In this paper I argue for an association between impurity and explanatory power in contemporary mathematics. This proposal is defended against the ancient and influential idea that purity and explanation go hand-in-hand (Aristotle, Bolzano) and recent suggestions that purity/impurity ascriptions and explanatory power are more or less distinct (Section 1). This is done by analyzing a central and deep result of additive number theory, Szemerédi’s theorem, and various of its proofs (Section 2). In particular, I focus upon the radically impure (...) (ergodic) proof due to Furstenberg (Section 3). Furstenberg’s ergodic proof is striking because it utilizes intuitively foreign and infinitary resources to prove a finitary combinatorial result and does so in a perspicuous fashion. I claim that Furstenberg’s proof is explanatory in light of its clear expression of a crucial structural result, which provides the “reason why” Szemerédi’s theorem is true. This is, however, rather surprising: how can such intuitively different conceptual resources “get a grip on” the theorem to be proved? I account for this phenomenon by articulating a new construal of the content of a mathematical statement, which I call structural content (Section 4). I argue that the availability of structural content saves intuitive epistemic distinctions made in mathematical practice and simultaneously explicates the intervention of surprising and explanatorily rich conceptual resources. Structural content also disarms general arguments for thinking that impurity and explanatory power might come apart. Finally, I sketch a proposal that, once structural content is in hand, impure resources lead to explanatory proofs via suitably understood varieties of simplification and unification (Section 5). (shrink)

The expression tode ti, commonly translated as ‘a this’, plays a key role in Aristotle’s metaphysics. Drawing lightly on theories of demonstratives in contemporary linguistics, I discuss the expres...

I contend that Adam Smith and David Hume offer re-interpretations of Aristotle’s notion of greatness of soul, focusing on the kind of magnanimity Aristotle attributes to Socrates. Someone with Socratic magnanimity is worthy of honor, responds moderately to fortune, and is virtuous—just and benevolent. Recent theorists err in claiming that magnanimity is less important to Hume’s account of human excellence than benevolence. In fact, benevolence is a necessary ingredient for the best sort of greatness. Smith’s “Letter to Strahan” attributes this (...) greatness to Hume. It encourages us to admire Hume as an exemplar of human excellence, to seek Hume’s virtues for ourselves, and to approve of the “love of literary fame” which Hume calls his “ruling passion.”. (shrink)

The article examines Aristotle’s seminal discussion of the fallacy of begging the question, reconstructing its complex articulation within a variety of different, but related, contexts. I suggest that close analysis of Aristotle’s understanding of the fallacy should prompt critical reconsideration of the scope and articulation of the fallacy in modern discussions and usages, suggesting how begging the question should be distinguished from a number of only partially related argumentative faults.

In this paper, Stoic epistemology is analysed in terms of how to achieve a stable grasping of reality through katalepsis. The paper argues that for the Stoics, this is a state accessible to any rational being because it is the upshot of a mental capacity we are necessarily bound to put into operation, namely that of experiencing and mentally ordering objects from the sensible world. The paper puts forward an original interpretation relying on a reconsidered notion of Stoic empeiria or (...) experience. It connects the Stoic theory of the development of reason and formation of conceptions with a more fluid oscillation between belief and knowledge so as to establish a peculiar relation between reason and the Stoic notion of experience, articulating a form of rational empiricism. (shrink)

This paper discusses some issues about Aristotle’s theory of scientific investigation in Posterior Analytics II 8. Aristotle says that scientific investigation comes in three stages. My point is that Aristotle’s theory of scientific investigation cannot avoid Meno’s paradox – the paradox about the impossibility of whatsoever sort of investigation – unless its second stage, the stage in which one establishes that an object exists, is understood in terms of establishing that the object is a legitimate explanandum in the domain of (...) a given science. (shrink)

In Posterior Analytics 2.19, Aristotle argues that we cannot have innate knowledge of first principles because if we did we would have the most precise items of knowledge without noticing, which is impossible. To understand Aristotle’s argument we need to understand why he thinks we cannot possess these items of knowledge without noticing. In this paper, I present three different answers to this question and three different readings of his argument corresponding to them. The first two readings focus on the (...) fact that we do not use the knowledge we allegedly possess innately. However, I argue that these readings fail to produce convincing arguments. I then offer a third reading, which focuses on the fact that we do not notice the knowledge we allegedly possess innately when we use it for the first time. I argue that this reading produces a more convincing argument than either of the first two. (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.

