In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but (...) have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. (shrink)

If a group is modelled as a single Bayesian agent, what should its beliefs be? I propose an axiomatic model that connects group beliefs to beliefs of group members, who are themselves modelled as Bayesian agents, possibly with different priors and different information. Group beliefs are proven to take a simple multiplicative form if people’s information is independent, and a more complex form if information overlaps arbitrarily. This shows that group beliefs can incorporate all information spread over the individuals (...) without the individuals having to communicate their (possibly complex and hard-to-describe) private information; communicating prior and posterior beliefs sufices. JEL classification: D70, D71.. (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)

A classification of the global catastrophic risks of AI is presented, along with a comprehensive list of previously identified risks. This classification allows the identification of several new risks. We show that at each level of AI’s intelligence power, separate types of possible catastrophes dominate. Our classification demonstrates that the field of AI risks is diverse, and includes many scenarios beyond the commonly discussed cases of a paperclip maximizer or robot-caused unemployment. Global catastrophic failure could happen at (...) various levels of AI development, namely, before it starts self-improvement, during its takeoff, when it uses various instruments to escape its initial confinement, or after it successfully takes over the world and starts to implement its goal system, which could be plainly unaligned, or feature-flawed friendliness. AI could also halt at later stages of its development either due to technical glitches or ontological problems. Overall, we identified around several dozen scenarios of AI-driven global catastrophe. The extent of this list illustrates that there is no one simple solution to the problem of AI safety, and that AI safety theory is complex and must be customized for each AI development level. (shrink)

Automatic detection of diseases and anatomical landmarks in medical images by the use of computers is important and considered a challenging process that could help medical diagnosis and reduce the cost and time of investigational procedures and refine health care systems all over the world. Recently, gastrointestinal (GI) tract disease diagnosis through endoscopic image classification is an active research area in the biomedical field. Several GI tract disease classification methods based on image processing and machine learning techniques have (...) been proposed by diverse research groups in the recent past. However, yet effective and comprehensive deep ensemble neural network-based classification model with high accuracy classification results is not available in the literature. In this thesis, we review ways and mechanisms to use deep learning techniques to research on multi-disease computer-aided detection about gastrointestinal and identify these images. We re-trained five state-of-the-art neural network architectures, VGG16, ResNet, MobileNet, Inception-v3, and Xception on the Kvasir dataset to classify eight categories that include an anatomical landmark (pylorus, z-line, cecum), a diseased state (esophagitis, ulcerative colitis, polyps), or a medical procedure (dyed lifted polyps, dyed resection margins) in the Gastrointestinal Tract. Our models have showed results with a promising accuracy which is a remarkable performance with respect to the state-of-the-art approaches. The resulting accuracies achieved using VGG, ResNet, MobileNet, Inception-v3, and Xception were 98.3%, 92.3%, 97.6%, 90% and 98.2%, respectively. As it appears, the most accurate result has been achieved when retraining VGG16 and Xception neural networks with accuracy reache to 98% due to its high performance on training on ImageNet dataset and internal structure that support classification problems. (shrink)

The neutrosophic automorphisms of a neutrosophic groups G (I) , denoted by Aut(G (I)) is a neu-trosophic group under the usual mapping composition. It is a permutation of G (I) which is also a neutrosophic homomorphism. Moreover, suppose that X1 = X(G (I)) is the neutrosophic group of inner neutrosophic auto-morphisms of a neutrosophic group G (I) and Xn the neutrosophic group of inner neutrosophic automorphisms of Xn-1. In this paper, we show that if any neutrosophic group of the (...) sequence G (I), X1, X2, … is the identity, then G (I) is nilpotent. (shrink)

At the heart of chemistry lies the periodic system of chemical elements. Despite being the cornerstone of modern chemistry, the overall structure of the periodic system has never been fully understood from an atomic physics point of view. Group-theoretical models have been proposed instead, but they suffer from several limitations. Among others, the identification of the correct symmetry group and its decomposition into subgroups has remained a problem to this day. In an effort to deepen our limited understanding of the (...) periodic law, we have extended the traditional Lie algebraic framework to account for the peculiar degeneracy structure of the periodic system. Starting from the four-dimensional hidden symmetry and accidental degeneracy of the hydrogen atom, as first revealed by Fock in 1935, our research has mainly focussed on the way this SO(4) symmetry of the Coulomb potential gets broken in the periodic system as a consequence of the transformation of the hydrogenic (n, l) filling order to the Madelung (n+l, n) order due to electronic repulsions, relativistic effects and spin-orbit couplings. In this PhD dissertation, a new left-step format of the periodic table is first proposed on the basis of the Madelung rule. Following the particle physics tradition, the chemical elements are then considered as various states of some 'atomic matter', which is described by a non-compact spectrum-generating dynamical Lie group. The chemical elements are shown to form a basis for a single infinite-dimensional degeneracy space of the SO(4,2) ⊗ SU(2) group. An explanation for the period doubling is then proposed in terms of a particular symmetry breaking of the SO(4,2) group to the anti de Sitter SO(3,2) group. The Madelung rule is rationalised on the basis of nonlinear Lie algebras which reflect the screening of the Coulomb hole. This opens new perspectives for a symmetry-based understanding of how the periodic law emerges from its quantum mechanical foundations, and holds the future promise of complementing our current phenomenological approach by a direct atomic physics approach. (shrink)

One of the crucial problems of argumentation schemes as illustrated in (Walton, Reed & Macagno 2008) is their practical use for the purpose of analyzing texts and producing arguments. The high number and the lack of a classification criterion make this instrument extremely difficult to apply practically. The purpose of this paper is to analyze the structure of argumentation schemes and outline a possible criterion of classification based on alternative and mutually-exclusive possibilities. Such a criterion is based not (...) on what an argument is, but how it can be understood and interpreted. The schemes are grouped according to an end-means principle, which is strictly bound to the ontological structure of the conclusion and the premises. On this view, a scheme can be selected according to the intended or reconstructed purpose of an argument and the possible strategies that can be used to achieve it. (shrink)

