According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done within first-order (...) logic, but it can be done within second-order one. Now let us consider the following question: can we axiomatically capture the semantic concept of conjunction? Such question, to be sure, does not make sense within the standard framework: we cannot construe it as asking whether we can form a first-order (or, for that matter, whatever-order) theory with an extralogical binary propositional operator so that its only model (up to isomorphism) maps the operator on the intended binary truth-function. The obvious reason is that the framework of standard logic does not allow for extralogical constants of this type. But of course there is also a deeper reason: an existence of a constant with this semantics is presupposed by the very definition of the framework1. Hence the question about the axiomatic capturability of concunction, if we can make sense of it at all, cannot be asked within the framework of standard logic, we would have to go to a more abstract level. To be able to make sense of the question we would have to think about a propositional ‘proto-language’, with uninterpreted logical constants, and to try to search out axioms which would fix the denotations of the constants as the intended truth-functions. Can we do this? It might seem that the answer to this question is yielded by the completeness theorem for the standard propositional calculus: this theorem states that the axiomatic delimitation of the calculus and the semantic delimitation converge to the same result.. (shrink)

This paper intends to explain key differences between Aristotle’s understanding of the relationships between nous, epistêmê, and the art of syllogistic reasoning(both analytic and dialectical) and the corresponding modern conceptions of intuition, knowledge, and reason. By uncovering paradoxa that Aristotle’s understanding of syllogistic reasoning presents in relation to modern philosophical conceptions of logic and science, I highlight problems of a shift in modern philosophy—a shift that occurs most dramatically in the seventeenth century—toward a project of construction, a pervasive desire for (...) rational certainty, and a general insistence on the reducibility of the sciences. The major motivation of this analysis is my intention to show that modern attempts to reduce science/epistêmê to a single science/method of inquiry occlude dialectical and ethico-political dimensions of “reason” and, hence, also impoverish philosophy’s critical capacities. (shrink)

We investigate the philosophical significance of the existence of different semantic systems with respect to which a given deductive system is sound and complete. Our case study will be Corcoran’s deductive system D for Aristotelian syllogistic and some of the different semantic systems for syllogistic that have been proposed in the literature. We shall prove that they are not equivalent, in spite of D being sound and complete with respect to each of them. Beyond the specific case of syllogistic, the (...) goal is to offer a general discussion of the relations between informal notions—in this case, an informal notion of deductive validity—and logical apparatuses such as deductive systems and (model-theoretic or other) semantic systems that aim at offering technical, formal accounts of informal notions. Specifically, we will be interested in Kreisel’s famous ‘squeezing argument’; we shall ask ourselves what a plurality of semantic systems (understood as classes of mathematical structures) may entail for the cogency of specific applications of the squeezing argument. More generally, the analysis brings to the fore the need for criteria of adequacy for semantic systems based on mathematical structures. Without such criteria, the idea that the gap between informal and technical accounts of validity can be bridged is put under pressure. (shrink)

ABSTRACT: 'Aristotelian logic', as it was taught from late antiquity until the 20th century, commonly included a short presentation of the argument forms modus (ponendo) ponens, modus (tollendo) tollens, modus ponendo tollens, and modus tollendo ponens. In late antiquity, arguments of these forms were generally classified as 'hypothetical syllogisms'. However, Aristotle did not discuss such arguments, nor did he call any arguments 'hypothetical syllogisms'. The Stoic indemonstrables resemble the modus ponens/tollens arguments. But the Stoics never called them 'hypothetical syllogisms'; nor (...) did they describe them as ponendo ponens, etc. The tradition of the four argument forms and the classification of the arguments as hypothetical syllogisms hence need some explaining. In this paper, I offer some explanations by tracing the development of certain elements of Aristotle's logic via the early Peripatetics to the logic of later antiquity. I consider the questions: How did the four argument forms arise? Why were there four of them? Why were arguments of these forms called 'hypothetical syllogisms'? On what grounds were they considered valid? I argue that such arguments were neither part of Aristotle's dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle's logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some 20th century theories. (shrink)