What role, if any, does dialectic play in Aristotle’s epistemology in the Topics? In this paper I argue that it does play a role, but a role that is independent of endoxa. In the first section, I sketch the case for thinking that dialectic plays a distinctively epistemological role—not just a methodological role, or a merely instrumental role in getting episteme. In the second section, I consider three ways it could play that role, on two of which endoxa play at (...) least a co-starring role. But in the third section, I argue that endoxa don’t play that co-starring role, by arguing for a deflationary account of what endoxa are. For that reason, the third view emerges as the most plausible; but the third view is, I’ll suggest, Platonic. (shrink)

This paper examines Aristotle’s notion of priority with the specific aim of capturing the sort of priority that characterizes the primacy of substances in his metaphysics. I reject the traditional interpretation, which understands the ontological priority of substances in terms of independent existence. But there are rather two sorts of priority: the ontological priority of substances should be understood in terms of completeness, whereas the ontological priority of “substances-of-something” (the essences) is a causal-explanatory priority. Furthermore, an important piece of Aristotle’s (...) argument against Platonism is that these two sorts of priority – namely, the completeness priority and the causal-explanatory priority – should be kept distinct. (shrink)

This book investigates Aristotle’s views on abstraction and explores how he uses it. In this work, the author follows Aristotle in focusing on the scientific detail first and then approaches the metaphysical claims, and so creates a reconstructed theory that explains many puzzles of Aristotle’s thought. Understanding the details of his theory of relations and abstraction further illuminates his theory of universals. Some of the features of Aristotle’s theory of abstraction developed in this book include: abstraction is a relation; perception (...) and knowledge are types of abstraction; the objects generated by abstractions are relata which can serve as subjects in their own right, whereupon they can appear as items in other categories. The author goes on to look at how Aristotle distinguishes the concrete from the abstract paronym, how induction is a type of abstraction which typically moves from the perceived individuals to universals and how Aristotle’s metaphysical vocabulary is "relational.’ Beyond those features, this work also looks at how of universals, accidents, forms, causes and potentialities have being only as abstract aspects of individual substances. An individual substance is identical to its essence; the essence has universal features but is the singularity making the individual substance what it is. These theories are expounded within this book. One main attraction in working out the details of Aristotle’s views on abstraction lies in understanding his metaphysics of universals as abstract objects. This work reclaims past ground as the main philosophical tradition of abstraction has been ignored in recent times. It gives a modern version of the medieval doctrine of the threefold distinction of essence, made famous by the Islamic philosopher, Avicenna. (shrink)

The hypothesis of a theoretical unity between On Sophistical Refutations and Prior Analytics presents a major challenge to scholars attempting to unify the criteria of analysis. This paper examines this problem and proposes a middle ground between the perspectives of Woods and Boger to address this crucial question: If a unitary and coherent theory of deduction exists, why does not the technical apparatus of syllogistic modes for analyzing fallacies appear in SE? This paper makes useful contributions to the discussion on (...) the internal unity of Aristotle's logical writings on around categorical syllogistics and the doctrine of sophistical refutations. (shrink)

RésuméEn expliquant la relation entre hypothèses et images dans l'analogie de la ligne du livre Vl de laRépubliquede Platon, je m'attarde d'abordsur l'élucidation platonicienne de la nature des mathématiques telle que la conçoit le mathématicien lui-même. Je poursuis avec une critique des interprétations traditionnelles de cette relation, qui partent de l'assomption douteuse que les mathématiques s'occupent des Formes platoniciennes. Pour formuler mon point de vue sur cette relation, j'exploite la notion de «structure». Je montre comment les «hypothèses» comme principes de (...) preuve peuvent déterminer les structures des «images» par lesquelles les structures intelligibles correspondantes des objets mathématiques en viennent à être connues. (shrink)

RésuméEn expliquant la relation entre hypothèses et images dans l'analogie de la ligne du livre Vl de laRépubliquede Platon, je m'attarde d'abordsur l'élucidation platonicienne de la nature des mathématiques telle que la conçoit le mathématicien lui-même. Je poursuis avec une critique des interprétations traditionnelles de cette relation, qui partent de l'assomption douteuse que les mathématiques s'occupent des Formes platoniciennes. Pour formuler mon point de vue sur cette relation, j'exploite la notion de «structure». Je montre comment les «hypothèses» comme principes de (...) preuve peuvent déterminer les structures des «images» par lesquelles les structures intelligibles correspondantes des objets mathématiques en viennent à être connues. (shrink)

