This book presents a detailed analysis of three ancient models of spatial magnitude, time, and local motion. The Aristotelian model is presented as an application of the ancient, geometrically orthodox conception of extension to the physical world. The other two models, which represent departures from mathematical orthodoxy, are a "quantum" model of spatial magnitude, and a Stoic model, according to which limit entities such as points, edges, and surfaces do not exist in (physical) reality. The book is unique in its (...) discussion of these ancient models within the context of later philosophical, scientific, and mathematical developments. (shrink)

From Greek antiquity to the latest theories, this historical survey of political philosophy not only covers the major thinkers in the field but also explores the theme of how political philosophy relates to the nature of man. It illustrates how the great political thinkers have always grounded their political thought in what the author terms a "normative anthropology," which typically has not only ethical but metaphysical and/or theological components. Starting with the ancient Greek Sophists, author Michael J. White (...) examines how thinkers over the centuries have approached such political and philosophical concerns as justice, morality, and human flourishing, offering substantial studies of--among others--Plato, Aristotle, Cicero, Locke, Hobbes, Rousseau, Marx, and J. S. Mill. White highlights the impact of Christianity on political philosophy, illustrating the diversity of that impact by studies of Augustine, Aquinas, and Marsilius of Padua. Concluding with an in-depth analysis of John Rawls and contemporary liberal political philosophy, this text blends insight and information in a refreshing and useful manner. A brief Epilogue considers both the value and the limitations of political philosophy and its study. (shrink)

It is not very surprising that it was no less true in antiquity than it is today that adult human beings are held to be responsible for most of their actions. Indeed, virtually all cultures in all historical periods seem to have had some conception of human agency which, in the absence of certain responsibility-defeating conditions, entails such responsibility. Few philosophers have had the temerity to maintain that this entailment is trivial because such responsibility-defeating conditions are always present. Another not (...) very surprising fact is that ancient thinkers tended to ascribe integrality to "what is" (to on). That is, they typically regarded "what is" as a cosmos or whole with distinguishable parts that fit together in some coherent or cohesive manner, rather than either as a "unity" with no parts or as a collection containing members (ta onta or "things that are") standing in no "natural" relations to one another. 1 The philoso phical problem of determinism and responsibility may, I think, best be characterized as follows: it is the problem of preserving the phenomenon of human agency (which would seem to require a certain separateness of individual human beings from the rest of the cosmos) when one sets about the philosophical or scientific task of explaining the integrality of "what is" by means of the development of a theory of causation or explanation ( concepts that came to be lumped together by the Greeks under the term "aitia"). (shrink)

This paper begins by pointing out that the Aristotelian conception of continuity (synecheia) and the contemporary topological account share the same intuitive, proto-topological basis: the conception of a ?natural whole? or unity without joints or seams. An argument of Aristotle to the effect that what is continuous cannot be constituted of ?indivisibles? (e.g., points) is examined from a topological perspective. From that perspective, the argument fails because Aristotle does not recognize a collective as well as a distributive concept of a (...) multiplicity of points. It is the former concept that allows contemporary topology to identify some point sets with spatial regions (in the proto-topological sense of this term). This identification, in turn, allows contemporary topology to do what Aristotle was unwilling to do: to conceive the property of continuity, as well as the properties of having measure greater than zero and having n- dimension, as emergent properties. Thus, a point set can be continuous (connected) although none of its subsets of sufficiently smaller cardinality can be. Finally, the paper discusses the manner in which a topological principle, viz., the principle that none of the singletons of points of a continuum can be open sets of that continuum, captures certain aspects of the Aristotelian proto-topological conception of the relation between points and continua. E.g., for both Aristotle and contemporary topology, points in a continuum exist simple as limits of the remainder of the continuum: their singletons have empty ?interiors? and, hence, they are not ?chunks? (topologically, regular closed set) of the continuum. (shrink)

This chapter contains sections titled: Aristotle on the Infinite (to apeiron): From Cosmological Principle to Mathematical Operation Aristotle on Space: Magnitude (megethos) and Place (topos) Aristotle on Time: The “Number of Motion” and “Ever‐rolling Stream” Bibliography.

