We discuss saturating ultrafilters on N, relating them to other types of nonprincipal ultrafilter. (a) There is an (ω,c)-saturating ultrafilter on $\mathbf{N} \operatorname{iff} 2^\lambda \leq \mathfrak{c}$ for every $\lambda and there is no cover of R by fewer than c nowhere dense sets. (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafilter on N is (ω,κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω,κ)-saturating, and (ii) the set of (ω,κ)-saturating ultrafilters is invariant under (...) homeomorphisms of $\beta\mathbf{N\backslash N}$ . (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω,c)-saturating. (shrink)

This paper and its predecessor () are about the question: 'Are the events in the entire universe encoded in and predictable from any of its parts?' To approach a positive answer in classical physics, the following result is proved and commented on: in Newton's theory of gravitation, the entire trajectory of a particle can be predicted given any segment of it, regardless of how the other particles are moving-provided that there is only a finite number of particles and that their (...) speeds remain bounded. (It is this condition, together with a set of parameters characterising the motion of the other particles, which enables us to estimate the effect of the other particles on the trajectory of the given particle.) The extension of this result to other theories, in particular to special relativity, is discussed. (shrink)

Orestes has returned to Argos, . For him to brandish at his father's murderers is natural there, where he is delivering a sort of general manifesto as to his aims, and where the strong word is justified and alleviated by the jingle with juxtaposed . But there is no reason for Orestes to go on insisting on the bloodthirstiness of these aims, and reads oddly in 100, where he is explaining soberly his plan of campaign.

In Posterior Analytics II 19 Aristotle likens the way in which sense perception gives rise to knowledge of the universal to the way in which one soldier’s ceasing his flight from the enemy leads other soldiers to do the same ‘heôs epi archên êlthen.’ Although the remark seems intended to characterize knowledge as the end result of an accumulative process, the concluding reference to ‘a starting point’ or archê has no clear meaning. I argue that the phrase can be plausibly (...) understood in light of Aristotle’s explanation of how a single calm thought can spread calm to other thoughts (Problems 917a29-33) and a single cloud can serve as a starting point (archê) for a revival of stormy weather (Problems 941a9-13). In addition, Aristotle elsewhere (e.g. Generation of Animals 788a13-16) speaks of the archê as the person or thing which initiates a process. Thus, when one soldier’s stopping causes another soldier to stop, this process can continue ‘until they come to the one who initiated the process.’ . (shrink)

The application of λκθος ànd its derivatives and the Latin terms ampullae and ampullari to the turgid or elevated style of poetry or oratory has provoked such a variety of explanations amongst modern and ancient commentators that it would be a tedious business to examine them all in detail. The ancient commentators on Horace, Ars Poetica, 11. 93–7 interdum tamen et vocem comoedia tollit, iratusque Chremes tumido delitigat ore; et tragicus plerumque dolet sermone pedestri Telephus et Peleus, cum pauper et (...) exsul uterque proicit ampullas et sesquipedalia verba… were not even in agreement on the question of the precise correspondence of the two words, for while Porphyrion said that proicit ampullas was imitated directly from the ληκθειος Mοcomflex;σα of Callimachus, Pseudacron's note makes no reference to ληκθζειν at all, but merely connects the metaphor with the inflated appearance of the Roman ampulla. And though modern scholars do not question the fact that the two words do correspond in some degree, there has been a wide divergence of opinion on the derivation and meaning of the metaphor. Ritter and Wilkins, for example, held that the meaning ‘paint-pots’ was indicated by the conjunction of ληκθος with pingere in one of Cicero's letters to Atticus, and that in Horace, Epistles 1. 3. 12–14. (shrink)

All commentators on these lines make two assumptions about the first clause, that means ‘sitting in judgement’, ‘punishing’, or the like, that the which is its subject as well as that of is the second in a series of two: the subsequent slaying punishes or sits in judgement on the previous; thus the slaying of Cly taemnestra herself will sit in judgement upon that of Agamemnon, just as that had sat in judgement upon the of Iphigenia. Then opinions differ as (...) to whether is to be taken as the object of both verbs, or of one but not the other. (shrink)

One of the earliest and best-known of Bentham's works, the Introduction to the Principles of Morals and Legislation sets out a profound and innovative philosophical argument. This definitive edition includes both the late H. L. A. Hart's classic essay on the work and a new introduction by F. Rosen.

La science des nombres infinis, introduite par Cantor, semble être en conflit direct avec la philosophie, qui pose que le nombre infini est contradictoire.Nous examinerons les objections contre le nombre transfini. Nous arriverons au résultat, que le nombre infini, conçu comme symbole, satisfaisant à un certain nombre de postulats, n’est pas contradictoire, tandis que le nombre infini, déterminé, actuel et entier, a bien des propriétés contradictoires.Les objections contre le nombre infini reviennent à ce qu’on suppose tacitement, que le nombre infini (...) doit avoir les propriétés des nombres finis ; et ensuite, on démontre facilement que le nombre infini est contradictoire.Il n’est pas permis d’appliquer la logique classique, qui est une résultante des propriétés des ensembles finis, aux nombres infinis, ce qui serait contraire à une logique plus générale, qui prescrit que l’application de certaines lois de la pensée ne doit pas excéder le domaine propre de ccs lois. (shrink)