Edited by Richard McKeon, with an introduction by C.D.C. Reeve Preserved by Arabic mathematicians and canonized by Christian scholars, Aristotle’s works have shaped Western thought, science, and religion for nearly two thousand years. Richard McKeon’s The Basic Works of Aristotle—constituted out of the definitive Oxford translation and in print as a Random House hardcover for sixty years—has long been considered the best available one-volume Aristotle. Appearing in paperback at long last, this edition includes selections from the Organon, On the Heavens, (...) The Short Physical Treatises, Rhetoric, among others, and On the Soul, On Generation and Corruption, Physics, Metaphysics, Nicomachean Ethics, Politics, and Poetics in their entirety. (shrink)

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; (...) they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (shrink)

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism, Grothendieck–Zermelo potentialism, transitive-set potentialism, forcing potentialism, countable-transitive-model potentialism, countable-model potentialism, and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, (...) and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account. (shrink)

The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.

We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically (...) different: the latter is a major fragment of the former. (shrink)

The species of indeterminist tense logic called Peircean by A. N. Prior is proved to be recursively decidable.

This chapter is a detailed study of predicativity in mathematics. It presents a number of historical versions predicativity requirements, looking for unifying ideas. The further development of the notions and requirements up to the present is traced, articulating connections among the different ideas. One underlying theme of the chapter is the motivations for the various requirements for rejecting impredicativity and the various ways of stating the requirement.

In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant (...) and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing. (shrink)

The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?

Ignacio Jane has argued that second-order logic presupposes some amount of set theory and hence cannot legitimately be used in axiomatizing set theory. I focus here on his claim that the second-order formulation of the Axiom of Separation presupposes the character of the power set operation, thereby preventing a thorough study of the power set of infinite sets, a central part of set theory. In reply I argue that substantive issues often cannot be separated from a logic, but rather must (...) be presupposed. I call this the logic-metalogic link. There are two facets to the logic-metalogic link. First, when a logic is entangled with a substantive issue, the same position on that issue should be taken at the meta- level as at the object level; and second, if an expression has a clear meaning in natural language, then the corresponding concept can equally well be deployed in a formal language. The determinate nature of the power set operation is one such substantive issue in set theory. Whether there is a determinate power set of an infinite set can only be presupposed in set theory, not proved, so the use of second-order logic cannot be ruled out by virtue of presupposing one answer to this question. Moreover, the legitimacy of presupposing in the background logic that the power set of an infinite set is determinate is guaranteed by the clarity and definiteness of the notions of all and of subset. This is also exactly what is required for the same presupposition to be legitimately made in an axiomatic set theory, so the use of second-order logic in set theory rather than first-order logic does not require any new metatheoretic commitments. (shrink)

We investigate axiomatizability of various fragments of first order computational tree logic showing that the fragments with the modal operator F are non axiomatizable. These results shows that the only axiomatizable fragment is the one with the modal operator next only.