ABSTRACT: In antiquity we encounter a distinction of two types of hypothetical syllogisms. One type are the ‘mixed hypothetical syllogisms’. The other type is the one to which the present paper is devoted. These arguments went by the name of ‘wholly hypothetical syllogisms’. They were thought to make up a self-contained system of valid arguments. Their paradigm case consists of two conditionals as premisses, and a third as conclusion. Their presentation, either schematically or by example, varies in different authors. For (...) instance, we find ‘If (it is) A, (it is) B; if (it is) B, (it is) C; therefore, if (it is) A, (it is) C’. The main contentious point about these arguments is what the ancients thought their logical form was. Are A, B, C schematic letters for terms or propositions? Is ‘is’, where it occurs, predicative, existential, or veridical? That is, should ‘A esti’ be translated as ‘it is an A’, ‘A exists’, ‘As exist’ or ‘It is true/the case that A’? If A, B, C are term letters, and ‘is’ is predicative, are the conditionals quantified propositions or do they contain designators? If one cannot answer these questions, one can hardly claim to know what sort of arguments the wholly hypothetical syllogisms were. In fact, all the above-mentioned possibilities have been taken to describe them correctly. In this paper I argue that it would be mistaken to assume that in antiquity there was one prevalent understanding of the logical form of these arguments - even if the ancients thought they were all talking about the same kind of argument. Rather, there was a complex development in their understanding, starting from a term-logical conception and leading to a propositional-logical one. I trace this development from Aristotle to Philoponus and set out the deductive system on which the logic of the wholly hypothetical syllogisms was grounded. (shrink)

ABSTRACT: For the Stoics, a syllogism is a formally valid argument; the primary function of their syllogistic is to establish such formal validity. Stoic syllogistic is a system of formal logic that relies on two types of argumental rules: (i) 5 rules (the accounts of the indemonstrables) which determine whether any given argument is an indemonstrable argument, i.e. an elementary syllogism the validity of which is not in need of further demonstration; (ii) one unary and three binary argumental rules which (...) establish the formal validity of non-indemonstrable arguments by analysing them in one or more steps into one or more indemonstrable arguments. The function of these rules is to reduce given non-indemonstrable arguments to indemonstrable syllogisms. Moreover, the Stoic method of deduction differs from standard modern ones in that the direction is reversed. The Stoic system may hence be called an argumental reductive system of deduction. In this paper, a reconstruction of this system of logic is presented. (shrink)

Three distinctly different interpretations of Aristotle?s notion of a sullogismos in Prior Analytics can be traced: (1) a valid or invalid premise-conclusion argument (2) a single, logically true conditional proposition and (3) a cogent argumentation or deduction. Remarkably the three interpretations hold similar notions about the logical relationships among the sullogismoi. This is most apparent in their conflating three processes that Aristotle especially distinguishes: completion (A4-6)reduction(A7) and analysis (A45). Interpretive problems result from not sufficiently recognizing Aristotle?s remarkable degree of metalogical (...) sophistication to distinguish logical syntax from semantics and, thus, also from not grasping him to refine the deduction system of his underlying logic. While it is obvious that Aristotle most often uses ?sullogimos? to denote a valid argument of a certain kind, we show that at Prior Analytics A4-6, 7, 45 Aristotle specifically treats a sullogismos as an elemental argument pattern having only valid instances and that such a pattern then serves as a rule of deduction in his syllogistic logic. By extracting Aristotle?s understanding of three proof-theoretic processes, this paper provides new insight into what Aristotle thinks reasoning syllogistically is and, moreover, it resolves three problems in the most recent interpretation that takes a sullogismos to be a deduction. (shrink)

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)

