Stephen Read

This chapter is a short note, co-written with Stephen Read, reacting to Hilary Putnam’s observation in his ‘Vagueness and Alternative Logic’ that intuitionistic logic would block the transition from the negation of the usual universally quantified conditional form of major premise for a Sorites to the assertion of a sharp boundary to the target predicate in the series concerned, and would thus allow the paradox to be reconceived as a straightforward _reductio_ of its major premise. It is pointed out that a Sorites need not employ that form of major premise but can instead proceed, in intuitionistic logic, from the negation of the existential claim that the series in question contains a sharp boundary and that, while an intuitionistically suspect double negation elimination step would still be needed to enforce the unpalatable conclusion that the predicate in question indeed has a sharp boundary, nothing like the semantic motivation that the Intuitionists have favoured in mathematics for a restriction on double negation elimination can be operative in this context.

The theory of properties of terms (proprietates terminorum) was the basis of the medievals’ semantic theory. It embraced those properties of linguistic expressions necessary to explain truth, fallacy and inference, the three central concepts of logical analysis. The theory evolved out of the work of Anselm and Abelard at the turn of the twelfth century, developed steadily through the thirteenth and fourteenth centuries and was still undergoing changes in the fifteenth and sixteenth centuries. It is generally agreed that its early stages were closely bound up with the theory of fallacies, but as a general semantic theory, it developed in response to a variety of needs, and one mistake of modern attempts at interpretation is to seek a unique rationale of one notion or another. Each notion evolved continually, satisfying one need at one time and another at a later date, and often several conflicting needs at the same time. Another mistake is to try to map each notion seriatim onto corresponding notions in contemporary semantic theory, but although one can see analogies and similarities, none of the medieval “properties” matches exactly any modern notion.

The idea of proof-theoretic validity originated in the work of Gentzen, when he suggested that the meaning of each logical expression was encapsulated in its introduction-rules. The idea was developed by Prawitz and Dummett, but came under attack by Prior under the soubriquet 'analytic validity'. Logical truths and logical consequences are deemed analytically valid by virtue of following, in a way which the present chapter clarifies, from the meaning of the logical constants. But different logics are based on different rules, confer different meanings and so validate different theorems and consequences, some of which are arguably not true or valid at all. It seems to follow that some analytic statements are in fact false. The moral is that we must be careful what rules we adopt and what meanings we use our rules to determine.

It is often suggested that truth-preservation is insufficient for logical consequence, and that consequence needs to satisfy a further condition of relevance. Premises and conclusion in a valid consequence must be relevant to one another, and truth-preservation is too coarse-grained a notion to guarantee that. Thus logical consequence is the intersection of truth-preservation and relevance. This situation has the absurd consequence that one might concede that the conclusion of an argument was true (since the argument had true premises and was truth-preserving); yet should refuse to infer the conclusion from the premises, in the absence of demonstration of the relevance of the premises to the conclusion. The error lies in giving insufficient attention to the notion of truth-preservation. Relevance is no separable ingredient in the analysis of logical consequence, but a necessary condition of it. If an argument really is truth-preserving, then that in itself is enough to show that the premises are (logically) relevant to the conclusion. © 2011 Elsevier B.V., All rights reserved.

Forty years ago, Niels Green-Pedersen listed five different accounts of valid consequence, variously promoted by logicians in the early fourteenth century and discussed by Niels Drukken of Denmark in his commentary on Aristotle’s Prior Analytics, written in Paris in the late 1330s. Two of these arguably fail to give defining conditions: truth preservation was shown by Buridan and others to be neither necessary nor sufficient; incompatibility of the opposite of the conclusion with the premises is merely circular if incompatibility is analysed in terms of consequence. Buridan proposed to define consequence in terms of preservation of signifying as things are. John Mair pinpointed a sophism which threatens to undermine this proposal. Speaking anachronistically, Bradwardine turned it around: he suggested that a necessary condition on consequence was that the premises signify everything the conclusion signifies. Dumbleton gave counterexamples to Bradwardine’s postulates in which the conclusion arguably signifies more than, or even completely differently from the premises. Yet a long-standing tradition held that some species of validity depend on the conclusion being in some way contained in the premises. We explore the connection between signification and consequence and its role in solving the insolubles.

Paul of Venice joined the Austin Friars at an early age and was sent by them from Padua to study at Oxford in 1390. When he returned, full of ideas and laden with books, he began his prodigious writing career with several books on logic, including the Logica Magna, which runs to some half a million words. The current volume contains the final treatise, on insolubles - that is, logical paradoxes. After surveying fifteen previous solutions, Paul develops his own, based on the idea that such propositions falsify themselves. Besides a critical edition of the Latin text, the volume also contains an English translation, a detailed commentary, excerpts from two other logical works of Paul, and a substantial introduction. The introduction describes the fourteenth-century background to Paul's treatise; it also gives a detailed rebuttal of a recent claim that the Logica Magna is not by Paul because its content clashes with genuine works of his. All in all, the volume greatly enhances our understanding of the development of logic, in particular of the semantics of propositions, during a crucial century in its history.

A Chinese translation of Thinking about Logic. In this book, Stephen Read sets out to rescue logic from its undeserved reputation as an inflexible, dogmatic discipline by demonstrating that its technicalities and processes are founded on assumptions which are themselves amenable to philosophical investigation. He examines the fundamental principles of consequence, logical truth and correct inference within the context of logic, and shows that the principles by which we delineate consequences are themselves not guaranteed free from error. Central to the notion of truth is the beguiling issue of paradox. Its philosophical value, Read shows, lies in exposing the invalid assumption on which the paradox is built. Thinking About Logic also discusses logical puzzles which introduce questions relating to language, the world, and their relationship.