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Content: 1. Pre-Aristotelian Logic 1.1 Syntax and Semantics 1.2 Argument Patterns and Valid Inference 2. Aristotle 2.1 Dialectics 2.2 Sub-sentential Classifications 2.3 Syntax and Semantics of Sentences 2.4 Non-modal Syllogistic 2.5 Modal Logic 3. The early Peripatetics: Theophrastus and Eudemus 3.1 Improvements and Modifications of Aristotle's Logic 3.2 Prosleptic Syllogisms 3.3 Forerunners (...) of Modus Ponens and Modus Tollens 3.4 Wholly Hypothetical Syllogisms 4. Diodorus Cronus and Philo the Logician 5. The Stoics 5.1 Logical Achievements Besides Propositional Logic 5.2 Syntax and Semantics of Complex Propositions 5.3 Arguments 5.4 Stoic Syllogistic 5.5 Logical Paradoxes 6. Epicurus and the Epicureans 7. Later Antiquity. (shrink)

In 1928, Friedrich Solmsen argued that Aristotle'sPosterior Analyticswas largely composed before thePrior Analytics. Ross rejected Solmsen's position in 1939, and a rather lengthy series of rebuttals and counter-attacks between the two scholars followed. Quite recently, Jonathan Barnes has revived this issue with arguments in favour of something very close to Solmsen's thesis: that Aristotle first developed a theory of demonstration (‘apodeictic’) before he had worked out the syllogistic, and that thePosterior Analyticswas originally conceived against this background. Subsequently, when Aristotle formulated (...) a syllogistic, he is supposed by Barnes to have revised or added to the contents of thePosterior Analyticsso as to make syllogistic the logic of Aristotelian science. Thus, Barnes says: ‘the syllogism is in fact an incidental adjunct to the theory of demonstration: the theory can be formulated without reference, explicit or implicit, to Syllogistic, and itcould havebeen discovered by someone who knew nothing whatever about the Syllogism’ (pp. 33–4). (shrink)

Aristotle’s modal syllogistic is his study of patterns of reasoning about necessity and possibility. Many scholars think the modal syllogistic is incoherent, a ‘realm of darkness’. Others think it is coherent, but devise complicated formal modellings to mimic Aristotle’s results. This volume provides a simple interpretation of Aristotle’s modal syllogistic using standard predicate logic. Rini distinguishes between red terms, such as ‘horse’, ‘plant’ or ‘man’, which name things in virtue of features those things must have, and green terms, such as (...) ‘moving’, which name things in virtue of their non-necessary features. By applying this distinction to the Prior Analytics, Rini shows how traditional interpretive puzzles about the modal syllogistic melt away and the simple structure of Aristotle’s own proofs is revealed. The result is an applied logic which provides needed links between Aristotle’s views of science and logical demonstration. The volume is particularly valuable to researchers and students of the history of logic, Aristotle’s theory of modality, and the philosophy of logic in general. (shrink)

Este livro marca o início da Série Investigação Filosófica. Uma série de livros de traduções de textos de plataformas internacionalmente reconhecidas, que possa servir tanto como material didático para os professores das diferentes subáreas e níveis da Filosofia quanto como material de estudo para o desenvolvimento pesquisas relevantes na área. Nós, professores, sabemos o quão difícil é encontrar bons materiais em português para indicarmos. E há uma certa deficiência na graduação brasileira de filosofia, principalmente em localizações menos favorecidas, com relação (...) ao conhecimento de outras línguas, como o inglês e o francês. Tentamos, então, suprir essa deficiência, ao introduzirmos traduções de textos importantes ao público de língua portuguesa, sem nenhuma finalidade comercial e meramente pela glória da filosofia. O presente volume é constituído de três traduções de verbetes importantes sobre lógica, da Enciclopédia de Filosofia da Stanford: (1) A Lógica de Aristóteles, (2) Lógica Clássica, (3) Lógica Modal. (shrink)

Aristotle’s theory of spontaneous generation offers many puzzles to those who wish to understand his theory both within the context of his biology and within the context of his more general philosophy of nature. In this paper, I approach the difficult and vague elements of Aristotle’s account of spontaneous generation not as weaknesses, but as opportunities for an interesting glimpse into the thought of an early scientist struggling to reconcile evidence and theory. The paper has two goals: to give as (...) charitable and full an account as possible of what Aristotle’s theory of spontaneous generation was, and to examine some of its consequences; and to reflect on Aristotle as a scientist, and what his comments reveal about how he approached a difficult problem. In particular, I propose that the well-recognized problem of the incompatibility between Aristotle’s concept of spontaneity and his theory of spontaneous generation presents an opportunity for insight into his scientific methodology when approaching ill-understood phenomena. (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)

O texto busca apresentar a discussão em torno de Ciência e Dialética em Aristóteles, de Porchat.