This paper develops an interpretation of the fourth account of conditionals in Sextus Empiricus's Outlines of Pyrrhonism that conceptually links it with contemporary ?relevance? interpretations of entailment. It is argued that the third account of conditionals, which analyzes the truth of a conditional in terms of the joint impossibility of antecedent and denial of consequent, should not be interpreted in terms of a relative incompatibility of antecedent and denial of consequent because of Stoic acceptance of the truth of some conditionals (...) of the form p ? ?p and its converse. Rather, it is suggested, ancient attempts to avoid the so-called paradoxes of implication involve the fourth account of conditionals. I hypothesize that this account is related to Stoic attempts to define truth conditions for conditionals in terms of a theory of the concludency (validity) of arguments in opposition to the more common procedure (represented by the first three accounts of conditionals) of specifying truth conditions for conditionals ?semantically? and using those truth conditions in the development of a theory of argument validity. (shrink)

There is what might be called a ‘majority position’ in the history of Western philosophy according to which causes are sufficient for or ‘necessitate’ their effects. However, there is also a singificant ‘minority position’ according to which causes are necessary relative to their effects. The second/third century A.D. Peripatetic Alexander of Aphrodisias is an ancient representative of the minority position. He attributes his own view — with some justification, I shall suggest – to Aristotle. This paper has two, somewhat loosely (...) connected purposes. The first is to explore the origin of the conception of ‘causes’ as necessary conditions in Aristotle, particularly in On Generation and Corruption 2.11 and Posterior Analytics 2.12, and the development and use of the conception in Alexander's De fato. The second purpose of the paper is to explore and criticize a sophisticated contemporary version of the conception of causes as ‘necessary in the circumstances,’ that of J.L. Mackie. (shrink)

Partisan or Neutral? critically examines the Rawlsian ideal of a public, supposedly neutral, political theory meant to justify contemporary constitutional democracies. Placing this ideal-appealed to by neo-natural law theorists and advocates of "public theology" as well as by political theorists-against the background of the history of political liberalism, White shows its contradictory nature. He argues that any such legitimating theory will be 'partisan,' in the sense of appealing to convictions concerning the human good that will not be universally accepted. (...) He concludes that all politics must be imperfect-a matter of pragmatism and prudence in forming the most workable compromises possible and in acquiescing, where our principles allow us to do so, in situations that are often far from optimal. (shrink)

There is what might be called a ‘majority position’ in the history of Western philosophy according to which causes are sufficient for or ‘necessitate’ their effects. However, there is also a singificant ‘minority position’ according to which causes are necessary relative to their effects. The second/third century A.D. Peripatetic Alexander of Aphrodisias is an ancient representative of the minority position. He attributes his own view — with some justification, I shall suggest – to Aristotle. This paper has two, somewhat loosely (...) connected purposes. The first is to explore the origin of the conception of ‘causes’ as necessary conditions in Aristotle, particularly in On Generation and Corruption 2.11 and Posterior Analytics 2.12, and the development and use of the conception in Alexander's De fato. The second purpose of the paper is to explore and criticize a sophisticated contemporary version of the conception of causes as ‘necessary in the circumstances,’ that of J.L. Mackie. (shrink)

DESPITE THE WELL-KNOWN historical significance of Aristotle’s doctrine of the productive or active intellect it is not unusual to find contemporary discussions treating the doctrine as an excrescence on the text of the De anima, a work, it is frequently nowadays supposed, in which an otherwise securely naturalistic epistemology and rational psychology are developed. Although the doctrine of the intellectus agens is found only in one place in Aristotle’s extant texts, the third book of the De anima, I shall nonetheless (...) maintain that an argument can be ferreted out of Aristotle’s discussion that establishes that the doctrine can be seen to follow from principles that are fundamental to Aristotle’s thought. I shall call this an “Aristotelian” argument for nous poiêtikos because of the fact that the argument obviously is not found, as I state it, on the surface of Aristotle’s text. I claim that there is a significant sense in which it is there, but that one must dig for it or, as I just put it, ferret it out. In a schematic form, the Aristotelian argument goes as follows. (shrink)

