Reflections on some aspects of Richard McKirahan's 2019 lectures on "An Aristotelianizng Parmenides." This chapter offers a reconstruction of Aristotle's representation of Melissus.

Aristotle is said to have held that any kind of actual infinity is impossible. I argue that he was a finitist (or "potentialist") about _magnitude_, but not about _plurality_. He did not deny that there are, or can be, infinitely many things in actuality. If this is right, then it has implications for Aristotle's views about the metaphysics of parts and points.

In contemporary metaphysics, the Eleatic Principle (EP) is a causal criterion for reality. Articulating the EP with precision is notoriously difficult. The criterion purportedly originates in Plato’s Sophist, when the Eleatic Visitor articulates the EP at 247d-e in the famous Battle of the Gods and the Giants. There, the Visitor proposes modifying the ontologies of both the Giants (who are materialists) and the Gods (who are friends of the many forms), using a version of the EP according to which only (...) items which have the capacity to affect or to be affected are real. Recently, it has been argued that while there are some genetic connections between the EP and the views of some of the historical Eleatics (Parmenides and Zeno), the views of Melissus are generally incompatible with the EP. This raises the following question: What, exactly, is Eleatic about the Eleatic Principle? In this paper, I look to the dialectical context in which the Visitor’s appeal to the EP appears, and I propose that there are at once three relevant senses of affecting and being affected in the text: (1) tangible contact (as in the interactions between the bodies of the materialists), (2) Cambridge change (as in the form's being affected by a soul via the act of coming to know), and (3) the notion of a cause that does not itself change (as in a form’s affecting another form, a soul, or any sensible item). I argue that these have clear parallels to the metaphysics of all three main Eleatic philosophers, and that the views of Parmenides, Zeno, and Melissus are therefore compatible with the EP. This has interesting consequences for narratives about the reception of the Eleatics in Plato, the inclusion of an Eleatic Visitor as an interlocutor in the Sophist, and for the metaphysics of subsequent schools (such as the Stoics) who are widely thought to have been influenced by the EP. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other hand, (...) Aristotle says that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

This work deals with problems involving infinities and infinitesimals. It explores the ideas behind zero, its relationship to ontological nothingness, finititude (such as finite numbers and quantities), and the infinite. The idea of infinity and zero are closely related, despite what many perceive as an intuitive inverse relationship. The symbol 0 generally refers to nothingness, whereas the symbol infinity refers to ``so much'' that it cannot be quantified or captured. The notion of finititude rests somewhere between complete nothingness (...) and something having no end. -/- My concern is that many of the philosophers arguing for or against the ontology (or possibility) of an actual infinite set are unaware or unfamiliar with the mathematical literature attempting to clearly and rigorously define these terms. I believe it is a mistake to leave mathematicians out of this conversation, as analysts in particular have defined (and used) infinity in a way that is relevant to the ongoing debate between philosophers. -/- For this paper, I have selected examples from Principles of Analysis by Walter Rudin, which has become a standard text for Real Analysis classes taken by pure mathematicians and engineers. I will also restrict the scope of examples to Euclidean spaces for simplicity. Finally, I will be using addition, + and multiplication * in their usual way. (shrink)

“Why did God create the World?” is one of the traditional questions of theology. In the twentieth century this question was rephrased in a secularized manner as “Why is there something rather than nothing?” While creation - at least in its traditional, temporal, sense - has little place in Spinoza’s system, a variant of the same questions puts Spinoza’s system under significant pressure. According to Spinoza, God, or the substance, has infinitely many modes. This infinity of modes follow from (...) the essence of God. If we ask: “Why must God have modes?,” we seem to be trapped in a real catch. On the one hand, Spinoza’s commitment to thoroughgoing rationalism demands that there must be a reason for the existence of the radical plurality of modes. On the other hand, the asymmetric dependence of modes on the substance seems to imply that the substance does not need the modes, and that it can exist without the modes. But if the substance does not need the modes, then why are there modes at all? Furthermore, Spinoza cannot explain the existence of modes as an arbitrary act of grace on God’s side since Spinoza’s God does not act arbitrarily. Surprisingly, this problem has hardly been addressed in the existing literature on Spinoza’s metaphysics, and it is my primary aim here to draw attention to this problem. In the first part of the paper I will present and explain the problem of justifying the existence of infinite plurality modes in Spinoza’s system. In the second part of the paper I consider the radical solution to the problem according to which modes do not really exist, and show that this solution must be rejected upon consideration. In the third and final part of the paper I will suggest my own solution according to which the essence of God is active and it is this feature of God’s essence which requires the flow of modes from God’s essence. I also suggest that Spinoza considered radical infinity and radical unity to be roughly the same, and that the absolute infinity of what follow from God’s essence is grounded in the absolute infinity of God’s essence itself. (shrink)