In Prior Analytics 1.15, Aristotle attempts to give a proof through the impossible of Barbara, Celarent, Darii, and Ferio with an assertoric first premiss, a contingent second premiss, and a possible conclusion. These proofs have been controversial since antiquity. I shall show that they are valid, and that Aristotle is able to explain them by relying on two meta-syllogistic lemmas on the nature of possibility interpreted as syntactic consistency. It will turn out that Aristotle's proofs are not of the intended (...) schemata. I shall investigate some of the results that the impact of this reconstruction on the modal syllogistic has: the relationship between Aristotle's syllogistic and the logics of relevance; the value of Aristotle's requirement that universal affirmative propositions be taken ‘absolutely’; the destruction of many Aristotelian proofs; the recovery of certain principles of modal opposition from a charge of inconsistency. (shrink)

This paper offers a novel interpretation of the nature and role of logical argumentation in Aristotle’s natural philosophy. In contrast to the standard domain interpretation, which makes logikôs argumentation the contrary of phusikôs, relying on principles drawn from outside the domain of natural science, I propose that the essential or defining feature of logikôs argumentation is the use of principles that are general relative to the question under investigation. My interpretation is developed and illustrated with a close textual analysis of (...) Aristotle’s explanation of mule sterility in Generation of Animals II 8. (shrink)

ABSTRACT: I argue that Socrates employs both elenchus and the method of hypothesis in the examination of the slave. I show that the elenchus is a necessary step of the inquiry. Being reduced to ignorance, Socrates tacitly uses the method of hypothesis to move the slave from ignorance to correct opinion. I tease this out from the questions Socrates asks. Although the method of hypothesis begins from a question distinct from elenchus, the solution to the problem leads the slave to (...) answer the original question. The examination shows that the two methods can be employed in a sustained inquiry. (shrink)

I shall argue that, according to Aristotle, the knowledge we may attain is profoundly qualified by our status as human knowers. Throughout the corpus, Aristotle maintains a separation of knowledge at the broadest level into two kinds, human and divine. The separation is not complete—human knowers may enjoy temporarily what god or the gods enjoy on a continuous basis; but the division expresses a fact about humanity's place in the cosmos, one that imposes strict conditions on what we may know, (...) with what degree of certainty, and in what areas. While passages bearing on human knowledge are familiar, looking at them collectively and in comparison with certain other well known Aristotelian doctrines may significantly affect how we understand the goals of his philosophy and why our hopes for reaching them must be limited. (shrink)

In Metaphysics A 6, Aristotle claims that Plato only recognises formal and material causes. Yet, in various dialogues, Plato seems to use and distinguish efficient and final causes too. Consequently, Harold Cherniss accuses Aristotle of being an unfair, forgetful, or careless reader of Plato. Since then, scholars have tried to defend Aristotle’s exegetical skills. I offer textual evidence and arguments to show that their efforts still fall short of the desired goal. I argue, instead, that we can reject Cherniss’ assertation (...) by re-examining Aristotle’s exegetical and methodological assumptions. (shrink)