Aristotle classifies opposition (ἀντικεῖσθαι) into four groups: relatives (τὰ πρός τι), contraries (τὰ ἐναντία), privation and possession (στρέσις καὶ ἓξις) and affirmation and negation (κατάφασις καὶ ἀπόφασις). (Cat. , 10, 11b15-23) His example of relatives are the double and the half. Aristotle’s description of relatives as a kind of opposition is as such: ‘Things opposed as relatives are called just what they are, of their opposites (αὐτὰ ἃπερ ἐστι τῶν ἀντικειμένων λέγεται) or in some other way in relation to (...) them. For example, the double is called just what it is double of the other (οἷον τὸ διπλάσιον, αὐτὸ ὃπερ ἐστίν, ἑτέρου διπλάσιον λέγεται). Again, knowledge and the knowable are opposed as relatives, and knowledge is called just what it is, of the knowable, and the knowable too is called just what it is, in relation to its opposite, knowledge; for the knowable is called knowable by something-by knowledge.’ (Cat., 10, 11b24-30) 2) Senses of relatives In Met., Δ, 1020b26-32 Aristotle distinguishes three senses of relatives: i) That which contains something else many times to that which is contained many times in something else, and that which exceeds to that which is exceeded. E.g. double to half ii) The active to the passive; e.g. that which can heat to that which can be heated iii) The measurable to the measure, e.g. the knowable to knowledge and the perceptible to perception. 3) Relatives and contraries Aristotle’s discussion of relatives is unbelievably ambiguous. While he enumerates relatives besides contraries, privation-possession and affirmation-negation as four types of opposition (Cat., 10, 11b15-23), not only does not he restrict relatives to oppositions but he also does not totally differentiate it from contraries. Aristotle distinguishes between two senses of relatives one of which is contraries: ‘We have distinguished elsewhere the two senses in which relatives are so called-some as contraries (ὡς ἐναντία), others as knowledge to things known, a term being called relative because what is said one to the other is said by the other to itself (τῷ λέγεσθαί τι ἄλλο πρὸς αὐτό). (Met., I, 1056b34-1057a1) It seems that this differentiation is the same as, and maybe the one he himself refers to, the differentiation between the second and the third senses of relatives in Met., Δ, 1021a27-32: ‘Relative terms which imply number or capacity, therefore, are all relative because their very essence includes in its nature a reference to something else, not because something else is related to it (πρός τι πὰντα ἐστὶ πρός τι τῷ ὃπερ ἐστὶν ἄλλου λέγεσθαι αὐτὸ ὃ ἐστιν, ἀλλὰ μὴ τῷ ἄλλο πρὸς ἐκεινο); but that which is measurable or knowable or thinkable is called relative because something else is related to it. For the thinkable implies that there is thought of it, but the thought is not relative to that of which it is the thought; for we should then have said the same thing twice.’ Although Aristotle confirms contrariety in relatives (e.g. virtue to vice and knowledge to ignorance), he denies that there is a contrary ‘to every relative’ as there can be no contrary to what is double or treble. (Cat., 7, 6b15-19) The fact that Aristotle dedicates one sense of relatives to contraries means that the fourfold division of oppositions is not such a strict division without any kind of community between them. In his example of the first sense, however, Aristotle says: ‘One and number are in a sense opposed, not as contrary, but as we have said some relative terms are opposed; for inasmuch as one is measure and the other measurable, they are opposed.’ (Met., I, 1057a4-6) Now, the question is: Is Aristotle’s first sense of relative a contrary or not? Aristotle tells us that the opposition in this sense of relatives is the opposition of measure and measurable. The second sense has two differences with the first one: i) it is not a contrary and ii) what is said by one of the relatives to the other is also said by the latter to the former. To differentiate it from the first sense, when the one is the measure of the other, the latter will be the measure of the former as well: ‘But though knowledge is similarly spoken of as related to the knowable, the relation does not work out similarly for while knowledge might be thought to be the measure, and the knowable the thing measured, the fact is that all knowledge is knowable, but not all that is knowable is knowledge, because in a sense knowledge is measured by the knowable.’ (Met., I, 1057a7-12) Aristotle’s assertion at Met., I, 1057a36-37 that ‘of relative terms, those which are not contrary have no intermediate’ ensures us that we must take the first sense as contraries. Here he mentions a criterion of differentiation between two senses: while the first sense accepts intermediates (as, for example, great and small do), the second one does not. (Met., I, 1057a37-b1) It seems, however, that it is only the second sense that is the essential sense of relatives because the first sense, Aristotle says, is an accidental sense: ‘The one is opposed then to the many in numbers as measure to things measureable; and these are opposed as relatives which are not from their very nature relatives.’ (Met., I, 1056b32-34) 4) Definition of relative Aristotle’s first definition of the category of relative (πρός τι) is as such: ‘We call relative all such things as are said to be just what they are, of or than other things, or in some other way in relation to something else (Πρός τι δὲ τὰ τοιαῦτα λέγεται, ὃσα αὐτὰ ἃπερ ἐστὶν ἓτερων εἶναι λέγεται, ἢ ὁπωσοῦν ἄλλως πρὸς ἓτερον). (Cat., 7, 6a36-37; repeated almost without any change in: Cat., 7, 6b6-8) Aristotle’s examples are larger (because it is what it is than something else (τοῦθ᾿ ὃπερ ἐστὶν ἑτέρου λέγεται) that means it is called larger than something) and double. (Cat., 7, 6a37-b1) Aristotle’s aporia regarding relative is about their relation with substances: whether no substance is spoken of as a relative, or whether this is possible with regard to some secondary substances. (Cat., 7, 8a13-15) In the case of primary substances, it seems that he is confident, at least at first, that neither themselves nor their parts are spoken of in relation to anything: neither an individual man is called someone’s individual man nor an individual hand is called someone’s individual hand (but someone’s hand). (Cat., 7, 8a15-21) Although it is obvious that most of secondary substances are not spoken of as relatives (a man is not called someone’s man) (Cat., 7, 8a21-25), there are some with them there is room for dispute: a head is someone’s head and a hand is called someone’s hand, which seem to be relatives. (Cat., 7, 8a25-28) Aristotle thinks this must be due to the previous definition’s being problematic: with that definition ‘it is either exceedingly difficult or impossible to reach the solution that no substance is spoken of as a relative.’ (Cat., 7, 8a28-32) Thus, Aristotle changes his previous definition to a new one: ‘Those things are relatives for which being is the same as being somehow related to something’ (ἔστι τὰ πρός τι οἷς τὸ εἶναι ταὐτόν ἐστι τῷ πρός τί). (Cat., 7, 8a32-34) Although the first definition does indeed apply to all relations, its problem is that it does not take ‘their own being relative’ (τῷ πρός τι αὐτοῖς εἶναι) the same as ‘their being what they are of other things’ (ἃπερ ἐστὶν ἑτέρων λέγεσθαι). (Cat., 7, 8a34-37) Although this change of definition is to exclude all substances from being relative, what Aristotle says in Topics (To., Z, 8, 146a39- ), seems to ignore this: ‘For of everything relative the substance is relative to something else, seeing that the being of every relative term is identical with being in a certain relation to something.’ 5) Necessary knowledge of the related Knowledge of that in relation to which a relative is spoken of is necessary when one knows the relative: it is impossible to know a relative and at the same time not to know that in relation to which it is spoken of. (Cat., 7, 8a37-b3) To prove this, Aristotle adheres to the definition of relatives: ‘If someone knows of a certain ‘this’ that it is a relative, and being for relatives is the same as being somehow related to something, he knows that also to which this is somehow related.’ (Cat., 7, 8a37-b3; cf. 7, 8b3-15 for Aristotle’s examples) This necessary knowledge of the related, if we call it so, dedicates Aristotle an epistemological reason, besides the ontological one mentioned in his definition, for the exclusion of substances. As we noted in our discussion of Aristotle’s definition of relatives, he changed his first definition to exclude all substances from being relatives. Therefore, it is evident that for him relatives must not include substances, either primary or secondary. Aristotle does have no problem with primary substances simply because it is evident for him that they cannot be relatives. The same can be said about most of the secondary substances as well. The problem for which he changed the definition of substances was about some secondary substances like ‘head’ or ‘hand’: the fact that it is possible to know a hand or head without necessarily knowing definitely that in relation to which it is spoken of proves that they are not relatives. (Cat., 7, 8b15-21) 6) Reciprocation Aristotle regards reciprocation (ἀντιστρέφειν) necessary in all relations: ‘All relatives are spoken of in relation to correlatives that reciprocate.’ (Cat., 7, 6b28-29; cf. Cat., 10, 12b21-24) The sense of reciprocation is clear by his own examples: ‘The slave is called slave of a master and the master is called master of a slave; the double double of a half, and the half half of a double.’ (Cat., 7, 6b29-31) Reciprocation of relatives is so necessary that if there seems that we do not have reciprocation, it must necessarily be due to a mistake and that in relation to which something is spoken of must have not been given properly. (Cat., 7, 6b36-7a1) Aristotle’s example is this: If a wing is given as of a bird, ‘bird of a wing’ does not reciprocate because it has not been given properly: a wing is of a winged and not of a bird; a wing is wing of a winged and a winged is winged with a wing. (Cat., 7, 7a1-5) Even if we do not have a proper name to have a proper reciprocation, Aristotle points, we must invent names. (Cat., 7, 7a5-15 and 7b10-14) Thus, having proper names is the condition of necessary reciprocation: ‘All relatives are spoken of in relation to correlatives that reciprocate, provided they are properly given.’ (Cat., 7, 7a22-23) This condition is also said in another way: there is no reciprocation if a relation is given as related to some chance thing or to something that is accidentally the related thing like when, for example, a slave is given as of a man or a biped instead of being given as of a master. (Cat., 7, 7a25-31) 7) Simultaneity of relatives Aristotle believes that in most cases relatives are simultaneous: double and half or master and slave must exist at the same time: when there is a half, there is a double and when there is a slave, there is a master. (Cat., 7, 7b10-14) However, this receives some exceptions like knowable, which is prior and can, thus, exist before and without knowledge. (Cat., 7, 7b22-27) What approves this non-simultaneousness for Aristotle is that the destruction of knowledge does not carry the knowable to destruction. (Cat., 7, 7b27-31) The same is said about perception and perceptible. (Cat., 7, 7b35-8a9) Aristotle attaches simultaneity to reciprocation. This reciprocation, however, is not a simple reciprocation but ‘reciprocation as to implication of existence’ (ἀντιστρέφει μὲν κατὰ τὴν τοῦ εἶναι) with the condition that neither be in any way the cause of the other’s existence. (Cat., 13, 14b27-29; the same is said also at: Cat., 13, 15a4-11) Aristotle’s examples are the double and the half because when there is a double there is a half and when there is a half there is a double but neither is the cause of the other’s existence. (Cat., 13, 14b29-32) (We have ‘reciprocation as to implication of existence’ also in relation between genus and species but it seems that this has a totally different sense there. 8) Having no independent reality That a relative cannot be a substance, either a primary or a secondary substance, is discussed in our review of both the definition of relatives and their reciprocation. The differentiation of relatives and substances are so deep for Aristotle that after questioning the possibility of any common element or principle for substances and relatives (Met., Λ, 1070a33-36), he asserts that no substance, on the one hand, is the element of relatives and none of the relatives, on the other hand, is the element of substances. (Met., Λ, 1070b3-4) Aristotle does not, however, suffice to this. For him, relatives have the least substantiality, which is, for him, almost the same as reality: ‘The relative is least of all categories a real thing (φύσις τις) or substance, and less than quality and quantity; and the relative is an affection of quantity.’ (Met., M, 1088a22-25) Aristotle believes that an evidence of this least reality (ὄν) and substantiality is that relatives have no proper generation, destruction or movement while each of substance, quality and quantity have them. (Met., M, 1088a29-35) Moreover, neither motion (Phy., E, 1) nor any kind of change is applicable to relatives so that relatives are neither themselves alterations nor the subject of alterations (Phy., Z, 3) and the process of losing and acquiring states cannot be considered as alterations because these are the result of the alterations of non-relative things of which they are states. (Phy., Z, 3) Relative is not even matter but is something different. (Met., M, 1088a24-25) The reason is that the matter is potentially of a nature but relative is neither potentially nor actually of a nature. (Met., M, 1088b1-2) To understand the status of relatives in Aristotle’s world, we have to make a tripartite classification; a classification that though Aristotle himself did not make, his assertions approve it. Based on this division, there are three kinds of being in the world: i) Independent beings: This class includes only substances. ii) Dependent beings: This class includes qualities and quantities. Although they are real, they are ‘in’ substances and cannot exist without them. iii) Super dependent beings: This class includes at least relatives. The fact that relatives must be considered in a third class different from substances, on the one hand, and qualities and quantities, on the other hand, is obvious not only from the previously mentioned texts (Met., M, 1088a22-25, 29-35 and b1-2) in which relatives’ reality is considered less than all substances, qualities and quantities and different from matters but from this text: ‘For there is nothing either great or small, many or few, or, in general, relative which is not something (τι ὄν) different (ἓτερον) [that is also] many or few or great or small’. (Met, N, 1088a27-30) Therefore, super dependent beings are those beings that must necessarily be also something else, i.e. an independent or dependent being. Ackrill (1963, 99) points that some of the words used by Aristotle to exemplify relational entities, e.g. slave, are endowed with a complete sense and do not need to be supplemented by a correlate. (Thus, the linguist criterion of incompleteness would be deficient. Some other instances of relative entities like ‘state,’ ‘knowledge’ and perception are not necessarily followed by the genitive case. 9) Extent of relatives What is the extent of relatives? What are or are not included in relatives either among categories or among other concepts? Which concepts or things Aristotle regard as relative? i) Aristotle includes state (ἓξις) and position (θέσις) among relatives. (Cat., 7, 6b2-6; Cat., 7, 6b11-14; Cat., 8, 11a20-24) ii) Although relative is not matter but is something different (Met., M, 1088a24-25), matter is ‘non-being’ only in virtue of an attribute (Phy., A, 9) and is, thus, a relative term. (Phy., B, 2, 194b9) 10) Property and relativity Aristotle draws a contrast between two types of giving a property, absolute and relative: ‘A property is given either in its own right and for always or relative to something else and for a time.’ (To., E, 1, ^128b15-18) Being a civilized man, for example, is an absolute property for man while to command is a relative property for the soul. The absolute property, however, is considered not only potential to be discussed or observed in relation to many things or periods of time, it in fact ‘belongs to its subject relatively to every single thing that there is, so that if the subject is not distinguished relatively to everything else, the property will not have been given correctly.’ (To., E, 1, 129a18-20) Therefore, an absolute property is a property that ‘is ascribed to a thing in comparison with everything else and distinguishes it from everything else.’ (To., E, 1, 128b33- ) An absolute property is, then, absolutely relative and not conditionally relative, i.e. relative to a specific thing. This is what differentiates it from a relative property, which is relative to a certain thing: ‘A property relative to something else is one which separates its subject off not from everything else but from a particular definite thing.’ (To., E, 1, 128b33- ) One important difference between absolute and relative properties is that while an accident can be a relative property, it can never be an absolute property. (To., I, 5, ^102b20-36) For example, sitting, which is an accident, is also a property relatively to those who are not sitting. (To., I, 5, ^102b20-36) 11) Platonic theories as relatives At least three of Platonic theories Aristotle attaches to relatives: i) Forms ‘It seems that a Form is always spoken of in relation to a Form-this desire itself is for the pleasure itself, and wishing itself is for the good itself.’ (To., Z, 8, 147a^10) Moreover, the theory of Forms takes the relative prior to the absolute as, for example, it takes not the dyad but the number as first. (Met., A, 990b15-17) Aristotle believes that this is against not only the necessities of the case but only the Platonists’ opinion because Forms must be substances only, if they can be shared. (Met., A, 990b27-29) There are, however, things like ‘equal’ which are only relative: they are only in relation to something else. Now the theory that ideas are supposed to be substances only should entail the substantiality of merely relatives. The reason being that Forms are not shared incidentally but each thing shares in that which is not predicated of a subject. (Met., A, 990b29-31) ii) Unequal The Platonic theory of great and small or unequal, which we know more from Aristotle and has, in Platonic philosophy, a role like that of matter in Aristotelian philosophy, is also called relative by Aristole. (Met., M, 1089b4-15) iii) Potentiality Aristotle says that Platonists take what potentially is a ‘this’ and a substance but not actually so a relative because it is ‘neither potentially the one or being, nor the contradictory of the one nor of being, but one among beings.’ (Met., N, 1089b15-20) 12) All things as relative Aristotle attaches the theory that ‘everything is true’ to relativity and thinks the consequence of believing in this theory is that everything is true: ‘He who says all things that appear are true, makes all things relative.’ (Met., Γ, 1011a17-20) 13) Genera and species of relatives There are some genera, Aristotle hints, that are relative but their species are not relative. In such cases, the genera are spoken of in relation to something, but none of the particular cases are so spoken. (Cat., 8, 11a20-36) Aristole’s examples are knowledge and grammar of which the former is a relative but the latter is a quality. The consequence of this is that there is nothing absurd for a thing to be in both genera of relative and quality. (Cat., 8, 11a37- ) However, when the species is a relative, the genera will be a relative too. (To., Δ, 4, 124b15- ) Moreover, Aristotle asserts that ‘the differentiae of relative terms are themselves relative.’ (To., Z, 6, 145a14-16) Things that are called relative are called so, Aristotle says, ‘because the classes that include them are of this sort, e.g. medicine is thought to be relative bcs its genus, knowledge, is thought to be relative,’ (Met., Δ, 1021b4-6) 14) Relatives as indefinite? Fabio Morales takes 1088a29-b1 as a textual evidence supporting the assumption that Aristotle regarded relational terms as indefinite. (shrink)

