Aristotle: Demonstration

This is a translation, made by myself, of the paper to be published in Portuguese in the journal Discurso, 2020, in honour of the late professor Oswaldo Porchat. I discuss what Aristotle was trying to encode when he said that the object of scientific knowledge is necessary, or that what we know (scientifically) cannot be otherwise etc. The paper is meant as a continuation of previous papers—orientated towards a book on the Posterior Analytics—and thus does not discuss in much detail (...) key passages, as the very definition of scientific knowledge in APo I.2, or passages from APo I.4 and I.6 (for these, I refer to my previous papers). This paper is mainly focused on Aristotle’s references to his notion of scientific knowledge both in other passages from the APo and in other treatises. I intend to show that there is a progressive, intrinsic relation between the two requirements by which scientific knowledge is defined. It is not true that each of these requirements stems from a different source. The Causal-Explanatory requirements gives Aristotle the general heading. Then, the Necessity Requirement ranges over the explanatory relation between explanans and explanandum and thereby specifies what sort of cause is sctricly required for having scientific knowledge of a given explanandum. Now, Aristotle was also concerned with the necessary truth of the elemental predications that constitute a demonstration. My claim that the Necessity Requirement ranges over the explanatory relation does not ignore that concern, and does not deny it. My claim is that Aristotle’s main focus, and main concern, consists in stressing that the explanatory factor to be captured in scientific knowledge of a given explanandum is such that cannot be otherwise. (shrink)

This presentation includes a complete bibliography of John Corcoran’s publications devoted at least in part to Aristotle’s logic. Sections I–IV list 20 articles, 43 abstracts, 3 books, and 10 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article that antedates Corcoran’s Aristotle’s studies and the Journal of Symbolic Logic article first reporting his original results; it ends with works published in 2015. A few of the items are annotated with endnotes connecting them with (...) other work. In addition, Section V “Discussions” is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 8 items published in the 1970s, 22 in the 1980s, 39 in the 1990s, 56 in the 2000s, and 65 in the current decade. The secondary bibliography is annotated with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. As is evident from the Acknowledgements sections, all of Corcoran’s publications benefited from correspondence with other scholars, most notably Timothy Smiley, Michael Scanlan, and Kevin Tracy. All of Corcoran’s Greek translations were done in consultation with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. REQUEST: Please send errors, omissions, and suggestions. I am especially interested in citations made in non-English publications. (shrink)

The article aims to show how, for Maimonides, the exercise of reason is by no means natural. It involves, in addition to a personal predisposition, a long philosophical training, which leads from logic to metaphysics. Nevertheless, there are truths that cannot be known through the demonstrative method: in such cases, dialectic comes to the rescue, which yet Maimonides always criticizes in the arguments of the theologians.

This book looks at the very beginning of the philosophy of mathematics in Western thought. It covers the first reflections on attempts to untie mathematics from its practical usage in administration, commerce, and land-surveying and discusses the first ideas to see mathematical structures as constituents underlying the physical world in the Pythag-oreans. The first two sections focus on the epistemic status of mathematical knowledge in relation to philosophical knowledge and on the various ontological positions ancient Greek philosophers in early and (...) classical times ascribe to mathematical objects – from independent and separate entities to mere abstractions and idealizations. Section 3 dis-cusses the paradigmatic role mathematical deductions have played for philosophy, the role of mathematical diagrams, and mathematical methods of interest for philosophers. Section 4, finally, investigates a couple of individual concepts that are fundamental for both philosophy and mathematics, such as infinity. (shrink)

If the Aristotelian division has above all an epistemic and heuristic purpose, this article aims to examine the ontological appropriation of division by Aristotle in the Metaphysics, too often neglected by secondary literature. Division is also an ontological operation, insofar it governs the relationship between the eidos (which is identified, in the context of the central books of the Metaphysics, with the primary substance, i.e. what is properly speaking), its genus and the final differentia of this genus. More precisely, we (...) will show that division is a case of the hylemorphic relation: the genus is an (intelligible) matter to which the final differentia obtained by division is predicated, and which then acts as the formal cause of this genus being such a determined eidos. However, this ontological and hylemorphic dimension of Aristotelian division is largely compatible with its methodological use. (shrink)

Does Aristotle offer a definition of the soul? In fact, he rejects the possibility of defining the soul univocally. Because “life” is a homonymous concept, so too is “soul”. Given the specific causal role that Aristotle envisages for form and essence, the soul requires multiple different definitions to capture how it functions as a cause in each form of life. Aristotle suggests demonstrations can be given which express these causal definitions; I reconstruct these demonstrations.