In the opening sentences of Book 2 of Posterior Analytics, Aristotle defines the four types of question that one can pose within the demonstrative science. In the edition by William D. Ross, the text reads as follows : τὰ ζητούμενά ἐστιν ἴσα τὸν ἀριθμὸν ὅσαπερ ἐπιστάμεθα. ζητοῦμεν δὲ τέτταρα, τὸ ὅτι, τὸ διότι, εἰ ἔστι, τί ἐστιν. ὅταν μὲν γὰρ πότερον τόδε ἢ τόδε ζητῶμεν, εἰς ἀριθμὸν θέντες, οἷον πότερον ἐκλείπει ὁ ἥλιος ἢ οὔ, τὸ ὅτι ζητοῦμεν. σημεῖον δὲ τούτου· (...) εὑρόντες γὰρ ὅτι ἐκλείπει πεπαύμεθα· καὶ ἐὰν ἐξ ἀρχῆς εἰδῶμεν ὅτι ἐκλείπει, οὐ ζητοῦμεν πότερον. ὅταν δὲ εἰδῶμεν τὸ ὅτι, τὸ διότι ζητοῦμεν, οἷον εἰδότες ὅτι ἐκλείπει καὶ ὅτι κινεῖται ἡ γῆ, τὸ διότι ἐκλείπει ἢ διότι κινεῖται ζητοῦμεν.The things we seek are equal in number to those we understand. We seek four things: the fact, the reason why, if it is, what it is. For when we seek whether it is this or this, putting it into a number, we seek the fact. Evidence for this: on finding that it is eclipsed we stop; and if from the start we know that it is eclipsed, we do not seek whether it is. When we know the fact we seek the reason why. (shrink)

At Topics 8.11 161b19–33 Aristotle lists five criticisms () which may be leveled against a dialectical argument ‘in itself’ (). The five criticisms correspond in many respects to the familiar conditions Aristotle places on syllogism and refutation. However, begging the question —the violation of the condition that the conclusion of a syllogism be something different () from the premises—seems not to appear on the list of five criticisms. That this omission is only apparent becomes clear once it is seen that (...) the five criticisms are not independent but rather amount to five successive filters through which an argument may be passed before its non-genuineness is exposed. This result sheds light on Aristotle's curious insistence, in the face of the heterogeneity of his examples of BTQ in Topics 8.13, that every question-begging refutation contains a premise that is ‘the same’ as the conclusion. The fact that the five criticisms expose BTQ as a defect in syllogistic reasoni.. (shrink)

Aristotle frequently describes essence as a “cause” or “explanation”, thus ascribing to essence some sort of causal or explanatory role. This explanatory role is often explicated by scholars in terms of essence “making the thing be what it is” or “making it the very thing that it is”. I argue that this is problematic, at least on the assumption that “making” expresses an explanatory relation, since it violates certain formal features of explanation. I then consider whether Aristotle is vulnerable to (...) this problem by examining the explanatory role of essence in Posterior Analytics and Metaphysics Z 17. (shrink)

In Aristotle's views on cognition a series of terms – hupolêpsis, doxa, and epistêmê – play key roles. But it has not been noticed that each of these comes in two kinds – one unqualified and the other qualified. When these and their interrelations are properly explored, a deeply systematic picture of cognition emerges, in which doxa is best understood as ‘belief’, hupolêpsis as ‘supposition’, and epistêmê as a sort of belief, so that – contrary to orthodoxy – we can (...) have belief and knowledge of the same things at the same time. Many of these conclusions are shown to mark a continuity with Plato, in that neither thinker, it is argued, holds a so-called ‘two-worlds’ picture of cognition. (shrink)

The paper provides an interpretation of Aristotle’s view on scientific inquiry as problem solving. It tackles passages where Aristotle emphasises the role that the problem-solving activity has in science, and where he describes the history of humans’ problem-solving activity and the historical development of natural science as a problem-solving activity. Further, the paper examines Aristotle’s practice of raising, assessing and solving problems as well as the heuristic procedures he employs to move from ignorance to scientific knowledge. Finally, it raises a (...) question about Aristotle’s view on the epistemic status of scientific definitions: does Aristotle conceive of the definitions, which are the results of scientific inquiries, as truths or rather as plausible hypotheses that may or may not be true? (shrink)

Something Aristotle calls ‘right logos’ plays a crucial role in his theory of virtue. But the meaning of ‘logos’ in this context is notoriously contested. I argue against the standard translation ‘reason’, and—drawing on parallels with Plato’s work, especially the Laws—in favor of its being used to denote what transforms an inferior epistemic state into a superior one: an explanatory account. Thus Aristotelian phronēsis, like his and Plato’s technē and epistēmē, is a matter of grasping explanatory accounts: in this case, (...) accounts that identify the right action and say why it is right. Arguably, Aristotelian rationality is a matter of being able to grasp accounts in general. (shrink)

This paper discusses how Hegel understands the principle of contradiction. It is analysed, first, the way Aristotle understands this principle, emphasizing the so called semantic and ontological readings. Then, it is analysed whether it is correct to argue that Hegel rejects this principle, holding here that, strictly speaking, this is not correct. To do this, we present a dialetheist reading of Hegelian logic, which allows to state that: i) Hegel does not accept trivialism and, from i), ii) is not feasible (...) a full rejection of the principle of contradiction in the Hegelian logic. The so-called cognitive contextualism can dismiss some contradictions as mere parameterisations, but does not lead to the strong thesis that Hegel rejects the principle of contradiction. (shrink)

