Aristotle, though not the first Greek virtue ethicist, was the first to establish virtue ethics as a distinct philosophical discipline. His exposition of the subject in his Nicomachean Ethics set the terms of subsequent debate in the European and Arabic traditions by proposing a set of plausible assumptions from which virtue ethics should proceed. His conception of human well-being and virtue as well as his brand of ethical naturalism were influential from antiquity through the Middle Ages and continue to (...) be influential today. (shrink)

This presentation of Aristotle's natural deduction system supplements earlier presentations and gives more historical evidence. Some fine-tunings resulted from conversations with Timothy Smiley, Charles Kahn, Josiah Gould, John Kearns,John Glanvillle, and William Parry.The criticism of Aristotle's theory of propositions found at the end of this 1974 presentation was retracted in Corcoran's 2009 HPL article "Aristotle's demonstrative logic".

Prior Analytics by the Greek philosopher Aristotle (384 – 322 BCE) and Laws of Thought by the English mathematician George Boole (1815 – 1864) are the two most important surviving original logical works from before the advent of modern logic. This article has a single goal: to compare Aristotle’s system with the system that Boole constructed over twenty-two centuries later intending to extend and perfect what Aristotle had started. This comparison merits an article itself. Accordingly, this article (...) does not discuss many other historically and philosophically important aspects of Boole’s book, e.g. his confused attempt to apply differential calculus to logic, his misguided effort to make his system of ‘class logic’ serve as a kind of ‘truth-functional logic’, his now almost forgotten foray into probability theory, or his blindness to the fact that a truth-functional combination of equations that follows from a given truth-functional combination of equations need not follow truth-functionally. One of the main conclusions is that Boole’s contribution widened logic and changed its nature to such an extent that he fully deserves to share with Aristotle the status of being a founding figure in logic. By setting forth in clear and systematic fashion the basic methods for establishing validity and for establishing invalidity, Aristotle became the founder of logic as formal epistemology. By making the first unmistakable steps toward opening logic to the study of ‘laws of thought’—tautologies and laws such as excluded middle and non-contradiction—Boole became the founder of logic as formal ontology. (shrink)

It is sometimes asked whether virtue ethics can be assimilated by Kantianism or utilitarianism, or if it is a distinct position. A look atAristotle’s ethics shows that it certanly can be distinct. In particular, Aristotle presents us with an ethics of aesthetics in contrast to themore standard ethics of cognition: A virtuous agent identifies the right actions by their aesthetic qualities. Moreover, the agent’s concernwith her own aesthetic character gives us a key to the important role the emotions play (...) for Aristotle, which further distinguishes him from the other two theories we have mentioned. (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)

This paper is an essay in what Austin (_Proc Aristotel Soc_ 57: 1–30, 1956–1957) called "linguistic phenomenology". Its focus is on showing how the grammatical features of ordinary causal verbs, as revealed in the kinds of linguistic constructions they can figure in, can shed light on the nature of the processes that these verbs are used to describe. Specifically, drawing on the comprehensive classification of English verbs founds in Levin (_English verb classes and alternations: a preliminary investigation_, University of Chicago (...) Press, Chicago, 1993), I divide the forms of productive causal processes into five classes, corresponding to Aristotle's ontological categories: there are processes that cause change in location, in state, and in quantity; and that lead to the creation and destruction of substances. These broader categories are then subdivided into other ones, corresponding to different Levin classes, according to further differences in the causal processes they involve. I conclude by discussing the relevance of this argument to research in metaphysics and experimental philosophy. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination (...) and skill. Several attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)

Friendship is an important part of the good life. While many roboticists are eager to create friend-like robots, many philosophers and ethicists are concerned. They argue that robots cannot really be our friends. Robots can only fake the emotional and behavioural cues we associate with friendship. Consequently, we should resist the drive to create robot friends. In this article, I argue that the philosophical critics are wrong. Using the classic virtue-ideal of friendship, I argue that robots can plausibly be considered (...) our virtue friends - that to do so is philosophically reasonable. Furthermore, I argue that even if you do not think that robots can be our virtue friends, they can fulfil other important friendship roles, and can complement and enhance the virtue friendships between human beings. (shrink)

In De Anima II 5, 417a21-b16, Aristotle makes a number of distinctions between types of transitions, affections, and alterations. The objective of this paper is to sort out the relationships between these distinctions by means of determining which of the distinguished types of change can be coextensive and which cannot, and which can overlap and which cannot. From the results of this analysis, an interpretation of 417a21-b16 is then constructed that differs from previous interpretations in certain important respects, chief (...) among which is its characterization of transitions from first potentiality to first actuality, e.g., learning, not as `ordinary', but rather as acquisitions of natural dispositions or faculties. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, (...) on the epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (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)

In Physics 4.11, Aristotle discusses a sophistical puzzle in which "being Coriscus-in-the-Lyceum is different from being Coriscus-in-the-market-place." I take this puzzle to threaten the persistence of changing entities. Aristotle's answer to the puzzle is that the changing thing "is the same in respect of that, by (means of) being which at any time it is (what it is), S but in definition it is different." That is, Coriscus may be described as either a persisting substrate or as one (...) or more accidental unities. Described as the former, Coriscus persists, but described as the latter, he does not. (shrink)