A well known and referenced global result is the nilpotent characterisation of the finite p-groups. This un doubtedly transends into neutrosophy. Hence, this fact of the neutrosophic nilpotent p-groups is worth critical studying and comprehensive analysis. The nilpotent characterisation depicts that there exists a derived series (Lower Central) which must terminate at {ϵ} (an identity), after a finite number of steps. Now, Suppose that G(I) is a neutrosophic p-group of class at least m ≥ 3. We (...) show in this paper that Lm−1(G(I)) is abelian and hence G(I) possesses a characteristic abelian neutrosophic subgroup which is not supposed to be contained in Z(G(I)). Furthermore, If L3(G(I)) = 1 such that pm is the highest order of an element of G(I)/L2(G(I)) (where G(I) is any neutrosophic p-group) then no element of L2(G(I)) has an order higher than pm. (shrink)

In this paper, we provide a detailed review of previous classifications of 2 × 2 games and suggest a mathematically simple way to classify the symmetric 2 × 2 games based on a decomposition of the payoff matrix into a cooperative and a zero-sum part. We argue that differences in the interaction between the parts is what makes games interesting in different ways. Our claim is supported by evolutionary computer experiments and findings in previous literature. In addition, we provide (...) a method for using a stereographic projection to create a compact 2-d representation of the game space. (shrink)

Propositional logics in general, considered as a set of sentences, can be undecidable even if they have “nice” representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple—at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented (...) in various ways can be approximated by finite-valued logics. It is shown that the minimal m-valued logic for which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics. (shrink)

Society is a composite of interacting people and groups. These groups play a significant role in maintaining social status, establishing group identity and social identity, and enforcing norms. As such, groups are essential for understanding human behavior. Nevertheless, the study of groups in everyday group life yields many diverse and sometimes contradicting theories of group behavior, and researchers tend to agree that we have yet to understand the emergence of groups out of aggregates of individuals. (...) The current paper aims to shed new light on the convoluted interrelation between groups and individuals by focusing on individuals’ social identities and group categorization. It does so by exploring the dynamic nature of the self and its implications on identity and group membership, and introducing a framework recognizing the fluidity of groups and group categorization. Incorporating historical insights with contemporary theories, this paper argues for a flexible understanding of group dynamics that surpasses rigid in-group and out-group classifications, proposing instead that group affiliations exist along a continuum that reflects the ever-changing social landscape. (shrink)

