Charles Sayward

A type theory constructed with reference to a particular language will associate with each monadic predicate P of that language a class of individuals C(P) of which it is categorically significant to predicate P (or which P spans, for short). The extension of P is a subset of C(P), which is a subset of the language’s universe of discourse. The set C(P) is a category discriminated by the language. The relation 'is spanned by the same predicates as' divides the language’s universe of discourse into equivalence classes. These are the types discriminated by the language. This paper criticizes an attempt by Peter Strawson to explain terms peculiar to type theory in terms of other notions not peculiar to type theory

An account of the logic of bivalent languages with truth-value gaps is given. This account is keyed to the use of tables introduced by S. C. Kleene. The account has two guiding ideas. First, that the bivalence property insures that the language satisfies classical logic. Second, that the general concepts of a valid sentence and an inconsistent sentence are, respectively, as sentences which are not false in any model and sentences which are not true in any model. What recommends this approach is (1) its relative simplicity, and (2) the fact that it leaves the fundamental features of classical logic intact.

If a native of India asserts "Killing cattle is wrong" and a Nebraskan asserts "Killing cattle is not wrong", and both judgments agree with their respective moralities and both moralities are internally consistent, then the moral relativist says both judgments are fully correct. At this point relativism bifurcates. One branch which we call content relativism denies that the two people are contradicting each other. The idea is that the content of a moral judgment is a function of the overall moral point of view from which it proceeds. The second branch which we call truth value relativism affirms that the two judgments are contradictory. Truth value relativism appears to be logically incoherent. How can contradictory judgments be fully correct? For though there will be a sense of correctness in which each judgment is correct — namely by that of being correct relative to the morality relative to which each was expressed — if contradictory, the judgments cannot both be true, and thus cannot both be correct in this most basic sense of correctness. We defend truth value relativism against this sort of charge of logical incoherence by showing it can be accommodated by the existing semantical metatheories of deontic logic. Having done this we go on to argue that truth value relativism is the best version of relativism.

Russell held that the theory of natural numbers could be derived from three primitive concepts: number, successor and zero. This leaves out multiplication and addition. Russell introduces these concepts by recursive definition. It is argued that this does not render addition or multiplication any less primitive than the other three. To this it might be replied that any recursive definition can be transformed into a complete or explicit definition with the help of a little set theory. But that is a point about set theory, not number theory. We have learned more about the distinction between logic and set theory than was known in Russell's day, especially as this affects logicist aspirations.

As a way of dealing with the semantical paradoxes Quine has suggested: that semantical expressions such as ‘true’ and ‘true of’ be used with numerical subscripts; that when a truth locution T is applied to a sentence S, the subscript on T is greater than any within S; otherwise, the result of applying T to S is ill formed. A problem is that this introduces infinitely many semantical primitives. The paper suggests a way around the problem. The paper raises a further problem, leaving it open whether this further problem has a satisfactory solution.

The idea underlying the Begriffsschrift account of identities was that the content of a sentence is a function of the things it is about. If so, then if an identity a=b is about the content of its contained terms and is true, then a=a and a=b have the same content. But they do not have the same content; so, Frege concluded, identities are not about the contents of their contained terms. The way Frege regarded the matter is that in an identity the terms flanking the symbol for identity do not have their ordinary contents, but instead have themselves as their contents. In ?Uber Sinn und Bedeutung? Frege became convinced that if an identity a=bis about the signs aand b, then it expresses no proper knowledge. So, since it is evident that many such identities do express proper knowledge, Frege concluded that identities are not about their contained signs. So he became convinced that his Begriffsschrift account was incorrect. What was the error in the argument that led Frege to that account? It was thinking that the content of a sentence is a function of the contents of its constituent signs, that is, the things it is about

This title introduces students to non-classical logic, syllogistic, to quantificational and modal logic. The book includes exercises throughout and a glossary of terms and symbols. Taking students beyond classical mathematical logic, "Philosophical Logic" is a wide-ranging introduction to more advanced topics in the study of philosophical logic. Starting by contrasting familiar classical logic with constructivist or intuitionist logic, the book goes on to offer concise but easy-to-read introductions to such subjects as quantificational and syllogistic logic, modal logic and set theory. Chapters of this title include: Sentential Logic; Quantificational Logic; Sentential Modal Logic; Quantification and Modality; Set Theory; Incompleteness; An Introduction to Term Logic; and, Modal Term Logic. In addition, the book includes a list of symbols and a glossary of terms for ease of reference and exercises throughout help students master the topics covered in the book.

A case against Prior’s theory of propositions goes thus: (1) everyday propositional generalizations are not substitutional; (2) Priorean quantifications are not objectual; (3) quantifications are substitutional if not objectual; (4) thus, Priorean quantifications are substitutional; (5) thus that Priorean quantifications are not ontologically committed to propositions provides no basis for a similar claim about our everyday propositional generalizations. Prior agrees with (1) and (2). He rejects (3), but fails to support that rejection with an account of quantification on which there could be quantifications that are neither substitutional nor objectual. The paper draws from the work of Lorenzen an alternative conception of quantification in terms of which that needed account can be given.