In Metaphysics Λ.6-7 Aristotle argues that an unmoved substance causes the outermost sphere to rotate. His argument has puzzled and divided commentators from ancient Greece to the present. I offer a novel defense of Aristotle's argument by highlighting the logic of classification that Aristotle deploys. The core of Aristotle's argument is the identification of the unmoved substance on the 'table of opposites' as simple and purely actual. With this identification in place, Aristotle argues that the (...) outermost sphere activates its capacity to relocate its body by rotating because this is how it can be a simple and actual substance. But unlike traditional attempts to rehabilitate Aristotle's argument, I argue that the sphere does not rotate in order to imitate the Prime Mover. Rather, the sphere rotates because that is good for it to do, and the Prime Mover is a metaphysically necessary source of simple and actual substantial being, which the sphere acts for the sake of. (shrink)

This paper defends Aristotle’s project of deriving the order of time from the order of change in Physics 4.11, against the objection that it contains a vicious circularity arising from the assumption that we cannot specify the direction of a change without invoking the temporal relations of its stages. It considers and rejects a solution to this objection proposed by Ursula Coope, and proposes an alternative solution. It also considers the related problem of how the temporal orders and directions (...) derived from individual changes can together constitute a single, globally consistent order and direction of time. (shrink)

Aristotle recognizes two modes of apprehending time, viz., perceiving time and grasping time intellectually. This chapter clarifies what is and is not involved in these two modes of apprehending time. It also clarifies the way in which they interact, and argues that, according to Aristotle, one’s intellectual grasp of time has an effect on one’s perception of time for those beings who have intellect.

The reason for physics’ failure to find a final theory of the universe is examined. Problems identified are: the lack of unequivocal definitions for its fundamental elements (time, length, mass, electric charge, energy, work, matter-waves); the danger of relying too much on mathematics for solutions; especially as philosophical arguments conclude the universe cannot have a mathematical basis. It does not even need the concept of number to exist. Numbers and mathematics are human inventions arising from the human predilection for measurement. (...) Following Aristotle, a single fundamental cause is proposed to explore the efficacy of using pure non-mathematical philosophy to explain the universe, contrary to current quantum physical views. The cause is taken as Time, which is then defined, surprisingly leading automatically to a definition of a three-dimensional space answering the question into what can a universe be placed (space before space seems non-sensical). This enables the philosophical arguments to derive a universal rule giving clear-cut descriptions (definitions) of force (both gravitational and electromagnetic), motion, energy, and particles. The formation of atomic nuclei and atomic properties together with an unexpected role for neutrinos follow. In particular, a Popper-test can be prepared by introducing the concept of measurement to allow the philosophy arguments to be tested in another discipline - mathematics. The solution agrees with human physical observations to a remarkable degree of accuracy, automatically explaining and predicting values for Planck’s and the fine-structure constants. Although mathematical, it is included as confirming the efficacy of philosophy in attaining a final theory. (shrink)

In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of the system (...) is deducible by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)

JUNE 2015 UPDATE: A BIBLIOGRAPHY: JOHN CORCORAN’S PUBLICATIONS ON ARISTOTLE 1972–2015 By John Corcoran -/- This presentation includes a complete bibliography of John Corcoran’s publications relevant to his research on Aristotle’s logic. Sections I, II, III, and IV list 21 articles, 44 abstracts, 3 books, and 11 reviews. It starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article from Corcoran’s Philadelphia period that antedates his Aristotle studies and the Journal (...) of Symbolic Logic article from his Buffalo period first reporting his original results; it ends with works published in 2015. A few of the items are annotated as listed or with endnotes connecting them with other work and pointing out passages that in-retrospect are seen to be misleading and in a few places erroneous. 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, 23 in the 1980s, 42 in the 1990s, 56 in the 2000s, and 69 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally, of the Lukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration 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. Also, let me know of passages I should comment on. (shrink)

Aristotle said that induction (epagōgē) is a proceeding from particulars to a universal, and the definition has been conventional ever since. But there is an ambiguity here. Induction in the Scholastic and the (so-called) Humean tradition has presumed that Aristotle meant going from particular statements to universal statements. But the alternate view, namely that Aristotle meant going from particular things to universal ideas, prevailed all through antiquity and then again from the time of Francis Bacon until the (...) mid-nineteenth century. Recent scholarship is so steeped in the first-mentioned tradition that we have virtually forgotten the other. In this essay McCaskey seeks to recover that alternate tradition, a tradition whose leading theoreticians were William Whewell, Francis Bacon, Socrates, and in fact Aristotle himself. The examination is both historical and philosophical. The first part of the essay fills out the history. The latter part examines the most mature of the philosophies in the Socratic tradition, specifically Bacon’s and Whewell’s. After tracing out this tradition, McCaskey shows how this alternate view of induction is indeed employed in science, as exemplified by several instances taken from actual scientific practice. In this manner, McCaskey proposes to us that the Humean problem of induction is merely an artifact of a bad conception of induction and that a return to the Socratic conception might be warranted. (shrink)

At De Anima II 5, 417b17, Aristotle says, ‘The first transition (πρώτη μεταβολή) in that which can perceive is brought about by the parent, and when it is born it already has [the faculty of] sense-perception in the same way as it has knowledge. Actual sense-perception is so spoken of in the same way as contemplation.’ The purpose of this paper is to determine the nature of first transitions.

