Stephen Read

Jan Lukasiewicz's treatise on Aristotle's Syllogistic, published in the 1950s, has been very influential in framing contemporary understanding of Aristotle's logical systems. However, Lukasiewicz's interpretation is based on a number of tendentious claims, not least, the claim that the syllogistic was intended to apply only to non-empty terms. I show that this interpretation is not true to Aristotle's text and that a more coherent and faithful interpretation admits empty terms while maintaining all the relations of the traditional square of opposition

Logical pluralism is the claim that different accounts of validity can be equally correct. Beall and Restall have recently defended this position. Validity is a matter of truth-preservation over cases, they say: the conclusion should be true in every case in which the premises are true. Each logic specifies a class of cases, but differs over which cases should be considered. I show that this account of logic is incoherent. Validity indeed is truth-preservation, provided this is properly understood. Once understood, there is one true logic, relevance logic. The source of Beall and Restall’s error is a recent habit of using a classical metalanguage to analyse non-classical logics generally, including relevance logic.

General-elimination harmony articulates Gentzen’s idea that the elimination-rules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies’ test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction- and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett’s notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable.

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.

Gabriel Nuchelmans, Dilemmatic arguments. Towards a history of their logic and rhetoric. Amsterdam, New York, Oxford, Tokyo:North-Holland, 1991. 152pp. No price stated Francis P. Dinneen, Peter of Spain:language in dispute. Amsterdam and Philadelphia: John Benjamins Publishing Company, 1990. xxxix + 271 pp. Hfl. 110/$58.00 Charles H. Manekin, The logic of Gersonides. A translation of Sefer ha-heqqesh ha-yashar of Rabbi Levi ben Gershom with introduction, commentary and analytical glossary. Dordrecht, Boston and London:Kluwer Academic Publishers, 1992. xii + 341 pp. £61.00 F. P. Ramsey, Notes on philosophy, probability and mathematics. Edited by Maria Carla Galavotti. Naples:Bibliopolis, 1991 [published 1992]. 349 pp. No price stated Thomas Drucker, Perspectives on the History of Mathematical Logic. Boston, Basel, Berlin; Birkhäuser, 1991. xxiii 4- 195 pp. 108 SFr./128 DM J.P. Cleave, A study of logics. Oxford:Clarendon Press, 1991. xiii + 417 pp. £65.00 Hartry Field, Realism, mathematics and modality. Oxford and Cambridge :Blackwell, 1989. viii + 290 pp. £14.95 T.E. Forster, Set theory with a universal set, exploring an untyped universe. Oxford:Clarendon Press, 1992. viii + 152 pp., £25.00 Mark Sainsbury, Logical forms:an introduction to philosophical logic. Cambridge, Mass, and Oxford: Basil Blackwell Publishers, 1991. viii + 398 pp. £40.00, £11.95 L.C. Burns, Vagueness. An investigation into natural languages and the sorites paradox. Dordrecht, Boston and London:Kluwer, 1991. xii + 202 pp. dollar;130.00/£74.00/Dfl 130 Douglas Walton, Slippery slope arguments. Oxford:The Clarendon Press, 1992. xi + 296 pp. £35.00 Bas Van Fraassen, Quantum mechanics. An empiricist view. Oxford:Clarendon Press, 1991. xvi + 541 pp. £50.00 /£17.50 R. Giuntini, Quantum logic and hidden variables. Mannheim, Vienna and Zürich:BI-Wissenschaf ts-verlag, 1991. x + 184 pp. No price stated W. Spohn, B.C. Van Fraassn And B. Skyrms, Existence and explanation:essays presented in honor of Karel Lambert. Dordrect, Boston and London: Kluwer Academic Publishers, 1991. xii 4- 243 pp. £56.00 Michele Di Francesco, Il realismo analitico. Logica, ontologia e significato nel primo Russell Milan:Guerini, 1991. 269 pp. 30,000 Lire Kenneth Blackwell and Carl Spadont, A detailed catalogue of the second archives of Bertrand Russell. Bristol:Thoemmes Press, 1992. xxvi + 433 pp. £60.00