Predication is a lingual relation. We have this relation when a term is said (λέγεται) of another term. This simple definition, however, is not Aristotle’s own definition. In fact, he does not define predication but attaches his almost in a new field used word κατηγορεῖσθαι to λέγεται. In a predication, something is said of another thing, or, more simply, we have ‘something of something’ (ἓν καθ᾿ ἑνὸς). (PsA. , A, 22, 83b17-18) Therefore, a relation in which two terms are posited (...) to each other in a way that one is said (predicated) of the other is a predication. The term of which the other term is said is called a subject (ὑποκειμένον) and the other, which is said about the subject, is called the predicate. Thus, in a predication the predicate is predicated of the subject; given that being predicated is almost as the same as being said. The relation between being said and being predicated is so close that ‘if something is said of a subject both its name and its definition are necessarily predicated of the subject.’ (Cat., 5, 2a19-26) This, however, is true only about the second genera and not the accidents. a) Nature of relation in predication What is Aristotle’s theory about the nature of the relation in a predication? How does he fundamentally understand this relation? Phil Corkum distinguishes between predication logic and traditional term logic and argues that the relation between subject and predicate in Aristotle is of the latter kind. While in predicate logic, subjects and predicates have distinct roles, they have the same role in traditional term logic. In predication logic, subjects refer, but predicates characterize. Thereupon, a sentence expresses a truth if the object to which the subject refers is correctly characterized by the predicate. In traditional term logic, both subjects and predicates refer and a sentences expresses a truth if both name one and the same thing. He concludes that Aristotle ‘problematically conflates prediction and identity claims’ because while he thinks both subjects and predicates refer, he would deny that a sentence is true just in case the subject and the predicate name one and the same thing. Based on this, Corkum believes that Aristotle’s core semantic is not identity but the weaker relation of constitution, which is a mereological interpretation: ‘All men are mortal’ is true just in case the mereological sum of humans is part of the mereological sum of mortals. b) Aristotle’s theory of predication: one or two theories? Frank Lewis finds an inconsistent gap between the theory of predication in the Categories and that in the later books of the Metaphysics VII, 6. So too Joan Kung, Terry Irwin, Daniel Graham. c) Distinction of ‘being said of’ and ‘being in’ Owen enumerates the texts in which Aristotle’s distinction between ‘being said of’ and ‘being in’ is asserted: OI. 11b38-12a17; To., 127b1-4; Cat. 1a20-b9; 2a11-14; 2a27-b6; 2b15-17; 3a7-32; 9b22-24 d) Predication in Aristotle and the standard ‘S is P’ Marie De Rijk Lambertus thinks that the ‘S is P’ pattern is misleading when it comes to express predication in Aristotle. In his view, ‘The Aristotelian procedure should be described in terms of appositively assigning an attribute (κατηγορούμενον) to a substrate (ὑποκείμενον), rather than ascribing a predicate to a subject by means of a copula…. The comment should be considered an attribute which is said to fall to a substrate, without understanding this procedure in terms of sentence predication.’ e) Aristotle’s predication: bipartite or tripartite? It is a difficulty to make Aristotle’s theories of name-verb predication and tripartite predication consistent. The reason is that tripartite is not consistant with his name-verb structure based on which he says that the predicate term is the ‘verb.’ (20a31; 20b1-2; 16a13-5) The problem is that in a tripartite form like ‘Socrates is white’ we cannot take ‘is white’ as the verb because, based on Aristotle himself, no part of a verb can be significant itself. (16b6-7) However, his assertions about the equivalences between the two structures (OI. 12, 21b9-10; PrA. 51b13-16; Met., 1017b22-30) must mean that they are not inconsistent in his own view. Thus, as Allen Bāck points, ‘Aristotle seems to think that he has a single, consistent theory.’ The sense of saying or speaking as the root sense of rhema makes the relation between predication (saying something of something) and verb more interesting. The use of rhema in the sense of a long expression approves this. In Plato’s work (399ac) where Socrates claims that the name anthropos derives from anthron ha opopen (‘he who examines what he has seen’) we have an onoma from, or in place of, a rhema. 1) Subject The subject (ὑποκειμένον) is that of which another term is said or predicated. Aristotle’s own definition is of a much more philosophical value: ‘By subject I mean that which is expressed by an affirmative term (λέγω δὲ ὑποκείμενον τὸ κάταφάσει δηλούμενον).’ (Met., K, 1067b18) Throughout Aristotle’s work, three kinds of subjects can be found: possible, prime and absolute subject. a) Possible subject Those things that can take the position of a subject in a predication we call possible subjects, no matter they can or cannot take the position of predicate in any predication. All terms except the ultimate predicates are possible subjects because they can take the position of subject. b) Prime subject Those that in any predication can only take the position of a subject and not the position of a predicate are prime subjects. (Cat., 5, 3a36-b2; Met., Z, 1028b36-1029a1) This is the truest sense of subject and belongs only to substances. (Met., Z, 1029a1-2) In fact, ‘that which is not predicated of a subject, but of which all else is predicated’ is a substance. (Met., Z, 1029a7-9) In Categories, besides primary substances (Cat., 2, 1b3-6), individual qualities are also said not to be able to be said of anything else. (Cat., 2, 1a23-29) Aristotle’s examples are a certain ‘knowledge-of-grammar’ (τὶς γράμματικὴ) and ‘an individual white’ (τὸ τὶ λευκὸν). Thus, although substances are indeed in the truest sense individuals and, thereby, in the truest sense primary subjects, other individuals can take the position of subject as well. c) Absolute subject While substances are prime subjects of all other things, there is still something of which substances can be predicated and this predication is not an accidental predication: matter. (Met., Z, 1029a21-24) This predication is, indeed, the predication of a ‘form or this’ on matter and material substance. (Met., Θ, 1049a34-36) The only thing that is absolutely a subject, therefore, and can be a predicate is prime matter (πρώτη ὓλη). (Met., Θ, 1049a24-27) d) Subject and substance The most important thing in being a substance is being a subject: ‘It is because the primary substances are subjects for all the other things and all the other things are predicated of them or are in them, that they are called substances most of all.’ (Cat., 5, 2b15-17) And what is nearer to this position is more a substance as a species is more a substance than a genus.’ (Cat., 5, 2b7-8) e) Primary versus accidental subject A subject is a primary subject in a predication when the predicate is predicated of it as itself. In such a case, the subject is the subject of the predicate qua itself and not qua something else. A subject, on the one hand, is an accidental subject when it is not the primary subject of the predicate that is predicated on it. It means that a subject is an accidental subject when the predicate is predicated of it not qua itself but qua something else. An accidental subject is, therefore, anything other than a substance. 2) Predicate as universal It is evident from our discussion of kinds of subjects that except matter, primary substances (if we ignore both accidental predication and cases of absolute subjects) and other individuals, everything else can take the position of a predicate. Since primary substances are individuals, it results that everything except matter and individuals are capable to take the position of predicate. The immediate consequence of this is that the predicate must be a universal. When individuals cannot take the position of predicate, this position cannot be taken by any particular. Therefore, only universals can be predicated of others. Moreover, albeit universals can be a simple subject (for another universal), they are not prime subjects. The closeness of universal and predicate is to the extent that Aristotle differentiates universal from subject. (Met., Θ, 1049a27-30) 3) Categories or classes of predicates Aristotle distinguishes ten highest classes into one of which each predicate must necessarily fall: substance (or what a thing is), quality, quantity, relation, place, time, position, state, activity and passivity. (Cat., 4, 1b25-27; To., I, 9, 103b20- ; PsA., A, 22, 83b13-23) The substance counted among predicates must refer to secondary and not primary substances not only because Aristotle insists that primary substances cannot be predicated but also from his own examples of the category of substance: man and animal. (To., I, 9, 103b25) However, if a predicate be asserted of itself or its genus be asserted of it, the predicate signifies substance or what is; but if ‘one kind of predicate is asserted of another kind,’ it signifies one of the other nines. (To., I, 9, 103b30) These categories are so comprehensive that no predicate can remain outside them so that even non-being has as many senses as categories. (Met., M, 1089a26-30) a) That whether Aristotle’s categories are merely linguistic or fundamentally ontological is a so much controversial dispute. Some like G. E. R. Lloyd believe that ‘categories are primarily intended as a classification of reality … rather than of the signifying terms themselves.’ b) Although Aristotle’s categories have been historically regarded as a classification of predications, there are some recent commentators like Jonathan Barnes that think it is classifying predicates and not predications. 4) Kinds of predicates In his introduction of Categories, as J. W. Thorp truly points out , Aristotle distinguishes between the category of substance and the other categories using μὲν ... δὲ construction. This construction illustrates that beside tenfold classification of categories, there is an even more crucial differentiation between two kinds of predicates: substance or what is on the one hand and the other nine ones on the other hand: the distinction between what is predicated of the subject as what it is and what is predicated of it as an accident. Therefore, we have three kinds of classification - a twofold, a fourfold and a tenfold-complicatedly classifying the same thing, viz. the predicate. Predicates can be divided also to four kinds: property, definition, genus and accident. (To., I, 4, 101b11-20) A property is that predicate that is convertible with its subject and does not signify its essence. A definition is that predicate that is convertible with its subject and signifies its essence. It is a genus if it is not predicated convertibly and is contained in the definition of its subject. And it is an accident if it is neither convertible nor contained in the definition of its subject. (To., I, 8, ^103b3) All of these four kinds are among categories. (To., I, 9, 103b20) It seems that there might be some kind of relation between the twofold and the fourfold distinction: whereas genus and definition are predicated as what it is, accidents are not so predicated. There seems to be some rules that determine to each of these four kinds each predicate belongs. At To., IV, 123b30ff. Aristotle asserts: ‘If B has a contrary and A does not, then B does not belong to A as its genus.’ There is a dispute around per se accidents (τὰ καθ’ αὑτὰ συμβεβηκότα) (Topics., 102a18) whether they must be regarded as either of the four kinds or as a fifth kind. Barnes (1970) argued that they ‘do not fit at all neatly into the fourfold classification.’ He argues that based on the definition of accidents at A, 5, 102b4-5, they must be accidents but based on another definition, A, 5, 102b6-7, they cannot be accidents. He also defends the view that they are not properties. Barnes concludes: ‘This shows that the two definitions are not equivalent, and hence that the ‘predicables’ are not well-defined.’ Demetris J. Hadgopoulos defends the view that the two definitions of accidents are equivalent and per se accidents are properties. 5) Five types of predications We have five kinds of predications based on our division of subjects and our discussion of predicates: simple, primary (itself divided to substantial and accident), accidental (itself divided to primary and secondary), aoincidental and absolute predication. a) Simple predication. This is a predication in which a universal is predicated of either a particular or a universal subject. This sense of predication includes all other senses except the first kind of accidental predication. b) Primary predication. This is a predication in which a universal is predicated of a primary subject, namely a substance. A primary predication is of two kinds: i) Substantial predication. A primary predication in which the predicate is part of, or is included in, the definition of the subject. The universal that is the predicate here is the definition or the genus or the species or the diferentia of the subject. It is this predication that Aristotle calls a unity: ‘A statement may be called a unity … because it exhibits a single predicate as inhering not accidentally in a single subject.’ (PsA., B, 10, 93b35-37) It is only substantial predicates that can be predicated of each other: ‘Predicates which are not substantial are not predicated of one another.’ (PsA., A, 22, 83b18-19) Aristotle defines a substantial predication also in another way. A predication in which a higher genus is predicated on a lower genus, species, a substance or an individual, (or a species is predicated on an individual) is a substantial predication. Therefore, a substantial predication is a predication in a series that has individuals on one of its ends and a general category on its other end. In fact, only substantial predicates can be predicated of one another. (PsA., A, 22, 83b17-19) This series cannot be an infinite series because otherwise not only substances would not be definable but also a genus would be equal to one of its own species. (PsA., A, 22, 83b7-10) -/- ii) Accident predication. A primary predication in which the predicate is not part of, or is not included in, the definition of its subject. The universal that is the predicate here is the property or the accident of the subject. c) Accidental predication. This is a predication in which the subject is not a primary subject and its predicate is a substance. This predication is of two kinds: i) Primary accidental predication. This is a predication in which an accident is predicated of substance. Such a predication can go ad infinitum, which is inferable from Aristotle’s assertion that the secondary accidental predication cannot go ad infinitum. ii) Secondary accidental predication. It is a predication in which an accident is predicated of another accident. Such a predication, Aristotle asserts, cannot go ad infinitum because even more than two accidents cannot be combined. Now what is the meaning of Aristotle’s assertion that we cannot continue this ad infinitum? Does it mean that we cannot continue the predication ‘The musician is white’ and say ‘The white is Athenian?’ But why not? Or maybe he means that by adding any predication, we are still in a condition of predicating two accidents of a substance on each other. -/- As primary subjects, substances can also take the position of predicate though not essentially but accidentally. In propositions like ‘the white is a log’ or ‘That big thing is a log’ we observe substance taking the position of predicate but this is only accidentally: ‘When I affirm ‘the white is a log,’ I mean that something which happens to be white is a log- not that white is the subject in which log inheres (οὐχ ὡς τὸ ὑποκείμενον τῷ ξύλῳ τὸ λευκόν ἐστι), for it was not qua white or qua a species of white that the white (thing) came to be a log (οὔτε λευκὸν ὄν οὔθ᾿ ὃπερ λευκόν τι ἐγἐνετο ξύλον), and the white (thing) is consequently not a log except incidentally.’ (PsA., 22, 83a3-9) Aristotle compares this accidental use of substance in the position of predicate with the substantial use of it in the place of a subject: ‘On the other hand, when I affirm ‘the log is white,’ I do not mean that something else, which happens to be a log, is white (οὐχ ὃτι ἓτερόν τί ἐστι λευκόν, ἐκείνῳ δὲ συμβέβηκε ξύλῳ εἶναι), (as I should if I said ‘the musician is white,’ which would mean ‘The man who happens also to be a musician is white’); on the contrary, log is here the subject- the subject which actually came to be white, and did so qua wood or qua a species of wood and qua nothing else.’ (PsA., 22, 83a8-14) Aristotle asks us not to call propositions like ‘The white is a log’ a predication at all or at least call them accidental predication instead of simple (ἁπλως) predication. (PsA., A, 22, 83a14-17) An accidental predication, in which we have a substance in the place of predicate, is different from an essential predication in that while the subject of an accidental predication is said to be the predicate not by itself but as something different with which it coincides, the subject of an essential predicate is said to be the predicate by itself and not because it is something else: ‘Since there are attributes which are predicated of a subject essentially and not accidentally- not, that is, in the sense in which we say ‘That white (thing) is a man,’ which is not the same mode of predication as when we say ‘The man is white’: the man is white not because he is something else but because he is man, but the white is man because ‘being white’ coincides with ‘humanity’ within one subject. Therefore, there are terms such as are essentially subjects of predicates. (PsA., A, 19, 81a24-29) This capability of ‘non-essentially and only accidentally’ being a subject does belong, in fact, to all sensible things. (PsA., A, 27a32-36) d) Coincidental predication. this is a predication in which none of the subject and predicate are a substance or an individual. In this predication, one of the accidents of a substance or individual is predicated of one of the other accidentals of the same substance or individual. For example, when it is said that ‘The white is musical,’ there is an individual, say Socrates, for which both of white and musical are accidents. e) Absolute predication. This is a predication in which a form or a substance is predicated of a matter or a material substance. Some commentators like Loux and Lewis regarded this predication as close to accident predication, both based on inherence. As R.M. Dancy points out, these predications make Aristotle’s theory of predication immanentist: not only accident predications are explained by the immanence of accidents in substances, some of the substantial predications, which were not explained in Categories, are also explained by the immanence of form in matter. It is interesting that in this kind of predication, it is the predicate and not the subject, which is in the strictest sense substance. As the central books of Metaphysics claim, the form is the substance. Thus, in this predication, substance gets away from the sense of ὑποκείμενον. Corkum mentiones 68a19 as the only instance where Aristotle explicitly claims that a term may be predicated of itself, a passage that is, in Corkum’s view, problematic. 6) Demonstrable predicates Those predicates are demonstrable that are so related to their subjects that there are other predicates prior to them predicable of their subjects (ἔτι δ᾿ ἄλλος, εἰ ὧν πρότερα ἄττα κατηγορεῖται). (PsA., A, 22, 83b32-34) 7) Series of predications Since it is possible for a predicate to be itself the subject of another predication, we can have a series of predications in which the predicate of a predication is the subject of the next predication. This series has the following features: a) It has a limit (in number) (PsA., A, 22, 83b14-16) on the side of subjects: there are subjects that cannot themselves be predicated (PsA., A, 27, 43a39-41), which are, as we noted, prime subjects: particulars, i.e. prime substances and other individuals. (PsA., A, 19-22) Aristotle calls them ultimate (ὓστατον). (PsA., 21, 82a36-b1) b) It has a limit (in number of kinds) (PsA., A, 22, 83b14-16) also on the side of predicates: there must be predicates that cannot be subjects. (PsA., A, 19-22; PsA., A, 27, 43a36-39) Aristotle calls them primary (πρῶτον). (PsA., A, 21, 82b1-4) These are the highest categories or the highest genera of categories. (PsA., A, 22, 83b14-16) c) The series has an escalating shape from mere subjects at its downside to mere predicates at its upside. It is Aristotle himself whos uses the words up and down in this sense. (e.g. PsA., A, 22, 83b2-3) d) The predications that lie between lowest and highest ones must be finite in number. (PsA., A, 19-22 especially: 20, 82a21-35) These have subjects and predicates, each capable of both of the roles of being subject and being predicated. A result of this fitness is that neither demonstration can go to infinity nor everything is demonstrable, the two points Aristotle always insist on. e) The upward side includes the more universal ones and the downward the more particular ones. (PsA., 20, 82a21-23) f) It follows from the above features that ‘neither the ascending nor the descending series of predications … are infinite.’ (PsA., A, 22, 83b24-25) g) Reciprocation and convertibility. Except in case of terms (i.e. subjects and predicates) that are at each of the ends of series of predications, namely ultimates and primaries, it is possible to reciprocate (ἀντιστρέφειν) terms and convert the predication. (PsA., A, 19, 82a15-20) h) Antipredication. Antipredication means that in a predication (S is P), the subject becomes the predicate of its predicate (P is S) in the same category its predicate was predicated on it. For example, if P is in the category of quality, antipredication means that S be predicated of P in the category of quality. In other words, P is a quality of S and S is a quality of P. Aristotle rejects this. (PsA., A, 22, 83a36-b3) i) Self-predication versus other-predication. Aristotle distinguishes between a predication in which a term is said of itself and a predication in which a term is said of another. (Phy., Δ, 2, 209a31-33) j) Predicablity (Cat., 5, 3a36-b2): 1) Primary substances and individuals are not predicable. 2) Secondary substances are of two kinds: genus and species i) Genus is predicable able both of the species and of the individuals. ii) Species is predicable of the individual. 3) Differentiae are predicable both of the species and of the individuals. 8) Predication as classification a) Richard Patterson believes that Aristotle’s so repeated construction using hoper (estin A hoper B) in Topics (120a23 sq., 122b19, 26sq., 123a, 124a18, 125a29, 126a21, 128a35. Also in Posterior Analytics (83a24-30) (Brunschwig’s list. ‘expresses the fact that A is a kind of B (esti A B tis), that A is a species of the genus B.’ 9) Universal predication Aristotle defines universal predication (κατὰ παντὸς κατηγορεῖσθαι) as such: ‘wherever no instance of the subject (τῶν τοῦ ὑποκειμέμνου) can be found of which the other term cannot be asserted.’ (PrA., A, 24b27-29) In this predication, the subject is included in another as in a whole (ἐν ὃλῳ εἶναι) and the predicate is predicated of all of the subject (κατὰ παντὸς κατηγορεῖσθαι). (PrA., A, 24b26-27) Attach another discussion we had about ἐν ὃλῳ here. 10) Quantity of predication A predication, truly stated, has a quantity, which is the number of objects under the name of the subject of which the predicate is predicated. The quantity of subject can be stated in four ways: a. Indefinite: when the predicate belongs or does not belong to the subject without any mark to show to haw many of the particulars under the name of the subject it does or does not belong. E.g. ‘Man is white’; ‘Man is not white.’ (PrA., A, 24a20; OI., I, 7, 17b8-12) b. Universal quantity: When the predicate belongs to all or none of the subject. E.g. ‘Every man is animal’; ‘No man is animal.’ (PrA., A, 24a18-19; OI., I, 7, 17b5-6) The contrary of a predication of a universal quality is a predication of a universal quality. E.g. the contrary of ‘Every man is white’ is ‘No man is white.’ (OI., I, 7, 17b20-23) c. Particular quantity: When the predicate belongs (or does not belong) to some of the subject. E.g. ‘Some men are white’; ‘Some men are not white.’ (PrA., A, 24a19-20) d. Single quantity: when the subject is a proper name of only one object. E.g. ‘Socrates is white’; ‘Socrates is not white.’ The contrary of this predication is of a single quantity: ‘Socrates is not white.’ (OI., I, 7, 17b38-18a4) 11) Convertibility of predication. Some predications are convertible, that is, it is possible to change the place of subject and predicate in a true proposition so that the converted proposition remains true. This is supposed to mean that given the truth of a predication, the truth of the converted predication is inferable. The convert form of a predication depends on its quantity. a. Indefinite quantity: This predication has no strictly true convert because its quantity is not stated. b. Universal quantity: It can be converted in two ways: i) A negative universal can be converted to a negative universal in which the terms are changed. E.g. ‘No pleasure is good’ can be converted to ‘No good is pleasure.’ (PrA., A, 2, 25a5-8 and a14-19) ii) An affirmative universal can be converted to an affirmative particular in which the terms are changed. E.g. ‘Every pleasure is good’ can be converted to ‘Some good is pleasure.’ (PrA., A, 2, 25a7-9) c. Particular quantity: If it is negative, it cannot be converted but if it is affirmative, it can be converted to an affirmative predication with particular quantity. E.g. ‘Some pleasure is good’ can be converted to ‘Some good is pleasure.’ (PrA., A, 2, 25a10-13 and a20-24) d. Single quantity: It cannot be converted. 12) Characteristics of relations between subject and predicate 1. A predicate is of a wider range than its subject. It is based on this fact that Aristotle: a) Prevents individuals to be predicate because there is nothing of which an individual be of a wider range. In other words, it is due to the fact that since an individual is only ‘one particular’ thing and, thus, cannot be of a wider extent than anything that Aristotle prevents them of being a predicate. Matthews, however, thinks we cannot use Socrates, a substance and an individual, in the place of predicate and say it of Socrates because ‘Socrates does not classify Socrates: it names him.’ b) Prevents differentia, species and things under species to be predicated of genus. (To., Z, 6, 144a27-) c) Prevents the species and the things under it to be predicated of the differentia. (To., Z, 6, 144b1-4) d) Also about the effect because it is wider than its subject. (PsA., B, 17, 99a) e) Each attribute is wider than every individual it is predicated on, though several attributes, collectively considered, might not be wider but exactly the substance of a thing. (PsA., B, 13, 96a32-b1) 2. The predicate of a predicate of a subject will be predicated of the subject too: ‘whenever one thing is predicated of another as of a subject, all things said of what is predicated will be said of the subject too.’ (Cat., 3, 1b10-15) In fact, it is due to its predication of the subject that it is predicated of its predicate. (Cat., 5, 2a36-b1) Moreover, what is not predicated of the predicate of a subject cannot be predicated of it as well. (PrA., A, 27, 43b22-27) Thus, what, for example, is not predicated of animal, cannot be predicated of man. 3. The predicate of a subject can be predicated of its predicates as well. (Cat., 5, 3a1-6) For example, you call the individual man grammatical and, thence, you call both a man and an animal grammatical. Nonetheless, the predicate of a subject belongs to it more properly than to its higher predicates. (PrA., A, 27, 43a27-32) 4. It is only the subject that can be distributed and not the predicate. (PrA., A, 27, 43b16-22; OI, I, 7, 17b12-16) Therefore, we can say e.g. ‘Every man is animal’ but we cannot say ‘Every man is every animal.’ 5. It is the reason of the relation between subjects and predicates, that is the reason of predication, which is the subject of inquiry. (Met., Z, 1041a20-24) In other words, since it is a meaningless inquiry to ask why a thing is itself (Met., Z, 1041a14-15), the only remaining meaningful inquiry is to ask why something is something else, i.e. to ask about the reason of predication. 6. A subject cannot categorize its predicate in the same category in which it is categorized by it. If e.g. A is a quality of B, B cannot be a quality of A. Therefore, there is no reciprocation in the same category. (PsA., A, 22, 83a36-39) 13) Characteristics of series of predications 1. A series of secondary accidental predication cannot go ad infinitum for not even more than two terms can be comnbined. For an accident is not an accident of an accident, unless it be because both are accidents of the same subject. (Met., Γ, 1007b1-4) 2. Infinite series cannot be traversed in thought. (PsA., A, 22, 83b6-7) 3. The predications of genera on each other must be ended and cannot go to infinity because otherwise not only substances would not be definable but also a genus would be equal to one of its own species. (PsA., A, 22, 83b7-10) Therefore, a series of predication of genera on each other must be limited on both sides. There must be an upward limit in general categories as well as a downward limit in individual because they cannot be predicated of others. Whatever lies between these limits can both be predicated of others and others be predicated of them. (PrA., A, 27, 43a36-43) 4. The order of predicates matters: it makes a difference whether the series be ABC or BAC. (PsA., B, 13, 96b25-32) . (shrink)

