Stephen Read

One of the manuscripts of Buridan’s Summulae contains three figures, each in the form of an octagon. At each node of each octagon there are nine propositions. Buridan uses the figures to illustrate his doctrine of the syllogism, revising Aristotle's theory of the modal syllogism and adding theories of syllogisms with propositions containing oblique terms (such as ‘man’s donkey’) and with ‘propositions of non-normal construction’ (where the predicate precedes the copula). O-propositions of non-normal construction (i.e., ‘Some S (some) P is not’) allow Buridan to extend and systematize the theory of the assertoric (i.e., non-modal) syllogism. Buridan points to a revealing analogy between the three octagons. To understand their importance we need to rehearse the medieval theories of signification, supposition, truth and consequence.

I. SUPPOSITION AND SIGNIFICATIONIn an insightful article on the medieval theory of supposition, Elizabeth Karger noted a remarkable development in the characterization of the material mode of supposition between William of Ockham and his contemporaries in the early fourteenth century and Paul of Venice and others at the turn of the fifteenth century.1. E. Karger, “La Supposition Materielle comme Supposition Significative: Paul de Venise, Paul de Pergula,” in A. Maierú, ed., English Logic in Italy in the 14th and 15th Centuries (Naples, 1982), pp. 331–41. For William, material supposition is explicitly ‘non-significative’, while for Paul, every mode of supposition is a kind of signification.2. Paul of Venice, Logica Magna (Tractatus de Suppositionibus), ed. and trans. A. Perreiah (St. Bonaventure, 1971), p. 4: “nam omnis suppositio est significatio.” In particular, we can set these definitions in opposition.

_ Source: _Page Count 21 Ian Rumfitt has recently drawn our attention to a couple of paradoxes of signification, claiming that although Thomas Bradwardine’s “multiple-meanings” account of truth and signification can solve the first of them, it cannot solve the second. The paradoxes of signification were in fact much discussed by Bradwardine’s successors in the fourteenth century. Bradwardine’s solution appears to turn on a distinction between the principal and the consequential signification of an utterance. However, although such a distinction played an important role in his successors’ theories, it is shown that Bradwardine’s account of signification does not admit any such distinction, no part being prior to the others. Accordingly his solution, unlike those of his successors, does not fall prey to Rumfitt’s paradoxes.

Thomas Bradwardine makes much of the fact that his solution to the insolubles is in accordance with Aristotle's diagnosis of the fallacy in the Liar paradox as that of secundum quid et simpliciter. Paul Spade, however, claims that this invocation of Aristotle by Bradwardine is purely "honorary" in order to confer specious respectability on his analysis and give it a spurious weight of authority. Our answer to Spade follows Bradwardine's response to the problem of revenge: any proposition saying of itself that it is false says more than does Bradwardine's proposition saying of it that it is false, and so follows from that other proposition only in respect of part of what it says, and not simpliciter.