Bradwardine’s solution to the the logical paradoxes depends on the idea that every sentence signifies many things, and its truth depends on things’ being wholly as it signifies. This idea is underpinned by his claim that a sentence signifies everything that follows from what it signifies. But the idea that signification is closed under entailment appears too strong, just as logical omniscience is unacceptable in the logic of knowledge. What is needed is a more restricted closure principle. A clue can be found in speech act pluralism, whereby indirect speech reports are closed under intersubstitutivity of co-referential terms. The conclusion is that solving the semantic paradoxes does not require revision of logic, thus saving logic from paradox

Intentional verbs create three different problems: problems of non-existence, of indeterminacy, and of failure of substitutivity. Meinongians tackle the first problem by recognizing non-existent objects; so too did many medieval logicians. Meinongians and the medievals approach the problem of indeterminacy differently, the former diagnosing an ellipsis for a propositional complement, the latter applying their theory directly to non-propositional complements. The evidence seems to favour the Meinongian approach. Faced with the third problem, Ockham argued bluntly for substitutivity when the intentional complement is non-propositional; Buridan developed a novel way of resisting substitutivity. Ockham's approach is closer to the Meinongian analysis of these cases; Buridan's seems to raise difficulties for a referential semantics. The comparision between the Meinongian and medieval approaches helps to bring out merits and potential pitfalls of each.

t 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. What we show is that if an argument really is truth-preserving, then that in itself is enough to show that the premises are (logically) relevant to the conclusion.

Thomas Bradwardine's solution to the semantic paradoxes, presented in his Insolubilia written in Oxford in the early 1320s, turns on two main principles: that a proposition is true only if things are wholly as it signifies; and that signification is closed under consequence. After exploring the background in Walter Burley's account of the signification of propositions, the question is considered of the extent to which Bradwardine's theory is compatible with the distribution of truth over conjunction, disjunction, negation and the conditional.

Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions.

This book presents the very latest research on the medieval use of sophisms in logical and grammatical investigation by twenty-three of the leading experts in Europe and beyond. Important insights into the genre of sophismatic treatises have been gained only very recently, and the organisation of the European Symposium on this topic in 1990 led to a concentration of research and evaluation of insights. The papers are divided into three groups: one covers textual study and analysis of the role of sophisms in the medieval curriculum; another deals with grammatical sophisms; and the third covers particular logical sophisms, from 'Man is the worthiest of creatures' and problems in the theory of reference to the Liar paradox and the work of William of Ockham.

In his article "Verdades antiguas y modernas" (in the same issue, pp. 207-27), David Miller criticised Thomas Bradwardine’s theory of truth and signification and my defence of Bradwardine’s application of it to the semantic paradoxes. Much of Miller’s criticism is sympathetic and helpful in gaining a better understanding of the relationship between Bradwardine’s proposed solution to the paradoxes and Alfred Tarski’s. But some of Miller’s criticisms betray a misunderstanding of crucial aspects of Bradwardine’s account of truth and signification.

In recent years, speech-act theory has mooted the possibility that one utterance can signify a number of different things. This pluralist conception of signification lies at the heart of Thomas Bradwardine’s solution to the insolubles, logical puzzles such as the semantic paradoxes, presented in Oxford in the early 1320s. His leading assumption was that signification is closed under consequence, that is, that a proposition signifies everything which follows from what it signifies. Then any proposition signifying its own falsity, he showed, also signifies its own truth and so, since it signifies things which cannot both obtain, it is simply false. Bradwardine himself, and his contemporaries, did not elaborate this pluralist theory, or say much in its defence. It can be shown to accord closely, however, with the prevailing conception of logical consequence in England in the fourteenth century. Recent pluralist theories of signification, such as Grice’s, also endorse Bradwardine’s closure postulate as a plausible constraint on signification, and so his analysis of the semantic paradoxes is seen to be both well-grounded and plausible.