It is commonly assumed that Aristotle denies any real existence to infinity. Nothing is actually infinite. If, in order to resolve Zeno’s paradoxes, Aristotle must talk of infinity, it is only in the sense of a potentiality that can never be actualized. Aristotle’s solution has been both praised for its subtlety and blamed for entailing a limitation of mathematic. His understanding of the infinite as simply indefinite (the “bad infinite” that fails to reach its accomplishment), his conception of (...) the cosmos and even its prime mover as finite (in the sense of autarchic/self-contained) have been contrasted with the subsequent claim of God as ens infinitum (understood as a “positive” infinity). The goal of this essay is to reexamine the major texts (notably De caelo) and to demonstrate that (1) Aristotle’s claim according to which there is no actual infinite concerns only substances, not processes. (2) That Aristotle does not deny an actual infinite as such. (3) That when considering time and God (qua eternal) Aristotle acknowledges an actual infinite. (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)

[ES] El presente artículo estudia el influjo de los tratados físicos de Aristóteles sobre la concepción tomista en torno al lugar del infinito en el cosmos creado. Se analiza la posición sostenida por el Aquinate respecto a cuatro aspectos fundamentales de la teoría aristotélica en torno al infinito: existencia de una sustancia infinita, existencia de un cuerpo infinito, existencia de un infinito en acto y la infinitud del tiempo. Asimismo se expone el empleo de la teoría aristotélica del movimiento y (...) los lugares naturales, por parte del Doctor angélico, para la refutación de toda posición que conciba el acto de creación como una mutación temporalmente sucesiva, así como su caracterización de la divinidad como sustancia perfecta cuya infinitud no puede ser comprendida bajo la noción de cantidad. [EN] This article studies the influence of Aristotle’s physical treatises on the Thomist conception on the place of infinity in the created cosmos. It analizes the position held by Aquinas on four fundamental aspects of the Aristotelian theory about infinity: existence of an infinite substance, existence of an infinite body, existence of an infinite in act and the infinity of time. Is also exposed the use of the Aristotelian theory of motion and natural places by the Angelic Doctor for the refutation of every position that presents the act of creation as a temporally successive mutation and his characterization of divinity as a perfect substance whose infinity can not be understood under the notion of quantity. (shrink)