To this day, a hundred and fifty years after Mendeleev's discovery, the overal structure of the periodic system remains unaccounted for in quantum-mechanical terms. Given this dire situation, a handful of scientists in the 1970s embarked on a quest for the symmetries that lie hidden in the periodic table. Their goal was to explain the table's structure in group-theoretical terms. We argue that this symmetry program required an important paradigm shift in the understanding of the nature of chemical elements. The (...) idea, in essence, consisted of treating the chemical elements, not as particles, but as states of a superparticle. We show that the inspiration for this came from elementary particle physics, and in particular from Heisenberg's suggestion to treat the proton and neutron as different states of the nucleon. We provide a careful study of Heisenberg's last paper on the nature of elementary particles, and explain why the Democritean picture of matter no longer applied in modern physics and a Platonic symmetry-based picture was called for instead. We show how Heisenberg's Platonic philosophy came to dominate the field of elementary particle physics, and how it found its culmination point in Gell-Mann's classification of the hadrons in the eightfold way. We argue that it was the success of Heisenberg's approach in elementary particle physics that sparked the group-theoretical approach to the periodic table. We explain how it was applied to the set of chemical elements via a critical examination of the work of the Russian mathematician Abram Ilyich Fet the Turkish-American physicist Asim Orhan Barut, before giving some final reflections. (shrink)

Politicians select their words meticulously, never losing sight of their ultimate communicative goal. Sometimes, their objective may be that of not being fully understood by a large portion of the audience. They can achieve this by means of dogwhistles; linguistic expressions that, in addition to their literal meaning, convey a concealed message to a specific sub-group of the audience. This paper focuses on the distinction between overt and covert dogwhistles introduced by J. Saul (2018). I argue that, even if the (...) distinction successfully captures a genuine divide within the category of dogwhistles, the account proposed by Saul to explain the distinction is unsatisfactory. In response to this state of affairs, I illustrate how the distinction between overt and covert dogwhistle can be refined and illuminated by incorporating it into the 'Simple Theory' of dogwhistles advanced by J. Khoo (2017). (shrink)

There is a way of talking that would appear to involve ascriptions of purpose, goal directed activity, and intentional states to groups. Cases are familiar enough: classmates intend to vacation in Switzerland, the department is searching for a metaphysician, the Democrats want to minimize losses in the upcoming elections, and the US intends to improve relations with such and such country. But is this talk to be understood just in terms of the attitudes and actions of the individuals involved? (...) Is the talk, to take an overly simple proposal as an example, a mere summary of familiar individual attitudes of the group members? Or is the ascription of attitudes and actions to groups to be taken more literally, as suggesting that the group for example believes that P, or intends to A, over and above what the members individually think and do? In short, are there groups with minds of their own? Philip Pettit has deployed the “discursive dilemma” to defend the thesis that there are such group minds. In what follows, I explore the relationship between the group allegedly with a mind of its own and the individuals it comprises, and I consider just how this relationship must be understood in order to give Pettit’s argument for group minds its best chance for success. As I understand it, the discursive dilemma has to be used in conjunction with what might be called an indispensability argument for group minds. It is useful to distinguish two forms of this argument. The explanatory version of the indispensability argument is, very schematically, as follows: there is a compelling explanatory theory T concerning the social, certain indispensable elements of T entail the group mind thesis, so the group mind thesis is true. Several questions immediately arise: What sort of theory is T? In what sense is it indispensable? Are there other forms of indispensability? I don’t have definitive answers to these questions. But how we settle them will have implications for the interaction and support the discursive dilemma provides the indispensability argument. In particular, using the discursive dilemma to defend what I characterize below as a practical version of the indispensability argument commits us to the rationality of individual participants in a way that the explanatory version of the indispensability argument does not. My point in the first part of the paper is that if Pettit wants to avoid the weaknesses of the explanatory indispensability argument and pursue the practical version, then he owes us a story about the rationality of individual participation in groups. Pettit also owes us a story about the agency an individual exercises as part of a group. If it takes the actions of individuals to execute the intentions of the group, how are we to understand those actions in order for the group to count as having a mind of its own? How must group intentions figure in the practical or deliberative perspective of individuals who execute those intentions? I will argue that the proponent of the group mind thesis must proceed with some care here, because some natural ways of answering these questions will undermine the thesis. But in the end, I think that these questions are interesting independently of whether Pettit is right to think that groups do have minds of their own. That’s because investigating Pettit’s arguments might lead to new ideas about how the rationality and agency of individuals can be exercised, and suggests new ways of understanding how individuals can act together, irrespective of whether the groups they compose ever have minds of their own. (shrink)

The representation and classification of the structure of natural arguments has been one of the most important aspects of Aristotelian and medieval dialectical and rhetorical theories. This traditional approach is represented nowadays in models of argumentation schemes. The purpose of this article is to show how arguments are characterized by a complex combination of two levels of abstraction, namely, semantic relations and types of reasoning, and to provide an effective and comprehensive classification system for this matrix of semantic (...) and quasilogical connections. To this purpose, we propose a dichotomous criterion of classification, transcending both levels of abstraction and representing not what an argument is but how it is understood and interpreted. The schemes are grouped according to an end-means criterion, which is strictly bound to the ontological structure of the conclusion and the premises. On this view, a scheme can be selected according to the intended or reconstructed purpose of an argument and the possible strategies that can be used to achieve it. (shrink)

Scientists represent their world, grouping and organizing phenomena into classes by means of concepts. Philosophers of science have historically been interested in the nature of these concepts, the criteria that inform their application and the nature of the kinds that the concepts individuate. They also have sought to understand whether and how different systems of classification are related and more recently, how investigative practices shape conceptual development and change. Our aim in this paper is to provide a critical overview (...) of some of the key developments in this philosophical literature and identify some interesting issues it raises about the prospects of the so-called “special sciences”, including psychiatry, psychology, and the mind-brain sciences more generally, to discover natural kinds. (shrink)

Our aim in this chapter is to draw attention to what we see as a disturbing feature of conciliationist views of disagreement. Roughly put, the trouble is that conciliatory responses to in-group disagreement can lead to the frustration of a group's epistemic priorities: that is, the group's favoured trade-off between the "Jamesian goals" of truth-seeking and error-avoidance. We show how this problem can arise within a simple belief aggregation framework, and draw some general lessons about when the problem is (...) most pronounced. We close with a tentative proposal for how to solve the problem raised without rejecting conciliationism. (shrink)

According to certain kinds of semantic dispositionalism, what an agent means by her words is grounded by her dispositions to complete simple tasks. This sort of position is often thought to avoid the finitude problem raised by Kripke against simpler forms of dispositionalism. The traditional objection is that, since words possess indefinite (or infinite) extensions, and our dispositions to use words are only finite, those dispositions prove inadequate to serve as ground for what we mean by our words. (...) I argue that, even if such positions (emphasizing simple tasks) avoid the traditional form of Kripke's charge, they still succumb to special cases of the finitude problem. Furthermore, I show how such positions can be augmented so as to avoid even these special cases. Doing so requires qualifying the dispositions of interest as those possessed by the abstracted version of an actual agent (in contrast to, say, an idealized version of the agent). In addition to avoiding the finitude problem in its various forms, the position that results provides new materials for appreciating the role that abstracting models can play for a dispositionalist theory of meaning. (shrink)

I argue that a group can have normative commitments, and that the commitment of a group is not merely a sum or aggregate of the commitments of individual group members. I begin with a set of simple cases which illustrate two structurally different ways that group commitments can go wrong. These two kinds of potential failure correspond to two different levels of commitment: one at the individual level, owed to the other group members, and one at the group level, (...) which the group as a single body owes either to itself or to some third party. I distinguish the content of a commitment (what must be done for the commitment to be fulfilled) from the holder of that commitment: the party to whom the content is owed. I then discuss examples which support the two-level view of group commitment and show that, even when individual-level and group-level commitments have the same content, they are understood to have different holders. Finally I return to my original cases and argue that a two-level structure of group commitment allows us to make sense of the problems that occur in them. (shrink)

A. W. F. Edwards famously named “Lewontin’s fallacy” after Richard Lewontin, the geneticist who showed that most human genetic diversity can be found within any given racialized group. “Lewontin’s fallacy” is the assumption that uncorrelated genetic data would be sufficient to classify genotypes into conventional “racial” groups. In this chapter, I argue that Lewontin does not commit the fallacy named after him, and that it is not a genuine fallacy. Furthermore, I argue that when Edwards assumes that stable (...) class='Hi'>classification is all it takes to vindicate racial naturalism, he commits a fallacy of his own: Edwards’ fallacy. The ability to create a classificatory system, and then reliably sort things within that system, is not enough to make that system scientifically respectable. I show that Edwards’ fallacy is rife in debates about the existence of human biological races. (shrink)

We argue that the logical outcome of the cladistics revolution in biological systematics, and the move towards rankless phylogenetic classification of nested monophyletic groups as formalized in the PhyloCode, is to eliminate the species rank along with all the others and simply name clades. We propose that the lowest level of formally named clade be the SNaRC, the Smallest Named and Registered Clade. The SNaRC is an epistemic level in the classification, not an ontic one. Naming stops (...) at that level because there is no currently acceptable evidence for clades within it, not because no smaller clades exist. Later, included clades may be named. They would then become the SNaRCs, while the original SNaRC would keep its original name. We argue that all theoretical tasks of biology, in evolution and ecology, as well as practical tasks such as conservation assessment, are better approached using this rankless phylogenetic approach. (shrink)