Physics IV.10 (217b30–218a30) is pivotal in Aristotle’s discussion of time, preceding his own account from IV.11 onward. Aristotle presents three puzzles about the existence of time with reference to the “Now”. Modern interpretations often view this section as an aporetic prelude with Aristotle’s failure to provide explicit solutions. This paper examines Simplicius’ alternative interpretation, which draws upon the theory of proof and the syllogistic model from the Posterior Analytics. Simplicius contends that the arguments’ failure lies in their inability to fit (...) within the suitable syllogistic framework to establish a demonstrable definition of time, not in their aporetic nature. Every science has to prove the relation between (i) establishing whether X exists and (ii) showing what X is by establishing what the cause of X is. In evaluating Simplicius’ interpretation, this paper addresses two key aspects of the exegesis of IV.10: firstly, Simplicius can show why the “Now” is not part of the definition of time, and secondly, the ancient commentator underscores the close connection between the arguments in Physics IV.10 and the broader context of Aristotle’s discussion of time. Modern interpreters fail to address both of these issues. (shrink)

I discuss a short string of five sentences in Metaphysics V.5, 1015b6-9 relating demonstration to necessity. My proposal is that Aristotle focuses his attention on the demonstration as a demonstration. Other interpretations reduce the necessity in question to the modality of the component sentences of the demonstrations (the conclusion and the premises). My view does not deny that the modality of the component sentences is important, but takes seriously the idea that a demonstration itself should be understood as necessary—as not (...) capable of being otherwise. A demonstration cannot be different from what it is in the sense that [i] its components cannot be different from what they are, [ii] its components must be related to each other exactly in the way they are related. Demonstrations aim at the fully appropriate explanation of a given explanandum—and each demonstration is individuated by the explanandum it takes. Thus, the basic idea is that, for the target explanandum that individuates a given demonstration, the premises delivering the fully appropriate explanation cannot be replaces with different ones. I show how this proposal, which explains Aristotle’s language in 1015b6-9 accurately, does not make demonstrations ‘melt down into conditional necessity’, first, because the modality of the component sentences is still importantly involved, second, because the explanatory relation expressed in a demonstration is a necessary fact in the real world, so that the demonstration itself is also necessary (in the way I have explained) inasmuch as it captures that fact. (shrink)

In the Posterior Analytics Aristotle contrasts demonstrations with syllogisms through signs. In the Prior Analytics he defines a sign as a demonstrative premise. One is thus led to ask: is a sign a demonstration? This book reconstructs the history of the notion of "demonstration through signs" from roughly the third through to the thirteenth century. It examines the work of Aristotle's Greek, Arabic, and Latin commentators, both within and outside the tradition of the Posterior Analytics.

Most interpretations of Aristotle’s Posterior Analytics believe that the term ‘ameson’ is used to describe the principles or foundations of a given system of justification or explanation as epistemically prior to or more fundamental than the other propositions in the system. Epistemic readings (as I shall call them) arguably constitute a majority in the secondary literature. This predominant view has been challenged by Robin Smith (1986) and Michael Ferejohn (1994; 2013), who propose interpretations that should be classified as non-epistemic according (...) to the definition above. My aim in this article is purely negative. I intend to show that these non-epistemic interpretations are liable to serious objections and are in conflict with some important features of Aristotle’s theory of demonstration. (shrink)

Nos Segundos Analíticos I. 14, 79a16-21 Aristóteles afirma que as demonstrações matemáticas são expressas em silogismos de primeira figura. Apresento uma leitura da teoria da demonstração científica exposta nos Segundos Analíticos I (com maior ênfase nos capítulo 2-6) que seja consistente com o texto aristotélico e explique exemplos de demonstrações geométricas presentes no Corpus. Em termos gerais, defendo que a demonstração aristotélica é um procedimento de análise que explica um dado explanandum por meio da conversão de uma proposição previamente estabelecida. (...) Em uma estrutura silogística, a proposição previamente estabelecida é a premissa maior e o termo mediador deve ser comensurado ao explanandum. O conjunto da premissa maior e da premissa menor (o explanans) é coextensivo ao explanandum, mas há uma assimetria intensional entre o explanans e o explanandum, de modo que apenas o primeiro explique o último. Por fim, defendo que o elemento identificado no termo mediador deve ser o mais apropriado para explicar precisamente o que certo explanandum é. (shrink)

Aristotle’s aitiai are middle terms in Aristotelian syllogisms. I argue that stating the aitia of a thing therefore amounts to re-describing this same thing in an alternative and illuminating way. This, in turn, means that a thing and its aitiai really are one and the same thing under different descriptions. The purpose of this paper is to show that this view is implied by Aristotle’s account of explanation, and that it makes more sense than one might expect.

