Greek and Roman Logic

In ancient philosophy, there is no discipline called “logic” in the contemporary sense of “the study of formally valid arguments.” Rather, once a subfield of philosophy comes to be called “logic,” namely in Hellenistic philosophy, the field includes (among other things) epistemology, normative epistemology, philosophy of language, the theory of truth, and what we call logic today. This entry aims to examine ancient theorizing that makes contact with the contemporary conception. Thus, we will here emphasize the theories of the “syllogism” in the Aristotelian and Stoic traditions. However, because the context in which these theories were developed and discussed were deeply epistemological in nature, we will also include references to the areas of epistemological theorizing that bear directly on theories of the syllogism, particularly concerning “demonstration.” Similarly, we will include literature that discusses the principles governing logic and the components that make up arguments, which are topics that might now fall under the headings of philosophy of logic or non-classical logic. This includes discussions of problems and paradoxes that connect to contemporary logic and which historically spurred developments of logical method. For example, there is great interest among ancient philosophers in the question of whether all statements have truth-values. Relevant themes here include future contingents, paradoxes of vagueness, and semantic paradoxes like the liar. We also include discussion of the paradoxes of the infinite for similar reasons, since solutions have introduced sophisticated tools of logical analysis and there are a range of related, modern philosophical concerns about the application of some logical principles in infinite domains. Our criterion excludes, however, many of the themes that Hellenistic philosophers consider part of logic, in particular, it excludes epistemology and metaphysical questions about truth. Ancient philosophers do not write treatises “On Logic,” where the topic would be what today counts as logic. Instead, arguments and theories that count as “logic” by our criterion are found in a wide range of texts. For the most part, our entry follows chronology, tracing ancient logic from its beginnings to Late Antiquity. However, some themes are discussed in several eras of ancient logic; ancient logicians engage closely with each other’s views. Accordingly, relevant publications address several authors and periods in conjunction. These contributions are listed in three thematic sections at the end of our entry

What Is a Perfect Syllogism in Aristotelian Syllogistic?

The question as to what makes a perfect Aristotelian syllogism a perfect one has long been discussed by Aristotelian scholars. G. Patzig was the first to point the way to a correct answer: it is the evidence of the logical necessity that is the special feature of perfect syllogisms. Patzig moreover claimed that the evidence of a perfect syllogism can be seen for Barbara in the transitivity of the a-relation. However, this explanation would give Barbara a different status over the other three first figure syllogisms. I argue that, taking into account the role of the being-contained-as-in-a-whole formulation, transitivity can be seen to be present in all four first figure syllogisms. Using this wording will put the negation sign with the predicate, similar to the notation in modern predicate calculus

The Problem of Modal Upgrading in Aristotle’s Apodictic Syllogistic

This is another contribution to the unending controversy over the two Barbaras. My approach to the problem is hopefully quite new: I wish to view the issue through the prism of modal upgrading. Modal upgrading occurs when a subject term that has only been predicated of assertorically in the premises is predicated of apodictically either: i) in the conclusion of a given syllogism, or; ii) in some proposition that is derived from either the premises of the given syllogism alone or the premises in combination with other propositions that do not refer to the proposition’s subject term. I call the proposition after it has been upgraded the upgraded proposition. When a conclusion is the upgraded proposition, it is obviously a different predicate being predicated than was predicated in the premises. Aristotle endorses this kind of upgrading; it is effectively what happens in any valid mixed modal syllogism when the minor premise is not apodictic (e.g., Barbara LXL). In other cases the upgraded proposition is not a conclusion but still follows from the premises alone. In these cases it is the same predicate being predicated in the upgraded proposition as in the premises, although the quantity of the propositions are different (one is universal, another particular). Aristotle rejects this kind of upgrading and takes its occurrence as sufficient to deny the validity of the given syllogism (e.g., Barbara XLL). I will describe a third type where both the predicate remains the same and the quantity of the proposition remains the same as in the premise, e.g., the upgrading of “All C are B” to “All C are necessarily B”. In these cases it will turn out that the upgraded proposition is not derived from the premises alone, or at least, not syllogistically from the premises alone. This kind of upgrading too is reason for denying the validity of any syllogism from which the upgraded proposition follows as a consequence. I will show that Barbara LXL entails this kind of modal upgrading and should be rejected for this reason. Armed with this notion of modal upgrading I want to attack the problem of the two Barbaras in Aristotle’s apodictic syllogistic. Aristotle himself endorses mixed modal Barbara when the major is necessary and the minor is assertoric, thereby endorsing the first kind of modal upgrading, but rejects Barbara when the minor is necessary and the major is assertoric on the grounds that it leads to the second kind of modal upgrading. Theophrastus endorses the peioram rule which rejects both Barbaras on the grounds that the conclusion can only be as strong as the weakest premise. Łukasiewicz endorses both Barbaras. I will argue that both Barbaras lead to unacceptable modal upgrading and should be taken to be invalid for that reason. Hence, I agree with Theophrastus about the two Barbaras; however, I do not endorse the peioram rule because I think that the negative mixed modal syllogisms generally avoid this problem and is mostly correct

Essence and Necessity, and the Aristotelian Modal Syllogistic: A Historical and Analytical Study

The following is a critical and historical account of Aristotelian Essentialism informed by recent work on Aristotle’s modal syllogistic. The semantics of the modal syllogistic are interpreted in a way that is motivated by Aristotle, and also make his validity claims in the Prior Analytics consistent to a higher degree than previously developed interpretative models. In Chapter One, ancient and contemporary objections to the Aristotelian modal syllogistic are discussed. A resolution to apparent inconsistencies in Aristotle’s modal syllogistic is proposed and developed out of recent work by Patterson, Rini, and Malink. In particular, I argue that the semantics of negation is distinct in modal context from those of assertoric negative claims. Given my interpretive model of Aristotle’s semantics, in Chapter Two, I provide proofs for each of the mixed apodictic syllogisms, and propose a method of using Venn Diagrams to visualize the validity claims Aristotle makes in the Prior Analytics. Chapter Three explores how Aristotle’s syllogistic fits within Aristotle’s philosophy of science and demonstration, particularly within the context of the Posterior Analytics. Consideration is given to the Aristotelian understanding of the relationship among necessity, explanation, definition, and essence. Chapter Four applies Aristotelian modal logic in contemporary contexts. I contrast Aristotelian modality and essentialism with contemporary modalism based upon the semantics of possible worlds, e.g. Kripke and Putnam. I also develop an account of how Aristotelian modal logic can ground a sortal dependent theory of identity, as discussed by Wiggins

Per Se Modality and Natural Implication – an Account of Connexive Logic in Robert Kilwardby

We present a formal reconstruction of the theories of the medieval logician Robert Kilwardby, focusing on his account of accidental and natural inferences and the underlying modal logic that gives rise to it. We show how Kilwardby’s use of an essentialist modality underpins his connexive account of implication

Išvedimai tarp Buridano modalinių teiginių

In recent years modal syllogistic provided by 14th century logician John Buridan has attracted increasing attention of historians of medieval logic. The widespread use of quantified modal logic with the apparatus of possible worlds semantics in current analytic philosophy has encouraged the investigation of the relation of Buridan’s theory of modality with the modern developments of symbolic modal logic. We focus on the semantics of and the inferential relations among the propositions that underlie Buridan’s theory of modal syllogism. First, we review all inferences between propositions of necessity, possibility, contingency, and non-contingency, with or without quod est locution, that are valid in Buridan’s semantics, and offer a comprehensive diagrammatic representation that includes them all. We then ask the question if there is a way to model those results in first order modal logic. Three ways of formalizing Buridan’s propositions in quantified modal logic are considered. Comparison of inferences between the quantified formulas and Buridan’s propositions reveals that, when supplied with a suitable formalization, Buridan’s semantics of categorical statements and immediate inferences among them can be fully captured by the quantified modal system T.Straipsnyje nagrinėjama XIV a. logiko Jono Buridano modalinė logika, pastaraisiais metais sulaukianti vis didesnio viduramžių logikos tyrėjų dėmesio. Šiuolaikinėje analitinėje filosofijoje plačiai naudojama modalinė predikatų logika, paremta galimų pasaulių semantika, paskatino kelti klausimus apie Buridano modalumų teorijos ir šiuolaikinės simbolinės modalinės logikos santykį. Straipsnyje tiriama Buridano modalinės silogistikos pagrindą sudarančių teiginių semantinė interpretacija ir tarp šių teiginių galiojantys išvedimo ryšiai. Pirmiausia aptariami išvedimai tarp būtinumo, galimumo, atsitiktinumo ir neatsitiktinumo teiginių (su ir be frazės quod est), kurie yra logiškai taisyklingi pagal Buridano aprašytą semantiką. Pateikiama išsami diagraminė visų išvedimo sąryšių reprezentacija. Siekiant nustatyti, ar esama tinkamo būdo Buridano gaunamus rezultatus išreikšti predikatų logikoje su modalumais, aptariamos trys literatūroje aptinkamos Buridano teiginių formalizacijos versijos. Palyginus taisyklingus išvedimus tarp kvantifikuotų formulių ir tarp Buridano modalinių teiginių, galima tvirtinti, kad Buridano teiginių semantika ir galiojantys tiesioginiai išvedimai gali būti adekvačiai perteikiami predikatų logikos su aletiniais modalumais (sistemos T) priemonėmis

Aristotle's Theory of the Assertoric Syllogism

Although the theory of the assertoric syllogism was Aristotle's great invention, one which dominated logical theory for the succeeding two millenia, accounts of the syllogism evolved and changed over that time. Indeed, in the twentieth century, doctrines were attributed to Aristotle which lost sight of what Aristotle intended. One of these mistaken doctrines was the very form of the syllogism: that a syllogism consists of three propositions containing three terms arranged in four figures. Yet another was that a syllogism is a conditional proposition deduced from a set of axioms. There is even unclarity about what the basis of syllogistic validity consists in. Returning to Aristotle's text, and reading it in the light of commentary from late antiquity and the middle ages, we find a coherent and precise theory which shows all these claims to be based on a misunderstanding and misreading

Truth in Practical Reason: Practical and Assertoric Truth in Aristotle’s Nicomachean Ethics

Truth has always been a controversial subject in Aristotelian scholarship. In most cases, including some well-known passages in the Categories, De Interpretatione and Metaphysics, Aristotle uses the predicate ‘true’ for assertions, although exceptions are many and impossible to ignore. One of the most complicated cases is the concept of practical truth in the sixth book of Nicomachean Ethics: its entanglement with action and desire raises doubts about the possibility of its inclusion to the propositional model of truth. Nevertheless, in one of the most extensive studies on the subject, C. Olfert has tried to show that this is not only possible but also necessary. In this paper, we explain why trying to fit practical truth into the propositional model comes with insurmount­able problems. In order to overcome these problems, we focus on multiple aspects of practical syllogism and correlate them with Aristo­tle’s account of desire, happiness and the good. Identifying the role of such concepts in the specific steps of practical reasoning, we reach the conclusion that practical truth is best explained as the culmination of a well-executed practical syllogism taken as a whole, which ultimately explains why this type of syllogism demands a different approach and a different kind of truth than the theoretical one

A BIBLIOGRAPHY: JOHN CORCORAN’S PUBLICATIONS ON ARISTOTLE 1972–2015

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