Rumfitt on truth-grounds, negation, and vagueness

Abstract

the date of receipt and acceptance should be inserted later Classical logic is characterized by the familiar truth-value semantics, in which an interpretation assigns one of two truth values to any propositional letter in the language (in the propositional case), and a function from a power of the domain to the set of truth values in the predicate case.1 Truth values of composite sentence are assigned on the basis of the familiar truth functions. This abstract semantics immediately yields an applied semantics in the sense that the truth value of an interpreted sentence is given by the truth value of that sentence in an interpretation in which the propositional variables are given the truth values of the statements that interpret them. So if p is interpreted as the statement "Paris is in France" and q as "London is in Italy" then the truth value of "p ∨ q" is |p ∨ q| where the interpretation | | is given by |p| = T and |q| = F . And since the truth value of |A ∨ B| is defined aswe have that |p ∨ q| = T , and so that "Paris is in France or London is in Italy" is true.On the basis of this semantics, we can, as is done in any introductory logic textbook, define an implication relation: if X is a set of sentences, then X ⇒ A if, for every interpretation | | such that |B| = T for all B ∈ X, also |A| = T . This formal entailment relation can likewise be used to define a logical entailment relation on statements. A statement A entails a statement B iff A ⇒ B , where A and B are correct symbolizations of the statements A and B, respectively.The question of whether classical logic is the correct logic is the question of whether the implication relation so defined agrees with the pre-theoretic notion of implication between statements. Typically, and reasonably, we gloss over the intermediate step of symbolizing statements in English into the formal language Richard Zach University of Calgary, Department of Philosophy, 2500 University Dr NW, Calgary, AB T2N0A9, Canada, http://richardzach.org/, E-mail: rzach@ucalgary.ca 1 That is, a one-place predicate symbol is assigned a function from D to {T, F }, a two-place predicate a function from D 2 , etc.