The paper provides some introductory comments and a preliminary translation of Avicenna’s Burhān, IV, 2. I shall first set the stage by outlining the structure of the book (sec. 1). I will then briefly introduce (sec. 2) a number of notions that are dealt with in the first treatise of the Burhān (e.g. definition, description). Burhān, IV, 2 is split into two parts: the first focuses mainly on Aristotle’s An. Post., B, 4, whereas the second covers some of the topics (...) of B, 5 and B, 6. Accordingly, sec. 3 will be devoted to a cursory presentation of Aristotle’s arguments in An. Post., B, 4 along with a more detailed discussion of its Avicennan counterpart, focusing on the indemonstrability of definition; sec. 4, finally, will be a presentation of the second part of the chapter, concerning the relationship between definition and division. An English translation of the entire chapter is appended to the paper and is accompanied by some notes. (shrink)

Since Mates’ seminal Stoic Logic there has been uncertainty and debate about how to treat the term anapodeiktos when used of Stoic syllogisms. This paper argues that the customary translation of anapodeiktos by ‘indemonstrable’ is accurate, and it explains why this is so. At the heart of the explanation is an argument that, contrary to what is commonly assumed, indemonstrability is rooted in the generic account of the Stoic epistemic notion of demonstration. Some minor insights into Stoic logic ensue.

ABSTRACT: Alexander of Aphrodisias’ commentaries on Aristotle’s Organon are valuable sources for both Stoic and early Peripatetic logic, and have often been used as such – in particular for early Peripatetic hypothetical syllogistic and Stoic propositional logic. By contrast, this paper explores the role Alexander himself played in the development and transmission of those theories. There are three areas in particular where he seems to have made a difference: First, he drew a connection between certain passages from Aristotle’s Topics and (...) Prior Analytics and the Stoic indemonstrable arguments, and, based on this connection, appropriated at least four kinds of Stoic indemonstrables as Aristotelian. Second, he developed and made use of a specifically Peripatetic terminology in which to describe and discuss those arguments – which facilitated the integration of the indemonstrables into Peripatetic logic. Third, he made some progress towards a solution to the problem of what place and interpretation the Stoic third indemonstrables should be given in a Peripatetic and Platonist setting. Overall, the picture emerges that Alexander persistently (if not always consistently) presented passages from Aristotle’s logical œuvre in a light that makes it appear as if Aristotle was in the possession of a Peripatetic correlate to the Stoic theory of indemonstrables. (shrink)

ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules (...) which establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments (cut type rules and antilogism). The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed (similar to tableau methods). The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented, and similarities to relevance logic are pointed out. (shrink)

ABSTRACT: A detailed presentation of Stoic theory of arguments, including truth-value changes of arguments, Stoic syllogistic, Stoic indemonstrable arguments, Stoic inference rules (themata), including cut rules and antilogism, argumental deduction, elements of relevance logic in Stoic syllogistic, the question of completeness of Stoic logic, Stoic arguments valid in the specific sense, e.g. "Dio says it is day. But Dio speaks truly. Therefore it is day." A more formal and more detailed account of the Stoic theory of deduction can be found (...) in S. Bobzien, Stoic Syllogistic, OSAP 1996. (shrink)

according to aristotle's posterior analytics, scientific expertise is composed of two different cognitive dispositions. Some propositions in the domain can be scientifically explained, which means that they are known by "demonstration", a deductive argument in which the premises are explanatory of the conclusion. Thus, the kind of cognition that apprehends those propositions is called "demonstrative knowledge".1 However, not all propositions in a scientific domain are demonstrable. Demonstrations are ultimately based on indemonstrable principles, whose knowledge is called "comprehension".2 If the knowledge (...) of all scientific propositions were... (shrink)

Some of Aristotle’s statements about the indemonstrability of the Principle of Non-Contradiction (PNC) in Metaphysics Γ 4 merit more attention. The consensus seems to be that Aristotle provides two arguments against the demonstrability of the PNC, with one located in Γ 3 and the other found in the first paragraph of Γ 4. In this article, I argue that Aristotle also relies upon a third argument for the same conclusion: the argument from truth. Although Aristotle does not explicitly state this (...) argument, it is the best argument that he could use to defend some of his statements in the second paragraph of Γ 4. Since the argument relies on only a few of Aristotle’s core views about truth, I propose that it is faithful to his considered position throughout his corpus, and it may be the strongest argument he could offer for the indemonstrability of the PNC. (shrink)