Compare two conceptions of validity: under an example of a modal conception, an argument is valid just in case it is impossible for the premises to be true and the conclusion false; under an example of a topic-neutral conception, an argument is valid just in case there are no arguments of the same logical form with true premises and a false conclusion. This taxonomy of positions suggests a project in the philosophy of logic: the reductive analysis of the modal conception (...) of logical consequence to the topic-neutral conception. Such a project would dispel the alleged obscurity of the notion of necessity employed in the modal conception in favour of the clarity of an account of logical consequence given in terms of tractable notions of logical form, universal generalization and truth simpliciter. In a series of publications, John Etchemendy has characterized the model-theoretic definition of logical consequence as truth preservation in all models as intended to provide just such an analysis. In this paper, I will argue that Aristotle intends to provide an account of a modal conception of logical consequence in topic-neutral terms and so is engaged in a project comparable to the one described above. That Aristotle would be engaged in this sort of project is controversial. Under the standard reading of the Prior Analytics, Aristotle does not and cannot provide an account of logical consequence. Rather, he must take the validity of the first figure syllogisms (such as the syllogism known by its medieval mnemonic ‘Barbara’: A belongs to all B; B belongs to all C; so A belongs to all C) as obvious and not needing justification; he then establishes the validity of the other syllogisms by showing that they stand in a suitable relation to the first figure syllogisms. I will argue that Aristotle does attempt to provide an account of logical consequence—namely, by appeal to certain mereological theorems. For example, he defends the status of Barbara as a syllogism by appeal to the transitivity of mereological containment. There are, as I will discuss, reasons to doubt the success of this account. But the attempt is not implausible given certain theses Aristotle holds in semantics, mereology and the theory of relations. (shrink)

Much of the last fifty years of scholarship on Aristotle’s syllogistic suggests a conceptual framework under which the syllogistic is a logic, a system of inferential reasoning, only if it is not a theory or formal ontology, a system concerned with general features of the world. In this paper, I will argue that this a misleading interpretative framework. The syllogistic is something sui generis: by our lights, it is neither clearly a logic, nor clearly a theory, but rather exhibits certain (...) characteristic marks of logics and certain characteristic marks of theories. In what follows, I will present a debate between a theoretical and a logical interpretation of the syllogistic. The debate centers on the interpretation of syllogisms as either implications or inferences. But the significance of this question has been taken to concern the nature and subject-matter of the syllogistic, and how it ought to be represented by modern techniques. For one might think that, if syllogisms are implications, propositions with conditional form, then the syllogistic, in so far as it is a systematic taxonomy of syllogisms, is a theory or a body of knowledge concerned with general features of the world. Furthermore, if the syllogistic is a theory, then it ought to be represented by an axiomatic system, a system deriving propositional theorems from axioms. On the other hand, if syllogisms are inferences, then the syllogistic is a logic, a system of inferential reasoning. And furthermore, it ought to be represented as a natural deduction system, a system deriving valid arguments by means of intuitively valid inferences. I will argue that one can disentangle these questions—are syllogisms inferences or implications, is the syllogistic a logic or a theory, is the syllogistic a body of worldly knowledge or a system of inferential reasoning, and ought we to represent the syllogistic as a natural deduction system or an axiomatic system—and that we must if we are to have a historically accurate understanding of Aristotle. (shrink)

This paper is chiefly aimed at individuating some deep, but as yet almost unnoticed, similarities between Aristotle's syllogistic and the Stoic doctrine of conditionals, notably between Aristotle's metasyllogistic equimodality condition (as stated at APr. I 24, 41b27–31) and truth-conditions for third type (Chrysippean) conditionals (as they can be inferred from, say, S.E. P. II 111 and 189). In fact, as is shown in §1, Aristotle's condition amounts to introducing in his (propositional) metasyllogistic a non-truthfunctional implicational arrow '', the truth-conditions of (...) which turn out to be logically equivalent to truth-conditions of third type conditionals, according to which only the impossible (and not the possible) follows from the impossible. Moreover, Aristotle is given precisely this non-Scotian conditional logic in two so far overlooked passages of (Latin and Hebraic translations of) Themistius' Paraphrasis of De Caelo (CAG V 4, 71.8–13 and 47.8–10 Landauer). Some further consequences of Aristotle's equimodality condition on his logic, and notably on his syllogistic (no matter whether modal or not), are pointed out and discussed at length. A (possibly Chrysippean) extension of Aristotle's condition is also discussed, along with a full characterization of truth-conditions of fourth type conditionals. (shrink)

Some of the outstanding masters of Kurdish historical schools (Medresê) are usually and rightly seen as belonging to the Aristotelian tradition. In this introductory study I briefly present some manuscripts of Kurdish glosses on Aristotelian logical texts, and show that the Aristotelian logical tradition, as inherited from early Islamic philosophers, also formed an important strand in Kurdish schools. Kurdish students' peculiar approach to Aristotelian logic affected the way in which Categories, De Interpretatione and Isagoge were studied in Kurdish schools from (...) the fifteenth century onwards. (shrink)

