The taxonomy and analysis of fallacies in Aristotle's Sophistical Refutations pre-date the formal logic of his Prior Analytics A4-6. Of the 64 fully described examples of ?sophistical refutations? which are fallacious because they are only apparently valid, 49 have the wrong number of premisses or the wrong form of premiss or conclusion for analysis by the Prior Analytics theory of the categorical syllogism. The rest Aristotle either frames so that they do not look like categorical syllogisms or analyses in a (...) quite different way than by categorical syllogistic. (shrink)

Prawitz has recently developed a theory of epistemic grounding that differs in many respects from his earlier semantics of arguments and proofs. An innovative approach to inferences yields a new conception of the intertwinement of the notions of valid inference and proof. We aim at singling out three reasons that may have led Prawitz to the ground-theoretic turn, i.e.: a better order in the explanation of the relation between valid inferences and proofs; a notion of valid inference based on which (...) valid inferences and proofs are recognisable as such; a reconstruction of the deductive activity that makes inferences capable of yielding justification per se. These topics are discussed by Prawitz with reference to a very general and ancient question: why and how correct deduction has the epistemic power to compel us to accept its conclusions, provided its premises are justified? We conclude by remarking that, in spite of some improvements, the ground-theoretic approach shares with the previous one a problem of vacuous validity which, as Prawitz himself points out, blocks in both cases a satisfactory explanation of epistemic compulsion. (shrink)

In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristotle’s Prior Analytics, with a view to motivate a new approach, which builds upon previous work by Marion & Rückert (2016) on the dictum de omni. This approach sets Aristotle’s work within the context of dialectic and uses Lorenzen’s dialogical logic, hereby reframed with use of Martin-Löf's constructive type theory as ‘immanent reasoning’. We then provide rules of syllogistic for the latter, and provide proofs of (...) e-conversion, Darapti and Bocardo and e-subalternation, while showing how close to Aristotle’s text these proofs remain. (shrink)

It has often been claimed that (i) Aristotle's expression 'protasis' means 'premiss' in syllogistic contexts and (ii) cannot refer to the conclusion of a syllogism in the Prior Analytics. In this essay we produce and defend a counter-example to these two claims. We argue that (i) the basic meaning of the expression is 'proposition' and (ii) while it is often used to refer to the premisses of a syllogism, in Prior Analytics 1.29, 45b4-8 it is used to refer to the (...) conclusion of a syllogism. In our view, the best explanation of Aristotle's use of the expression 'protasis' is that it means proposition throughout but is frequently used without change of meaning (in certain specific contexts) to refer to the premisses from which a conclusion follows. In Prior Analytics 1.29, 45b4-8 he uses 'protasis' to refer to the conclusion when he needs a single expression to refer to both the conclusion and one of the premisses of the syllogism that constitutes the core of a syllogism through the impossible. If we are correct, we have shown that the view that the expression 'the final protasis' in EN 7.3, 1147b9ff must mean 'the final premiss' and so cannot refer to the conclusion of the relevant syllogism is mistaken. (shrink)

I show that in his De motu animalium Aristoteles proposes a teleology of the practical on the most general zoological level, i.e. on the level common to humans and self-moving animals. A teleology of the practical is a teleological account of the highest practical goals of animal and human self-motion. I argue that Aristotle conceives of such highest practical goals as goals that are contingently related to their realizations. Animal and human self-motion is the kind of action in which certain (...) state of affairs that realize values are mechanized. (shrink)

A predicate logic typically has a heterogeneous semantic theory. Subjects and predicates have distinct semantic roles: subjects refer; predicates characterize. A sentence expresses a truth if the object to which the subject refers is correctly characterized by the predicate. Traditional term logic, by contrast, has a homogeneous theory: both subjects and predicates refer; and a sentence is true if the subject and predicate name one and the same thing. In this paper, I will examine evidence for ascribing to Aristotle the (...) view that subjects and predicates refer. If this is correct, then it seems that Aristotle, like the traditional term logician, problematically conflates predication and identity claims. I will argue that we can ascribe to Aristotle the view that both subjects and predicates refer, while holding that he would deny that a sentence is true just in case the subject and predicate name one and the same thing. In particular, I will argue that Aristotle's core semantic notion is not identity but the weaker relation of constitution. For example, the predication ‘All men are mortal’ expresses a true thought, in Aristotle's view, just in case the mereological sum of humans is a part of the mereological sum of mortals. (shrink)

Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning showing (...) by deductively evident steps that its conclusion is a consequence of its premises. In particular, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. (shrink)

