This detailed monograph deals with such problems as "The Unity of the Finite and Infinite," "Logic and the Concept of Function," "Mathematical Logic," "Formal and Dialectical Logic." The author mentions the work of Reichenbach and Lukasiewicz.--R. L. J.

This detailed monograph deals with such problems as "The Unity of the Finite and Infinite," "Logic and the Concept of Function," "Mathematical Logic," "Formal and Dialectical Logic." The author mentions the work of Reichenbach and Lukasiewicz.--R. L. J.

Originally published in Proceedings of the British Academy 60 (1974), 339-359.

J. L. Ackrill; VIII*—Aristotle's Definitions of Psuche, Proceedings of the Aristotelian Society, Volume 73, Issue 1, 1 June 1973, Pages 119–134, https://doi.org.

A comprehensive one-year graduate (or advanced undergraduate) course in mathematical logic and foundations of mathematics. No previous knowledge of logic is required; the book is suitable for self-study. Many exercises (with hints) are included.

J. L. Ackrill; VIII*—Aristotle's Definitions of Psuche, Proceedings of the Aristotelian Society, Volume 73, Issue 1, 1 June 1973, Pages 119–134, https://doi.org.

The traditional verbal/nonverbal dichotomy is inadequate for completely describing cerebral lateralization. Musical functions are not necessarily mediated by the right hemisphere; evidence for a specialist left-hemisphere mechanism dedicated to the encoded speech signal is weakening, and the right hemisphere possesses considerable comprehensional powers. Right-hemisphere processing is often said to be characterized by holistic or gestalt apprehension, and face recognition may be mediated by this hemisphere partly because of these powers, partly because of the right hemisphere's involvement in emotional affect, and (...) possibly through the hypothesized existence of a specialist face processor or processors in the right. The latter hypothesis may, however, suffer the same fate as the one relating to a specialist encodedness processor for speech in the left. Verbal processing is largely the province of the left because of this hemisphere's possession of sequential, analytic, time-dependent mechanisms. Other distinctions are special cases of an analytic/holistic dichotomy. More fundamentally, however, the left hemisphere is characterized by its mediation of discriminations involving duration, temporal order, sequencing, and rhythm, at thesensory level, and especially at themotorlevel. Spatial aspects characterize the right, the mapping of exteroceptive body space, and the positions of fingers, limbs, and perhaps articulators, with respect to actual and target positions. Thus there is a continuum of function between the hemispheres, rather than a rigid dichotomy, the differences being quantitative rather than qualitative, of degree rather than of kind. (shrink)

_The Foundations of Arithmetic_ is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.

In a single volume intended for philosophy students of all levels as well as their teachers, this reader provides modern, accurate translations of the texts necessary for a careful study of most aspects of Aristotle's philosophy. Professor Ackrill has drawn on his broad experience of teaching graduate classes in selecting the texts, and his choice reflects issues of current philosophical interest as well as the perennial themes. Only recent translations which achieve a high level of accuracy have been chosen: the (...) aim is to place the reader without Greek, as nearly as possible, in the position of a reader of Greek. As an aid to study, a valuable guide to the key topics covered is also suppied, giving references to works or passages contained in the reader, an indication of their interrelation, and a currnet bibliography. (shrink)

Aristotle is widely regarded as the greatest of all philosophers; indeed, he is traditionally referred to simply as `the philosopher'. Today, after more than two millennia, his arguments and ideas continue to stimulate philosophers and provoke them to controversy. In this book J.L. Ackrill conveys the force and excitement of Aristotle's philosophical investigations, thereby showing why contemporary philosophers still draw from him and return to him. He quotes extensively from Aristotle's works in his own notably clear English translation, and a (...) picture emerges of a lucid, lively, subtle and tough-minded thinker of astonishing range and penetration. Professor Ackrill identifies many striking connections between Aristotle's ideas and ideas in recent philosophy; he also raises philosophical questions of his own, and exemplifies the way in which Aristotle can still be argued with and learned from. (shrink)

The idea of a 'logic of quantum mechanics' or quantum logic was originally suggested by Birkhoff and von Neumann in their pioneering paper [1936]. Since that time there has been much argument about whether, or in what sense, quantum 'logic' can be actually considered a true logic (see, e.g. Bell and Hallett [1982], Dummett [1976], Gardner [1971]) and, if so, how it is to be distinguished from classical logic. In this paper I put forward a simple and natural semantical framework (...) for quantum logic which reveals its difference from classical logic in a strikingly intuitive way, viz. through the fact that quantum logic admits (suitably formulated versions of) the characteristic quantum-mechanical notions of superposition and incompatibility of attributes. That is, precisely the features that distinguish quantum from classical physics also serve, within this framework, to distinguish quantum from classical logic. Some light is shed on the question of whether quantum logic is a genuine logical system by introducing a natural entailment relation for quantum-logical formulas with the implication symbol. The novelty is that, although implication behaves as it should (i.e. the 'deduction theorem' holds), the order of introduction of premises is significant. The fact that a reasonable entailment relation can be formulated for quantum logic supports the view that it is a genuine logical system and not merely an algebraic formalism. (shrink)

In this paper (a sequel to [4]) I put forward a "local" interpretation of mathematical concepts based on notions derived from category theory. The fundamental idea is to abandon the unique absolute universe of sets central to the orthodox set-theoretic account of the foundations of mathematics, replacing it by a plurality of local mathematical frameworks - elementary toposes - defined in category-theoretic terms.