For Aristotle, demonstrative knowledge is the result of what he calls ‘intellectual learning’, a process in which the knowledge of a conclusion depends on previous knowledge of the premises. Since demonstrations are ultimately based on indemonstrable principles (the knowledge of which is called ‘νοῦς’), Aristotle is often described as advancing a foundationalist doctrine. Without disputing the nomenclature, I shall attempt to show that Aristotle’s ‘foundationalism’ should not be taken as a rationalist theory of epistemic justification, as if the first principles (...) of science could be known as such independently of their explanatory connections to demonstrable propositions. I shall argue that knowing first principles as such involves knowing them as explanatory of other scientific propositions. I shall then explain in which way noetic and demonstrative knowledge are in a sense interdependent cognitive states – even though νοῦς remains distinct from (and, in Aristotle’s words, more ‘accurate’ than) demonstrative knowledge. (shrink)

One of the most important ideas of our times is the conviction that, despite their diversity, all humans have an equal basic moral status (or an equal fundamental worth and dignity), that they are, as Thomas Jefferson famously stated, “created equals” and they should (because they are entitled to) be treated as equals. In this article I defend the suggestion – stated by philosophers like Brian Barry, Ronald Dworkin or Joel Feinberg – that the principle of equal human worth is (...) indemonstrable and has no other basis than itself. I argue, essentially, that all other theories about the basis of equality are false or at least problematic. In the last section I also try to show that, most probably, the truth of the principle of basic human equality cannot be established through any of the usual methods of argumentation and/or justification. (shrink)

In disagreement with Claudia Baracchi’s controversial thesis that there is a “simultaneity and indissolubility” if not an “identity” of intelligence (nous) and perception (aisthēsis) at the core of Aristotle’s philosophy, I will argue that Aristotle maintains a fundamental distinction between these cognitive faculties. My goal in this paper is to examine specific parts of two central and complex passages, VI.8, 1142a12-30 and VI.11, 1143a33-b15, from the Nicomachean Ethics to show that Baracchi’s view is unpersuasive. I will show that Aristotle considers (...) nous to be a different capacity than mere perception and one through which particulars and indemonstrable principles become intelligible. Moreover, I will show that Aristotle considers that the objects of nous differ in kind from those that sensation (our senses) and perception (inference from our senses) grasp. After examining critically Baracchi’s thesis in light of a close reading of those two relevant passages, I will conclude the paper by showing the significance of Aristotle’s claim that a state is defined in terms of its objects of apprehension for understanding the distinction between nous and aisthēsis. (shrink)

ABSTRACT: This paper traces the earliest development of the most basic principle of deduction, i.e. modus ponens (or Law of Detachment). ‘Aristotelian logic’, as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as ‘hypothetical syllogisms’. However, Aristotle did not discuss such arguments, nor did he call any (...) arguments ‘hypothetical syllogisms’. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them ‘hypothetical syllogisms’; nor did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle’s logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called ‘hypothetical syllogisms’? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle’s dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle’s logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories. (shrink)

The primary sense of the word ‘hypothesis’ in modern colloquial English includes “proposition not yet settled” or “open question”. Its opposite is ‘fact’ in the sense of “proposition widely known to be true”. People are amazed that Plato [1, p. 1684] and Aristotle [Post. An. I.2 72a14–24, quoted below] used the Greek form of the word for indemonstrable first principles [sc. axioms] in general or for certain kinds of axioms. These two facts create the paradoxical situation that in many cases (...) it is impossible to translate the Greek form of the word using the English form: the primary sense of the word ‘hypothesis’ in modern colloquial English is diametrically opposed to one sense used by Plato and by his most accomplished student Given current colloquial English usage it is impossible to get the word hypothesis to carry the connotation of “settled truth” much less “axiomatic truth”. The ‘hypo-’ [under] in the Plato-Aristotle use of ‘hypothesis’ might carry the sense of “basis” or “foundational” as opposed to “less than usual or normal”. This paradox parallels the one pointed out by Robin Smith: it is impossible for the English word ‘syllogism’ to carry the meaning of its Greek form Aristotle intended. There are other cases as well: it is impossible for the English biological term ‘genus’ to carry the meaning of its Greek form the Greek genos refers to family as in our ‘genealogy’, not to “higher species” as in our ‘generic’. (shrink)

ABSTRACT: An introduction to Stoic logic. Stoic logic can in many respects be regarded as a fore-runner of modern propositional logic. I discuss: 1. the Stoic notion of sayables or meanings (lekta); the Stoic assertibles (axiomata) and their similarities and differences to modern propositions; the time-dependency of their truth; 2.-3. assertibles with demonstratives and quantified assertibles and their truth-conditions; truth-functionality of negations and conjunctions; non-truth-functionality of disjunctions and conditionals; language regimentation and ‘bracketing’ devices; Stoic basic principles of propositional logic; 4. (...) Stoic modal logic; 5. Stoic theory of arguments: two premisses requirement; validity and soundness; 6. Stoic syllogistic or theory of formally valid arguments: a reconstruction of the Stoic deductive system, which consisted of accounts of five types of indemonstrable syllogisms, which function as nullary argumental rules that identify indemonstrables or axioms of the system, and four deductive rules (themata) by which certain complex arguments can be reduced to indemonstrables and thus shown to be formally valid themselves; 7. arguments that were considered as non-syllogistically valid (subsyllogistic and unmethodically concluding arguments). Their validity was explained by recourse to formally valid arguments. (shrink)