When in classical formal logic the notions of deduction, valid inference and logical consequence are defined, causal language plays no role. The founder of western logic, Aristotle, identified ‘non-cause’, or ‘positing as cause what is not a cause’, as a logical fallacy. I argue that a systematic re-examination of Aristotle's analysis of NCF, and the related language of logical causality, in the Sophistical Refutations, Topics, Analytics and Rhetoric, helps us to understand his conception of. It reveals that Aristotle's syllogismhood is (...) non-monotonic, and thus narrower than classical validity as consisting in, or coextensive with, necessary truth-preservation. It also supports the conclusion that a syllogism is a form of relevant consequence: the premisses must be causes of the conclusion in the sense that they are not only sufficient but also necessary for the conclusion to follow. Close analysis of NCF also suggests that syllogistic relevance is not a semantic matter, but a syntactic.. (shrink)

In this article I analyze the Peirce’s concepts of abduction, insight, logica utens and logica docens in order to propose, finally, a vision in which “genuine” abductions are generated only after an exercise in logica docens. These “genuine” abductions are characterized in contrast to those generated by means of logica utens.

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)

Although he does not have an explicit theory of supposition as is found in the works of Latin medieval philosophers, Avicenna has two doctrines giving something equivalent: the threefold distinction of quiddity, corresponding to a division of simple, personal and material supposition, and his analyses of truth conditions for categorical propositions, where sentential context determines in part the reference of their terms. While he does address which individuals are being referred to by the universal terms used there, Avicenna concentrates more (...) on the varied temporal durations of the predications being made. In Western terms, he has incorporated ampliation and restriction into the theory of supposition itself. (shrink)

According to the standard and largely traditional interpretation, Aristotle’s conception of nous, at least as it occurs in the Posterior Analytics, is geared against a certain set of skeptical worries about the possibility of scientific knowledge, and ultimately of the knowledge of Aristotelian first principles. On this view, Aristotle introduces nous as an intuitive faculty that grasps the first principles once and for all as true in such a way that it does not leave any room for the skeptic to (...) press his skeptical point any further. Thus the tradi- tional interpretation views Aristotelian nous as having an internalist justificatory role in Aristotelian epistemology. In contrast, a minority (empiricist) view that has emerged recently holds the same internalist justificatory view of nous but rejects its internally certifiable infallibility by stressing the connection between nous and Aristotelian induction. I argue that both approaches are flawed in that Aristotle’s project in the Posterior Analytics is not to answer the skeptic on internalist justificatory grounds, but rather lay out a largely externalist explication of scientific knowledge, i.e. what scientific knowledge consists in, without worrying as to whether we can ever show the skeptic to his satisfaction that we do ever possess knowledge so defined. (shrink)

This paper discusses the contrast between scientific knowledge and opinion as it is presented by Aristotle in Posterior Analytics A.33. Aristotle's contrast is formulated in terms of understanding or not understanding some "necessary items". I claim that the contrast can only be understood in terms of explanatory relevance. The "necessary items" are middle terms (or explanatory factors) that are necessary for the fully appropriate explanation. This approach gives a coherent interpretation of each step in the chapter.

I argue that Topics VI does not contain any serious theory about definitions, but only a collection of advices for formulating definitions in a dialectical context, namely, definitions aiming to catch what the opponent means. Topics VI is full of inconsistencies that can be explained away by this approach: the inconsistencies reflect "acceptable opinions about definitions" that distinct groups of interlocutors accept. I also argue that the "topoi" need not be pieces of serious theory Aristotle is commited to. The "topoi" (...) must also be considered as "endoxa", namely, as accepted opinions about how it is legitimate to draw an inference. (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)

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)

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)

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)

This is a brief note that looks at the problem presented by the traditional rendering of the modal syllogism Disamis XLL. In two recent articles, I argue that we should not attribute Disamis XLL to Aristotle. The purpose of this note is to provide textual support for my claim.

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)

Essentialists claim that at least some individuals or kinds have essences. This raises an important but little-discussed question: how do we come to know what the essence of something is? This paper examines Aristotle’s answer to this question. One influential interpretation (viz., the Explanationist Interpretation) is carefully expounded, criticized, and then refined. Particular attention is given to what Aristotle says about this issue in DA I.1, APo II.2, and APo II.8. It is argued that the epistemological claim put forward in (...) DA I.1 differs from that put forward in APo II.2 and II.8, contrary to what has been claimed by Explanationists, and that each of these distinct epistemological claims rests on a distinct non-epistemological thesis about essence. Consequently, an ‘Enriched Explanationist Interpretation’ is developed which takes into account both of the aforementioned elements in Aristotle’s epistemology of essence. The paper concludes by highlighting an insight the preceding exegetical discussion offers to contemporary essentialists seeking to explain how we come to know what something’s essence is. (shrink)

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (shrink)

It has been common in contemporary logic and philosophy of logic to identify the validity of an inference with its conclusion being a (logical) consequence of its premisses.