Francesco Patrizi da Cherso's Discussiones peripateticae (1581) are one of the most comprehensive analyses of the whole of Aristotelian philosophy to be published before Werner Jaeger's Aristoteles. The main thrust of the argument in the Discussiones is that whatever Aristotle had said that was true was not new, and that whatever he had said that was new was not true. The article shows how Patrizi proves this with respect to the Organon, and deals with the implications for the history af (...) ancient philosophy in general implied by his stance. (shrink)

The paper investigates the propriety of applying the form versus matter distinction to arguments and to logic in general. Its main point is that many of the currently pervasive views on form and matter with respect to logic rest on several substantive and even contentious assumptions which are nevertheless uncritically accepted. Indeed, many of the issues raised by the application of this distinction to arguments seem to be related to a questionable combination of different presuppositions and expectations; this holds in (...) particular of the vexed issue of demarcating the class of logical constants. I begin with a characterization of currently widespread views on form and matter in logic, which I refer to as ‘logical hylomorphism as we know it’—LHAWKI, for short—and argue that the hylomorphism underlying LHAWKI is mereological. Next, I sketch an overview of the historical developments leading from Aristotelian, non-mereological metaphysical hylomorphism to mereological logical hylomorphism (LHAWKI). I conclude with a reassessment of the prospects for the combination of hylomorphism and logic, arguing in particular that LHAWKI is not the only and certainly not the most suitable version of logical hylomorphism. In particular, this implies that the project of demarcating the class of logical constants as a means to define the scope and nature of logic rests on highly problematic assumptions. (shrink)

v. 1. The Greek tradition -- v. 2. Greek texts and fragments of the Latin translation of "Alexander's" commentary -- v. 3. Appendices, Danish summary, indices.

In the late nineteenth century there were two very active lines of research in the field of formal logic. First, logicians (mostly in English-speaking countries) were engaged in formulating a generally traditional logic as an algebra, a part of mathematics; second, logicians (mostly on the continent) were busy building a non-traditional logic that could serve, not as a part of, but as the foundation of, mathematics. By the end of the First World War the former line had been pretty well (...) abandoned while the second continued to expand. However, that old abandoned line, stretching from Aristotle, through the Scholastics and then Leibniz to the nineteenth century algebraists, had not been completely forgotten. One of those logicians who has recently worked on the restoration (and, importantly, the extension) of that line is Fred Sommers. His Term Logic preserves a number of traditional insights (especially involving the theory of logical syntax), while also enjoying a power to account for formal inference at least comparable to that of the standard logic now in place. (shrink)

One way to determine the quality and pace of change in a science as it undergoes a major transition is to follow some feature of it which remains relatively stable throughout the process. Following the chosen item as it goes through reinterpretation permits conclusions to be drawn about the nature and scope of the broader change in question. In what follows, this device is applied to the change which took place in logic in the mid-nineteenth century. The feature chosen as (...) the focal point is the categorical syllogism. (shrink)

A connection between Aristotle's syllogistic and the calculus of relations is investigated. Aristotle's and Gergonne's syllogistics are considered as some algebraic structures. It is proved that Gergonne's syllogistic is isomorphic to closed elements algebra of a proper approximation relation algebra. This isomorphism permits to evaluate Gergonne's syllogisms and also Aristotle's syllogisms, laws of conversion and relations in the square of oppositions by means of regular computations with Boolean matrices.

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. Further volumes will follow, including Mediaeval and (...) Renaissance Logic and Logic: A History of its Central. In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality. Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredly international enterprise, whose research programmes reach to every corner of the learned world. Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value. Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas. (shrink)

In his Mathesis rationis Leibniz discounted out of hand four categorical propositions that would have considerably broadened the resultant syllogistic logic. He did this despite the facts both that he had devised a suitable manner for expressing the latent quantification over terms, and that he had reasoned adequately to determine which of the syllogisms in the resulting broadened logic were valid. Leibniz's reasons for discounting these non-standard propositions are shown to be inadequate, and the resultant syllogistic logic is outlined.