I discuss the exact meaning of the thesis according to which the object of scientific knowledge is necessary. The thesis is expressed by Aristotle in the Posterior Analytics, in his definition of scientific knowledge. The traditional interpretation understands this definition as depending on two parallel and independent requirements, the causality requirement and the necessity requirement. Against this interpretation, I try to show, through the examination of several passages that refer to the definition of scientific knowledge, that the necessity requirement specifies (...) more exactly the causality requirement: what cannot be otherwise is the explanatory relation between the explanandum and the cause by which it is what it is. (shrink)

I defend an interpretation of Aristotle’s _Posterior Analytics _Book I which distinguishes between two projects in different passages of that work: to explain what a given science is and to explain what properly scientific knowledge is. I present Aristotle’s theory in answer to ii, with special attention to his definition of scientific knowledge in 71b9-12 and showing how this is developed on chapters 1.2-9 and 1.13 into a solid Theory of Scientific Demonstration. The main point of this theory is that (...) demonstrations need to capture relevant explanations. Some formal requirements of the demonstration are unfoldings of the main project, 1.e., to capture and present properly relevant causal-explanatory relations. (shrink)

Some have suggested that images can be arguments. Images can certainly bolster the acceptability of individual premises. We worry, though, that the static nature of images prevents them from ever playing a genuinely argumentative role. To show this, we call attention to a dilemma. The conclusion of a visual argument will either be explicit or implicit. If a visual argument includes its conclusion, then that conclusion must be demarcated from the premise or otherwise the argument will beg the question. If (...) a visual argument does not include its conclusion, then the premises on display must license that specific conclusion and not its opposite, in accordance with some demonstrable rationale. We show how major examples from the literature fail to escape this dilemma. Drawing inspiration from the graphical logic of C. S. Peirce, we suggest instead that images can be manipulated in a way that overcomes the dilemma. Diagrammatic reasoning can take one stepwise from an initial visual layout to a conclusion—thereby providing a principled rationale that bars opposite conclusions—and the visual inscription of this correct conclusion can come afterward in time—thereby distinguishing the conclusion from the premises. Even though this practical application of Peirce’s logical ideas to informal contexts requires that one make adjustments, we believe it points to a dynamic conception of visual argumentation that will prove more fertile in the long run. (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)

In Prior Analytics I.30, Aristotle seems too much optmistic about finding out the principles of sciences. For he seems to say that, if our empirical collection of facts in a given domain is exhaustive or sufficient, it will be easy for us to find out the explanatory principles in the domain. However, there is a distance between collecting facts and finding out the explanatory principles in a given domain. In this paper, I discuss how the key expression in the sentence (...) at 46a25 should be interpreted: “the true characteristics of things” (“τῶν ἀληθῶς ὑπαρχόντων τοῖς πράγμασιν”). I argue that, on a more accurate interpretation of the expression, Aristotle’s point would cease to look like a piece of naïve or even silly optimism. (shrink)

Aristotle contrasts episteme and doxa through the key notions of universal and necessary. These notions have played a central role in Aristotle’s characterization of scientific knowledge in the previous chapters of APo. They are not spelled out in APo I.33, but work as a sort of reminder that packs an adequate characterization of scientific knowledge and thereby gives a highly specified context for Aristotle’s contrast between episteme and doxa. I will try to show that this context introduces a contrast in (...) terms of explanatory claims: on the one hand, episteme covers those claims which capture explanatory connections that are universal and necessary and thereby deliver scientific understanding; on the other hand, doxa covers the explanatory attempts that fail at doing so. (shrink)

I argue against the standard interpretation of Aristotle’s account of ‘natural predication’ in Posterior Analytics 1.19 and 1.22 according to which only substances can serve as subjects in such predications. I argue that this interpretation cannot accommodate a number of demonstrations Aristotle sanctions. I propose a new interpretation that can accommodate them.

: In APo II 3-7 Aristotle discusses a series of difficulties concerning definition, deduction, and demonstration. In this paper I focus on two interrelated but distinct questions: firstly, what are exactly the difficulties emerging from or alluded to in the discussion in II 3-7; secondly, whether and in what sense the discussion in II 3-7 can be considered an aporetic discussion with a specific role to play in the development of the argument in APo II.

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)

My aim is to show that, in Posterior Analytics B 8, the conception of lunar eclipse brought about by pre-demonstrative knowledge (hoti) is deeply vague and radically different from the one obtained by demonstrative knowledge (dioti).

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

How many scientific demonstrations can a single phenomenon have? This paper argues that, according to Aristotle's theory of scientific knowledge as laid out in the Posterior Analytics, a single conclusion may be demonstrated via more than one explanatory middle term. I also argue that this model of multiple demonstration is put into practice in the biological writings. This paper thereby accomplishes two related goals: it clarifies certain relatively obscure passages of the Posterior Analytics and uses them to show how Aristotle (...) explains biological phenomena by reference to both final and material causes in the Parts of Animals. The first part of the paper explains the account of multiple demonstration present in the Posterior Analytics and distinguishes it from another kind of plural explanation rejected by Aristotle. The second part of the paper turns to the biological explanation in the Parts of Animals and shows how Aristotle's account of multiple demonstrations works in practice. The paper thus provides evidence for the claim that the ‘applied’ reasoning on display in the biological works is in harmony with the framework of the logical treatises, and thus may also shed light on questions of the unity of the Aristotelian corpus. (shrink)

I discuss an important feature of the notion of cause in Post. An. 1. 13, 78b13–28, which has been either neglected or misunderstood. Some have treated it as if Aristotle were introducing a false principle about explanation; others have understood the point in terms of coextensiveness of cause and effect. However, none offers a full exegesis of Aristotle's tangled argument or accounts for all of the text's peculiarities. My aim is to disentangle Aristotle's steps to show that he is arguing (...) in favour of a logical requirement for a middle term's being the appropriate cause of its explanandum. Coextensiveness between the middle term and the attribute it explains is advanced as a sine qua non condition of a middle term's being an appropriate or primary cause. This condition is not restricted either to negative causes or to middle terms in second‐figure syllogisms, but ranges over all primary causes qua primary. (shrink)

I present Aristotle’s theory of causation in a way that privileges a comparison with contemporary discussion on causation. I do so by selecting in Aristotle’s theory points that are interesting to contemporary discussion and by translating Aristotle in the contemporary philosophical terminology. I compare Aristotle’s views with Mackie’s (1993/1965) and Sosa’s (1993/1980). Mackie is a humean regularist regarding the metaphysics of causal necessity, but his theory postulates some formal aspects of the causal relation which are similar to the Aristotelian theory. (...) By introducing the notion of causal field as a third causal relatum – a sample in which the cause must be picked out as a explanation coextensive to the effect –, Mackie holds a point very similar to Aristotle, inasmuch as both defend that the causal relation is triadic and both have the extensional desideratum. Sosa (1993/1980) defends that every cause necessitates its effect, arguing against classical Humeanism, and he also proposes a causal pluralism. Both these theses are clearly Aristotelian. Aristotle comprehends that the causal relation has three relata: a subjacent C, a property of that subjacent A and a cause B that explains why such property is attributed to that subjacent. The subjacent is similar to Mackie’s notion of causal field, for in regard to it the cause must be coextensive to the property. But Aristotle also says that for each explanandum (property attributed to a subjacent of which one aims to investigate a cause) there is only one cause, which necessitates it. Sosa and Aristotle are similar because both develop a causal pluralism from a theory of causes as having intrinsic necessitation. For Aristotle, causes/explanations [aitiai] are central for attaining scientific knowledge, and its triadic structure determines the syllogistic structure of scientific demonstrations. Aristotle’s theory of causation also involves comprehending the actualisation of capacities (dispositional properties of objects) affirming that the causal relation is resultant from intrinsic properties of the objects, and the cause B is the essence of the explanandum. (shrink)