Aristotle’s subtle distinction between the forms of friendship and his concept of loving friend as one’s other self propose a solution to the fundamental objection to any eudaimonian theory of friendship, namely that friendship – as basically non-moral phenomenon – is but an egoistic device of one’s happy life. Aristotelian theorems are based on his concept of analogy and on a philosophically specific notion of “self”. Since both of these are rooted in Platonism, Aristotle has to evolve them dialectically in (...) a critical distance to Plato. Still, his dialectical theory of friendship needs to be rooted not in metaphysics but in political theory after all. Political friendship as a utopian perspective taken by each of the citizens in their pursuit of a close relationship with any other indicates a notion of “infinity as perfection” which presents the decisive step beyond Plato and toward the later course of the history of philosophy. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition (...) for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Spinoza: Une Lecture d'AristoteYitzhak MelamedFrédéric Manzini. Spinoza: Une Lecture d'Aristote. Paris: Presses Universitaires de France, 2009. Pp. 334. Paper, $39.95.The occasion that prompted the current study was the discovery of a tiny typo in the text of Spinoza's Cogitata Metaphysica—the appendix to his 1663 book, Descartes' Principle of Philosophy. As it turned out, this typo, a reference to Book XI instead of Book XII of Aristotle's Metaphysics, was (...) inadvertently reproduced by Spinoza from a contemporary edition of Aristotle's works, allowing Manzini to identify precisely the edition used by Spinoza as the 1548 edition of Aristotle's works printed in Basel by Johannes Oporinus. It also led him to conclude that, contrary to the belief of many, Spinoza read Aristotle's text very closely, and convinced him of the importance of studying the dialogue between the two great philosophers.The book is comprised of three parts, reversing the order of Spinoza's presentation of his philosophy in the Ethics. The first part discusses Spinoza's ethical and political theories—where, the author claims, the influence of Aristotle is most manifest—against the background of the Nicomachean Ethics and the Politics. Here, much attention is given to the notions of the summum bonum and beatitude. The second part discusses Aristotelian and Spinozistic theories of knowledge. In the third and final part, dedicated to metaphysics and first philosophy, the author compares the philosophers' absolutely impersonal conceptions of God, examines their conceptions of substance, and develops a reading of what he calls Spinoza's "ousio-theology" (305).According to Manzini, Aristotle was one of Spinoza's first major philosophical interlocutors, and there is no doubt he is right about this. This conclusion, however, is unsurprising, [End Page 126] since seventeenth-century philosophers, whether devoted Aristotelians or determined anti-Aristotelians, conversed in an Aristotelian philosophical language. Manzini shows that Spinoza engaged directly with Aristotle's text, but also tries to substantiate an even stronger claim by stressing the agreements between the two philosophers.There are two reasons that make it difficult to see Spinoza as being consciously and positively influenced by Aristotle, however. First, Spinoza's dialogue with his philosophical predecessors is, without exception, highly critical and polemical. He does not usually mention philosophers with whom he agrees (e.g. Crescas and Machiavelli). Secondly, several of Spinoza's explicit references to Aristotle are highly critical. Consider, for example, Spinoza's claim, in a letter to Hugo Boxel, that "the authority of Plato, Aristotle, and Socrates carries little weight with me" (Letter 56), or his mockery of the "delusions of the Aristotelians and Platonists" (Theological Political Treatise [TTP], Pref.; cf. ch. 13). Manzini suggests that in these passages Spinoza is not so much addressing Aristotle, but rather using him as a foil to target medieval Aristotelianism (13). Looking closely at the context, my impression is that he is right about the passages in the TTP, but not about the quote from Letter 56. More importantly, the claim that Spinoza uses Aristotle's name as a foil is a double-edged sword that may also serve to undermine Manzini's argument. Throughout the book, Manzini points out interesting and significant similarities between Spinoza's and Aristotle's texts, but to establish direct positive influence one must consider and rule out mediating agents. In the present case there is plenty of evidence that Spinoza read medieval Hebrew Aristotelians closely. Unfortunately, most of the Hebrew authors mentioned by Spinoza were never translated into modern European languages, and there are very few, if any, good studies of Spinoza's dialogue with these texts; in the absence of such translations and studies, some of Manzini's conclusions must remain tentative.Manzini's erudition and analysis of the text are impressive. For example, near the end of his celebrated Letter on the Infinite, Spinoza criticizes "recent Peripatetics" for misunderstanding the ancient method of proving God's existence, citing Crescas's correct formulation of the ancient proof. Manzini rightly points out that the proof Spinoza ascribes to the "recent Peripatetics" was in fact Aristotle's, while the alleged "ancient" proof (which allows for actual infinity and relies on Avicennian... (shrink)

Xenophanes of Colophon was a philosophically-minded poet who lived in various parts of the ancient Greek world during the late 6th and early 5th centuries BC. He is best remembered for a novel critique of anthropomorphism in religion, a partial advance toward monotheism, and some pioneering reflections on the conditions of knowledge. Many later writers, perhaps influenced by two brief characterizations of Xenophanes by Plato (Sophist 242c-d) and Aristotle (Metaphysics 986b18-27) identified him as the founder of Eleatic philosophy (the view (...) that despite appearances, what there is is a changeless, motionless, and eternal ‘one’. In fact, the Xenophanes who emerges from the surviving fragments defies simple classification. He was a traveling rhapsode who criticized the stories about the gods told by the poets, and he defended a novel conception of the divine nature. But he was also a reflective observer of the human condition, a practitioner of the special form of inquiry introduced by the Milesians philosopher-scientists, and a civic counselor who encouraged his fellow citizens to respect the gods and work to safeguard the well-being of their city. (shrink)

Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

The argumentative structure of this article can be summarized as follows: 1) Spinoza was repeatedly accused of eleaticism; 2) there is a rupture with the eleaticism when one admits the ‘multivocity’ of the logical operators “is” and “is not”; as can be seen, for example, 2.1) in the discussion introducing the Great Genera in Plato’s Sophist, and 2.2) in a certain use made by Aristotle of the category’s doctrine in order to soften the Parmenidean version of the Principle of Non-Contradiction. (...) 3) Spinoza admits the ‘multivocity’ of “is” and “is not”. Hence, Spinoza should not be placed among the Eleatics but inside the brotherhood of the suspects of the so-called parricide against Parmenides. This reasoning, which almost sounds as a truism, will, however, suggest two interesting further considerations: one on how an important tradition of interpretation of Spinoza has misplaced the problem of the relationship of Spinoza’s ontology with the logical principles; and another on the type of logic that is implied by the theory of modes of perception or kinds of knowledge. (shrink)