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

In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately (...) for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’. (shrink)

There have been and continue to be disagreements about how to consider the traditional square of opposition and the traditional inferences of obversion, conversion, contraposition and inversion from the perspective of contemporary quantificational logic. Philosophers have made many different attempts to save traditional inferences that are invalid when they involve empty classes. I survey some of these attempts and argue that the only satisfactory way of saving all the traditional inferences is to make the existential assumption that both the subject (...) and predicate classes and their complement classes are non-empty for all the propositions we admit. I briefly indicate the room for continued controversy over how properly to interpret Aristotle?s statements regarding these inferences, but find some plausibility in the views of Manley Thompson and A.N.Prior that Aristotle had in mind a particular arrangement of existential import unfamiliar to most contemporary logicians. (shrink)

Aristotle's syllogistic is extended to include denumerably many quantifiers such as more than 2/3 and exactly 2/3. Syntactic and semantic decision procedures determine the validity, or invalidity, of syllogisms with any finite number of premises. One of the syntactic procedures uses a natural deduction account of deducibility, which is sound and complete. The semantics for the system is non-classical since sentences may be assigned a value other than true or false. Results about symmetric systems are given. And reasons are given (...) for claiming that syllogistic validity is relevant validity. (shrink)

This paper introduces the reader to the medieval Hebrew tradition of logic by considering its treatment of Aristotelian syllogistic. Starting in the thirteenth century European Jews translated Arabic and Latin texts into Hebrew and produced commentaries and original compendia.Because they stood culturally and geographically at the cross-roads of two great traditions they were influenced by both.This is clearly seen in the development of syllogistic theory, where the Latin tradition ultimately replaces, though never entirely, its Arabic counterpart.Specific attention is devoted to (...) the debate about the so-called Galenian fourth figure.In medieval Hebrew logic one finds both defenders and detractors of the figure, the former appearing towards the beginning of the period in question.With the ascendancy of scholastic logic the fourth figure virtually disappears from Hebrew texts. (shrink)

John Corcoran?s natural deduction system for Aristotle?s syllogistic is reconsidered.Though Corcoran is no doubt right in interpreting Aristotle as viewing syllogisms as arguments and in rejecting Lukasiewicz?s treatment in terms of conditional sentences, it is argued that Corcoran is wrong in thinking that the only alternative is to construe Barbara and Celarent as deduction rules in a natural deduction system.An alternative is presented that is technically more elegant and equally compatible with the texts.The abstract role assigned by tradition and Lukasiewicz (...) to Barbara and Celarent is retained.The two ? perfect syllogisms? serve as ?basic elements? in the construction of an inductively defined set of valid syllogisms.The proposal departs from Lukasiewicz, and follows Corcoran, however, in construing the construction as one in natural deduction.The result is a sequent system with fewer rules and in which Barbara and Celarent serve as basic deductions.To compare the theory to Corcoran?s, his original is reformulated in current terms and generalized.It is shown to be equivalent to the proposed sequent system, and several variations are discussed.For all systems mentioned, a method of Henkin?style completeness proofs is given that is more direct and intuitive than Corcoran?s original. (shrink)

This paper examines the quest for the quantification of the predicate, as discussed by W.S. Jevons, and relates it to the discussion about universals and particulars between Plato and Aristotle. We conclude that the quest for the quantification of the predicate can only be achieved by stripping the syllogism from its metaphysical heritage.

This article investigates the prospect of giving de dicto- and de re-necessity a uniform treatment. The historical starting point is a puzzle raised by Aristotle's claim, advanced in one of the modal chapters of his Prior Analytics, that universally privative apodeictic premises simply convert. As regards the Prior and the Posterior Analytics, the data suggest a representation of propositions of the type in question by doubly modally qualified formulae of modal predicate logic that display a necessity operator in two distinct (...) positions. Can the N-operator occurring in these positions be given a unified semantical treatment (which would justify dispensing with a notational differentiation)? A positive answer, based on a suitably shaped truth condition for N-formulae, is given, and is supported in the final section with an alternative proof theoretically based conception of a property's essential belonging to an individual. (shrink)

