This book claims that Aristotle followed an aspect theory of predication.

Ways into the Logic of Alexander of Aphrodisias is intended to give an overview of the logic of Alexander of Aphrodisias (fl. early third century A D). Since ...

This book presents the first study of the development of the theory of modal syllogistic in the Middle Ages.

The Editor-in-Chief would like to thank the following colleagues who have helped maintain the standards ….

Aristotle created logic and developed it to a level of great sophistication. There was nothing there before; and it took more than two millennia for something better to come around. The astonishment experienced by readers of the Prior Analytics, the most important of Aristotle's works that present the discipline, is comparable to that of an explorer discovering a cathedral in a desert. This article explains and evaluates some of Aristotle's views about propositions and syllogisms. The most important omission is the (...) difficult subject of syllogisms involving modalities. Aristotle distinguishes two relations of opposition that can obtain between propositions with the same subject- and predicate-expressions: contrariety and contradiction. In every canonical syllogism, one term is common to the two premises: it is called “middle term,” or simply “middle.” The remaining two terms of the premises are the only ones occurring in the conclusion: they are called “extreme terms,” or simply “extremes.”. (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)

Not until the early 1920’s was it possible to distinguish Aristotelian or traditional logic from Aristotle’s own ancient logic. We can now recognize many aspects of his logical investigations that are themselves modern, in the sense that modern logicians are making discoveries that Aristotle had already made or had anticipated. Here we gather five salient features of Aristotle’s logical investigations that reveal a striking philosophical modernity: 1) Aristotle took logic to be that part of epistemology used to establish knowledge of (...) logical consequence; 2) Prior Analytics is a metalogical treatise on the syllogistic deduction system; 3) Aristotle recognized the epistemic efficacy of certain elemental argument patterns, and he explicitly formulated them as rules of natural deduction in corresponding sentences; 4) Prior Analytics is a proof-theoretic treatise in which Aristotle describes a natural deduction system and demonstrates certain of the logical relationships among syllogistic deduction rules ; and finally, 5) Aristotle worked with a notion of substitution sufficient for distinguishing logical syntax and semantics. In this connection he also distinguished validity from deducibility sufficiently well to note the completeness of his logic. (shrink)

Addresses various crucial approaches to the history of philosophy - narrative, philological, hermeneutic, and systematic. This book elaborates the principles of each approach and puts focus on their capacity to properly comprehend problems.

In this fast-paced, enlightening guide, Sharon M. Kaye takes us on a whistle-stop tour of medieval philosophy, revealing its astounding legacy to the discipline today.

Aristotle was the founder not only of logic but also of modal logic. In the Prior Analytics he developed a complex system of modal syllogistic which, while influential, has been disputed since antiquity--and is today widely regarded as incoherent. Combining analytic rigor with keen sensitivity to historical context, Marko Malink makes clear that the modal syllogistic forms a consistent, integrated system of logic, one that is closely related to other areas of Aristotle's philosophy. Aristotle's modal syllogistic differs significantly from modern (...) modal logic. Malink considers the key to understanding the Aristotelian version to be the notion of predication discussed in the Topics--specifically, its theory of predicables and the ten categories. The predicables introduce a distinction between essential and nonessential predication. In contrast, the categories distinguish between substantial and nonsubstantial predication. Malink builds on these insights in developing a semantics for Aristotle's modal propositions, one that verifies the ancient philosopher's claims of the validity and invalidity of modal inferences. While it acknowledges some limitations of this reconstruction, Aristotle's Modal Syllogistic brims with bold ideas, richly supported by close readings of the Greek texts. (shrink)

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

By delving into the history and development of logic from its beginnings to the modern era, George Englebretsen rehabilitates term logic and demonstrates that an enhanced traditional logic remains a viable possibility. Taking inspiration from Fred Sommers' work, he creates an updated and fascinating version of term logic; one he believes to be just as legitimate as, and in ways superior to, the currently predominant mathematical logic.

a single life-span. Philosophers, then, do not see more or know more, and they do not see less or know less. They aim to see less detail and more of the abstract. Their details, if you like, are abstractions. Walking on God’s earth as a pedestrian, as a farmer working his fields or as a passer-by, one’s picture of one’s surroundings is every bit as intelligent as that of the pilot riding the sky. The views of the field are radically (...) different, however. One sees only a specific field and in all lively detail: the exact pattern of the land, or even the exact outline of a given leaf, grasshopper, grain of sand even. Acquaintance with minute detail is not without its price: details may stand in the way of conjuring the big picture. It may be difficult to compare whichever field one happens to be in with far off fields, with respect to their size or shape or any other quality. One may wish to inquire if far off fields were already planted, harvested, or even if they exist. A pedestrian may find it hard or even impossible to do so. The pedestrian view contains fine points that the pilot’s map never would, but it does not necessarily contain more information, for it lacks the general context. After all, there are only so many items that one can observe and account for at a single glance, a single map, a single book, a single life-span. (shrink)

In der 1970 gegründeten Reihe erscheinen Arbeiten, die philosophiehistorische Studien mit einem systematischen Ansatz oder systematische Studien mit philosophiehistorischen Rekonstruktionen verbinden. Neben deutschsprachigen werden auch englischsprachige Monographien veröffentlicht. Gründungsherausgeber sind: Erhard Scheibe, Günther Patzig und Wolfgang Wieland. Von 1990 bis 2007 wurde die Reihe von Jürgen Mittelstraß mitherausgegeben.