The Role of Negation in the Description of the World According to Aristote’s Metaphysics. The notions of ‘being’ and ‘non-being’ have entered philosophical language, forming the basis of ontology and meontology, as the counterparts of the Greek expressions to on and to me on (nominalised forms, affirmative and negative, of the participle of the verb einai). Originally, however, these expressions did not have any objectifying meaning, but played the role of meta-language names, representing the copula einai in all its forms, (...) most generally in its affirmative and negative forms. The copula itself, which in later philosophy took on the existential meaning, had functioned only as a semantically empty connective of predicates. Over time the participle on has been used as a universal name of all predicates. The above-mentioned expressions became central in Greek philosophical terminology thanks to the debates, initiated by Parmenides, on the role of negation in the description of the world. Parmenides himself proposed a complete excision of negative sentences as describing by elimination, and created a positive-monistic system which abandoned multiplicity, divisibility, and variability. Later philosophers defended negation, fighting back against the paradoxes formulated by the Eleatics and later by the Sophists. Plato observed that without negation it is impossible to describe the multiplicity of things. He also distinguished a relative negation which does not eliminate anything but makes it possible to confront some things with others. According to the atomists, the divisibility of physical things forces us to accept that they consist of a positive element in the form of an impenetrable body, and of another element lacking any characteristics, i.e. the void. Finally, Aristotle, when analysing the process of change, justified the consistency of the statement that something comes out of “not being” and “being”, under the assumption that the former is understood as being actual, and the latter as being potential. In all these conceptions there is nothing non-existent, there are only proposals of how to identify the aspects of reality whose explanation justifies the use of negation. The affirmative and negative forms of the above expressions have also provoked reflection on the problem of truth and falsity. It has been observed that they are used in everyday language not only to state an agreement or disagreement with the actual facts of the matter, but also to affirm or deny something. (shrink)

This is a review of Monika Michalowska's critical edition of question 3 from Richard Kilvington's questions commentary on the Sentences by Peter Lombard. It is published in open access under a CC BY 4.0 license.

‘Substance’ (substantia, zelfstandigheid) is a key term of Spinoza’s philosophy. Like almost all of Spinoza’s philosophical vocabulary, Spinoza did not invent this term, which has a long history that can be traced back at least to Aristotle. Yet, Spinoza radicalized the traditional notion of substance and made a very powerful use of it by demonstrating – or at least attempting to demonstrate -- that there is only one, unique substance -- God (or Nature) -- and that all other things are (...) merely modes or states of God. Some of Spinoza’s readers understood these claims as committing him to the view that only God truly exists, and while this interpretation is not groundless, we will later see that this enticing and bold reading of Spinoza as an ‘acosmist’ comes at the expense of another audacious claim Spinoza advances, i.e., that God/Nature is absolutely and actually infinite. But before we reach this last conclusion, we have a long way to go. So, let me first provide an overview of our plan. In the first section of this paper we will examine Spinoza’s definitions of ‘substance’ and ‘God’ at the opening of his magnum opus, the Ethics. Following a preliminary clarification of these two terms and their relations to the other key terms defined at the beginning of the Ethics, we will briefly address the Aristotelian and Cartesian background of Spinoza’s discussion of substance. In the second section, we will study the properties of the fundamental binary relations pertaining to Spinoza’s substance: inherence, conception, and causation. The third section will be dedicated to a clarification of Spinoza’s claim that God, the unique substance, is absolutely infinite. This essential feature of Spinoza’s substance has been largely neglected in recent Anglo-American scholarship, a neglect which has brought about an unfortunate tendency to domesticate Spinoza’s metaphysics to more contemporary views. The fourth section will study the nature of Spinoza’s monism. It will discuss and criticize the interesting yet controversial views of the eminent Spinoza scholar, Martial Gueroult, about the plurality of substances in the beginning of the Ethics; address Spinoza’s claim in Letter 50 that, strictly speaking, it is improper to describe God as “one”; and, finally, evaluate Spinoza’s kind of monism against the distinction between existence and priority monism recently introduced into the contemporary philosophical literature. The fifth and final section will explain the nature, reality, and manner of existence of modes. We therefore have an ambitious plan; let’s get down to business. (shrink)

The paper proposes a novel understanding of how Aristotle’s theoretical works complement each other in such a way as to form a genuine system, and this with the immediate (and ostensibly central) aim of addressing a longstanding question regarding Aristotle’s ‘first philosophy’—namely, is Aristotle’s first philosophy a contribution to theology, or to the science of being in general? Aristotle himself seems to suggest that it is in some ways both, but how this can be is a very difficult question. My (...) answer is in some respects a version of one that goes back at least to the middle ages—i.e., that first philosophy is concerned with the gods (and to that extent offers a theology) because the gods are causes and principles of beings precisely insofar as they are beings. The more original aspect of my position lies in my claim that the sort of tension found in the Metaphysics is likewise to be found in many of Aristotle’s physical works. Thus, for example, the De caelo is (I argue) concerned generally with natural beings (= beings susceptible of change), but its discussions are focused largely on the heavenly bodies and the Aristotelian elements insofar as they admit of change with respect to place. Here I claim that the particular objects of discussion are dealt with precisely because they are causes and principles of natural beings as such. Something similar goes, I claim, for the De generatione et corruptione, the general concern of which is a particular species of natural being—i.e., natural beings susceptible of generation and corruption. In this way, I argue, Aristotle successively deals in his theoretical works with those causes and principles of (say) a horse which attach to it insofar as it is a being, those causes and principles of a horse which attach to it insofar as it is a natural being, those causes and principles of a horse which attach to it insofar as it is a perishable natural being, and so on for the lower genera under which the species horse is subsumed. (shrink)

Aristotle and William of Ockham both argue that existence or being is a predicate, though not a distinguishing predicate. I place Ockham’s argument in an Aristotelian context and discuss its merits. I then turn to empiricist criticisms of the view that we can coherently predicate being of things. I argue that while Ockham’s argument is cogent, his account of how we come to have the concept of being is inadequate. Ockham’s view needs to be supplemented with Kantian insights.

Aristotle explains existence through postulating essences that are intrinsic and percep- tion independent. I argue that his theory is more plausible than Hume’s and Russell’s theories of existence. Russell modifies Hume’s theory because he wants to allow for the existence of mathematical objects. However, Russell’s theory facilitates a problematic collapse of ontology into epistemology, which has become a feature of much analytic philosophy. This collapse obscures the nature of truth. Aristotle is to be praised for starting with a clear account (...) of ordinary objects rather than immediately reifying mathematical objects. He thus allows us to have a coherent account of existence and truth, and to easily resist collapsing ontology into epistemology. (shrink)

This dissertation is a study of the relationship between dunamis and energeia in Aristotle’s ontology. Throughout his writings, Aristotle employs these terms to uncover what I call a proto-phenomenological description of the different ways of being. While contemporary scholarship has suggested the significance of dunamis and energeia for Aristotle’s understanding of being, the relationship between these terms has often been interpreted as mutually exclusive. Accordingly, dunamis would be understood as subordinate to energeia, which would function as the sole primary term (...) of Aristotle’s ontology. I argue that it is a mistake to consider dunamis and energeia as non-reciprocal and subordinate terms. Furthermore, I suggest that this mistake often leads to an underestimation of the dynamic character of Aristotle’s proto-phenomenological account of being. To recover this dimension of Aristotle’s thinking, I claim that dunamis and energeia ought to be understood as reciprocal and co-constitutive terms characterized by a constant and dynamic interrelation. To defend this interpretation, I turn to Aristotle’s conceptions of nature, movement, and soul as discussed in the Metaphysics, Physics, and De Anima. I argue that each of these key terms provide a concrete illustration of how dunamis and energeia function as co-constitutive principles for the manifestation of the being of natural beings. I propose that this proto-phenomenological approach to being remains Aristotle’s most significant and enduring contribution to Western ontology. (shrink)