In 1980 L. M. de Rijk edited some texts connected with medieval disputation ( Die mittelaterlichen Traktate De modo opponendi et respondendi ), towards which he showed a strikingly contemptuous attitude. The reason for his contempt was that the treatises did not fit the obligationes and sophismata tradition. In this article I focus on the original version, the Thesaurus Philosophorum , to highlight the distinction of this family of treatises with respect to the “modern“ tradition. First, I study the features (...) of the disputation that can be recognised through the collection of fallacious arguments contained in the Thesaurus . Second, I briefly examine the contents of the treatise and their arrangement, showing that they are closely related to the kind of disputation in question. I hope to support the idea that neither the technique of disputation nor the contents and their arrangement deserve a straightforward rejection. (shrink)

The aim of this article is to help to clarify the role which Aristotle gives to definition in his theory of demonstration. I shall begin by examining his handling of the relations between definition and demonstration in chapters 8-10 of the second book of the Posterior Analytics, in order to provide an outline for an interpretation of Aristotle's thought. Secondly, I shall examine chapter 10 in more detail, bringing out the contrast between the commentary by Averroes and that of Grosseteste. (...) I have chosen these two commentators because, both being generally magnificent interpreters of Aristotle, as far as the nature and types of definition are concerned their understanding of Aristotle is strikingly different. (shrink)

To meet a dilemma between the axiomatic theory of demonstrative science in Posterior analyticsand the non-aximatic practice of demonstrative science in the physical treatises, Jonathan Barnes has proposed that the theory of demonstration was not meant to guide scientific research but rather scientific pedagogy. The present paper argues that far from contributing directly to oral instruction, the axiomatic account of demonstrative science is a model for the written expression of science.The paper shows how this interpretation accords with related theories in (...) the Organon, including the theories of dialectic in Topicsand of deduction in Prior analytics. (shrink)

It is argued that, in the traditional subject-predicate sentence, two interpretations of the subject term coexist, one intensional and the other extensional, which explains the superficial difference between the traditional S-P relation and the predication of predicate logic. Data from psychological studies of syllogistic reasoning support the view that the contrast between predicate and argument is carried over to the traditional S-P sentence.

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

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

If two statements are contraries if and only if they cannot both be true, but can both be false, then some corresponding A and E categorical statements are not contraries, even on the presupposition that something exists which satisfies the subject term. For some such statements are necessarily true and thus cannot be false. There is a similar problem with subcontraries.

This paper discusses the fallacies of combination and division as they are presented by Aristotle in chapter 4 of his Sophistici Elenchi. Aristotle's examples are concise, their discussion is unclear, and it is difficult to distinguish the cases of combination from those of division. I analyse the Aristotelian examples and the interpretations offered so far. I show that these interpretations suffer from a major defect: they fail to identify a common characteristic whereby the Aristotelian examples can be classified as instances (...) of combination or division. In my reconstruction of the examples, I repair this deficiency: I give a single pattern of explanation for the fallacy of combination and another (similar) pattern for the fallacy of division. Thus, it is possible to free Aristotle from the following charges: (i) he did not clearly distinguish between combination and division, and (ii) he reduced combination and division to a single fallacy. My explanation of the fallacies uses the notion of scope of an expression: in modern terminology, the fallacy of combination can be described as ?fallacy of the wide scope?, the fallacy of division as ?fallacy of the narrow scope? (shrink)

The goal of this paper is an interpretation of Aristotle's modal syllogistics closely oriented on the text using the resources of modern modal predicate logic. Modern predicate logic was successfully able to interpret Aristotle's assertoric syllogistics uniformly , that is, with one formula for universal premises. A corresponding uniform interpretation of modal syllogistics by means of modal predicate logic is not possible. This thesis does not imply that a uniform view is abandoned. However, it replaces the simple unity of the (...) assertoric by the complex unity of the modal. The complexity results from the fact that though one formula for universal premises is used as the basis, it must be moderated if the text requires . Aristotle introduces his modal syllogistics by expanding his assertoric syllogistics with an axiom that links two apodictic premises to yield a single apodictic sentence . He thus defines a regular modern modal logic. By means of the regular modal logic that is thus defined, he is able to reduce the purely apodictic syllogistics to assertoric syllogistics. However, he goes beyond this simple structure when he looks at complicated inferences. In order to be able to link not only premises of the same modality, but also premises with different modalities, he introduces a second axiom, the T-axiom, which infers from necessity to reality or - equivalently - from reality to possibility. Together, the two axioms, the axiom of regularity and the T-axiom, define a regular T-logic. It plays an important role in modern logic. In order to be able to account for modal syllogistics adequately as a whole, another modern axiom is also required, the so-called B-axiom. It is very difficult to decide whether Aristotle had the B-axiom. The two last named axioms are sufficient to achieve the required contextual moderation of the basic formula for universal propositions. (shrink)