The paper delves into manuscript sources connected with various disputations held at Prague University from around 1390 to 1420 and singles out a set of hitherto unknown quaestiones dealing with the nature of the human intellect and its relation to the body. Prague disputations from around 1400 arguably offer a unique vantage point on late medieval anthropological issues, since they encompass an entanglement of numerous doctrinal influences from Buridanian De anima commentaries to John Wyclif’s theories. The paper delineates several conceptual (...) tensions regarding the nature of the intellect, e.g., between materialism (entailed by the emphasis on the intellect’s inherence in the body) and personal immortality. It presents several strategies Prague masters employed to overcome these tensions. For example, an anonymous participant of the 1409 quodlibet develops the Buridanian distinction between the rationally demonstrable materialist tendency and the indemonstrable “catholic truth” about the intellect both inhering in and separable from the body. On the other hand, Wyclif’s adherents (Jacob of Mies and another anonymous master) postulate an immortal spirit hypostatically united to each human being beside the human soul educed from the potency of the matter. Yet, the boundaries between the doctrinal standpoints in question seem permeable, whereby a rigid definition of antagonistic groups in late medieval Prague intellectual milieu (e.g., Buridanians vs Wycliffites) is rendered ineffective. (shrink)

Paul Guyer’s Kant on Freedom, Law, and Happiness is a collection of essays written over a period of ten years on the roles of freedom, reason, law, and happiness in Kant’s practical philosophy. The centrality of these concepts has always been acknowledged, but Guyer proposes a different way to understand their interconnections. Kant extols respect for moral law and conformity to moral principle for its own sake while at the same time celebrating the value of human freedom and autonomy. Guyer (...) sees tensions between these two poles of Kant’s practical philosophy—obedience to law and the value of freedom. He argues: “A profound paradox can be avoided only if it can be shown that Kant intended obedience to universal law to be mandatory solely as the necessary condition for the realization of human freedom and through that freedom a systematic and unselfish distribution of happiness among all persons” and that “the sheer fact of adherence to universal law is not an end in itself but is rather the means to the realization of the human potential for autonomy or freedom in both choice and action” (p. 1). One guiding theme of Guyer’s book is that Kant’s practical philosophy is based on the fundamental, and hence indemonstrable, intrinsic value of freedom, which Guyer understands as a value that is prior to the moral law, providing both an end to be realized through conformity to universal law and the basis of the authority of moral principles. Conformity to universal law has no intrinsic value in itself; it is the means to the preservation, enhancement, and full realization of human freedom, and is required because of its instrumental connection to the prior value of freedom. As he says, “freedom of choice and its natural expression in action are what human beings value most, and the fundamental principle of morality and the rules for both po-. (shrink)

