Charles Sayward

Peter Geach proposed a substitutional construal of quantification over thirty years ago. It is not standardly substitutional since it is not tied to those substitution instances currently available to us; rather, it is pegged to possible substitution instances. We argue that (i) quantification over the real numbers can be construed substitutionally following Geach's idea; (ii) a price to be paid, if it is that, is intuitionism; (iii) quantification, thus conceived, does not in itself relieve us of ontological commitment to real numbers.

Hilary Putnam suggests that the essence of the realist conception of mathematics is that the statements of mathematics are objective so that the true ones are objectively true. An argument for mathematical realism, thus conceived, is implicit in Putnam's writing. The first premise is that within currently accepted science there are objective truths. Next is the premise that some of these statements logically imply statements of pure mathematics. The conclusion drawn is that some statements of pure mathematics are objectively true. A key principle assumed is that if one statement logically implies a second, then if the first is objectively true so is the second. A question about this principle is raised and answered. The problem with the argument is with the second premise.

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.

Consider the general proposition that normally when people pain-behave they are in pain. Where a traditional philosopher like Mill tries to give an empirical proof of this proposition (the argument from analogy), Malcolm tries to give a transcendental proof. Malcolm’s argument is transcendental in that he tries to show that the very conditions under which we can have a concept provide for the application of the concept and the knowledge that the concept is truly as well as properly applied. The natural basis for applying the concept of pain to someone else is pain-behavior like groaning and crying out. To know that a person pain-behaving is in pain is to rule out countervailing circumstances (smiles, exaggerated cries, winks, absence of plausible cause, and so on). The basic move by Malcolm is to make these special conditions a function merely of the concept of pain.

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 standard response is illustrated by E, J. Lemmon's claim that if all objects in a given universe had names and there were only finitely many of them, then we could always replace a universal proposition about that universe by a complex proposition. It is because these two requirements are not always met that we need universal quantification. This paper is partly in agreement with Lemmon and partly in disagreement. From the point of view of syntax and semantics we can replace a universal proposition about any universe (finite or infinite, countable or uncountable) by a complex proposition (= sentence built up from atomic sentences and the connectives). But from the point of view of communication such a replacement is not possible if the universe is infinite.

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.

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