‘Substance’ (substantia, zelfstandigheid) is a key term of Spinoza’s philosophy. Like almost all of Spinoza’s philosophical vocabulary, Spinoza did not invent this term, which has a long history that can be traced back at least to Aristotle. Yet, Spinoza radicalized the traditional notion of substance and made a very powerful use of it by demonstrating – or at least attempting to demonstrate -- that there is only one, unique substance -- God (or Nature) -- and that all other things are (...) merely modes or states of God. Some of Spinoza’s readers understood these claims as committing him to the view that only God truly exists, and while this interpretation is not groundless, we will later see that this enticing and bold reading of Spinoza as an ‘acosmist’ comes at the expense of another audacious claim Spinoza advances, i.e., that God/Nature is absolutely and actually infinite. But before we reach this last conclusion, we have a long way to go. So, let me first provide an overview of our plan. In the first section of this paper we will examine Spinoza’s definitions of ‘substance’ and ‘God’ at the opening of his magnum opus, the Ethics. Following a preliminary clarification of these two terms and their relations to the other key terms defined at the beginning of the Ethics, we will briefly address the Aristotelian and Cartesian background of Spinoza’s discussion of substance. In the second section, we will study the properties of the fundamental binary relations pertaining to Spinoza’s substance: inherence, conception, and causation. The third section will be dedicated to a clarification of Spinoza’s claim that God, the unique substance, is absolutely infinite. This essential feature of Spinoza’s substance has been largely neglected in recent Anglo-American scholarship, a neglect which has brought about an unfortunate tendency to domesticate Spinoza’s metaphysics to more contemporary views. The fourth section will study the nature of Spinoza’s monism. It will discuss and criticize the interesting yet controversial views of the eminent Spinoza scholar, Martial Gueroult, about the plurality of substances in the beginning of the Ethics; address Spinoza’s claim in Letter 50 that, strictly speaking, it is improper to describe God as “one”; and, finally, evaluate Spinoza’s kind of monism against the distinction between existence and priority monism recently introduced into the contemporary philosophical literature. The fifth and final section will explain the nature, reality, and manner of existence of modes. We therefore have an ambitious plan; let’s get down to business. (shrink)

The essay studies Aristotle’s critique of Parmenides in the light of the Heideggerian account of Platonic-Aristotelian metaphysics as an approach to being in terms of beings. Aristotle’s critique focuses on the presuppositions of the Parmenidean thesis of the unity of being. It is argued that a close study of the presuppositions of Aristotle’s own critique reveals an important difference between the Aristotelian metaphysical framework and the Parmenidean “protometaphysical” approach. The Parmenides fragments indicate being as such in the sense of the (...) pure, undifferentiated “is there” —as the intelligible accessibility of meaningful reality to thinking, prior to its articulation into determinate beings. For Aristotle, by contrast, “being itself” has no other plausible meaning than “being-something-determinate as such”, which itself remains equivocal. In this sense, Aristotle can indeed be said to conceive being in terms of beings, as the being-ness of determinate beings. (shrink)

This paper focuses on Aristotle’s discussion of PNC in Metaphysics Gamma and argues that the argument operates at three different levels: ontological, doxastic, and semantic through the invocation of three philosophical personae: the first one can only state what is otherwise unprovable, the second one can only confirm that we should trust PNC, the third one denies PNC and must be silenced. Aristotle cannot prove what is beyond proof. This situation results in a fundamental ambiguity in the figure of the (...) philosopher. The Metaphysics is written from the standpoint of an investigative thinker who admits her puzzlement before a question that will forever remain open and imagines another philosopher who has achieved a god-like insight into the first principles of all things. The path from the first figure to the second one, however, remains an enigmatic leap. (shrink)

In this essay, I offer a critical analysis of one of the most provocative aspects of Heidegger’s unfinished confrontation with Aristotle’s thinking. Over the course of his lifelong engagement with Aristotle’s texts, Heidegger rarely failed to notice the constitutive ambiguity of the ancient Greek philosopher’s position within the history of being. On the one hand, Aristotle appeared to be the founder of the Western metaphysical tradition of ontotheology, whereby God was understood as the supreme principle and being of all beings. (...) But on the other hand, there were traces or echoes of a more originary thinking in Aristotle’s texts that, according to Heidegger, could be developed for the sake of another beginning. In this essay, I draw on this constitutive ambiguity by focusing on Heidegger’s partial translation of Metaphysics Λ 6, where Aristotle most explicitly discusses the being (οὐσία) of the unmoved being. Using Heidegger’s translation as an aid, I argue that Aristotle’s account of the unmoved being in Metaphysics Λ 6 is perhaps best understood as of- fering an enigmatic description of the origin of movement and time as an act of pure temporalizing (reine Zeitigung), which is how Heidegger translates the notion of ἐνέργεια. In the conclusion of this essay, I suggest that the temporal account of being and time as pure temporalizing offers a possible way to reappropriate Aristotle’s thinking in a phenomenological context. (shrink)

I examine the reasons Aristotle presents in Physics VIII 8 for denying a crucial assumption of Zeno’s dichotomy paradox: that every motion is composed of sub-motions. Aristotle claims that a unified motion is divisible into motions only in potentiality (δυνάμει). If it were actually divided at some point, the mobile would need to have arrived at and then have departed from this point, and that would require some interval of rest. Commentators have generally found Aristotle’s reasoning unconvincing. Against David Bostock (...) and Richard Sorabji, inter alia, I argue that Aristotle offers a plausible and internally consistent response to Zeno. I defend Aristotle’s reasoning by using his discussion of what to say about the mobile at boundary instants, transitions between change and rest. There Aristotle articulates what I call the Changes are Open, Rests are Closed Rule: what is true of something at a boundary instant is what is true of it over the time of its rest. By contrast, predications true of something over its period of change are not true of the thing at either of the boundary instants of that change. I argue that this rule issues from Aristotle’s general understanding of change, as laid out in Phys. III. It also fits well with Phys. VI, where Aristotle maintains that there is a first boundary instant included in the time of rest, but not a “first in which the mobile began to change.” I then show how this rule underlies Aristotle’s argument that a continuous motion cannot be composed of actual sub-motions. Aristotle distinguishes potential middles, points passed through en route to a terminus, from actual middles. The Changes are Open, Rests are Closed Rule only applies to actual middles, because only they are boundaries of change that the mobile must arrive at and then depart from. On my reading, Aristotle argues that the instant of arrival, the first instant at which the mobile has come to be at the actual middle, cannot belong to the time of the subsequent motion. If it did, the mobile would already be moving towards the next terminus and thus, per Phys. VI 6, would have already left. But it cannot have moved away from the midpoint at the very same moment it has arrived there. This means that the instant of arrival must be separated from the time of departure by an interval of rest. I show how Aristotle’s reasoning applies generally to rule out any continuous reflexive motion or continuous complex rectilinear motion. On my interpretation, however, the argument does not apply to every change of direction. When, as in the case of projectile motion, multiple movers and their relative powers explain why the mobile changes directions, distinct sub-motions are not involved. Aristotle holds that such motions cannot be continuous, not because they involve intervals of rest, but because they involve multiple causes of motion. My interpretation of the Changes are Open, Rests are Closed Rule allows us to make better sense of Aristotle’s argument than any previous interpretation. (shrink)

A curious feature of Aztec philosophy is that the basic metaphysical question of the “Western” tradition cannot be formulated in their language, in Nahuatl. This did not, however, prevent the Aztecs from developing an account of 'reality', or whatever it is that might exist. The article is the first of its kind to compare the work of Aristotle on ousia (being) and the Aztecs on teotl and ometeotl. Through this analysis, it suggests that both of the Nahuatl terms are fundamental (...) for expressing the basic character of reality as process for the Aztecs. It also defends their notion as prima facia reasonable, despite its wide divergence from Aristotle's view. (shrink)