When reusing existing ontologies for publishing a dataset in RDF (or developing a new ontology), preference may be given to those providing extensive subcategorization for important classes (denoted as focus classes). The subcategories may consist not only of named classes but also of compound class expressions. We define the notion of focused categorization power of a given ontology, with respect to a focus class and a concept expression language, as the (estimated) weighted count of the categories that can be built (...) from the ontology’s signature, conform to the language, and are subsumed by the focus class. For the sake of tractable initial experiments we then formulate a restricted concept expression language based on existential restrictions, and heuristically map it to syntactic patterns over ontology axioms (so-called FCE patterns). The characteristics of the chosen concept expression language and associated FCE patterns are investigated using three different empirical sources derived from ontology collections: first, the concept expression pattern frequency in class definitions; second, the occurrence of FCE patterns in the Tbox of ontologies; and last, for class expressions generated from the Tbox of ontologies (through the FCE patterns); their ‘meaningfulness’ was assessed by different groups of users, yielding a ‘quality ordering’ of the concept expression patterns. The complementary analyses are then compared and summarized. To allow for further experimentation, a web-based prototype was also implemented, which covers the whole process of ontology reuse from keyword-based ontology search through the FCP computation to the selection of ontologies and their enrichment with new concepts built from compound expressions. (shrink)

This paper deals with Gärdenfors’ theory of conceptual spaces. Let \({\mathcal {S}}\) be a conceptual space consisting of 2-type fuzzy sets equipped with several kinds of metrics. Let a finite set of prototypes \(\tilde{P}_1,\ldots,\tilde{P}_n\in \mathcal {S}\) be given. Our main result is the construction of a classification algorithm. That is, given an element \({\tilde{A}}\in \mathcal {S},\) our algorithm classifies it into the conceptual field determined by one of the given prototypes \(\tilde{P}_i.\) The construction of our algorithm uses some (...) physical analogies and the Newton potential plays a significant role here. Importantly, the resulting conceptual fields are not convex in the Euclidean sense, which we believe is a reasonable departure from the assumptions of Gardenfors’ original definition of the conceptual space. A partitioning algorithm of the space \(\mathcal {S}\) is also considered in the paper. In the application section, we test our classification algorithm on real data and obtain very satisfactory results. Moreover, the example we consider is another argument against requiring convexity of conceptual fields. (shrink)

In real-time collaborative graphical editing systems, Object-based Group/Ungroup operations are frequently accessible and practically useful. However, the existing research on these operations of the graphical editing is rare and defective. In this paper, based on Multi-Version strategy and Address Space Transformation method, a new MVSDR algorithm, which is not only applied to simple operations (such as Create, Delete, ChangeATT, etc.), but also suitable for Group/Ungroup ones, is proposed to solve the consistency maintenance problem. The proposed algorithm abandons previous attempts (...) to divide conflict operations into Real-Conflict operations and Resolvable-Conflict ones and to deal with them separately, thus making the algorithm more simple and effective. In addition, an example analysis is also given in this paper to prove the algorithm’s correctness and effectiveness. (shrink)

This paper presents a systematic approach for analyzing and explaining the nature of social groups. I argue against prominent views that attempt to unify all social groups or to divide them into simple typologies. Instead I argue that social groups are enormously diverse, but show how we can investigate their natures nonetheless. I analyze social groups from a bottom-up perspective, constructing profiles of the metaphysical features of groups of specific kinds. We can characterize any (...) given kind of social group with four complementary profiles: its “construction” profile, its “extra essentials” profile, its “anchor” profile, and its “accident” profile. Together these provide a framework for understanding the nature of groups, help classify and categorize groups, and shed light on group agency. (shrink)

The Simple Question (SQ) asks: “What are the necessary and jointly sufficient conditions any x must satisfy in order for it to be true that x is a simple?” The main motivation for asking SQ stems from the hope that it could teach us important lessons for material-object ontology. It is universally accepted that a proper answer to it has to be finite, complete and devoid of mereological expressions. This paper argues that we should stop treating SQ (...) as the central question to be asked about simples; there are a plethora of questions about simples that may even be addressed without answering the original question. The paper closes with some tentative remarks on how this lesson could be extended to van Inwagen’s Special Composition Question. (shrink)

While this article sets out to advance our knowledge about the characteristics of Nigerian cybercriminals (Yahoo-Boys), it is also the first study to explore the narratives of the Economic and Financial Crimes Commission (EFCC) officers concerning them. It appraises symbolic interactionist insights to consider the ways in which contextual factors and worldview may help to illuminate officers’ narratives of cybercriminals and the interpretations and implications of such accounts. Semi-structured interviews of forty frontline EFCC officers formed the empirical basis of this (...) study and were subjected to a directed approach of qualitative content analysis. While prior studies, for example, indicated that only a group of cybercriminals deploy spiritual and magical powers to defraud victims (i.e. modus operandi), our data analysis extended this classification into more refined levels involving multiple features. In particular, analysis bifurcates cybercriminals and their operations based on three factors: [a] educational-attainment, [b] modus-operandi, and [c] networks-collaborators. Results also suggest that these cybercriminals and their operations are embedded in “masculinity-and-material-wealth”. These contributions thus have implications for a range of generally accepted viewpoints about these cybercriminals previously taken-for-granted. Since these criminals have victims all over the world, insights from our study may help various local and international agencies [a] to understand the actions/features of these two groups of cybercriminals better and develop more effective response strategies. [b] to appreciate the vulnerabilities of their victims better and develop more adequate support schemes. We also consider the limitations of social control agents’ narratives on criminals. (shrink)

Background Retraction is a mechanism for alerting readers to unreliable material and other problems in the published scientific and scholarly record. Retracted publications generally remain visible and searchable, but the intention of retraction is to mark them as “removed” from the citable record of scholarship. However, in practice, some retracted articles continue to be treated by researchers and the public as valid content as they are often unaware of the retraction. Research over the past decade has identified a number of (...) factors contributing to the unintentional spread of retracted research. The goal of the Reducing the Inadvertent Spread of Retracted Science: Shaping a Research and Implementation Agenda (RISRS) project was to develop an actionable agenda for reducing the inadvertent spread of retracted science. This included identifying how retraction status could be more thoroughly disseminated, and determining what actions are feasible and relevant for particular stakeholders who play a role in the distribution of knowledge. -/- Methods These recommendations were developed as part of a year-long process that included a scoping review of empirical literature and successive rounds of stakeholder consultation, culminating in a three-part online workshop that brought together a diverse body of 65 stakeholders in October–November 2020 to engage in collaborative problem solving and dialogue. Stakeholders held roles such as publishers, editors, researchers, librarians, standards developers, funding program officers, and technologists and worked for institutions such as universities, governmental agencies, funding organizations, publishing houses, libraries, standards organizations, and technology providers. Workshop discussions were seeded by materials derived from stakeholder interviews (N = 47) and short original discussion pieces contributed by stakeholders. The online workshop resulted in a set of recommendations to address the complexities of retracted research throughout the scholarly communications ecosystem. -/- Results The RISRS recommendations are: (1) Develop a systematic cross-industry approach to ensure the public availability of consistent, standardized, interoperable, and timely information about retractions; (2) Recommend a taxonomy of retraction categories/classifications and corresponding retraction metadata that can be adopted by all stakeholders; (3) Develop best practices for coordinating the retraction process to enable timely, fair, unbiased outcomes; and (4) Educate stakeholders about pre- and post-publication stewardship, including retraction and correction of the scholarly record. -/- Conclusions Our stakeholder engagement study led to 4 recommendations to address inadvertent citation of retracted research, and formation of a working group to develop the Communication of Retractions, Removals, and Expressions of Concern (CORREC) Recommended Practice. Further work will be needed to determine how well retractions are currently documented, how retraction of code and datasets impacts related publications, and to identify if retraction metadata (fails to) propagate. Outcomes of all this work should lead to ensuring retracted papers are never cited without awareness of the retraction, and that, in public fora outside of science, retracted papers are not treated as valid scientific outputs. (shrink)

The purpose of the article is to reveal the scientific approach that substantiates the impact of the creation of strategic alliances (SA) on the digital transformation of business and the development of their innovative power based on identified success factors. The aim was achieved using the following methods: abstract logic and typification (for classification of SA's success factors), generalization (to determine the peculiarities of SA's influence on their innovation development), analytical and ranking method (to determine the relationship between the (...) dynamics creating of SA and the degree of acceleration of digital transformation), expert evaluation (to determine the degree of influence of SA success factors by business areas). Implementing a business-integrated approach to understanding digital transformation will accelerate implementation of innovation and organizational change, identifies future prospects for market, customer and business relationships, highlighting the importance of researching the factors that ensure their success. An alternative vision of SA’s success factors is suggested, which are determined by the possibilities the organization of partnerships and organizational culture, integration of business sectors, compatibility of management goals, external relevance, obtaining synergies. Innovation is considered by the authors as an integration factor that reconciles all groups of SA success factors, giving them the necessary focus to solve business problems. The results of the study show that the creation of SA has a significant impact on accelerating and changes priority of digital transformation of business areas involved in strategic partnerships, and the impact of SA on the development of their innovation power is crucial. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The (...) previous attempts all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