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)

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)

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...) introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence. (shrink)

In an earlier article (see J Gen Philos Sei (2010) 41: 341-355) I have compared Aristotle's syllogistic with Kant's theory of "pure ratiocination". "Ratiocinia pura" („reine Vernunftschlüsse") is Kant's designation for assertoric syllogisms Aristotle has called 'perfect'. In Kant's view they differ from non-pure ratiocinia precisely in that their validity rests only on the validity of the Dictum de omni et nullo (which, however, in Kant's view can be further reduced to more fundamental principles) whereas the validity of non-pure ratiocinia (...) additionally presupposes the validity of inferences which Kant calls consequentiae immediatae. I have argued that Kant's view is in some (not in all) essential features in accordance with Aristotle's view concerning perfect syllogisms and certainly leading to a tenable and interesting logical theory. As a result I have rejected not only the interpretation of Aristotle adopted by Theodor Ebert, but also the objections he has raised against Kant's logical theory. As far as Aristotle is concerned, Ebert has attempted to defend his position in the first part of his reply to my article published in J Gen Philos Sei (2009) 40: 357-365, and I have argued against this defence in issue 1 of the J Gen Philos Sei (2010) 41: 199-213 (cf. Ebert's answer in the same issue pp. 215-231). In the following discussion I deal with Eberts defence of his criticism of Kant published in the second part of his reply to my article (see J Gen Philos Sei (2009) 40: 365-372). I shall argue, that Kant's principle 'nota notae est nota rei ipsius' and his use of technical vocabulary stand up to the objections raised by Ebert. His attempts to prove that Kant's logical theory is defective are based on several misinterpretations. (shrink)

This paper explores Aristotle's remarks in Posterior Analytics on certain special disciplines that are subordinate to pure mathematical sciences. Optics, harmonics and mechanics prove their own contents by means of premises belonging to arithmetic or geometry. Even though subaltern sciences are exceptions to the prohibition on kind crossing, the premises to their demonstrations are legitimately appropriate to the relative conclusions. In order to delineate the demonstrative structure of subordinate sciences, Aristotle introduces the distinction between knowledge of a fact and knowledge (...) of the reason for it. In his view, these two different levels of knowledge, characterizing respectively empirical and theoretical approaches, are closely related. Nautical astronomy, for instance, deals with the observation and recording of the astral motions, which are made intelligible by mathematical astronomy. The Aristotelian ideal of an explanatory connection between appearances and mathematical principles seems to be the main aspect in the treatment of subordinate sciences in Posterior Analytics. (shrink)

Aristotle presents four causes in Posterior Analytics 2.11, but where we expect matter we find instead the confusing formula, ‘what things being the case, necessarily this is the case’, and an equally confusing example. Some commentators infer that Aristotle is not referring to matter, others that he is but in a non-standard way. I argue that APo. 94a20-34 presents not matter, but determination by general features or facts, including facts about something’s genus. The closest connection to matter is Aristotle’s view (...) that the relation between genus and species is analogous to that between matter and a hylomorphic compound. (shrink)

This paper sets out to evaluate the claim that Aristotle’s Assertoric Syllogistic is a relevance logic or shows significant similarities with it. I prepare the grounds for a meaningful comparison by extracting the notion of relevance employed in the most influential work on modern relevance logic, Anderson and Belnap’s Entailment. This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, that the premises (...) of a proof are actually used to derive the conclusion. Turning to Aristotle’s Prior Analytics, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the Prior Analytics. In a final section I consider the view that Aristotle distinguished between validity on the one hand and syllogistic validity on the other. Following this line of reasoning, Aristotle’s logic might not be a relevance logic, since relevance is part of syllogistic validity and not, as modern relevance logic demands, of general validity. I argue that the reasons to reject this view are more compelling than the reasons to accept it and that we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic. (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)

Modern logicians have sought to unlock the modal secrets of Aristotle's Syllogistic by assuming a version of essentialism and treating it as a primitive within the semantics. These attempts ultimately distort Aristotle's ontology. None of these approaches make full use of tests found throughout Aristotle's corpus and ancient Greek philosophy. I base a system on Aristotle's tests for things that can never combine (polarity) and things that can never separate (inseparability). The resulting system not only reproduces Aristotle's recorded results for (...) the apodictic syllogistic in the Prior Analytics but it also generates rather than assumes Aristotle's distinctions among 'necessary', 'essential' and 'accidental'. By developing a system around tests that are in Aristotle and basic to ancient Greek philosophy, the system is linked to a history of practices, providing a platform for future work on the origins of logic. (shrink)

The notion of identity is investigated through Aristotle and Hegel as supporters of two different ontological conceptions: pluralism of substances and relational holism. Through Aristotle, I examine both the thesis according to which the identity of an object is constituted by its properties and the difficulties which this thesis encounters. Aristotle easily defines the identity in species, in genus, and in number; some problems arise regarding the identity of individuals: for these, it is not enough to indicate the definition and (...) the proper qualities, but matter is needed. Matter cannot, however, be a criterion for identifying duplicate objects: in this case, it plays at most the role of a “weak individuator.” A weak individuator involves relations with other entities. The use of relations in determining the identity of an entity is extensively treated by Hegel, according to whom, in order to define the identity of an object a multiplicity of particular objects is required and, therefore, relations among entities. I conclude by proposing a notion of the object understood not as an independent, separate, and autonomous item, but as a portion of the world, which is given in a phenomenological context and identifies a dialectical context. (shrink)

We may try to explain proofs as chains of valid inference, but the concept of validity needed in such an explanation cannot be the traditional one. For an inference to be legitimate in a proof it must have sufficient epistemic power, so that the proof really justifies its final conclusion. However, the epistemic concepts used to account for this power are in their turn usually explained in terms of the concept of proof. To get out of this circle we may (...) consider an idea within intuitionism about what it is to justify the assertion of a proposition. It depends on Heyting’s view of the meaning of a proposition, but does not presuppose the concept of inference or of proof as chains of inferences. I discuss this idea and what is required in order to use it for an adequate notion of valid inference. (shrink)

Ever since ?ukasiewicz, it has been opinio communis that Aristotle's modal syllogistic is incomprehensible due to its many faults and inconsistencies, and that there is no hope of finding a single consistent formal model for it. The aim of this paper is to disprove these claims by giving such a model. My main points shall be, first, that Aristotle's syllogistic is a pure term logic that does not recognize an extra syntactic category of individual symbols besides syllogistic terms and, second, (...) that Aristotelian modalities are to be understood as certain relations between terms as described in the theory of the predicables developed in the Topics. Semantics for modal syllogistic is to be based on Aristotelian genus-species trees. The reason that attempts at consistently reconstructing modal syllogistic have failed up to now lies not in the modal syllogistic itself, but in the inappropriate application of modern modal logic and extensional set theory to the modal syllogistic. After formalizing the underlying predicable-based semantics (Section 1) and having defined the syllogistic propositions by means of its term logical relations (Section 2), this paper will set out to demonstrate in detail that this reconstruction yields all claims on validity, invalidity and inconclusiveness that Aristotle maintains in the modal syllogistic (Section 3 and 4). (shrink)

In Posterior Analytics 1.3, Aristotle advances three arguments against circular proof. The third argument relies on his discussion of circular proof in Prior Analytics 2.5. This is problematic because the two chapters seem to deal with two rather disparate conceptions of circular proof. In Posterior Analytics 1.3, Aristotle gives a purely propositional account of circular proof, whereas in Prior Analytics 2.5 he gives a more complex, syllogistic account. My aim is to show that these problems can be solved, and that (...) Aristotle’s third argument in 1.3 is successful. I argue that both chapters are concerned with the same conception of circular proof, namely the propositional one. Contrary to what is often thought, the syllogistic conception provides an adequate analysis of the internal deductive structure of the propositional one. Aristotle achieves this by employing a kind of multiple-conclusion logic. (shrink)

This is my first professional philosophical essay. I wrote "The twofold meaning of being" in 1996 when I was a student of Nunzio Incardona at University of Palermo (Italy) and before my degree thesis, "The difference in Aristotle’s Metaphysics". It still waits to be published.

Although there are many questions to be asked about philosophy of mathematics, the fundamental questions to be asked will be questions about what the mathematical object is in view of being and what the mathematical reasoning is in view of knowledge. It is clear that other problems will develop in parallel within the framework of the answers to these questions. For this reason, when we approach Aristotle's philosophy of mathematics over these two basic problems, we come up with the concept (...) of abstraction. In our work, I will try to explain the mathematical abstraction that Aristotle has developed to understand mathematical philosophy. (shrink)

The present paper aims to show that the reconstruction of the formal framework of the proofs in Pr. An. 1.15, as proposed by Malink and Rosen 2013 (‘Proof by Assumption of the Possible in Prior Analytics 1.15’, Mind, 122, 953-85) is due to affront a double impasse. Malink and Rosen argue convincingly that Aristotle operates with two different modal frameworks, one as found in the system of modal logic presented in Prior Analytics 1.3 and 8-22, and one occurring in many (...) of Aristotle’s works, such as the Physics, De Caelo and the Metaphysics. However, they misconstrue the latter framework. More precisely, they misconstrue the domain of significance of what they call the ‘Principle of Necessitation’. As a consequence, bringing the two frameworks into one results into a contradictory modal logic. On the other hand, if the Principle of Necessitation is rectified, the proofs put forward by Malink and Rosen in the same paper are no longer available. (shrink)