □ is to be “read” as “it is provable in P(eano) A(rithmetic) that…” and that the formulae of predicate provability logic are to be interpreted via formulae of PA as follows. Pr( x ), alias Bew( x ), is the standard provability predicate of PA. For any formula F of PA, Pr[ F ] is the formula of PA that expresses the PA-provability of F “of” the values of the variables free in F , i.e., it is the formula of PA with the same free variables as F that expresses the PA-provability of the result of substituting for each variable free in F the numeral for the value of that variable. For the details of the construction of Pr[ F ], the reader may consult [B2, p. 42]. If F is a sentence of PA, then Pr[ F ] = Pr(‘ F ’), the sentence that expresses the PA-provability of F . Let υ 1 , υ 2 ,… be an enumeration of the variables of PA. An interpretation * of a formula ϕ of PPL is a function which assigns to each predicate symbol P of ϕ a formula P * of the language of arithmetic whose free variables are the first n variables of PA, where n is the degree of P .

Particulars, Individual Qualities, and Universals

We intend to develop an account of the relation between particulars and universals. Loosely derived from the work of Thomas Reid,’ the account will be empiricist, in that it has our understanding of general concepts dependent upon our prior acquaintance with particular individuals, and it will be nominalist, in that it does not require that universals actually exist.

Finite matrices and the logic of conditionals

An adequate mark for a deductive system of sentential logic is a system of many-valued logic such that the question whether an inference is deductively valid is equivalent to the question whether every truth assignment which gives the premisses designated values also gives the conclusion a designated value. What I intend to show here is that no finite matrix is adequate for the probabilistic logic studied by Adams.’ Since Adams has shown’ that, for the restricted language to which the probabilistic theory applies (conditionals as main connectives only; no logically impossible antecedents), probabilistic validity, validity in Stalnaker’s system,“ and validity in Lewis’s system5 all coincide, none of these systems has an adequate finite matrix. Suppose, for reductio ad absurdum, that the finite matrix A = (M, D, +, * , -, t) is adequate for probabilistic logic. Here D G M is the set of designated values, i-, ., -, and t are operations on M, and a truth assignment (7) satisfies

TARSKI'S STAGGERING EXISTENTIAL ASSUMPTIONS

Tarski (1935) developed a theory of truth, then, looking around for a way to apply his new ideas, he developed a relative notion of truth in a model, and used it (in (1936)) to characterize logical truth and logical consequence. A model of a theory, on Tarski’s account, is obtained by assigning semantic values to all the extralogical constants that appear in the theory in such a way as to make the axioms true. A theory is consistent iff (if and only if) it has a model. A sentence is a logical consequence of a theory iff it is true in every model of the theory, and a sentence is logically true or valid iff it’s a logical consequence of the empty theory, that is, iff it’s true in every model. None of the notions of truth, truth in a model, consistency, validity, or logical consequence are original with Tarski, nor did Tarski intend to use these old notions in a novel way. He hoped that the clarity and precision of his semantic theory would enable us better to understand the connections among familiar notions. Tarski’s paper was an important landmark on the way to modern model theory, but the notion of model Tarski employed differs in important ways from the notion we use today. We shall have no trouble convincing ourselves that any sentence that’s true in every model according to Tarski’s notion of model is also true in every model according to the contemporary notion, and that every logically valid sentence is true in every model according to Tarski’s notion of model. However, the converse implications remain controversial, requiring, as we shall see, ontological commitment on a massive scale. One difference between what Tarski’s says and the usual notion of model was almost surely unintended. Tarski defines a model of a theory K to be a sequence assigning values to variables that satisfies the formulas obtained from the members of K by uniformly replacing all the extralogical constants by variables of appropriate types. What’s odd about this definition is that it doesn’t make allowance for changing the universe of discourse from one model to another. If K is a first-order theory of the real

Gödel, Lucas, and the Soul-Searching Selfie

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 Godel 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.