This book treats ancient logic: the logic that originated in Greece by Aristotle and the Stoics, mainly in the hundred year period beginning about 350 BCE. Ancient logic was never completely ignored by modern logic from its Boolean origin in the middle 1800s: it was prominent in Boole’s writings and it was mentioned by Frege and by Hilbert. Nevertheless, the first century of mathematical logic did not take it seriously enough to study the ancient logic texts. A renaissance in (...) ancient logic studies occurred in the early 1950s with the publication of the landmark Aristotle’s Syllogistic by Jan Łukasiewicz, Oxford UP 1951, 2nd ed. 1957. Despite its title, it treats the logic of the Stoics as well as that of Aristotle. Łukasiewicz was a distinguished mathematical logician. He had created many-valued logic and the parenthesis-free prefix notation known as Polish notation. He co-authored with Alfred Tarski’s an important paper on metatheory of propositional logic and he was one of Tarski’s the three main teachers at the University of Warsaw. Łukasiewicz’s stature was just short of that of the giants: Aristotle, Boole, Frege, Tarski and Gödel. No mathematical logician of his caliber had ever before quoted the actual teachings of ancient logicians. -/- Not only did Łukasiewicz inject fresh hypotheses, new concepts, and imaginative modern perspectives into the field, his enormous prestige and that of the Warsaw School of Logic reflected on the whole field of ancient logic studies. Suddenly, this previously somewhat dormant and obscure field became active and gained in respectability and importance in the eyes of logicians, mathematicians, linguists, analytic philosophers, and historians. Next to Aristotle himself and perhaps the Stoic logician Chrysippus, Łukasiewicz is the most prominent figure in ancient logic studies. A huge literature traces its origins to Łukasiewicz. -/- This Ancient Logic and Its Modern Interpretations, is based on the 1973 Buffalo Symposium on Modernist Interpretations of Ancient Logic, the first conference devoted entirely to critical assessment of the state of ancient logic studies. (shrink)

Imagine an equilateral triangle “pointing upward”—its horizontal base under its apex angle. A semiotic triangle has the following three “vertexes”: (apex) an expression, (lower-left) one of the expression’s conceptual meanings or senses, and (lower-right) the referent or denotation determined by the sense [1, pp. 88ff]. One example: the eight-letter string ‘coleslaw’ (apex), the concept “coleslaw” (lower-left), and the salad coleslaw (lower-right) [1, p. 84f]. Using Church’s terminology [2, pp. 6, 41]—modifying Frege’s—the word ‘coleslaw’ expresses the concept “coleslaw”, the word ‘coleslaw’ (...) denotes or names the salad coleslaw, and the concept “coleslaw” determines the salad coleslaw—recalling Frege’s principle that sense determines denotation. Church [2, p. 6] wrote: -/- We shall say that a name denotes or names its denotation and expresses its sense. […] Of the sense we say that it determines its denotation, or is a concept of the denotation. -/- Aristotle seems cognizant of distinctions going beyond those in semiotic triangles. The expression Aristotle’s semiotic pyramids seem warranted by Aristotle’s Categories, 1a1: -/- When [two] things have a name (onoma) in common and the concept (logos) of being (ousia) which corresponds to the name in each case is different, they are called same-named (homonuma). Thus, for example, both a man and a picture [of an animal] are called animals. These have only a name in common. In each case the name’s concept of being [an animal] is different; for if one says what being an animal is for each of them, one will give two distinct concepts. -/- Semiotic triangles and pyramids in Aristotle’s logic are compared to those in Church’s [2]. [1] JOHN CORCORAN, Sentence, proposition, judgment, statement, and fact, Many Sides of Logic, College Publications, 2009. [2] ALONZO CHURCH, Introduction to Mathematical Logic, Princeton, 1956. -/- The semiotic pyramid in Categories, 1a1 has a square base under the vertex ‘animal’. On the corners of the square are: the concept “animal”, the concept “animal picture”, the animals, and the animal pictures. The animals are homonymous with the animal pictures. People find Aristotle’s example far-fetched or inept even if the experience of pointing to a picture while saying “That is Tarski” is familiar. Imagine looking at a painting while thinking “That is an animal”. Without putting too fine a point on this, notice that in Aristotle’s sense it is individual things that are homonymous, not words. It would be natural to say also in his sense that two things are homonyms if one is homonymous with the other. In contrast, we use the words homonym and homonymous to relate words that are spelled the same and pronounced the same but have different meaning. Consider the noun ‘center’ and the verb ‘center’. Consider the noun ‘smell’ and the verb ‘smell’. The spelling of two homonyms is an ambiguity, or an ambiguous spelling. We need appropriate adjectives to distinguish the Categorical senses of ‘homonym’ and ‘homonymous’ from the current English sense just mentioned. I propose ‘ontological’ for the sense relating things and ‘linguistic’ for that relating words. Given that all words are things but not all things are words, we ask are words that are linguistically homonymous also ontologically homonymous? END OF POST ABSTRACT. (shrink)

This publication offers the first critical edition of John Buridan’s second set of questions on Aristotle's “De generatione et corruptione”. The edition was made by Michiel Streijger, Paul Bakker and Hans Thijssen. First published as a printed book in 2010, the publication has been converted to open access with the CC BY-NC-ND license as of September 2023.

This contribution offers an interpretation of the last half of chapter 1 of book 5 of Aristotle’s Physics in the form of a commentary. Among other things, it attempts an explanation of why Aristotle calls the termini of changes ‘something underlying’ (ὑποκείμενον) and ‘something not underlying’ (μὴ ὑποκείμενον). It also provides an analysis of Aristotle’s argument for the claim that what is not simpliciter does not change in the light of this interpretation.

The premise-fact confusion in Aristotle’s PRIOR ANALYTICS. -/- The premise-fact fallacy is talking about premises when the facts are what matters or talking about facts when the premises are what matters. It is not useful to put too fine a point on this pencil. -/- In one form it is thinking that the truth-values of premises are relevant to what their consequences in fact are, or relevant to determining what their consequences are. Thus, e.g., someone commits the premise-fact fallacy (...) if they think that a proposition has different consequences were it true than it would have if false. C. I. Lewis said that confusing logical consequence with material consequence leads to this fallacy. See Corcoran’s 1973 “Meanings of implication” [available on Academia. edu]. -/- The premise-fact confusion occurs in a written passage that implies the premise-fact fallacy or that suggests that the writer isn’t clear about the issues involved in the premise-fact fallacy. Here are some examples. -/- E1: If Abe is Ben and Ben swims, then it would follow that Abe swims. -/- Comment: The truth is that from “Abe is Ben and Ben swims”, the proposition “Abe swims” follows. Whether in fact Abe is Ben and Ben swims is irrelevant to whether “Abe swims” follows from “Abe is Ben and Ben swims”. -/- E1 suggests that maybe “Abe swims” wouldn’t follow from “Abe is Ben and Ben swims” if the latter were false. -/- E2: The truth of “Abe is Ben and Ben swims” implies that Abe swims. -/- E3: Indirect deduction requires assuming something false. -/- Comment: If the premises of an indirect deduction are true the conclusion is true and thus the “reductio” assumption is false. But deduction, whether direct or indirect, does not require true premises. In fact, indirect deduction is often used to determine that the premises are not all true. -/- Anyway, the one-page paper accompanying this abstract reports one of dozens of premise-fact errors in PRIOR ANALYTICS. In the session, people can add their own examples and comment on them. For example, is the one at 25b32 the first? What is the next premise-fact error after 25b32? Which translators or commentators discuss this? -/- . (shrink)

This presentation includes a complete bibliography of John Corcoran’s publications relevant on Aristotle’s logic. The Sections I, II, III, and IV list respectively 23 articles, 44 abstracts, 3 books, and 11 reviews. Section I starts with two watershed articles published in 1972: the Philosophy & Phenomenological Research article—from Corcoran’s Philadelphia period that antedates his discovery of Aristotle’s natural deduction system—and the Journal of Symbolic Logic article—from his Buffalo period first reporting his original results. It ends with works (...) published in 2015. Some items are annotated as listed or with endnotes connecting them with other work and pointing out passages that, in retrospect, are seen to be misleading and in a few places erroneous. In addition, Section V, “Discussions”, is a nearly complete secondary bibliography of works describing, interpreting, extending, improving, supporting, and criticizing Corcoran’s work: 10 items published in the 1970s, 24 in the 1980s, 42 in the 1990s, 60 in the 2000s, and 70 in the current decade. The secondary bibliography is also annotated as listed or with endnotes: some simply quoting from the cited item, but several answering criticisms and identifying errors. Section VI, “Alternatives”, lists recent works on Aristotle’s logic oblivious of Corcoran’s research and, more generally in some cases, even of the Łukasiewicz-initiated tradition. As is evident from Section VII, “Acknowledgements”, Corcoran’s publications benefited from consultation with other scholars, most notably George Boger, Charles Kahn, John Mulhern, Mary Mulhern, Anthony Preus, Timothy Smiley, Michael Scanlan, Roberto Torretti, and Kevin Tracy. All of Corcoran’s Greek translations were done in collaboration with two or more classicists. Corcoran never published a sentence without discussing it with his colleagues and students. (shrink)

We begin with an introductory overview of contributions made by more than twenty scholars associated with the Philosophy Department at the University of Buffalo during the last half-century to our understanding and evaluation of Aristotle's logic. More well-known developments are merely mentioned in..

A revisionist account of how philosophical induction was conceived in the ancient world and how that conception was transmitted, altered, and then rediscovered. I show how philosophers of late antiquity and then the medieval period came step-by-step to seriously misunderstand Aristotle’s view of induction and how that mistake was reversed by humanists in the Renaissance and then especially by Francis Bacon. I show, naturally enough then, that in early modern science, Baconians were Aristotelians and Aristotelians were Baconians.

One innovation in this paper is its identification, analysis, and description of a troubling ambiguity in the word ‘argument’. In one sense ‘argument’ denotes a premise-conclusion argument: a two-part system composed of a set of sentences—the premises—and a single sentence—the conclusion. In another sense it denotes a premise-conclusion-mediation argument—later called an argumentation: a three-part system composed of a set of sentences—the premises—a single sentence—the conclusion—and complex of sentences—the mediation. The latter is often intended to show that the conclusion follows from (...) the premises. The complementarity and interrelation of premise-conclusion arguments and premise-conclusion-mediation arguments resonate throughout the rest of the paper which articulates the conceptual structure found in logic from Aristotle to Tarski. This 1972 paper can be seen as anticipating Corcoran’s signature work: the more widely read 1989 paper, Argumentations and Logic, Argumentation 3, 17–43. MR91b:03006. The 1972 paper was translated into Portuguese. The 1989 paper was translated into Spanish, Portuguese, and Persian. (shrink)

Essentialism in philosophy is the position that things, especially kinds of things, have essences, or sets of properties, that all members of the kind must have, and the combination of which only members of the kind do, in fact, have. It is usually thought to derive from classical Greek philosophy and in particular from Aristotle’s notion of “what it is to be” something. In biology, it has been claimed that pre-evolutionary views of living kinds, or as they are sometimes (...) called, “natu-ral kinds”, are essentialist. This static view of living things presumes that no tran-sition is possible in time or form between kinds, and that variation is regarded as accidental or inessential noise rather than important information about taxa. In contrast it is held that Darwinian, and post-Darwinian, biology relies upon varia-tion as important and inevitable properties of taxa, and that taxa are not, therefore, kinds but historical individuals. Recent attempts have been made to undercut this account, and to reinstitute essentialism in biological kind terms. Others argue that essentialism has not ever been a historical reality in biology and its predecessors. In this chapter, I shall outline the many meanings of the notion of essentialism in psychology and social science as well as science, and discuss pro- and anti-essentialist views, and some recent historical revisionism. It turns out that nobody was essentialist to speak of in the sense that is antievolutionary in biology, and that much confusion rests on treating the one word, “essence” as meaning a single notion when in fact there are many. I shall also discuss the philosophical implica-tions of essentialism, and what that means one way or the other for evolutionary biology. Teaching about evolution relies upon narratives of change in the ways the living world is conceived by biologists. This is a core narrative issue. (shrink)

Since at least late antiquity, Aristotle’s Prior Analytics B 23 has been misread. Aristotle does not think that an induction is a syllogism made good by complete enumeration. The confusion can be eliminated by considering the nature of the surviving text and watching very closely Aristotle’s moving back and forth between “induction” and “syllogism from induction.” Though he does move freely between them, the two are not synonyms.

After MacIntyre contains original essays by leading moral and political philosophers who assess both the merits and limitations of Alasdair MacIntyre's work. Among the themes explored here are MacIntyre's historical arguments about the sources of the failure of modernity; the validity and relevance of his attempt to reinstate the ideas of Aristotle and Aquinas as central to any satisfactory moral understanding; the effectiveness of his critique of modern liberalism; and the adequacy of key concepts, such as tradition and practice, (...) in his analyses and arguments. In addition, MacIntyre responds in a substantial essay to the issues raised by the other contributors. (shrink)

The discussions of conatus – force, tendency, effort, and striving – in early modern metaphysics have roots in Aristotle’s understanding of life as an internal experience of living force. This paper examines the ways that Spinoza’s conatus is consonant with Aristotle on effort. By tracking effort from his psychology and ethics to aesthetics, I show there is a conatus at the heart of the activity of the ψυχή that involves an intensification of power in a way which anticipates (...) many of the central insights of early modern and 20th century European philosophy. The first section outlines how Aristotle’s developmental conception of the soul as geometrically ordered lays the foundation for his understanding of effort. The developmental series of powers of the soul are analogous to the series of shapes in mathematics. The second section links the striving of the soul to the gradual acquisition of virtues as a directed activity unifying multiplicity. The third examines the paradigm of self-awareness that Aristotelian effort involves. In the final section I show how ancient Greek theories of music were founded on the experience of striving. The “nature” of music is defined by Aristoxenus, and Theophrastus, in relation to the passion and intentionality of the soul. The geometrical order, as a synthesis of elements in geometry, music, or ethics, is a generative process in which past elements are retained and reintegrated in later stages of development. It requires effort to think geometrically, and the progress of knowledge itself is an integral aspect of all effort. Effort is the lived and self-aware cause which, moving step by step in an orderly and deliberate way, grows and advances upon itself. For both Spinoza and Aristotle, effort is the immanent intelligence which accomplishes what is in the purview of its understanding. Thus, will, in this conception of effort, is not something we already possess innately, but emerges gradually by an effort aimed at improvement. (shrink)

It is one thing for a given proposition to follow or to not follow from a given set of propositions and it is quite another thing for it to be shown either that the given proposition follows or that it does not follow.* Using a formal deduction to show that a conclusion follows and using a countermodel to show that a conclusion does not follow are both traditional practices recognized by Aristotle and used down through the history of logic. (...) These practices presuppose, respectively, a criterion of validity and a criterion of invalidity each of which has been extended and refined by modern logicians: deductions are studied in formal syntax (proof theory) and coun¬termodels are studied in formal semantics (model theory). The purpose of this paper is to compare these two criteria to the corresponding criteria employed in Boole’s first logical work, The Mathematical Analysis of Logic (1847). In particular, this paper presents a detailed study of the relevant metalogical passages and an analysis of Boole’s symbolic derivations. It is well known, of course, that Boole’s logical analysis of compound terms (involving ‘not’, ‘and’, ‘or’, ‘except’, etc.) contributed to the enlargement of the class of propositions and arguments formally treatable in logic. The present study shows, in addition, that Boole made significant contributions to the study of deduc¬tive reasoning. He identified the role of logical axioms (as opposed to inference rules) in formal deductions, he conceived of the idea of an axiomatic deductive sys¬tem (which yields logical truths by itself and which yields consequences when ap¬plied to arbitrary premises). Nevertheless, surprisingly, Boole’s attempt to imple¬ment his idea of an axiomatic deductive system involved striking omissions: Boole does not use his own formal deductions to establish validity. Boole does give symbolic derivations, several of which are vitiated by “Boole’s Solutions Fallacy”: the fallacy of supposing that a solution to an equation is necessarily a logical consequence of the equation. This fallacy seems to have led Boole to confuse equational calculi (i.e., methods for gen-erating solutions) with deduction procedures (i.e., methods for generating consequences). The methodological confusion is closely related to the fact, shown in detail below, that Boole had adopted an unsound criterion of validity. It is also shown that Boole totally ignored the countermodel criterion of invalid¬ity. Careful examination of the text does not reveal with certainty a test for invalidity which was adopted by Boole. However, we have isolated a test that he seems to use in this way and we show that this test is ineffectual in the sense that it does not serve to identify invalid arguments. We go beyond the simple goal stated above. Besides comparing Boole’s earliest criteria of validity and invalidity with those traditionally (and still generally) employed, this paper also investigates the framework and details of THE MATHEMATICAL ANALYSIS OF LOGIC. (shrink)

This interesting and imaginative monograph is based on the author’s PhD dissertation supervised by Saul Kripke. It is dedicated to Timothy Smiley, whose interpretation of PRIOR ANALYTICS informs its approach. As suggested by its title, this short work demonstrates conclusively that Aristotle’s syllogistic is a suitable vehicle for fruitful discussion of contemporary issues in logical theory. Aristotle’s syllogistic is represented by Corcoran’s 1972 reconstruction. The review studies Lear’s treatment of Aristotle’s logic, his appreciation of the Corcoran-Smiley paradigm, (...) and his understanding of modern logical theory. In the process Corcoran and Scanlan present new, previously unpublished results. Corcoran regards this review as an important contribution to contemporary study of PRIOR ANALYTICS: both the book and the review deserve to be better known. (shrink)

This chapter has two objectives. The first is to clarify Aristotle’s view of the first principles of the sciences. The second is to stake out a critical position with respect to this view. The paper sketches an alternative to Aristotle’s intuitionism based in part on the use of quantitative inductive logics.

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other (...) hand, Aristotle says that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

“Second-order Logic” in Anderson, C.A. and Zeleny, M., Eds. Logic, Meaning, and Computation: Essays in Memory of Alonzo Church. Dordrecht: Kluwer, 2001. Pp. 61–76. -/- Abstract. This expository article focuses on the fundamental differences between second- order logic and first-order logic. It is written entirely in ordinary English without logical symbols. It employs second-order propositions and second-order reasoning in a natural way to illustrate the fact that second-order logic is actually a familiar part of our traditional intuitive logical framework and (...) that it is not an artificial formalism created by specialists for technical purposes. To illustrate some of the main relationships between second-order logic and first-order logic, this paper introduces basic logic, a kind of zero-order logic, which is more rudimentary than first-order and which is transcended by first-order in the same way that first-order is transcended by second-order. The heuristic effectiveness and the historical importance of second-order logic are reviewed in the context of the contemporary debate over the legitimacy of second-order logic. Rejection of second-order logic is viewed as radical: an incipient paradigm shift involving radical repudiation of a part of our scientific tradition, a tradition that is defended by classical logicians. But it is also viewed as reactionary: as being analogous to the reactionary repudiation of symbolic logic by supporters of “Aristotelian” traditional logic. But even if “genuine” logic comes to be regarded as excluding second-order reasoning, which seems less likely today than fifty years ago, its effectiveness as a heuristic instrument will remain and its importance for understanding the history of logic and mathematics will not be diminished. Second-order logic may someday be gone, but it will never be forgotten. Technical formalisms have been avoided entirely in an effort to reach a wide audience, but every effort has been made to limit the inevitable sacrifice of rigor. People who do not know second-order logic cannot understand the modern debate over its legitimacy and they are cut-off from the heuristic advantages of second-order logic. And, what may be worse, they are cut-off from an understanding of the history of logic and thus are constrained to have distorted views of the nature of the subject. As Aristotle first said, we do not understand a discipline until we have seen its development. It is a truism that a person's conceptions of what a discipline is and of what it can become are predicated on their conception of what it has been. (shrink)

