Stephen Read

Frege’s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel’s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ‘complete’ it is clear from Dedekind’s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or complete, there are logical extensions of these theories into second-order and by the addition of generalized quantifiers which are categorical. Frege’s project really found success through Gödel’s completeness theorem of 1930 and the subsequent development of first- and higher-order model theory

MEDIEVAL AND RENAISSANCE LOGICSIMON OF FAVERSHAM, Quaestiones super Libro Elenchorum. Text in Latin with introduction and notes in English, edited by Sten Ebbesen, Thomas Izbicki, John Longeway, Francesco del Punta, Eileen Serene and Eleonore Stump. Toronto: Pontifical Institute of Mediaeval Studies, 1984. xiv + 270 pp. $3 1.OO.JACOPO ZABARELLA, De methodis libri quatuor; Liber de regressu. Edited by Cesare Vasoli. Bologna: Editrice CLUEB, 1985. xxxviii+ 193 pp. Lire 57,000.EDITIONSG. W. F. HEGEL, Wissenschafl der Logik. Erster Teil. Die objektive Logik, Erster Band. Die Lehre von Sein. Herausgegeben von F. Hogemann und W. Jaeschke. Hamburg: Felix Meiner Verlag, 1985. x + 448 S. DM 140.BERNARD BOLZANO, Wissenschaftslehre, Sects. 1-45. Edited and introduction Jan Berg. Stuttgart-Bad Cannstatt: Friedrich Fromman, 1985. 269 pp. No price stated.MAX H. FISCH, C. J. W. KLOESEL, E. C. MOORE and DON D.ROBERTS, LYNN A. ZIEGLER, Writings of Charles S. Peirce: a chronological edition. Bloomington, Indiana: Indiana University Press. Vol. I 1857-1866. 1982. xxxv + 698 pp. $32.50. Vo l. I1 1867-1871. 1984. xlviii + 649 pp. with introductory essays by Max H. Fisch, C. F. Delaney and Daniel D. -Merrill. $35.00.ALFRED TARSKI, Logic, semantics, metamathematics. Translated by J. H. Woodger. Second edition, edited and introduced by John Corcoran. Philadelphia: Hackett, 1983, xxx+ 506 pp. $50.00/£37.50 ; $29.50/&27.50.J. MITTELSTRASS, Enzyklopadie Philosophie und Wissenschaftstheorie, Bd.2: H-0. Mannheim, Wien, Zurich: Bibliographisches Institut, 1984. 1105 pp. DM 156.LOGIC AND MATHEMATICSA. GROTE, Anzahl, Zahl und Menge. Die phanomenologische Grundlagen der Arithmetik. Hamburg: Meiner, 1983. vii + 143 pp. DM 36.VICTOR LOWE, Alfred North Whitehead: the man and his work. Volume 1: 1861-1910. Baltimore and London: The Johns Hopkins University Press, 1985. xi+ 351 pp. $27.50/£26.40.MURIEL SELTMAN and PETER SELTMAN, Piaget's logic: a critique of genetic epistemology. London, Boston and Sydney: George Allen and Unwin, 1985. ix+ 353 pp. £30.00.MORRIS KLINE, Mathemafics: the loss of certainty. New York: Oxford University Press, 1980. vi + 366 pp. $19.95.RACHEL WALLACE GARDEN, Modern logic and quantum mechanics. Bristol: Adam Hilger, 1983. xii+ 178 pp. £18.00.LOGIC AND LANGUAGEH. LEBLANC, E. MENDELSON and A. ORESTEIN, Logic, language, and mathematics. Dordrecht and Boston: D. Reidel Publishing Company, 1984, 282 pp. Dfl.90/$34.50.R. M. MARTIN, Mind, modality, meaning and method. Albany, New York: State University of New York Press, 1983. xiv+ 225 pp. $44.50 /$16.95.HERBERT HOCHBERG, Logic, ontology, and language-essays on truth and reality. Miinchen and Wein: Philosophia Verlag, 1984. 448 pp., $74.00.

What binds the constituents of a state of affairs together and provides unity to the fact they constitute? I argue that the fact that they are related is basic and fundamental. This is the thesis of Factualism: the world is a world of facts. I draw three corollaries: first, that the Identity of truth is mistaken, in conflating what represents (the proposition) with what is represented (the fact). Secondly, a popular interpretation of Wittgenstein's Tractatus, due to Steinus, whereby false propositions are taken to picture non-existent state of affairs, cannot be right. For Wittgenstein, propositions had two poles, and a proposition and its negation picture the same fact. Finally, the metaphysics of modal realism must be wrong, for there are no non-actual states of affairs to constitute any world other than the actual world. (Published Online October 13 2005).

The recovery of Aristotle’s logic during the twelfth century was a great stimulus to medieval thinkers. Among their own theories developed to explain Aristotle’s theories of valid and invalid reasoning was a theory of consequence, of what arguments were valid, and why. By the fourteenth century, two main lines of thought had developed, one at Oxford, the other at Paris. Both schools distinguished formal from material consequence, but in very different ways. In Buridan and his followers in Paris, formal consequence was that preserved under uniform substitution. In Oxford, in contrast, formal consequence included analytic consequences such as ‘If it’s a man, then it’s an animal’. Aristotle’s notion of syllogistic consequence was subsumed under the treatment of formal consequence. Buridan developed a general theory embracing the assertoric syllogism, the modal syllogism and syllogisms with oblique terms. The result was a thoroughly systematic and extensive treatment of logical theory and logical consequence which repays investigation.

This volume, the first dedicated and comprehensive companion to medieval logic, covers both the Latin and the Arabic traditions, and shows that they were in fact sister traditions, which both arose against the background of a Hellenistic heritage and which influenced one another over the centuries. A series of chapters by both established and younger scholars covers the whole period including early and late developments, and offers new insights into this extremely rich period in the history of logic. The volume is divided into two parts, 'Periods and Traditions' and 'Themes', allowing readers to engage with the subject from both historical and more systematic perspectives. It will be a must-read for students and scholars of medieval philosophy, the history of logic, and the history of ideas.

Logical inferentialism claims that the meaning of the logical constants should be given, not model-theoretically, but by the rules of inference of a suitable calculus. It has been claimed that certain proof-theoretical systems, most particularly, labelled deductive systems for modal logic, are unsuitable, on the grounds that they are semantically polluted and suffer from an untoward intrusion of semantics into syntax. The charge is shown to be mistaken. It is argued on inferentialist grounds that labelled deductive systems are as syntactically pure as any formal system in which the rules define the meanings of the logical constants

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals.