Building on previous scholarly work on the mathematical roots of assertoric syllogistic we submit that for Aristotle, the semantic value of the copula in universal affirmative propositions is the relation of divisibility on positive integers. The adequacy of this interpretation, labeled here ‘arithmetical dictum’, is assessed both theoretically and textually with respect to the existing interpretations, especially the so-called ‘mereological dictum’.

We consider semantic and syntactic transformations of the concept of "the logical" in the ancient philosophy in the form of crypto-logos, para-logismos, dia-logos, and syl-logismos. We interpret Heraclitus' concept of Logos as a cryptologos through which intuitive insight (epístasthai gnóomen) reveals hidden or implicit harmony (harmoníe aphanés) in nature (phýsis) as a conceptual unity of ontic opposites (tà enantía). In Pramenides' paraconsistent concept of the identity of Being and thought, we point to para-logical hypotheses about the One that are carried (...) out through antithetical deductions of thought and which maintain the dynamics of the ontic determinations of being (ón) in the statics of the conceptual determinations of Being (tò eînai). As the beginning of the explicative granulation of ''the logical'' we consider Plato's concept of the dialectical skill (dialektikè tékhne) of dividing concepts of genus into species and sub-species that logically represent ontic opposites in problem-formulated questions. Finally Aristotle's concept of lógos as a statement-making sentence / proposition (lógos apophantikós) made explicit the Being (tò eînai), or the Being as Being (tò ón hê ón), in semantic and syntactic figures and modes of syllogistic inferences in which ontological (eînai), ontic (ón), conceptual (logikôos) and linguistic (légomenon) correspondence is shown. We conclude that with these changes in the concept of lógos, the path has been taken from the hidden or implicit Truth of the phenomena of nature and the world (pân) to explicit truthfulness of propositions as the unhiddeness (alétheia) of Being trough the semantical and syntactical visibility of the logical structures of being, thought and language in scientific knowledge based on demonstration (apoódeiksis). (shrink)

Discussions of Aristotle’s Metaphysics Z tend to treat it either as an independent treatise on substance and essence or as preliminary to the main conclusions of the Metaphysics. I argue instead that Z is central to Aristotle’s project of first philosophy in the Metaphysics: the first philosopher seeks the first causes of being qua being, especially substances, and in Z, Aristotle establishes that essences or forms are the first causes of being of perceptible substances. I also argue that the centrality (...) of Z to first philosophy is compatible with its status as theology. (shrink)

Aristotle’s theory of human happiness in the Nicomachean Ethics explicitly depends on the claim that contemplation (theôria) is peculiar to human beings, whether it is our function or only part of it. But there is a notorious problem: Aristotle says that divine beings also contemplate. Various solutions have been proposed, but each has difficulties. Drawing on an analysis of what divine contemplation involves according to Aristotle, I identify an assumption common to all of these proposals and argue for rejecting it. (...) This allows a straightforward solution to the problem and there is evidence that Aristotle would have adopted it. (shrink)

On Zeno Beach there are infinitely many grains of sand, each half the size of the last. Supposing Aristotle denied the possibility of Zeno Beach, did he have a good argument for the denial? Three arguments, each of ancient origin, are examined: the beach would be infinitely large; the beach would be impossible to walk across; the beach would contain a part equal to the whole, whereas parts must be lesser. It is attempted to show that none of these arguments (...) was Aristotle’s. Indeed, perhaps Aristotle’s finitism applied to magnitude only, not plurality. (shrink)

Ch.9 of Metaph. 12 gives no support to the common view (against which I have argued elsewhere) that in ch.7 Aristotle identifies his Prime Mover not only as a god but also as an intellect. Rather, ch.9 approaches the divinity of intellect as a common belief (ἔνδοξον) from the Greek philosophical and poetic tradition (as at ch.7, 1072b23) that now requires dialectical testing. Here Aristotle initially establishes that there is a most active intellect (proposed ch.7, 1072b18–19: demonstrated ch.9, 1074b17–21, b28–9), (...) and that it contemplates what is best (proposed loc. cit.: demonstrated ch.9, 1074b21–7). Thus ch.9 proceeds by deducing, as a new result, characteristics implying that this intellect is itself the Prime Mover, since its object, qua best, must be the god of ch.7, and divine intellect is essentially identical (1074b33–1074a5), and an indivisible unity (a5–10), with that object. (shrink)