In this article I first set out Aristotle's explanation of chance as a term that refers to an event that occurs unusually and that appears significant in the context of the human search to achieve a goal. On this basis Aristotle argues against Democritus that the order in the universe could not be due to chance. Aristotle argues that all natural beings strive for their full potential and greatest possible development, and this is their way of striving (...) for the goodness of God. And they strive for survival and to remain in their best condition for as long as possible, and this is their way of striving for the eternity of God. Chance abnormalities occur accidentally in this process. This view of Aristotle enables us to give a much more satisfactory explanation of the evolution of species than that put forward by Darwin and Neodarwinians. In the field of ethics Aristotle argues that a certain measure of good fortune is required for happiness and even for the performance of virtuous acts. Finally, Aristotle rejects determinism and supports his belief in free choice by means of the reality of accidental occurrences and indeterminism in the field of physics. (shrink)

Fifty years ago this year a legal practitioner turned military intelligencer turned philosopher, Herbert Hart, published The Concept of Law, still deservedly best-seller in thought about law. It presents law, especially common law and constitutionally ordered systems such as ours, as a social reality which results from the sharing of ideas and making of decisions that, for good or evil, establish rules of law which are what they are, whether just or unjust. But right at its centre is a chapter (...) on justice, informed by Hart’s professional knowledge of Plato and Aristotle and the tradition of civilized thought about justice, thought which he sums up like this: “the general principle latent in [the] diverse applications of the idea of justice is that individuals are entitled in respect of each other to a certain relative position of equality or inequality.” “Hence”, he goes on, “[the] leading precept [of justice] is often formulated as ‘Treat like cases alike’; though we need to add … ‘and treat different cases differently’”. This article will say something about three aspects of this vast topic: (i) about the factual basis and normative grounds of equality; (ii) about the proposed principle of equal concern; and (iii) about laws and social policies that pursue equality by selective prohibition of direct and indirect discrimination, and of harassment or vilification, victimisation and offence. (shrink)

Because wars are fought by human beings and not merely machines, a strong virtue ethic is an essential prerequisite for those engaged in combat. From a philosophical perspective, war has historically been seen as separate and outside of the commonly accepted forms of morality. Yet there remains a general, though not well-thought out, sense that those human beings who fight wars should act ethically. Since warfighters are often called upon to contemplate and complete tasks during war that are not normally (...) required of non-warfighters (civilians), and it is believed these warfighters should act ethically, it becomes necessary to ask: What does soldierly virtue look like? In order to answer this question, I will examine essential elements of moral responsibility and military authority. Once complete, I will propose an Aristotelian-based virtue ethics system, one that is equally valuable to warfighter and civilian. To do this, I will examine the conversation in academia regarding virtue ethics and the foundations of virtue. I will provide both a cursory and deep reading of elements of Aristotle’s conception of virtue, but ultimately I will provide a slightly more rigorous reading and reinterpretation of an Aristotelian-based virtue ethic, with a definitive final recommendation for this interpretation being best for everyone, in war or walking down the street. Upon completing a study of this work, one will walk away with a next-generation interpretation of Aristotelian-based Virtue Ethics that denies eudaimonia as the central principle, relying instead on phronesis as its central theme. Phronesis, when properly developed in the warfighter will lead to moral responsibility, better decision-making, and a seamless re-integration into civilian life once has returned from the horrors of war. (shrink)

Easy to understand philosophy papers in all areas. Table of contents: Three Short Philosophy Papers on Human Freedom The Paradox of Religions Institutions Different Perspectives on Religious Belief: O’Reilly v. Dawkins. v. James v. Clifford Schopenhauer on Suicide Schopenhauer’s Fractal Conception of Reality Theodore Roszak’s Views on Bicameral Consciousness Philosophy Exam Questions and Answers Locke, Aristotle and Kant on Virtue Logic Lecture for Erika Kant’s Ethics Van Cleve on Epistemic Circularity Plato’s Theory of Forms Can we trust our senses? (...) Yes we can Descartes on What He Believes Himself to Be The Role of Values in Science Modern Science Kant’s Moral Philosophy Plato’s Republic as Pol Potist Bureaucracy Schopenhauer on Human Suffering Bertrand Russell on the Value of Philosophy The Philosophical Value of Uncertainty Logic Homework: Theorems and Models Searle vs. Turing on the Imitation Game Hume, Frankfurt, and Holbach on Personal Freedom Manifesto of the University of Wisconsin, Madison Secular Society Michael’s Analysis of the Limits of Civil Protections Bentham and Mill on Different Types of Pleasure Set Theory Homework Aristotle on Virtue Nagel On the Hard Problem Wittgenstein on Language and Thought Camus and Schopenhauer on the Meaning of Life Camus’ Hero as Rebel without a Cause My Little Finger: Camus’ Absurdism Illustrated Are Late-term Abortions Ethical? Does Mathematics Assume the Truth of Platonism? The Self-defeating Nature of Utilitarianism and Consequentialism Generally What is The Good Life? Bentham and Mill regarding types of pleasures Kant’s Moral Philosophy Five Short Papers on Mind-body Dualism Tracy Latimer’s Father had the Right to Kill Her: Towards a doctrine of generalized self-defense Arguments Concerning God and Morality Goldman, Rousseau and von Hayek on the Ideal State J.S Mill on Liberty and Personal Freedom A Kantian Analysis of a Borderline Date-rape Situation Living Well as Flourishing: Aristotle’s Conception of the Good Life Three Essays on Medical Ethics: Answers to Exam Questions on Elective Amputation, Vaccination, and Informed Consent Hobbes, Marx, Rousseau, Nietzsche: Their Central Themes De Tocqueville on Egoism Mill vs. Hobbes on Liberty Exam-Essays on the Moral Systems of Mill, Bentham, and Kant Kant’s Moral System Aristotle on Virtue Plato’s Cave Allegory An Ethical Quandary Superorganisms The Tuskegee Experiment A Rawlsian Analysis Why Moore’s Proof of an External World Fails A Defense of Nagel’s Argument Against Materialism A Utilitarian Analysis of a Case of Theft The Paradox of the Self-aware Wretch: An Analysis of Pascal’s Moral Philosophy Jean-Paul Sartre: Decline and Fall of a Marxist Sell-out A One Page Proof of Plato’s Theory of Forms Plato’s Republic as Pol Potist Bureaucracy The problem of the one and the many Four Short Essays on Truth and Knowledge What is ‘the Good Life?’ The Ontological Argument Different Political Philosophies: Plato, Locke, Madison, Rousseau, Hayek, and Mill on the State What do I know with certainty? Skepticism about skepticism Neuroscience and Freewill Operant Conditioning What makes us special? Are Late-term Abortions Ethical? No Two Papers on Epistemology: Gettier and Bostrom Examination Nietzsche on Punishment God’s Foreknowledge and Moral Responsibility . (shrink)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows (...) from the premises: there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