Wrong beliefs, known by some as ‘alternative facts’, have proliferated lately in important areas of human life, including social, political, and public health domains. This can be and has been damaging. This brief article proposes an epistemological category classification of these wrong beliefs, with the following mappings: a) ‘No-Information’ marked by willful blindness produces ‘Empty Beliefs’; b) ‘Mis-Information’ yields ‘Mis(taken) Beliefs’; and c) ‘Dis-Information’ predicated on blatant distortions produces ‘Dis(torted) Beliefs’. This simple classification system, is perhaps epistemologically (...) satisfying, and moreover could have positive policy implications. (shrink)

In this paper I defend the existence of a Dialectical school proper against criticisms brought forward by Klaus Döring and by Jonathan Barnes. Whereas Döring claims that there was no Dialectical school separate from the Megarians, Barnes takes issue with my claim (argued for in “Dialektiker und frühe Stoiker bei Sextus Empiricus”) that most of the reports in Sextus on the dialecticians refer to members of the Dialectical school. Barnes contends that these dialecticians are in fact Stoic logicians. As against (...) Döring, I argue that the passage in Diogenes Laertius II 113 (first drawn attention to by David Sedley) which talks of a the Megarian Stilpo winning over disciples from the Dialecticians is not refuted by Döring’s arguments. It clearly shows that the Dialecticians and the Megarians at the time were taken to be different philosophical sects. As against Barnes I insist on the differences between the report in Ps.-Galen’s Historia 9 and the Sextan report on the theory of sign in AM VIII. These reports offer incompatible definitions of the indicative sign. Moreover, the classification of simple propositions reported by Sextus at AM VIII 96f. cannot be a truncated version of the (Stoic) list to be found in Diogenes Laertius VII 69f. since in Sextus’ report one of the three classes of simple propositions is labelled middle (meson). This is a certain sign that we are dealing with a triad, and hence that this list is meant to be complete. Therefore the classification found in Sextus and attributed to the dialecticians and the one in Diogenes Laertius reporting Stoic material do come from different sources. (shrink)

This essay is part of larger project in which I attempt to show that Western formal logic, from its inception in Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. More specifically, (a) racist/quasi-racist/proto-racist political forces were part of the impetus for logic’s attempt to classify the world into mutually exclusive, hierarchically-valued categories in the first place; and (b) these classifications, in turn, have been deployed throughout history to justify and empower (...) racism/quasi-racism/proto-racism. In other words, (a) an important part of the chaos and messiness that so troubles logic has been the natural bio-cultural diversity of human beings as described by concepts such as race and ethnicity; and (b) once these concepts were historically in place, taken for granted, buttressed with pseudo-science, and had their origins forgotten, they became tools for the oppression of diverse human groups. The central thesis of this essay is that close readings of a variety of Leibniz’s writings (including several essays and one letter) will reveal a pattern of mutual dependency and creation in regard to logic and racism in Leibniz. My conclusion will be that there is a meaningful connection between logic and racism (or at least between logic and politics in general) in Leibniz such that one should be skeptical of his logic’s claim to offer transparent and neutral descriptions of the world as well as imperatives for action. Put differently, I want to ask the following specific question: how do Leibniz’s own racially-informed political goals, along with the emergence of race-centric European colonialism, both invest and disguise themselves within Leibniz’s mathematizing of logic? (shrink)

This article describes the peculiarities of the formation and modeling of the enterprise risk management system in modern conditions. The main purpose of the research is further development of theoretical and methodical principles and development of practical recommendations aimed at improvement of management of business risks of wholesale enterprises. The critical analysis of literature sources and approaches to solving problems of enterprise risk management testifies to the lack of attention of scientists of the present-day to the investigated problem in the (...) sphere of activity of enterprises of wholesale trade. The urgency of this problem is caused by the existing losses of wholesale companies due to imperfect management as a result of the influence of various business risks. The methodological basis of the research is the system of methods used to obtain the final results: theoretical generalization, scientific abstraction and comparison – to deepen the essence of the concept of "business risk"; analysis and synthesis, grouping – for classification of business risks in the sphere of wholesale trade; methods of analysis, synthesis and logical generalization – to obtain conclusions during determination of the state and problems of realization of business activity in the sphere of wholesale trade of enterprises of Ukraine; systematic approach – for improvement of system of management of business risks of enterprises of wholesale trade on the basis of commercial values. The object of the research is the business risk in the wholesale trade environment. In the article the further development of the risk-based apparatus has been made, the negative consequences of the impact of business risks on the activities of the wholesale trade enterprises have been clarified, the enterprise risk management system of the enterprises has been built, which will reflect an integrated approach to its realization, practical recommendations on the formation of a model of management of business risks of the wholesale trade enterprises on the basis of commercial values. The results of the survey can be useful for business structures, which are focused on improving the management of business risks. (shrink)

This article concerns the criteria for when a group can collectively hold a belief. By proposing a cognitive non-summative account of group belief (GB), I highlight three necessary features at the individual level: commonality, mutuality, and group-based considerations. My account asserts that group G believes proposition p if and only if a sufficient majority of its members believe (1) pG, where pG is “Given some G-based considerations, p,” and (2) “the majority of G also believe that pG.” The article critiques (...) three significant accounts of GB: Anthony Quinton’s simple summative account, Margaret Gilbert’s and Raimo Tuomela’s conative non-summative views, and Jennifer Lackey’s modified summativism. (shrink)

This study focuses on the step-to-step procedure in creating Facebook Social Learning Group (FBSLG) and the perception of students on using FBSLG as learning management tool. Descriptive method was employed in this study participated by two hundred eighty (280) teacher education students in Nueva Ecija University of Science and Technology – College of Education during the academic year 2020-2021 who were purposively selected based on the criteria set by the researcher. Five simple steps on creating FBSLG were discussed by (...) the researcher. Using the adapted survey questionnaire, the data revealed that the students strongly agreed on the four aspects of perception about FBSLG namely: pedagogical affordance, social affordance, technological affordance, and their overall perception on the platform. Some worries of the students on the platform were addressed like the safety of sharing ideas in the FBSLG and some technical difficulties were encountered. It is concluded that the FBSLG is a good platform of conducting classes during this time of pandemic, so the use of this platform to other subjects is highly recommended. (shrink)

We conducted an experiment to determine the impact of short-term pressure on 1,228 Grade 8 students’ outcomes when performing simple math exercises. We required all students to complete 100 simple math questions for 90 seconds. We analysed students’ results and then divided them into three groups: (i) a control group who did nothing; (ii) a group who performed an easy task; and (iii) a group who performed a difficult task. Finally, we required all students to solve another (...) 100 simple math questions in 90 seconds and used a Bayesian model to compare the results of the three groups. We discovered that students who successfully solved complex tasks received higher outcomes within short periods than those who succeeded in the easy task. However, students who failed to solve either the easy or the problematic tasks received lower results than those who did nothing between the two attempts. Also, we found no differences between the results of male and female students. The findings shed further light on the Yerkes-Dodson law about the influence of stress and distress on students’ performances. (shrink)

Experiments are presented showing that visual search for Mueller-Lyer (ML) stimuli is based on complete configurations, rather than component segments. Segments easily detected in isolation were difficult to detect when embedded in a configuration, indicating preemption by low-level groups. This preemption—which caused stimulus components to become inaccessible to rapid search—was an all-or-nothing effect, and so could serve as a powerful test of grouping. It is shown that these effects are unlikely to be due to blurring by simple spatial (...) filters at early visual levels. It is proposed instead that they are due to more sophisticated processes that rapidly bind contour fragments into spatially-extended assemblies. These results support the view that rapid visual search cannot access the primitives formed at the earliest stages of visual processing; rather, it can access only higher-level, more ecologically-relevant structures. (shrink)