I consider the proper interpretation of the process of ecthesis which Aristotle uses several times in the Prior analytics for completing a syllogistic mood, i.e., showing how to produce a deduction of a conclusion of a certain form from premisses of certain forms. I consider two interpretations of the process which have been advocated by recent scholars and show that one seems better suited to most passages while the other best fits a single remaining passage. I also argue that ecthesis (...) for Aristotle really means ?setting out? the case to be proved using letters. Aristotle?s remarks about the use of letters in mathematical proofs suggest that he had some understanding of rules equivalent to universal generalization and existential instantiation; the ?proofs through ecthesis? are so called because they rest on the latter rule, with which use of letters is involved in a special way. (shrink)

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.

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)

In Prior Analytics A7 Aristotle points out that all valid syllogistic moods of the second and third figures as well as the two particular moods of the first figure can be reduced to the two universal first-figure moods Barbara and Celarent. As far as the third figure is concerned, it is argued that Aristotle does not want to say, as the transmitted text suggests, that only those two valid moods of this figure whose premisses are both universal statements are directly (...) reducible to Barbara and Celarent, but rather that it is those four valid moods of this figure whose respective minor premisses are universal statements of which this is true. It is shown that in order to carry this sense the transmitted text has to be corrected by inserting just one word, which seems to have dropped out. (shrink)

The paper elaborates two points: i) There is no principal opposition between predicate logic and adherence to subject-predicate form, ii) Aristotle's treatment of quantifiers fits well into a modern study of generalized quantifiers.

This paper begins by pointing out that the Aristotelian conception of continuity (synecheia) and the contemporary topological account share the same intuitive, proto-topological basis: the conception of a ?natural whole? or unity without joints or seams. An argument of Aristotle to the effect that what is continuous cannot be constituted of ?indivisibles? (e.g., points) is examined from a topological perspective. From that perspective, the argument fails because Aristotle does not recognize a collective as well as a distributive concept of a (...) multiplicity of points. It is the former concept that allows contemporary topology to identify some point sets with spatial regions (in the proto-topological sense of this term). This identification, in turn, allows contemporary topology to do what Aristotle was unwilling to do: to conceive the property of continuity, as well as the properties of having measure greater than zero and having n- dimension, as emergent properties. Thus, a point set can be continuous (connected) although none of its subsets of sufficiently smaller cardinality can be. Finally, the paper discusses the manner in which a topological principle, viz., the principle that none of the singletons of points of a continuum can be open sets of that continuum, captures certain aspects of the Aristotelian proto-topological conception of the relation between points and continua. E.g., for both Aristotle and contemporary topology, points in a continuum exist simple as limits of the remainder of the continuum: their singletons have empty ?interiors? and, hence, they are not ?chunks? (topologically, regular closed set) of the continuum. (shrink)

The incompleteness and artificiality of the ?traditional logic? of the textbooks is reflected in the way that syllogisms are commonly enumerated. The number said to be valid varies, but all the numbers given are of a kind that logicians should find irritating. Even the apparent harmony of what is almost invariably said to be the total number of syllogisms, 256, turns out to be illusory. In the following, it is shown that the concept of a distribution-value, which is related to (...) the traditional theory of distribution, and the familiar concept of quantity together suffice to produce a far better way of enumerating syllogisms and a more complete understanding of the systematic features of syllogistic logic. (shrink)