Aristotle analyses a large range of objects as composites of matter and form. But how exactly should we understand the relation between the matter and form of a composite? Some commentators have argued that forms themselves are somehow material, that is, forms are impure. Others have denied that claim and argued for the purity of forms. In this paper, I develop a new purist interpretation of Metaphysics Z.10-11, a text central to the debate, which I call 'hierarchical purism'. I argue (...) that hierarchical purism can overcome the difficulties faced by previous versions of purism as well as by impurism. Roughly, on hierarchical purism, each composite can be considered and defined in two different ways: From the perspective of metaphysics, composites are considered only insofar as they have forms and defined purely formally. From the perspective of physics, composites are considered insofar as they have forms and matter and defined with reference to both. Moreover, while the metaphysical definition is a definition in the strict sense of 'definition', the physical definition is a definition in a loose sense. Analogous points hold for intelligible composites and geometry. Finally, neither sort of definitional practice implies that, for Aristotle, forms are impure. (shrink)

Aristotle’s treatment of tactility is at odds with the hierarchical order of psyche’s faculties. Touching is the commonest and lowest power; it is possessed by all sentient beings; thinking is, on the contrary, the highest faculty that distinguishes human beings. Yet, while Aristotle maintains against some of his predecessors that to think is not to sense, he nevertheless posits a causal link between practical intelligence and tactility and even describes noetic activity as a certain kind of touch. This essay elucidates (...) Aristotle’s analysis of the sense of touch in De anima and argues that tactility provides a paradigm for sensitivity in general and in particular for the reflexivity of sensation whereby the senses disclose not only what they are sensing but also that they are sensing. This feature, it is argued, has epistemological and ontological consequences. The sense of touch testifies to the physical presence of material beings and provides an empirical verification of substance’s essential feature, namely, self-reference. (shrink)

It is commonly assumed that Aristotle thinks that his claim that being exhibits a category-based pros hen structure, which he introduces to obviate the problem of categorial heterogeneity, is sufficient to defend the possibility of a science of being qua being. We, on the contrary, argue that Aristotle thinks that the pros hen structure is necessary only, but not sufficient, for this task. The central thesis of our paper is that Aristotle, in what follows 1003b19, raises a second problem for (...) the possibility of the science of being qua being; and that he does not think that the resolution of the first, the category-based problem, is either necessary or sufficient for resolving this problem. This is the problem: how can a plurality of apparently primary kinds and their opposites (they include to hen, to on, to auto, to homoion, to heteron and to anhomoion) be the subject-matter of the science of being qua being? It has been argued that these kinds are per se attributes of ousia and that, therefore, this problem is not different from the first problem. This, we argue, is mistaken; for nowhere in Gamma 2 does Aristotle claim that unity is a per se attribute of ousia. Rather, he says that identity, similarity, etc. are per se attributes of being qua being and unity qua unity. Aristotle’s resolution of the second problem, we argue, is that most of these kinds are reducible to a single compound principle: being-and-unity. Being and unity, moreover, are themselves related to each other as primary ousia and consequent ousia; but, we argue, Aristotle leaves it open, in Gamma 2, which of the two is primary, and which is consequent ousia. (shrink)

In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.

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)

I discuss the argument Aristotle ascribes to Parmenides at Physics 186a23-32. I discuss (i) the reasons why Aristotle considers it as eristic and inconclusive and (i) the solution (lusis) Aristotle proposes against it.

This article considers whether and how there can be for Aristotle a genuine science of ‘pure’ psychology, of the soul as such, which amounts to considering whether Aristotle’s model of science in the Posterior Analytics is applicable to the de Anima.

Usage of the Greek verb ‘to be’ is generally divided into three broad categories — the predicative use, the existential and the veridical—and these usages often inform the way we understand Being in ancient philosophy. This article challenges this approach by arguing that Being is not the product of linguistic reflection in Parmenides, Plato and Aristotle; rather, these thinkers treat Being as the ontological and epistemological primary. Though this may overlap with the linguistic senses, it is not the same thing. (...) The article is divided into three sections: the first one raises several basic issues with the predicative interpretation of Being, the second argues that Being is unified and singular in a significant sense and the third brings out the special pre-immanence of Being. (shrink)

This book investigates what change is, according to Aristotle, and how it affects his conception of being. Mark Sentesy argues that change leads Aristotle to develop first-order metaphysical concepts such as matter, potency, actuality, sources of being, and the teleology of emerging things. He shows that Aristotle’s distinctive ontological claim—that being is inescapably diverse in kind—is anchored in his argument for the existence of change. -/- Aristotle may be the only thinker to have given a noncircular definition of change. When (...) he gave this definition, arguing that change is real was a losing proposition. To show that it exists, he had to rework the way philosophers understood reality. His groundbreaking analysis of change has long been interpreted through a Platonist lens, however, in which being is conceived as unchanging. Offering a comprehensive reexamination of the relationship between change and being in Aristotle, Sentesy makes an important contribution to scholarship on Aristotle, ancient philosophy, the history and philosophy of science, and metaphysics. (shrink)

Aristotle (384-322 BC) was an ancient Greek philosopher, pupil of Plato, and tutor of Alexander the Great. His works span the topics of biology, metaphysics, mind, logic, language, science, epistemology, ethics, and politics. Aristotle held that there are many divine beings, but a supremely divine being is the first cause of the universe and the goodness of all other beings. This divine being plays a fundamental explanatory role in Aristotle’s thought.

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...) part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems. (shrink)

Aristotle holds that individual substances are ontologically independent from nonsubstances and universal substances but that non-substances and universal substances are ontologically dependent on substances. There is then an asymmetry between individual substances and other kinds of beings with respect to ontological dependence. Under what could plausibly be called the standard interpretation, the ontological independence ascribed to individual substances and denied of non-substances and universal substances is a capacity for independent existence. There is, however, a tension between this interpretation and the (...) asymmetry between individual substances and the other kinds of entities with respect to ontological independence. I will propose an alternative interpretation: to weaken the relevant notion of ontological independence from a capacity for independent existence to the independent possession of a certain ontological status. (shrink)

According to Aristotle, phainomena or “appearances” provide the basis from which researches proceed. This shows that in spite of phainomena often corresponding to what falsely appears to be the case, there is genuine cognition through them. In this paper, I focus on two features of phainomenal cognition: accessibility and epistemological limitation. A phainomenal cognition of x is limited in the sense that there is always a stronger cognition of x to be attained. In this way, a research always aims at (...) surpassing the phainomenal cognition of its subject matter. On the other hand, phainomenal cognition is always somehow accessible. Resorting to the relation between phainomena and the distinction between the more intelligible to us and the more intelligible by nature, I intend to put forward a relative (as opposed to an absolute) understanding of both accessibility and epistemological limitation of phainomena. (shrink)

In 1794, Michael Wenzel Voigt, a professor of rhetoric in present-day Czechia, published the first German translation of Aristotle’s De anima. Voigt’s translation was explicitly intended to rescue Aristotle's views on the soul, and the bold strategy he adopts towards this end is to assert a direct connection between Aristotle’s doctrines and Kant’s Critical philosophy. Thus, he contends that Aristotle’s books on the soul can be read as an “appendix” or even as a “propadeutic” to Kant’s Critical works. Despite Voigt’s (...) rather hamfisted attempt to establish this connection, the conjecture at the core of his project—that a number of key Kantian claims about the mind or soul have a largely overlooked Aristotelian provenance—turns out to be correct, or so I argue in this chapter. Even so, we will see that this connection is not direct but is instead mediated by the reception of Aristotle in the late 17th and early 18th century. Moreover, it is not limited to the views Aristotle expresses in his texts on the soul but extends beyond his psychological and even metaphysical works to include views expressed in his logical and ethical thought. (shrink)

In what ways and how far does virtue shield someone against suffering evils? In other words, how do non-moral evils affect the lives of virtuous people and to what extent can someone endure evils while staying happy? The central purpose of this chapter is to answer these questions by exploring what Aristotle has to say about the effects of evils in human well-being in general and his treatment of extreme misfortunes.