MEDIEVAL LOGICCARLOS A. DUFOUR, Die Lehre der Proprietates Terminorum. Sinn und Referenz in mittelalterlicher Logik. München, Hamden, Wien: Philosophia, 1989. 312 pp. 148 DM.NORMAN KRETZMANN and BARBARA ENSIGN KRETZMANN The Sophismata of Richard Kilvington. Oxford: Published for the British Academy by Oxford University Press, 1990. xx + 156 pp. £27.50.LOGIC AND MATHEMATICSSOULEYMANE BACHIR DIAGNE, Boole. Paris: Editions Belin, 1989. 262pp. 75 Ffr.M.-M. TOEPELL, Über die Entstehung von David Hilberts “Grundlagen der Geometrie”. Göttingen: Vandenhoeck und Ruprecht, 1986. vii + 293 pp. 82 DM.VOLKER PECKHAUS, Hilbertprogramm und Kritische Philosophie. Das Göttinger Model1 interdisziplinarer Zusammenärbeit zwischen Mathematik und Philosophie. Gottingen: Vandenhoeck und Ruprecht, 1990. vi + 291 pp. 78 DM.E. WELTI, Die Philosophie des Strikten Finitismus, Entwicklungstheoretische und mathematische Untersuchungen iiber Unendlichkeitsbegriffe in Ideengeschichte und Mathematik, Bern, Frankfurt am Main und New York: Peter Lang, 1986. xxix + 652pp. $79.90.Philosophy of logicMARK A. NOTTURNO, Perspectives on psychlogism. Leiden, New York, Copenhagen and Cologne: E. J. Brill, 1989. viii + 504 pp. 225 Dfl/$112.50.ROBERT N. MCLAUGHLIN, On the logic of ordinary conditionals. Albany: State University of New York Press, 1990. xvii + 202 pp. $54.50 /$17.95.JAAKKO HINTIKKA and MERRILL HINTIKKA, The logic of epistemology and the epistemology of logic: selected essays. Dordrecht, Boston and London: Kluwer Academic Publishers, 1989.J. NORMAN and R. SYLVAN, Directions in relevant logic. Foreword by N. D. Belnap. Dordrecht, Boston and London: Kluwer Academic Publishers, 1989. xii + 453 pp. £72.00.LOGIC AND LANGUAGEC. J. F. WILLIAMS, What is identity? Oxford: Clarendon Press, 1989. xx + 207 pp. £25.00.GREGORY MCCULLOCH, The game of the name. Oxford: Oxford University Press, 1989. xvi + 320pp. £30.00, £9.95.GRAEME FORBES, Languages of possibility. Oxford: Basil Blackwell, 1989. x + 181 pp. £27.50.

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.

Kretzmann and Spade were led by Richard Kilvington’s proposed revisions to the rules of obligations in his discussion of the 47th sophism in his Sophismata to claim that the purpose of obligational disputations was the same as that of counterfactual reasoning. Angel d’Ors challenged this interpretation, realising that the reason for Kilvington’s revision was precisely that he found the art of obligation unsuited to the kind of reasoning which lay at the heart of the sophismatic argument. In his criticism, Kilvington focussed on a technique used by Walter Burley to force a respondent to grant an arbitrary falsehood and similar to Lewis and Langford’s famous defence of ex impossibili quodlibet. Kilvington observes that just as in obligational disputation, one may be obliged to grant a false proposition and deny a true one, so in counterfactual reasoning one may be obliged to doubt a proposition whose truth or falsity one knows, on pain of contradiction.

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

What makes necessary truths true? I argue that all truth supervenes on how things are, and that necessary truths are no exception. What makes them true are proofs. But if so, the notion of proof needs to be generalized to include verification-transcendent proofs, proofs whose correctness exceeds our ability to verify it. It is incumbent on me, therefore, to show that arguments, such as Dummett's, that verification- truth is not compatible with the theory of meaning, are mistaken. The answer is that what we can conceive and construct far outstrips our actual abilities. I conclude by proposing a proof-theoretic account of modality, rejecting a claim of Armstrong's that modality can reside in non-modal truthmakers.

Desde que Tarski publicó su estudio sobreel concepto de verdad en los años 30, hasido una práctica ortodoxa el considerarque t oda i nst anci a del esquema T esverdadera. Sin embargo, algunas instanciasdel esquema son falsas. Éstas incluyen lasi nst anci as paradój i cas ej empl i f i cadaspor la oración del mentiroso. Aquí sedemuestra que un esquema mejor permiteun tratamiento uniforme de la verdad enel que las paradojas semánticas resultanser simplemente falsas.Si nc e Ta r s ki publ i s he d hi s s t udy ofthe concept of truth in the 1930s, it hasbeen orthodox practice to suppose thatevery instance of the T-schema is true.However, some instances of the schemaare false. These include the paradoxicalinstances exemplified by the Liar sentence.It is shown that a better schema allows auniform treatment of truth in which thesemantic paradoxes turn out to be simplyfalse

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.