Two main claims are defended. The first is that negative categorical statements are not to be accorded existential import insofar as they figure in the square of opposition. Against Kneale and others, it is argued that Aristotle formulates his o statements, for example, precisely to avoid existential commitment. This frees Aristotle's square from a recent charge of inconsistency. The second claim is that the logic proper provides much thinner evidence than has been supposed for what appears to be the received (...) view, that is, for the view that insofar as they occur in syllogistic negative categoricals have existential import. At most there is a single piece of evidence in favor of the view?a special case of echthesis or the setting out of a case in proof. (shrink)

I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is (...) a fragment of my system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle. (shrink)

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (shrink)

In this article, the author attempts to explicate the notion of the best known Talmudic inference rule called qal wa-omer. He claims that this rule assumes a massive-parallel deduction, and for formalizing it, he builds up a case of massive-parallel proof theory, the proof-theoretic cellular automata, where he draws conclusions without using axioms.

Jonathan Lear has suggested that Aristotle attempts to demonstrate a proof-theoretic analogue of a compactness theorem in Posterior analyticsI, chs. 19?22. Aristotle argues in these chapters that there cannot be in finite series of predications of terms. Lear's analysis of Aristotle's arguments are shown to be based on confusions about the nature of infinite orderings. Three distinct confusions are identified. In final remarks, it is suggested that a compactness claim is irrelevant to the issues which motivate Aristotle's arguments.

I argue that Prior analyticsII.5?7, 8?10, and 1.45 actually contain studies of processes for transforming arguments into other arguments which Aristotle carried out before having completed the theory of perfecting syllogisms by reduction to first-figure moods as presented in Prior analytics1.4?7. This position rejects Ross's opinion that these passages are ?mental gymnastics?, and Patzig's view that some of these texts contain studies of alternative axiomatizations or other logical studies posterior to the completion of the basic theory of syllogisms.

We specify two problems in syllogistic: the lack of functoriality of predicates and the change of grammatical role of the middle term, from subject to predicate, in some syllogisms. The standard semantics, the class interpretation, by-passes these difficulties but, we argue, in a manner that is at odds with logical intuition. We propose a semantics that is category theoretic to handle these difficulties. With this semantics we specify when syllogisms are valid and we set limits to the class interpretation. To (...) perform this task we show how to construct the categorical notion of an entity in a system of kinds. We devote two brief sections to an argument that our approach is very much in the spirit of Aristotle. (shrink)

We extend the language of the classical syllogisms with the sentence-forms “At most 1 p is a q” and “More than 1 p is a q”. We show that the resulting logic does not admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed.

This paper undertakes a re-examination of Sir William Hamilton’s doctrine of the quantification of the predicate . Hamilton’s doctrine comprises two theses. First, the predicates of traditional syllogistic sentence-forms contain implicit existential quantifiers, so that, for example, All p is q is to be understood as All p is some q . Second, these implicit quantifiers can be meaningfully dualized to yield novel sentence-forms, such as, for example, All p is all q . Hamilton attempted to provide a deductive system (...) for his language, along the lines of the classical syllogisms. We show, using techniques unavailable to Hamilton, that such a system does exist, though with qualifications that distinguish it from its classical counterpart. (shrink)

The Aristotelian syllogistic cannot account for the validity of certain inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether reductio (...) ad absurdum is needed. Thus our main goal is to derive results on the existence (or nonexistence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments. (shrink)

A formal analysis is offered of Pseudo-Scotus's theory of the conversion of (i) propositions containing singular terms (including propositions with a singular term as predicate): and (ii) propositions with a quantified predicate. An attempt is made to steer a middle course between using the Aristotelian logic as a framework for the analysis, and using a Fregean framework.

Proceedings of an international research and development conference, Tuscon, Arizona, October 1985. One hundred and twenty-eight papers are presented in this hefty volume.