Aristotle notoriously begins his examination of being in the sense ofdunamisandenergeiainMetaphysicsTheta with what he describes as the sense that is ‘most dominant’ but not useful for his present aim. He proceeds to define the not-useful sense ofdunamisas “the principle of change in something else or in itself qua other”, along with other senses derived from this primary sense. But what then is the useful sense? All that Aristotle tells us at the outset is that it is a sense that extends (...) “beyond things spoken of only in relation to motion” and nowhere in Book Theta does he explicitly identify the useful sense as such. This has allowed for very different interpretations of the useful sense in the literature, the primary ones being that it is (1) ‘possibility’, (2) the potential to receive a form, (3) being-potentially-x understood modally as encompassing all specific senses ofdunamis, and (4) capacity for substantial change. The present paper argues that there are significant problems with all of these suggestions and defends an identification of the useful sense ofdunamiswith the sense that Aristotle explicitly opposes to the not-useful sense at the start of Theta 8:phusis(‘nature’). This goes hand-in-hand with an identification of the useful sense ofenergeiawith ‘activity’ as distinguished from motion/change at the end of Theta 6. What makes these senses ofdunamisandenergeiathe useful ones for Aristotle’s present aim is that they are required to explain fully the priority ofenergeiatodunamisin substance defended in Theta 8, they support the identification in Theta 9 ofenergeiawith the good, and they explain the unity of the only substances that Aristotle recognizes as being genuinely substances: natural living substances. (shrink)

The present PhD dissertation aims to examine the relation between modality and change in Aristotle’s metaphysics. -/- On the one hand, Aristotle supports his modal realism (i.e., worldly objects have modal properties - potentialities and essences - that ground the ascriptions of possibility and necessity) by arguing that the rejection of modal realism makes change inexplicable, or, worse, banishes it from the realm of reality. On the other hand, the Stagirite analyses processes by means of modal notions (‘change is the (...) actuality of what is virtual insofar as it is virtual’). In other words: to grasp what change is, one has to resort to the modal idiom of potentialities, while the fact that there is change is indicative of the fact that nature is full of modal properties. -/- Aristotle’s modal and kinetic realism finds a negative in the figure of the Megaric Diodorus Kronus. The polemical situation of Greek philosophy has indeed the dialectical advantage of not opposing the Aristotelian position to a sui generis straw man. Both in its reduction of modal judgements to temporal quantifications and in its dissolution of the reality of the state-of-being-in-motion in favour of a cinematographic view of processes, the philosophical figure of reductionist antirealism embodied by Diodorus constitutes an alternative that takes the opposite view of Aristotle’s realism. -/- The present study is structured as a discussion between these two positions. In the face of Diodorus’ antirealist challenges, Aristotle articulates modal and kinetic considerations: the answers he gives to Diodorus’ puzzles thus provide valuable insight into his metaphysics. Moreover, the examination of Aristotle’s and Diodorus’ metaphysics, because of their insight and originality, will not fail to interest the philosopher concerned with the metaphysical foundations of modern physics, insofar as the entanglement between modalities and processes is nowadays at the core of mechanics (phase spaces, path integrals, etc.). (shrink)

Already in ancient philosophy, there was a transition from the implicit and hidden action of the Logical ( lógos) in nature ( phýsis) to the scientific and explicit expression of the logical structures of thought, action, the world and language. Heraclitus' heno-logic with Logos as hidden implicit principle of homologization of opposites ( tà enantía) in nature differs from Parmenides' paraconsistent logic developed in a hypothetical hemidyalectics given in the formula ''All is One'' ( hén pánta eînai). Plato's concept of (...) dia-logic (dialektikè téchne ) with a new concept of Logos as the one genus of beings ( hén tí génon toôn ontoôn) in which the word not-Being (negation) got its place enabled production of dyadic logical structure by the granulation of genera into opposite species and sub-species that it contains. Aristotle's concept of triadic-logic as syl-logistics ( syllogismós) and demonstrative science (epistéme apodeiktikê ) give a new approach by new granulation of the concept of Logos into triadic logical structure: (1) the structure of being (substratum-attributes relation), (2) the structure of thought (substance-second substances relation), and (3) the structure of propositions (subject-predicate relation). Plato's dialectic and Aristotle's syllogistic both deconstructed the implicit ontological unity of the world (pan, kosmos, sphairos ) given through the concept of Logos in Pre-Socratic philosophy in order to make that unity in explicit form given by the logical and semantical structures of the propositions about the world, about the thought and about the language. The hidden implicit of the nature, which had to be known intuitively, was transformed into unhidden explicit inferential logical structures given in the semantics and pragmatics of scientific demonstration. (shrink)

In Metaphysics B 3 Aristotle sets out a famous argument for the thesis that being is not a genus. In his commentary on Metaphysics B, Syrianus criticizes this argument and explains in what sense being is to be regarded as a genus. I reconstruct both Syrianus’ criticisms and his own view. I bring out ways in which they can help us rethink key assumptions of Aristotle’s ontology and shed light both on Syrianus’ critical attitude towards Aristotle and on some of (...) Syrianus’ main views in metaphysics. (shrink)