This paper concerns Aristotle's kind‐crossing prohibition. My aim is twofold. I argue that the traditional accounts of the prohibition are subject to serious internal difficulties and should be questioned. According to these accounts, Aristotle's prohibition is based on the individuation of scientific disciplines and the general kind that a discipline is about, and it says that scientific demonstrations must not cross from one discipline, and corresponding kind, to another. I propose a very different account of the prohibition. The prohibition is (...) based on Aristotle's scientific and metaphysical essentialism, according to which a scientific demonstration must take as its starting point a set of per se properties of a subject, if these make up a single, unitary definition. The subject of demonstration here is a kind, although not the general kind associated with a discipline, but rather the particular kind that the particular demonstration is about. (shrink)

In Posterior Analytics II 16-17, Aristotle seems to claim that there cannot be more than one explanans of the same scientific explanandum. However, this seems to be true only for “primary-universal” demonstrations, in which the major term belongs to the minor “in itself” and the middle term is coextensive with the extremes. If so, several explananda we would like to admit as truly scientific would be out of the scope of an Aristotelian science. The secondary literature has identified a second (...) problem in II 16-17: the middle term of a demonstration is sometimes taken as the definition of the minor term (the subject), other times as the definition (or the causal part of the definition) of the major (the demonstrable attribute). I shall argue that Aristotle’s solution to the first problem involves showing that certain problematic attributes, which appear to admit more than one explanation, actually fall into the privileged scenario of primary-universal demonstrations. In addition, his solution suggests a conciliatory way-out to our second problem (or so I shall argue): the existence of an attribute as a definable unity depends on its subject having the essence it has, which suggests that both the essence of subjects and the essence of demonstrable attributes can play explanatory roles in demonstrations. (shrink)

In some influential histories of ancient philosophy, teleological explanation and mechanistic explanation are assumed to be directly opposed and mutually exclusive alternatives. I contend that this assumption is deeply flawed, and distorts our understanding both of teleological and mechanistic explanation, and of the history of mechanistic philosophy. To prove this point, I shall provide an overview of the first systematic treatise on mechanics, the short and neglected work Mechanical Problems, written either by Aristotle or by a very early member of (...) his school. I will argue that the work is thoroughly Aristotelian in methodology, and that taking it seriously can deepen our understanding of Aristotle’s discussion of animal and human self-motion in the Physics and On the Movement of Animals. (shrink)

ABSTRACT This is a review of David Bronstein's book "Aristotle on Knowledge and Learning: The Posterior Analytics" (Oxford: Oxford University Press, 2016).

In Posterior Analytics 71b9 12, we find Aristotle’s definition of scientific knowledge. The definiens is taken to have only two informative parts: scientific knowledge must be knowledge of the cause and its object must be necessary. However, there is also a contrast between the definiendum and a sophistic way of knowing, which is marked by the expression “kata sumbebekos”. Not much attention has been paid to this contrast. In this paper, I discuss Aristotle’s definition paying due attention to this contrast (...) and to the way it interacts with the two conditions presented in the definiens. I claim that the “necessity” condition ammounts to explanatory appropriateness of the cause. (shrink)

David Bronstein sheds new light on Aristotle's Posterior Analytics--one of the most important, and difficult, works in the history of western philosophy--by arguing that it is coherently structured around two themes of enduring philosophical interest: knowledge and learning. He argues that the Posterior Analytics is a sustained examination of scientific knowledge, an elegantly organized work in which Aristotle describes the mind's ascent from sense-perception of particulars to scientific knowledge of first principles. Bronstein goes on to highlight Plato's influence on Aristotle's (...) text, and argues against current orthodoxy that Aristotle is committed to the Socratic Picture of inquiry, according to which one should seek what a thing's essence is before seeking its demonstrable attributes and their causes. (shrink)

I argue that Aristotle’s account of scientific demonstrations in the Posterior Analytics is centred upon formal causation, understood as a demonstration in terms of essence (and as innocent of the distinction between form and matter). While Aristotle says that all four causes can be signified by the middle term in a demonstrative syllogism, and he discusses at some length efficient causation, much of Aristotle’s discussion is foremost concerned with the formal cause. Further, I show that Aristotle had very detailed procedures (...) for identifying the formal cause, and that he is aware of several problems which might lead one to erroneously identify the wrong form as the cause of a property. Finally, I show that Aristotle’s account can easily be adapted to material causation, and through some modifications (introduction of process universals related through parthood), hinted at in II 11-12 and 16-17, to efficient and final causation. (shrink)