We show that the first order theory of the lattice $\mathscr{L}^{ (S) of finite dimensional closed subsets of any nontrivial infinite dimensional Steinitz Exhange System S has logical complexity at least that of first order number theory and that the first order theory of the lattice L(S ∞ ) of computably enumerable closed subsets of any nontrivial infinite dimensional computable Steinitz Exchange System S ∞ has logical complexity exactly that of first order number theory. Thus, for example, the (...) lattice of finite dimensional subspaces of a standard copy of $\bigoplus_\omega$ Q interprets first order arithmetic and is therefore as complicated as possible. In particular, our results show that the first order theories of the lattice L(V ∞ ) of c.e. subspaces of a fully effective ℵ 0 -dimensional vector space V∞ and the lattice of c.e. algebraically closed subfields of a fully effective algebraically closed field F ∞ of countably infinite transcendence degree each have logical complexity that of first order number theory. (shrink)

R.I.G. Hughes presents a series of eight philosophical essays on the theoretical practices of physics. The first two essays examine these practices as they appear in physicists' treatises (e.g. Newton's Principia and Opticks ) and journal articles (by Einstein, Bohm and Pines, Aharonov and Bohm). By treating these publications as texts, Hughes casts the philosopher of science in the role of critic. This premise guides the following 6 essays which deal with various concerns of philosophy of physics such as laws, (...) disunities, models and representation, computer simulation, explanation, and the discourse of physics. (shrink)

R.I.G Hughes offers the first detailed and accessible analysis of the Hilbert-space models used in quantum theory and explains why they are so successful.

This book provides a collection of sources, many of them fragmentary and previously scattered and hard to access, for the development of Peripatetic philosophy in the later Hellenistic period and the early Roman Empire. It also supplies the background against which the first commentator on Aristotle from whom extensive material survives, Alexander of Aphrodisias (fl. c. AD 200), developed his interpretations which continue to be influential even today. Many of the passages are here translated into English for the first time, (...) including the whole of the summary of Peripatetic ethics attributed to 'Arius Didymus'. (shrink)

Reason and Value collects 15 new papers by leading contemporary philosophers on themes from the work of Joseph Raz. Raz has made major contributions in a wide range of areas, including jurisprudence, political philosophy, and the theory of practical reason; but all of his work displays a deep engagement with central themes in moral philosophy. The subtlety and power of Raz's reflections on ethical topics make his writings a fertile source for anyone working in this area. Especially significant are his (...) explorations of the connections between practical reason and the theory of value, which constitute a sustained and penetrating treatment of a set of issues at the very center of moral philosophy as it is practiced today. The contributors to the volume acknowledge the importance of Raz's contributions by engaging critically with his positions and offering independent perspectives on the topics that he has addressed. The volume aims both to honour Raz's accomplishments in the area of ethical theorizing, and to contribute to an enhanced appreciation of the significance of his work for the subject. (shrink)

The Nature of God explores a perennial problem in the philosophy of religion.

Classical positions on central topics--mind/body, epistemology, freedom/determinism--are presented in a series of imagined discussions between renowned philosophers and critical interlocutors.

Is pure mathematics - arithmetic as well as geometry - reducible to formal logic? Russell answered in the affirmative, considering this so significant as to constitute a fatal blow to Kant's synthetic-apriori philosophy of mathematics. But either pure arithmetic and pure geometry include the full, extra-logical content of their unique axioms and hence their unique theorems, or they do not. If they do, then this reductionism is trivially unsound. It they do not - if they include only the logic of (...) demonstration and exclude everything else - then it is trivially true, but insignificant. In fact this would accomplish no reduction at all, but rather a harmless formalization. (shrink)

Is pure mathematics - arithmetic as well as geometry - reducible to formal logic? Russell answered in the affirmative, considering this so significant as to constitute a fatal blow to Kant's synthetic-apriori philosophy of mathematics. But either pure arithmetic and pure geometry include the full, extra-logical content of their unique axioms and hence their unique theorems, or they do not. If they do, then this reductionism is trivially unsound. It they do not - if they include only the logic of (...) demonstration and exclude everything else - then it is trivially true, but insignificant. In fact this would accomplish no reduction at all, but rather a harmless formalization. (shrink)

According to Hume’s classical definition of causal relations, a cause must fulfill two distinct conditions: a) be a sufficient condition for the occurrence of its effect; b) be temporally prior to it. However, a careful logical analysis shows that the combination of sufficiency and temporality is impossible. This is because if a complete cause is a sufficient condition for its effect to occur-then the effect is a necessary condition for the occurrence of its own complete cause! Thus, there can be (...) no complete cause for anything to occur, which means that causal determination is logically impossible.The paper goes on to examine some possible responses to this argument, and then concludes with a nontemporal version of the paradox that shows that even the first of Hume’s above conditions cannot be fulfilled on its own. This is because if something (e.g., the cause) is a sufficient condition of something else (e.g., the effect), then it cannot be a sufficient condition of itself. (shrink)

According to Hume’s classical definition of causal relations, a cause must fulfill two distinct conditions: a) be a sufficient condition for the occurrence of its effect; b) be temporally prior to it. However, a careful logical analysis shows that the combination of sufficiency and temporality is impossible. This is because if a complete cause is a sufficient condition for its effect to occur-then the effect is a necessary condition for the occurrence of its own complete cause! Thus, there can be (...) no complete cause for anything to occur, which means that causal determination is logically impossible. The paper goes on to examine some possible responses to this argument, and then concludes with a nontemporal version of the paradox that shows that even the first of Hume’s above conditions cannot be fulfilled on its own. This is because if something is a sufficient condition of something else, then it cannot be a sufficient condition of itself. (shrink)

In this first paperback edition, a new introduction by Tom Burke establishes the ongoing importance of Sleeper's analysis of the integrity of Dewey's work and ...

This graduate-level_text by a master in the field builds a function theory of the rational field that combines aspects of classical and intuitionist analysis. Topics include recursive convergence, recursive and relative continuity, recursive and relative differentiability, the relative integral, elementary functions, and transfinite ordinals. 1961 edition.

This paper outlines the experience of four universities that collaborated on a pre-arrival shared reading project, the Big Read, in 2018/2019. They did so primarily to promote student engagement and retention and also to ease the transition into higher education, particularly for first-generation students, to promote staff connectedness, and to provide a USP for their institution. The paper covers all the associated processes, from isolating the respective aims of the collaborators to the choosing and sharing of a single agreed title. (...) In analysing the outcomes, recommendations are made for future cross-institutional projects of this kind. (shrink)

This book is not a study of Plato's philosophy, but a contribution to the literary interpretation of the dialogues, through analysis of their formal structure, ...