Although Aristotle’s Metaphysics received much attention in the nineteenth and the twentieth centuries, scholars and historians of science were not particularly interested in clarifying the aim of Aristotle’s appeal to astronomy in Λ 8. Read with monotheistic prejudices, this chapter was quickly abandoned by Aristotelian scholars as a gratuitous insertion, which downgrades Aristotle’s God for the sake of some supplementary principles, whose existence was dictated by celestial mechanics. On the other hand, historians of astronomy read the astronomical excursus as providing (...) a picture of Aristotle as an able astronomer, who made an important contribution to the theory of concentric celestial spheres of Eudoxus and Callippus by adding the counteracting spheres. The present article argues (a) that Aristotle purposefully turned to astronomy as the only mathematical science whose objects were correlative to the immaterial first substances or gods, the number of which had to be precisely determined by his own project of first philosophy; and (b) that he had no aspiration of improving the astronomical theory of his peers, contrary to an interpretation that first emerged with the Peripatetic exegete Sosigenes in the second century A.D. (shrink)

Metaphysics Θ treats potentiality (δύναμις) and actuality (ἐνέργεια), and many scholars think that Aristotle broaches these topics once he has answered his main questions in Ζ and Η. In Ζ he asked, what is primary being? After arguing in Ζ.1 that substance (οὐσία) is primary being—a being existentially, logically, and epistemologically prior to quantities and qualities and other categorial beings—he devotes the rest of the book to οὐσία itself, investigating what it is, to decide what entities count as primary substances. (...) I differ from the leading interpretative consensus that ΖΗ adequately answer the questions about primary substance, and contend instead that Metaphysics Θ continues the same investigation as ΖΗ and, using δύναμις and ἐνέργεια as tools, arrives at a striking new conception of hylomorphism, different from that in ΖΗ, which enables Aristotle to defend the substantial primacy of living organisms consisting of matter and form. (shrink)

In the first argument of Metaphysics Μ.2 against the Platonist introduction of separate mathematical objects, Aristotle purports to show that positing separate geometrical objects to explain geometrical facts generates an ‘absurd accumulation’ of geometrical objects. Interpretations of the argument have varied widely. I distinguish between two types of interpretation, corrective and non-corrective interpretations. Here I defend a new, and more systematic, non-corrective interpretation that takes the argument as a serious and very interesting challenge to the Platonist.

Chapter Θ10 of Aristotle's Metaphysics is traditionally taken to be about the truth of intuitions, namely episodes of an immediate and sub-propositional grasp of entities. This exegesis however saddles Aristotle with a broken-backed theory of truth because in other passages of his works he claims that truth and falsehood apply only to items of a propositional nature and denies that sub-propositional items can be true or false. An alternative exegesis is preferable which takes Θ10 to be about the truth of (...) existential statements and judgements concerning simple items. (shrink)

We critically reconsider an old hypothesis of the role of the dekad in Pythagorean philosophy. Unlike Zhmud, we claim that: 1) the dekad did play a role in Philolaus’ astronomical system, and 2) Aristotle did not project Plato’s theory of the ten eidetic numbers onto the Pythagoreans. We claim that the dekad, as the τέλειος ἀριθμός, should be understood in Philolaus’ philosophy as completeness and the basis of counting in Greek – as in most other languages – in a decimal (...) system. Additionally, we argue that the number ten is not even a candidate for the τέλειος ἀριθμός in Plato’s philosophy. As a final result of our discussion, we compare and contrast Philolaus’, Plato’s, and Speusippus’ accounts of completeness in relation to numbers. (shrink)

In this paper, I attempt to set out some of the main views about essence which Aristotle puts forward in Book Z of theMetaphysics.A central feature of Aristotle's account, as we shall see, is his distinction between primary and secondary cases of essence, and a similiar distinction in connection with the related notions of substance and definition. This division between primary and secondary cases means that Aristotle may well have two very different sorts of remark to make about essence, or (...) definition, or substance, according to which of the two kinds of case he has in mind. There is no reason to suppose the same logical properties apply to both the primary and the secondary cases. And when we come to define the various notions, we shall see that if our definitions are right, the logical properties in the two kinds of case are in fact quite different. (shrink)

InMetaphysicsΘ.6, 1048b14-17, Aristotle treats the problem of what it is for the infinite to exist potentially, i.e. to be potentially actual. According to my interpretation, Aristotle argues that to exist potentially is for the infinite to have a potentiality which cannot be actualized in reality but only in thought, because it is a potentiality the process of whose actualization cannot be brought to an end.

I discuss Aristotle’s opening argument against Platonic Forms in _Metaphysics_ A.9, ‘the Razor’, which criticizes the introduction of Forms on the basis of an analogy with a hypothetical case of counting things. I argue for a new interpretation of this argument, and show that it involves two interesting objections against the introduction of Forms as formal causes: one concerns the completeness and the other the adequacy of such an explanatory project.