We present a reading of the traditional syllogistics in a fragment of the propositional intuitionistic multiplicative linear logic and prove that with respect to a diagrammatic logical calculus that we introduced in a previous paper, a syllogism is provable in such a fragment if and only if it is diagrammatically provable. We extend this result to syllogistics with complemented terms à la De Morgan, with respect to a suitable extension of the diagrammatic reasoning system for the traditional case and a (...) corresponding reading of such De Morgan style syllogistics in the previously referred to fragment of linear logic. (shrink)

A diagrammatic logical calculus for the syllogistic reasoning is introduced and discussed. We prove that a syllogism is valid if and only if it is provable in the calculus.

The middle years of the nineteenth century saw two crucial develop ments in the history of modern logic: George Boole's algebraic treat ment of logic and Augustus De Morgan's formulation of the logic of relations. The former episode has been studied extensively; the latter, hardly at all. This is a pity, for the most central feature of modern logic may well be its ability to handle relational inferences. De Morgan was the first person to work out an extensive logic of (...) relations, and the purpose of this book is to study this attempt in detail. Augustus De Morgan (1806-1871) was a British mathematician and logician who was Professor of Mathematics at the University of London (now, University College) from 1828 to 1866. A prolific but not highly original mathematician, De Morgan devoted much of his energies to the rather different field of logic. In his Formal Logic (1847) and a series of papers "On the Syllogism" (1846-1862), he attempted with great ingenuity to reformulate and extend the tradi tional syllogism and to systematize modes of reasoning that lie outside its boundaries. Chief among these is the logic of relations. De Mor gan's interest in relations culminated in his important memoir, "On the Syllogism: IV and on the Logic of Relations," read in 1860. (shrink)

An investigation of Proclus' logic of the syllogistic and of negations in the Elements of Theology, On the Parmenides, and Platonic Theology. It is shown that Proclus employs interpretations over a linear semantic structure with operators for scalar negations (hypemegationlalpha-intensivum and privative negation). A natural deduction system for scalar negations and the classical syllogistic (as reconstructed by Corcoran and Smiley) is shown to be sound and complete for the non-Boolean linear structures. It is explained how Proclus' syllogistic presupposes converting the (...) tree of genera and species from Plato's diairesis into the Neoplatonic linear hierarchy of Being by use of scalar hyper and privative negations. (shrink)

The paper shows that in the Art of Thinking (The Port Royal Logic) Arnauld and Nicole introduce a new way to state the truth-conditions for categorical propositions. The definition uses two new ideas: the notion of distributive or, as they call it, universal term, which they abstract from distributive supposition in medieval logic, and their own version of what is now called a conservative quantifier in general quantification theory. Contrary to the interpretation of Jean-Claude Parienté and others, the truth-conditions do (...) not require the introduction of a new concept of ‘indefinite’ term restriction because the notion of conservative quantifier is formulated in terms of the standard notion of term intersection. The discussion shows the following. Distributive supposition could not be used in an analysis of truth because it is explained in terms of entailment, and entailment in terms of truth. By abstracting from semantic identities that underlie distribution, the new concept of distributive term is definitionally prior to truth and can, therefore, be used in a non-circular way to state truth-conditions. Using only standard restriction, the Logic’s truth-conditions for the categorical propositions are stated solely in terms of (1) universal (distributive) term, (2) conservative quantifier, and (3) affirmative and negative proposition. It is explained why the Cartesian notion of extension as a set of ideas is in this context equivalent to medieval and modern notions of extension. (shrink)

The mediaeval logic of Aristotelian privation, represented by Ockham's expositionof All A is non-P as All S is of a type T that is naturally P and no S is P, iscritically evaluated as an account of privative negation. It is argued that there aretwo senses of privative negation: (1) an intensifier (as in subhuman), the dualof Neoplatonic hypernegation (superhuman), which is studied in linguistics asan operator on scalar adjectives, and (2) a (often lexicalized) Boolean complementrelative to the extension of (...) a privative negation in sense (1) (e.g., Brute). Thissecond sense, which is the privative negation discussed in modern linguistics, isshown to be Aristotle's. It is argued that Ockham's exposition fails to capture muchof the logic of Aristotelian privation due to limitations in the expressive power of thesyllogistic. (shrink)

In his 1910 book On the principle of contradiction in Aristotle, Jan Łukasiewicz claims that syllogistic is independent of the principle of contradiction . He also argues that Aristotle would have defended such a thesis in the Posterior Analytics. In this paper, we first show that Łukasiewicz's arguments for these two claims have to be rejected. Then, we show that the thesis of the independence of assertoric syllogistic vis-à-vis PC is nevertheless true. For that purpose, we first establish that there (...) is no possibility to account for Aristotle's original views on syllogistic without using per impossibile deductions. At last, following an idea due to Smith 1983, we prove that there is a complete reconstruction of assertoric syllogistic using ecthesis instead of reductio ad absurdum. Contrary to what is claimed in Smith 1983, we show that Smith's system SE has to be improved for such a reconstruction. This results in an ecthetic system which is complete with respect to syllogistic arguments, not explosi.. (shrink)