Definition has the following features in Aristotle’s philosophy: 1. Each thing has only one definition and ‘it is impossible that there should be more than one definition for the same thing.’ (To., Z, 5, 142b^25; cf. To., Z, 4, 141a26) 2. Definition is ‘a formula of the essence’ (Met., H, 1042a17-18) and, thus, signifies the essence of the thing. (To., I, 5, ^101b30-) About the relation between definition and essence Aristotle regards three possibilities (PsA., B, 94a11-14): a) A definition as (...) an indemonstrable statement of essential nature. b) A definition as a syllogism of essential nature differing from demonstration in grammatical form. c) A definition as the conclusion of a demonstration giving essential nature. 3. ‘The definition is a formula and a formula has parts.’ (Met., H, 1042a18-20) Therefore, only the composite substances are definable and the primary parts of a composite substance are not definable. The reason is that ‘a definitory formula predicates something of something and one part of the definition must play the part of matter and the other that of form.’ (Met., H, 1043b28-32) Thus, definition is a sort of number: a divisible including indivisible parts. (Met., H, 1043b32-36) 4. To define a thing, one should give the species or the genera of the thing which are the only other things that can be substances of it and not its accidents. (Cat., 5, 2b31-36) Aristotle also brings differentiae into the definition: ‘There is nothing in the definition except the first-named genus and the differentiae. The other genera are the first genus and along with this the differentiae that are taken with, e.g. the first may be animal, the next animal which is footed, and again animal which is two-footed and featherless and …’ (Met., Z, 1037b28-33) Among genera the nearer genera are more informative because they are more distinctive and less general. (Cat., 5, 2b10-13 and b32-34) However, to give the species is always more informative and apt than giving the genus. (Cat., 5, 2b8-10) 5. Primary substances admit the definition of the species they are their individuals and of their genera. In the same way, species admit the definition of the genus. The reason is that everything said of the predicate can be said of the subject. (Cat., 5, 3b2-6) Therefore, everything admits the definition of its higher classes. 6. ‘Both the species and the individuals admit the definition of the differentia.’ (Cat., 5, 3b6-7) 7. A formula exhibiting the cause of a thing’s existence is a definition. (PsA., B, 10, 93b38-94a1) 8. It is possible to achieve the definition by division only if we keep three conditions in view (PsA., B, 97a23-26): a) The admission only of elements in the definable form. b) The arrangement of these elements in the right order c) The omission of no element in the definable form The process of division (PsA., B, 97a37-b6) is the process of dividing the genus by its differentia to the right species, the one that the subject accepts as its predicate, and again dividing the whole such searched by its right differentia to a predicable species. This process will be continued until we reach to that which is not further divisible, ‘i.e. that as soon as we have taken the last differentia to form the concrete totality, this totality admits of no division into species.’ In this way we achieve to a series of elements all in the definable forms without superfluous addition and without omission of any necessary element. This series includes the primary genus with all the differentia until it achieves to that which would admit of no division. The second condition, the right order, is the order in which that which is posited as primary be the one that is predicable of all of the other and not vice versa. The order must, then, be from more predicables to less predicables: ‘The right order will be achieved if the right term is assumed as primary, and this will be ensured if the term selected is predicable of all the other but not all they of it; since there must be one such term. Having assumed this, we at once proceed in the same way with the lower terms; for our second term will be the first of the remainder, our third the first of those which follow the second in a ‘contiguous’ series, since when the higher term is excluded, the term of the remainder which is ‘contiguous’ to it will be primary, and so on.’ (PsA., B, 97a26-34) 9. Things that are not the same must have different definitions. However, this does not mean that things that are the same must necessarily have the same definition. (To., I, 5, ^102a17-) 10. ‘Everything applicable to property and genus and accident will be applicable to definition as well.’ (To., I, 6, 102b27-) 11. ‘A name, e.g. ‘round’, means vaguely a sort of whole: its definition analyses this into its particular senses.’ Definition must be done through terms that are prior and more familiar and ‘anyone who has not defined a thing through terms that are prior and more familiar has not defined it at all.’ (To., Z, 4, 141a26-) It is indeed this very point that is the basis of definition by genus and differentiae: ‘A correct definition must define a thing through its genus and its differentiae, and these belong to the order of things which are without qualification more familiar than, and prior to, the species.’ (To., Z, 4, 141b^15-) This more familiarity, however, must be without qualification, i.e. not more familiar to certain people or in certain times. Otherwise, definition could not come always to be one and the same. (To., Z, 4, ^142a1-) . 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Aristotle’s points about circle and vicious circle are as follows: 1. Aristotle criticizes some thinkers because ‘they see no difficulty in holding that all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.’ (PsA., A, 3, 72b16-18) 2. ‘Not all knowledge is demonstrative’ and ‘knowledge of the immediate premises is independent of demonstration.’ Aristotle brings two reasons for this: ‘Since we must know the prior premises from which the demonstration is drawn, and since the regress must (...) end in immediate truths, those truths must be indemonstrable.’ (PsA., A, 3, 72b18-22) 3. Demonstration in the unqualified sense cannot be circular: ‘Demonstration must be based on premises prior to and better known than the conclusion; and the same things cannot simultaneously be both prior and posterior to one another: so circular demonstration is clearly not possible in the unqualified sense of demonstration, but only possible if ‘demonstration be extended to include that other method of argument which rests on a distinction between truths prior to us and truths without qualification prior, i.e. the method by which induction produces knowledge. But if we accept this extension of meaning, our definition of unqualified knowledge will prove faulty; for there seem to be two kinds of it. Perhaps, however, the second form of demonstration, that which proceeds from truths better known to us, is not demonstration in the unqualified sense of the term.’ (PsA., A, 3, 72b23-32) 4. The theory of circular demonstration ‘reduces to the mere statement that if a thing exists, then it does exist- an easy way of providing anything.’ (psA., A, 3, 72b32-34) 5. ‘To constitute the circle it makes no difference whether many terms or few or even only one is taken. Thus by direct proof, if A is, B must be; if B is, C must be; therefore, if A is, C must be. Since then- by the circular proof- if A is, B must be, and if B is, A must be. A may be substituted for C above. Then ‘if B is, A must be’ = ‘if B is, C must be.’ But C and A have been identified. Consequently, the upholders of circular demonstration are in the position of saying that if A is, A must be- a simple way of proving anything. Moreover, even such circular demonstration is impossible except in the case of attributes that imply one another, viz. ‘peculiar’ properties.’ (PsA, A, 3, 72b34-73a7) . (shrink)