In an earlier article (s. J Gen Philos Sci 40:341–355, 2009), I have rejected an interpretation of Aristotle’s syllogistic which (since Patzig) is predominant in the literature on Aristotle, but wrong in my view. According to this interpretation, the distinguishing feature of perfect syllogisms is their being evident. Theodor Ebert has attempted to defend this interpretation by means of objections (s. J Gen Philos Sci 40:357–365, 2009) which I will try to refute in part [1] of the following article. I (...) want to show that (1) according to Aristotle’s Prior Analytics perfect and imperfect syllogisms do not differ by their being evident, but by the reason for their being evident, (2) Aristotle uses the same words to denote proofs of the validity of perfect and imperfect syllogisms („ apodeixis “, “ deiknusthai ” etc.), (3) accordingly, Aristotle defines perfect syllogisms not as being evident, but as “requiring nothing beyond the things taken in order to make the necessity evident“, i.e. as not “requiring one or more things that are necessary because of the terms assumed, but that have not been taken among the propositions” ( APr. I. 1), (4) the proofs by which the validity of perfect assertoric syllogisms can be shown according to APr. I. 4 are based on the Dictum de omni et nullo , (5) the fact that Aristotle describes these proofs only in rough outlines corresponds to the fact that his proofs of the validity of other fundamental rules are likewise produced in rough outlines, e.g . his proof of the validity of conversio simplex in APr. I. 2, which usually has been misunderstood (also by Ebert): (6) Aristotle does not prove the convertibility of E -sentences by presupposing the convertibility of I -sentences; only the reverse is true. (shrink)

We explore the technical details and historical evolution of Charles Peirce's articulation of a truth table in 1893, against the background of his investigation into the truth-functional analysis of propositions involving implication. In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on ?The Philosophy of Logical Atomism? truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig (...) Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883?1884 in connection with the composition of Peirce's ?On the Algebra of Logic: A Contribution to the Philosophy of Notation? that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional. (shrink)

In this paper I argue that a truth functional account of conditional statements ‘if A then B’ not only is inadequate, but that it eliminates the very conditionality expressed by ‘if’. Focusing only on the truth-values of the statements ‘A’ and ‘B’ and different combinations of these, one is bound to miss out on the conditional relation expressed between them. But this is not a flaw only of truth functionality and the material conditional. All approaches that try to treat conditionals (...) as mere functions of their antecedents and consequents will end up in some sort of logical atomism where causal matters simply are reduced to the joint occurrence of A and B. What we need is a non-extensional approach to conditionals that can account for hypotheticality, potentiality, and dependency, none of which can be understood by looking to the antecedent or consequent per se. (shrink)

In this paper I present some of Robert N. McLaughlin's critique of a truth functional approach to conditionals as it appears in his book On the Logic of Ordinary Conditionals. Based on his criticism I argue that the basic principles of logic together amount to epistemological and metaphysical implications that can only be accepted from a logical atomist perspective. Attempts to account for conditional relations within this philosophical framework will necessarily fail. I thus argue that it is not truth functionality (...) as such that is the problem, but the philosophical foundation of modern logic. (shrink)

INTRODUCTION The main purpose of this work is to provide an English translation of and commentary on a recently published Arabic text dealing with ...

Most instantiations of the inference ‘y; so if x, y’ seem intuitively odd, a phenomenon known as one of the paradoxes of the material conditional. A common explanation of the oddity, endorsed by Mental Model theory, is based on the intuition that the conclusion of the inference throws away semantic information. We build on this explanation to identify two joint conditions under which the inference becomes acceptable: (a) the truth of x has bearings on the relevance of asserting y; and (...) (b) the speaker can reasonably be expected not to be in a position to assume that x is false. We show that this dual pragmatic criterion makes accurate predictions, and contrast it with the criterion defined by the mental model theory of conditionals, which we show to be inadequate. (shrink)

We present a series of arguments for logical nativism, focusing mainly on the meaning of disjunction in human languages. We propose that all human languages are logical in the sense that the meaning of linguistic expressions corresponding to disjunction (e.g. English or , Chinese huozhe, Japanese ka ) conform to the meaning of the logical operator in classical logic, inclusive- or . It is highly implausible, we argue, that children acquire the (logical) meaning of disjunction by observing how adults use (...) disjunction. Findings from studies of child language acquisition and from cross-linguistic research invite the conclusion that children do not learn to be logical—it comes naturally to them. (shrink)

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary (...) set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas and , C(X{{a}}) iff P(, ) AC(X). [P(, ) denotes the set of formulas which result by the simultaneous substitution of for p and for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian. (shrink)