This paper offers a new interpretation of Aristotle’s identification, in De Anima 2.5, of αἴσθησις with an ἀλλοίωσίς τις that is not ‘a kind of destruction of something by its contrary’. Drawing on a passage from Metaphysics Iota 5, it argues that when so described, what is referred to as an ἀλλοίωσίς τις is not a uniquely perceptual alteration.

JOHN CORCORAN AND WAGNER SANZ, Disbelief Logic Complements Belief Logic. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 USA E-mail: [email protected] Filosofia, Universidade Federal de Goiás, Goiás, GO 74001-970 Brazil E-mail: [email protected] -/- Consider two doxastic states belief and disbelief. Belief is taking a proposition to be true and disbelief taking it to be false. Judging also dichotomizes: accepting a proposition results in belief and rejecting in disbelief. Stating follows suit: asserting a proposition conveys belief and denying conveys disbelief. Traditional (...) logic implicitly focused on logical relations and processes needed in expanding and organizing systems of beliefs. Deducing a conclusion from beliefs results in belief of the conclusion. Deduction presupposes consequence: one proposition is a consequence of a set of a propositions if the latter logically implies the former. The role of consequence depends on its being truth-preserving: every consequence of a set of truths is true. This paper, which builds on previous work by the second author, explores roles of logic in expanding and organizing systems of disbeliefs. Aducing a conclusion from disbeliefs results in disbelief of the conclusion. Aduction presupposes contrequence: one proposition is a contrequence of a set of propositions if the set of negations or contradictory opposites of the latter logically implies that of the former. The role of contrequence depends on its being falsity-preserving: every contrequence of a set of falsehoods is false. A system of aductions that includes, for every contrequence of a given set, an aduction of the contrequence from the set is said to be complete. Historical and philosophical discussion is illustrated and enriched by presenting complete systems of aductions constructed by the second author. One such, a natural aduction system for Aristotelian categorical propositions, is based on a natural deduction system attributed to Aristotle by the first author and others. ADDED NOTE: Wagner Sanz reconstructed Aristotle’s logic the way it would have been had Aristole focused on constructing “anti-sciences” instead of sciences: more generally, on systems of disbeliefs. (shrink)

Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any point”; the last: the (...) parallel postulate. -/- Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. -/- There are other differences that are or might become important. -/- Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. -/- Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. -/- The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content. (shrink)

This work treats the correlative concepts knowledge and opinion, in various senses. In all senses of ‘knowledge’ and ‘opinion’, a belief known to be true is knowledge; a belief not known to be true is opinion. In this sense of ‘belief’, a belief is a proposition thought to be true—perhaps, but not necessarily, known to be true. All knowledge is truth. Some but not all opinion is truth. Every proposition known to be true is believed to be true. Some but (...) not every proposition believed to be true is known to be true. Our focus is thus on propositional belief (“belief-that”): the combination of propositional knowledge (“knowledge-that”) and propositional opinion (“opinion-that”). Each of a person’s beliefs, whether knowledge or opinion, is the end result of a particular thought process that continued during a particular time interval and ended at a particular time with a conclusive act—a judgment that something is the case. This work is mainly about beliefs in substantive informative propositions—not empty tautologies. We also treat objectual knowledge (knowledge of objects in the broadest sense, or “knowledge-of”), operational knowledge (abilities and skills, “knowledge-how-to”, or “know-how”), and expert knowledge (expertise). Most points made in this work have been made by previous writers, but to the best of our knowledge, they have never before been collected into a coherent work accessible to a wide audience. -/- Key words: belief, knowledge/opinion, propositional, operational, objectual, cognition, . (shrink)