A well-known epigram by Callimachus on the philosopher Diodorus Cronus reads as follows:The question of the third line, while perhaps recondite from a contemporary perspective, was clear in antiquity. The crows are asking ‘What follows ?’, in allusion to the Hellenistic disputes concerning the truth conditions of conditional propositions , disputes in which the views of Diodorus figured prominently.I agree with Sedley that the question of the last line is ‘much more problematic’. The common interpretation has been to read the (...) αθι as a form of αθις and to interpret it temporally. The result, in Pfeiffer's estimation, is ‘quomodo posthac erimus?’.This interpretation derives from Sextus Empiricus' discussion at M. 1.309–12 of the last two lines of the epigram. After crediting the grammarian with the ability to understand the allusion in the crows' first question , he proceeds to argue that the philosopher has a better chance than the grammarian of understanding the second question. But, to quote Sedley, Sextus ‘makes a ghastly mess of it’ when he attempts his own elucidation. According to an argument of Diodorus, a living thing does not die in the time in which it lives nor in a time in which it does not live. Hence, Sextus concludes, it must be the case that it never dies and, ‘if this is the case, we are always living and, according to him, we shall come to be hereafter ’. (shrink)

SummaryThis paper develops a semantical account of sentences containing performative principal verbs in which these verbs are analyzed as indexical expressions: the proposition picked out by a sentence containing a performative verb depends on aspects of the context of use of the sentence; and these same aspects of context of use also determine the truth value of the proposition picked out. A two‐dimensional modal operator is utilized in analyzing non‐ performative sentences that contain principal verb which, in other contexts, have (...) performative uses. Thus, the analysis developed accounts for the performative — non‐performative distinction within a formal semantic‐pragmatic framework; the more common approach of relegating the distinction to the somewhat vague realm of ‘language use’ is avoided.RésuméCet article développe un traitement sémantique des phrases contenant un verbe principal performatif. Ces verbes sont analysés comme des expressions indicées: la proposition sélectionnée par une phrase contenant un verbe performatif dépend du contexte d'usage de cette phrase; et ces mêmes aspects du contexte d'usage déterminent également la valeur de vérité de la proposition sélectionnée. Un opérateur modal à deux dimensions est utilisé pour l'analyse de phrases non‐performatives contenant un verbe principal performatif dans d'autres contextes. L'analyse développée rend ainsi compte de la distinction performatif — non‐performatif dans un cadre sémantico‐pragmatique; on évite ainsi la solution habituelle consistant à reléguer cette distinction dans le domaine plutôt vague de l'usage linguistique.ZusammenfassungEs wird eine semantische Darstellung von Sätzen entwickelt, die performative Hauptverben enthalten, indem diese Verben als indexikale Ausdrücke analysiert werden: die Proposition, die durch einen Satz mit einem performativen Verb bezeichnet wird, hängt von Aspekten des Gebrauchskontextes des Satzes ab; und gerade diese Aspekte bestimmen auch den Wahrheitswert der Proposition. Ein zweidimensionaler Modaloperator wird verwendet, um nicht‐performative Sätze zu analysieren, die Hauptverben enthalten, welche — in andern Kontexten — einen performativen Gebrauch haben. Auf diese Weise gibt die entwickelte Analyse innerhalb eines formalen semantisch‐syntaktischen Rahmenwerkes Rechenschaft über die Unterscheidung performativ — nicht‐performativ; man vermeidet dadurch das übliche Vorgehen, die Unterscheidung in den etwas vagen Bereich des sog. 〈Sprachgebrauchs〉 abzuschieben. (shrink)

This paper discusses the 'master argument' of diodorus cronos from a semantic perspective. An argument is developed which suggests that proposition (1), 'every proposition true about the past is necessary', May have provided the principal motivation for diodorus denial of proposition (3), I.E., His equation of possibility with present-Or-Future truth. It is noted that (1) and (3) are jointly inconsistent only given the assumption of a linear ordering of time. It is further noted that diodorus' fatalism "could" be employed to (...) justify this additional assumption. However, To then use the conclusion of the 'master' to argue for fatalism would obviously be circular. I suspect, Rather, That diodorus' assumption of temporal linearity was implicit and uncritical. (shrink)

The Megarians, as well as their Stoic heirs, are known to have been fatalists or logical determinists in the following, very broad sense of these terms: with respect to at least certain classes or kinds of nontautologous propositions, they held that the mere truth of a proposition entails its necessity. This paper explores, in a very tentative fashion, the relation between several versions of logical determinism and two passages in the Aristotelian corpus, one of which is specifically directed against Megarian (...) doctrine and the other of which is an argument purporting to establish a thesis concerning the necessity of future events which Aristotle wished to reject but which the Megarian logician Diodorus Cronus apparently accepted. (shrink)