A crucial socio-political challenge for our age is how to rede!ne or extend group membership in such a way that it adequately responds to phenomena related to globalization like the prevalence of migration, the transformation of family and social networks, and changes in the position of the nation state. Two centuries ago Immanuel Kant assumed that international connectedness between humans would inevitably lead to the realization of world citizen rights. Nonetheless, globalization does not just foster cosmopolitanism but simultaneously yields the (...) development of new group boundaries. Group membership is indeed a fundamental issue in political processes, for: “the primary good that we distribute to one another is membership in some human community” – it is within the political community that power is being shared and, if possible, held back from non-members. In sum, it is appropriate to consider group membership a fundamental ingredient of politics and political theory. How group boundaries are drawn is then of only secondary importance. Indeed, Schmitt famously declared that “[e]very religious, moral, economic, ethical, or other antithesis transforms into a political one if it is suffciently strong to group human beings e#ectively according to friend and enemy”. Even though Schmitt’s idea of politics as being constituted by such antithetical groupings is debatable, it is plausible to consider politics among other things as a way of handling intergroup di#erences. Obviously, some of the group-constituting factors are more easily discernable from one’s appearance than others, like race, ethnicity, or gender. As a result, factors like skin color or sexual orientation sometimes carry much political weight even though individuals would rather con!ne these to their private lives and individual identity. Given the potential tension between the political reality of particular groupmembership defnitions and the – individual and political – struggles against those definitions and corresponding attitudes, citizenship and civic behavior becomes a complex issue. As Kymlicka points out, it implies for citizens an additional obligation to non-discrimination regarding those groups: “[t]his extension of non-discrimination from government to civil society is not just a shift in the scale of liberal norms, it also involves a radical extension in the obligations of liberal citizenship”. Unfortunately, empirical research suggests that political intolerance towards other groups “may be the more natural and ‘easy’ position to hold”. Indeed, since development of a virtue of civility or decency regarding other groups is not easy, as it often runs against deeply engrained stereotypes and prejudices, political care for matters like education is justified. Separate schools, for example, may erode children’s motivation to act as citizens, erode their capacity for it and!nally diminish their opportunities to experience transcending their particular group membership and behave as decent citizens. This chapter outlines a possible explanation for such consequences. That explanation will be found to be interdisciplinary in nature, combining insights from political theory and cognitive neuroscience. In doing so, it does not focus on collective action, even though that is a usual focus for political studies. For example, results pertaining to collective political action have demonstrated that the relation between attitudes and overt voting behavior or political participation is not as direct and strong as was hoped for. Several conditions, including the individual’s experiences, self-interest, and relevant social norms, turned out to interfere in the link between his or her attitude and behavior. Important as collective action is, this chapter is concerned with direct interaction between agents and the in$uence of group membership on such interaction – in particular joint action. Although politics does include many forms of action that require no such physical interaction, such physical interaction between individuals remains fundamental to politics – this is the reason why separate schooling may eventually undermine the citizenship of its isolated pupils. This chapter will focus on joint action, de!ned as: “any form of social interaction whereby two or more individuals coordinate their actions in space and time to bring about a change in the environment”. Cognitive neuroscienti!c evidence demonstrates that for such joint action to succeed, the agents have to integrate the actions and expected actions of the other person in their own action plans at several levels of speci!city. Although neuroscienti!c research is necessarily limited to simple forms of action, this concurs with a philosophical analysis of joint action, which I will discuss below. (shrink)

This article introduces an experimental paradigm to selectively probe the multiple levels of visual processing that influence the formation of object contours, perceptual boundaries, and illusory contours. The experiments test the assumption that, to integrate contour information across space and contrast sign, a spatially short-range filtering process that is sensitive to contrast polarity inputs to a spatially long-range grouping process that pools signals from opposite contrast polarities. The stimuli consisted of thin subthreshold lines, flashed upon gaps between collinear inducers which (...) potentially enable the formation of illusory contours. The subthreshold lines were composed of one or more segments with opposite contrast polarities. The polarity nearest to the inducers was varied to differentially excite the short-range filtering process. The experimental results are consistent with neurophysiological evidence for cortical mechanisms of contour processing and with the Boundary Contour System model, which identifies the short-range filtering process with cortical simple cells, and the long-range grouping process with cortical bipole cells. (shrink)

In this paper, we review a general technique for converting the standard Lagrangian description of a classical system into a formulation that puts time on an equal footing with the system's degrees of freedom. We show how the resulting framework anticipates key features of special relativity, including the signature of the Minkowski metric tensor and the special role played by theories that are invariant under a generalized notion of Lorentz transformations. We then use this technique to revisit a classification (...) of classical particle-types that mirrors Wigner's classification of quantum particle-types in terms of irreducible representations of the Poincaré group, including the cases of massive particles, massless particles, and tachyons. Along the way, we see gauge invariance naturally emerge in the context of classical massless particles with nonzero spin, as well as study the massless limit of a massive particle and derive a classical-particle version of the Higgs mechanism. (shrink)

The Atlas Group appeals to philosophical thinking in multiple ways—both through its aesthetic figuration and its conceptual references. Presented as a foundation dedicated to the research and the compilation of documents on Lebanese contemporary history and organized in the form of an invented archive, this artistic project deliberately coalesces real and fictitious elements and confronts, subversively, Western views on the socio- political reality of the Middle East with implicit knowledge and experiences from the region. The singular constitution of this curious (...) archive subtly undermines the rational classification to which it alludes, thereby frustrating unilateral appropriations. Moreover, the question of the mediation of subjective experience and factual reality is consistently raised through the hysterical documents. This article aims to deploy how Walid Raad's project subtly criticises objectivist hegemonic claims by confronting divergent, often incommensurate approaches to the conflictual reality in Lebanon and its appropriation by media, historiography and politics. (shrink)

When our smartphones distract us, much more is at stake than a momentary lapse of attention. Our use of smartphones can interfere with the building-blocks of meaningfulness and the actions that shape our self-identity. -/- By analyzing social interactions and evolving experiences, Roholt reveals the mechanisms of smartphone-distraction that impact our meaningful projects and activities. Roholt’s conception of meaning in life draws from a disparate group of philosophers—Susan Wolf, John Dewey, Hubert Dreyfus, Martin Heidegger, and Albert Borgmann. Central to Roholt’s (...) argument are what Borgmann calls focal practices: dinners with friends, running, a college seminar, attending sporting events. As a recurring example, Roholt develops the classification of musical instruments as focal things, contending that musical performance can be fruitfully understood as a focal practice. -/- Through this exploration of what generates meaning in life, Roholt makes us rethink the place we allow smartphones to occupy in the everyday. But he remains cautiously optimistic. This thoughtful, needed interrogation of smartphones shows how we can establish a positive role for technologies within our lives. (shrink)

We present a number of modal logics to reason about group norms. As a preliminary step, we discuss the ontological status of the group to which the norms are applied, by adapting the classification made by Christian List of collective attitudes into aggregated, common, and corporate attitudes. Accordingly, we shall introduce modality to capture aggregated, common, and corporate group norms. We investigate then the principles for reasoning about those types of modalities. Finally, we discuss the relationship between group norms (...) and types of collective responsibility. (shrink)