Aristotle’s main thesis in Metaphysics Z.17, which takes substance to be a principle and a cause of some sort (1041a9–10, 1041b7–9, b30–31), is of a piece with the assumption that hylomorphic compounds are unified wholes (1041b11–12) – an assumption that proves critical to settling an important controversy about the form-matter relationship in that chapter, i. e. whether matter and form are mutually indistinguishable or rather just accidentally the same. By rejecting these interpretive options, this paper argues that form and matter (...) are bound together by an essential link which, nevertheless, is able to preserve both (i) the different explanatory roles that each of those principles play vis-à-vis specific substances and (ii) the compound’s proper unity. (shrink)

I reconstruct Aristotle’s analytical procedure in Prior Analytics I.45 and its metalogical implications. Aristotle’s analysis unfolds three groups of syllogisms: symmetrically analysable, asymmetrically analysable, and non-analysable syllogisms. From the first and the third group could be extracted 27 combinations of the two mutually non-derivable deductive rules. Aristotle’s reduced deductive system in APr. I.7 with the two moods in the first figure (traditionally called Barbara and Celarent) follows this pattern. I demonstrate that the deductive system with Barbara and Celarent is just (...) one of 27 possible complete deductive systems with the two mutually non-derivable syllogistic rules. On top of that, given the fact that there are two combinations of mutually non-derivable conversion rules, I conclude that Prior Analytics hides in itself 54 complete deductive systems with a minimal number of mutually non-derivable rules. Finally, it is commented what might be the role of analysis; to what extent Aristotle could be aware of possibilities presented here; why he still prefers only one deductive system to many others with the same deductive power. (shrink)

Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th (...) century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

In Prior Analytics 1.27–30, Aristotle develops a method for finding deductions. He claims that, given a complete collection of facts in a science, this method allows us to identify all demonstrations and indemonstrable principles in that science. This claim has been questioned by commentators. I argue that the claim is justified by the theory of natural predication presented in Posterior Analytics 1.19–22. According to this theory, natural predication is a non-extensional relation between universals that provides the metaphysical basis for demonstrative (...) science. (shrink)

European Journal of Philosophy, EarlyView.

Acceptance or rejection of factual assertions is a far more important process than logical validation of arguments. Not only are assertions more persuasive than arguments; this is desirable, since we want our beliefs and actions to be reasonable and not just rational. When do we resort to argument? Real speeches heavy on arguments aim to present the speaker as calm, serious, and knowledgeable. In public life, one argues not in order to demonstrate the claim for which one is arguing, but (...) to show that one shares the common prejudices or values that appear in the presuppositions and conclusions of one’s argument, and to demonstrate mastery of the subject matter by displaying relevant knowledge in coherently organized detail. Arguing is thus a way of presenting facts and principles so as to show one’s character as worthy of trust. (shrink)

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)

Each science has its own domain of investigation, but one and the same science can be formalized in different languages with different universes of discourse. The concept of the domain of a science and the concept of the universe of discourse of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppositions and implications of choices of domain and universe, this article discusses the treatment of omega arguments in three very different (...) formalizations of arithmetic. In Peano's formalization the domain is a restricted class of individuals, while the universe of discourse is the unrestricted class of all individuals. In Gödel's formalization the domain is a restricted class of individuals as in Peano's formalization, but the universe of discourse coincides with the domain. In Whitehead-Russell's formalization the domain is a class of logical notions in Tarski's sense, that are necessarily not individuals, whereas the universe of discourse is the unrestricted class of individuals as in Peano's formalization. The present approach emphasizes the viewpoint that the universe of discourse of a given discourse is important in determining which propositions are expressed by which sentences. (shrink)

Typically, Aristotle’s notion of an accidental unity is explained using our concept of identity, but doing so is fraught and liable to mislead. I argue that we should explain accidental unities in terms of sameness: doing so not only shows a coherence among texts thought to be in tension with one another, it reconciles the two competing conceptions of accidental unities in a satisfying way. I conclude by answering several Boolean questions that naturally arise in response to the inclusion of (...) accidental unities in Aristotle’s ontology. (shrink)