In his Essay concerning Human Understanding, John Locke explicitly refers to Newton’s Philosophiae naturalis principia mathematica in laudatory but restrained terms: “Mr. Newton, in his never enough to be admired Book, has demonstrated several Propositions, which are so many new Truths, before unknown to the World, and are farther Advances in Mathematical Knowledge” (Essay, 4.7.3). The mathematica of the Principia are thus acknowledged. But what of philosophia naturalis? Locke maintains that natural philosophy, conceived as natural science (as opposed to natural (...) history), would give us demonstrations of the necessary connection between the (ultimately, simple) ideas constitutive of our complex ideas of various natural kinds of substances (e.g., gold). Indeed Locke goes so far as to suggest that a completely adequate natural science would also realize (perhaps, per impossibile) the goal of transforming the corpuscularian hypothesis into knowledge by demonstrating the necessary connection between the ‘microstructure’ (primary qualities of insensible corpuscles) of a particular natural kind of substance (e.g., gold) and the ideas of secondary qualities constitutive of the complex idea of that kind of substance. Locke’s conclusion concerning the possibility of the development of a natural science thus conceived is pessimistic: In vain therefore shall we endeavor to discover by our Ideas, (the only true way of certain and universal knowledge,) what other Ideas are to be found constantly joined with that of our complex Idea of any Substance: since we neither know the real Constitution of the minute Parts, on which their Qualities do depend; not, did we know them could we discover any necessary connexion between them, and any of their Secondary Qualities: which is necessary to be done, before we can certainly know their necessary co-existence (Essay, 4.3.14). It is understandable that, with such a conception of the science of nature, Locke found little of it in Newton’s Principia. In this paper, I further explore what might, perhaps with some hyperbole, be termed Locke’s ‘disappointment’ with the Prinicipia as a contribution to natural science. In particular, I argue that Locke’s adherence to the idealist epistemology of the Way of Ideas entails that mathematics cannot lend its certainty as a scientia to natural philosophy. Consequently, he finds more mathematics than natural philosophy in the Principia. (shrink)

THIS PAPER PRESENTS THE SEMANTIC THEORY FOR A TEMPORAL-MODAL LOGIC WITH RIGIDLY REFERENTIAL TEMPORAL OPERATORS ('dtomorrow' AND 'dnow') IN WHICH THE 'TRADITIONAL' INDETERMINIST INTERPRETATION OF ARISTOTLE'S _DE INTERPRETATIONE 9 CAN BE MODELED. THIS LOGIC HAS, I BELIEVE, SOME INTRINSIC PHILOSOPHICAL INTEREST AND PLAUSIBILITY. HOWEVER, THE PRESENT PAPER IS PRINCIPALLY DEVOTED TO AN INITIAL EXAMINATION OF THE RELATION BETWEEN THE LOGIC AND SUCH TOPICS IN THE ANCIENT PHILOSOPHY OF THE TIME AND OF THE MODALITIES AS THE NECESSITY OF THE PAST, ABSOLUTE (...) VERSUS TEMPORALLY RELATIVE ALETHIC MODALITIES, THE 'PLENITUDE' PRINCIPLE AND UNACTUALIZED POSSIBILITIES, AND THE CYCLICAL NATURE OF TIME. (shrink)

This paper specifies classes of framesmaximally omnitemporally characteristic for Thomas' normal modal logicT 2 + and for each logic in the ascending chain of Segerberg logics investigated by Segerberg and Hughes and Cresswell. It is shown that distinct a,scending chains of generalized Segerberg logics can be constructed from eachT n + logic (n 2). The set containing allT n + and Segerberg logics can be totally- (linearly-) ordered but not well-ordered by the inclusion relation. The order type of this ordered (...) set is *( + 1). Throughout the paper my approach is fundamentally semantical. (shrink)

This chapter contains sections titled: Introduction: Mathematics and Philosophers – Plato, in Particular Mathematics and the Training of the Soul To Pythagoreanize or Not To Pythagoreanize Pythagoreanized Meta‐Mathematics and Ancient Mathematical Practice Mathematical Ontology Conclusion.