That i) there is a somehow determined chronology of Plato’s dialogues among all the chronologies of the last century and ii) this theory is subject to many objections, are points this article intends to discuss. Almost all the main suggested chronologies of the last century agree that Parmenides and Theaetetus should be located after dialogues like Meno, Phaedo and Republic and before Sophist, Politicus, Timaeus, Laws and Philebus. The eight objections we brought against this arrangement claim that to place the (...) dialogues like Meno, Phaedo and Republic both immediately after the early ones and before Parmenides and Theaetetus is epistemologically and ontologically problematic. -/- Key Words: Plato; chronology; knowledge; being; Parmenides -/- Introduction While the ancient philosophers, doxographers and commentators from Aristotle onward considered, more or less, the question of the date and arrangement of the dialogues (cf. Irwin 2008, 77 n. 69), they would not observe a firm necessity to consider the progress of Plato’s theories in dialogues, maybe because they did not think of any essential shift in there. We might be able to say, nevertheless, that the most prominent feature of the ancient attitude to Plato was its peculiar attention to the Republic and the Timaeus as the most mature works in his philosophy and also the consideration of Laws as a later work. This tendency can be discovered from the general viewpoint of the first chronologies of the early 19th century after starting to deal with the issue. That Schleiermacher observed Republic as the culmination of Plato’s philosophy and as one of the latest dialogues besides Laws and Timaeus could reflex the implicit chronology of the tradition in the first mirrors it found. Another tendency in Schleiermacher is taking the triology of Theaetetus, Sophist and Politicus as relatively early. From the last quarter of 19th century onward, stylometry helped scholars to establish a new framework to constructa new arrangement between the dialogues. Based on stylistic as well as literary findings, Campbell (1867, xxxff.) argued for the closeness of the style of Sophist and Politicus with Timaeus, Critias, Philebus and Lawsthat, especially because of the certain evidence about this last dialogue’s lateness, led to the consideration of all as late dialogues. Almost every other stylistic effort after Campbell approved the similarities between Sophist and Politicus with Timaeus, Critias, Philebus and Laws. The result of all such investigations led to a new chronology that, despite some differences, has a fixed structure in all its appearances. 1. The Standard Chronology of the Dialogues The chronologies that are now commonly accepted are mostly based on the arrangement of dialogues to three groups corresponding to three periods of Plato’s life, which became predominant after applying stylistic features in assessing the similarities between dialogues. The fact that all the stylometric considerations reached to the similar results about the date of dialogues while they were assessing different stylistic aspects helped the new chronology become prevailing not only among stylistic chronologies but also between those like Fine, Kahn and Vlastos who were inclined more to the content-based arrangements. Even this latter group could not neglect the apparently certain results of using the method of stylometry. This was the main reason, I think, that made what they called content-based chronology be under the domination of stylometry much more than they could expect. The division of the dialogues into three separate groups became something that most of the scholars took for granted so far as Kahn thinks this division 'can be regarded as a fixed point of departure in any speculation about the chronology of the dialogues' (1996, 44). Thereafter, all the chronologists are accustomed to divide the dialogues to three groups of early, middle and late corresponding to the three stages of Plato’s life. Nevertheless, some of them tried to make subdivisions among each group and introduce some of the dialogues as transitional between different periods and thus reached to a fourfold classification of the dialogues. Although theycould never achieve to a consensus about the place of some dialogues, about which we will discuss soon, the whole spirit of theirchronological arrangements is the same and thus compelling enough for us to unify all of them with the label of 'Standard Chronology of Dialogues' (SCD). We brought together some of the most famous chronologies in the table below to make a comparison easier and to show how all are approximately of the same opinion about the place of some dialogues. The following points must be noted about this table: 1. I divided the dialogues to eight groups of early, late early, transitional, early middle, late middle, post-middle, early late, and late. Although none of the chronologists applies this classification, it can be helpful to compare them. In this table, for example, if one of the chronologist’s beholds one of the dialogues as later than all the dialogues of middle group, it is considered here as late middle. Otherwise, if it is emphasized that it is after all of them, it is considered as post-middle. The same is true about the dialogues of the late group in which I regarded the first dialogues of that period as early late only in those who explicitly considered some dialogues as earlier than other ones in the late group. Though, therefore, some of the dialogues might have not been considered as forming a distinct class, they are distinguished here. A. SCD’s Early and Transitional Dialogues When we move from stylometric to content-based chronologies, the homogeneity between the dialogues of each group is more understandable. Guthrie (1975, v. 4, 50) distinguishes three groups, the first of them includes Apology, Crito, Laches, Lysis, Charmides, Euthyphro, Hippias Minor, Hippias Major, Protagoras, Gorgiasand Ion. In addition to Meno, Phaedo and Euthydems, his first group does not include Cratylus and Menexenus. Unlike Guthrie, almost all the other content-based chronologiesof our study desire to distinguish two categories inside the first group of which the latter must be considered as the transitional group leading to the dialogues of the middle period. Kahn distinguishes four groups of dialogues and arranges two of them before middle period dialogues. The first group including Apology, Crito, Ion, Hippias Minor, Gorgiasand Menexenus he calls 'early' or 'presystematic' dialogues (1998, 124). The second group he calls the 'threshold', 'pre-middle' or 'Socratic' dialogues including seven: Laches, Charmides, Lysis, Euthyphro, Protagoras, Euthydemus and Meno. Based on Vlastos’ arrangement we must distinguish Euthydemus, Hippias Major, Lysis, Menexenus and Meno as 'transitional' dialogues from the 'elenctic' dialogues that are the other dialogues of Kahn’smentioned first two groups plus the first book of theRepublic. Fine also distinguishes 'transitional' dialogues from early or 'Socratic' dialogues,but her transitional dialogues are Gorgias, Meno, Hippias Major, Euthydemus and Cratylus of which she thinks the last two dialogues are 'controversial' (2003, 1). Her Socratic dialogues are all the remaining dialogues of Kahn’s first two groups. In spite of all the differences between the mentioned chronologies, it can be seen that all of them are inclined to arrange the early dialogues in a way that: i) Besides the dialogues that are considered as late, it never includes Republic II-X, Theaetetus, Phaedrus and Parmenides. ii) It intends to consider the dialogues like Euthydemus and Hippias Majorthat look more critical as later among the earlier dialogues or as transitional group. iii) Those who do not consider Meno as a middle period dialogue place it in their second or transitional group. iv) None of the content-based chronologies considers Phaedo and Symposium as early. Stylometric chronologies also intend to put them either as last dialogues among their early ones or as middle. B. SCD’s Middle Period Dialogues Campbell listed Republic, Phaedrus, Parmenides and Theaetetus as his second group of dialogues, an idea thatwas accepted by Brandwood. Ledger’smiddle period dialogueshad Euthydemus, Symposium and Cratylus in addition to the dialogues that Campbell and Brandwood had mentioned as middle. Among content-based chronologies, Guthrie’s list of middle period dialogues did not include Parmenides but some dialogues which had been considered as early in stylometric ones: Meno, Phaedo, Republic, Symposium, Phaedrus, Euthydemus, Menexenus and Cratylus. In sofar as I know, Euthydemus and Menexenus have not been considered as middle by other content-based chronologists and Guthrie is an exception among them. That Phaedo, Phaedrus, Symposium and Republic were middle period dialogues almost all the philosophical chronologists like Kahn, Fine, Vlastos (except Rep.I), Irwin andKraut came along. The dialogues they do not string along about are Meno, Cratylus, Parmenides and Theaetetus. Those like Guthrie, Kraut and Irwin who did not consider Meno as early presumably posit it among middles. The same can be said about Cratylus in the suggested chronologies of Guthrie, Kahn, Vlastos and Kraut. Nonetheless, it is different in case of Parmenides and Theaetetus. Whereas all the mentioned stylometric chronologists like Campbell, Brandwood and Ledger set them among middle period dialogues,the philosophical chronologists, it might seem at first, did not arrive at a consensus about them. While Guthrie and Fine put them as the first dialogues of the late group, Vlastos and Kraut set them as the latest of the middle group, as well as Kahn who puts them as post-middle and amongst the late period dialogues. Regardless the way they classify their groups, their disagreement does not affect the arrangement of the dialogues: they all posit Parmenides and Theaetetus after the series of Meno, Phaedo, Phaedrus, Symposium and Republic and before Sophist, Politicus, Timaeus, Philebus and Laws. To sum up SCD’s arrangement of the middle period dialogues we can add: v) Republic and Phaedrus have been considered by all the mentioned chronologists as dialogues of the middle period. vi) All the philosophical chronologies have reached a consensus about setting Phaedo and Symposium alongside with Republic and Phaedrus as middle. vii) While Stylometric alongside some philosophic chronologists arrange Parmenides and Theaetetus among their middle period dialogues and mostly as the latest among them, other philosophical chronologists put them as the early among the late dialogues. We can conclude then that SCD intends to locate these two dialogues at the boundary between the middle and late period dialogues. -/- . (shrink)

We introduce a family of rules for adjusting one's credences in response to learning the credences of others. These rules have a number of desirable features. 1. They yield the posterior credences that would result from updating by standard Bayesian conditionalization on one's peers' reported credences if one's likelihood function takes a particular simple form. 2. In the simplest form, they are symmetric among the agents in the group. 3. They map neatly onto the familiar Condorcet voting results. 4. (...) They preserve shared agreement about independence in a wide range of cases. 5. They commute with conditionalization and with multiple peer updates. Importantly, these rules have a surprising property that we call synergy - peer testimony of credences can provide mutually supporting evidence raising an individual's credence higher than any peer's initial prior report. At first, this may seem to be a strike against them. We argue, however, that synergy is actually a desirable feature and the failure of other updating rules to yield synergy is a strike against them. (shrink)

Carleson’s celebrated theorem of 1965 [1] asserts the pointwise convergence of the partial Fourier sums of square integrable functions. The Fourier transform has a formulation on each of the Euclidean groups R , Z andΤ .Carleson’s original proof worked on Τ . Fefferman’s proof translates very easily to R . M´at´e [2] extended Carleson’s proof to Z . Each of the statements of the theorem can be stated in terms of a maximal Fourier multiplier theorem [5]. Inequalities for such (...) operators can be transferred between these three Euclidean groups, and was done P. Auscher and M.J. Carro [3]. But L. Carleson’s original proof and another proofs very long and very complicated. We give a very short and very “simple” proof of this fact. Our proof uses PNSA technique only, developed in part I, and does not uses complicated technical formations unavoidable by the using of purely standard approach to the present problems. In contradiction to Carleson’s method, which is based on profound properties of trigonometric series, the proposed approach is quite general and allows to research a wide class of analogous problems for the general orthogonal series. (shrink)

We motivate a picture of social epistemology that sees forgetting as subject to epistemic evaluation. Using computer simulations of a simple agent-based model, we show that how agents forget can have as large an impact on group epistemic outcomes as how they share information. But, how we forget, unlike how we form beliefs, isn’t typically taken to be the sort of thing that can be epistemically rational or justified. We consider what we take to be the most promising argument (...) for this claim and find it lacking. We conclude that understanding how agents forget should be as central to social epistemology as understanding how agents form beliefs and share information with others. (shrink)