A rejection system, also referred to as a complementary calculus, is a proof system axiomatising the invalid formulas of a logic, in contrast to traditional calculi which axiomatise the valid ones. Rejection systems therefore introduce a purely syntactic way of determining non-validity without having to consider countermodels, which can be useful in procedures for automated deduction and proof search. Rejection calculi have first been formally introduced by Łukasiewicz in the context of Aristotelian syllogistic and subsequently rejection systems for many well-known (...) logics have been proposed. In this paper, we deal with rejection systems for so-called non-deterministic finite-valued logics, a special case of non-deterministic many-valued logics which were introduced by Avron and Lev as a generalisation of traditional many-valued logics. More specifically, we introduce a systematic method for constructing sequent-style rejection systems for any given non-deterministic finite-valued logic. Furthermore, as special instances of our method, we provide concrete calculi for specific paraconsistent logics which can be characterised in terms of non-deterministic two- and three-valued semantics, respectively. (shrink)

Aristotle's explanation of what is said ‘of every’ and ‘of none’ has been interpreted either as involving individuals, or as regarding exclusively universal terms. I claim that Alexander of Aphrodisias endorsed this latter interpretation of the dictum de omni et de nullo. This interpretation affects our understanding of Alexander's syllogistic: as a matter of fact, Alexander maintained that the dictum de omni et de nullo is one of the core principles of syllogistic.

According to Philoponus, the activity of drawing syllogisms is a dynamic operation. Following the classical idea that actions are specified by their objects and habitual powers by their actions, Philoponus concludes that only a dynamic power can elicit the act of syllogizing. This power is identified with discursive reasoning (dianoia). Imagination, on the contrary, is a static power, that cannot elicit that particular motion of drawing a syllogistic inference. The issue, however, is not entirely uncontroversial, because Ammonius maintains that sophistical (...) syllogisms can only be formed by imagination, since they involve “empty concepts” as terms and only imagination can form such concepts. In this paper I will reconstruct Philoponus’ and Ammonius’ theories about the “activity” of syllogizing, and I shall explain how Philoponus can deal with sophistical syllogisms in a consistent way. (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)

Fred Sommers passed away in October of 2014 in his 92nd year. Having begun his teaching at Columbia University, he eventually became the Harry A. Wolfson Chair in Philosophy at Brandeis University, where he taught from 1963 to 1993. During his long and productive career, Sommers authored or co-authored over 50 books, articles, reviews, etc., presenting his ideas on numerous occasions throughout North America and Europe. His work was characterized by a commitment to the preservation and application of historical insights (...) and to the value of a well-articulated, coherent logical system. He was recognized for his independence and refusal to accept any view on the basis of authority alone. This made him a formidable critic but accounted in part for his many innovative and original ideas. In spite of his general contrariness in logic, Sommers earned the respect of the majority of his contemporaries, including Russell, Quine, van Benthem, Hacking, Suppes, and Strawson. In 2005, he was the subjec... (shrink)

This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the same properties (...) of the natural deduction counterpart ? that is, Corcoran's system. Moreover, the first-order logic framework that we work with allows us to understand how complicated the semantics of the syllogistic is in providing us with examples of bizarre, unexpected interpretations of the syllogistic rules. Finally, we provide a first attempt at finding the structure of that semantics, reducing the search to the characterization of the class of models of T RD. (shrink)

Some strategies to turn any logic into a paraconsistent system are examined. In the environment of universal logic, we show how to paraconsistentize logics at the abstract level using a transformation in the class of all abstract logics called paraconsistentization by consistent sets. Moreover, by means of the notions of paradeduction and paraconsequence we go on applying the process of changing a logic converting it into a paraconsistent system. We also examine how this transformation can be performed using multideductive abstract (...) logics. To conclude, the conceptual notion paraconsistent orbit of a logic is proposed. (shrink)

This article is intended to examine the structure and scope of the argumentation drawn in Metaphysics Γ 4, 1006a18-b34. As we shall see, though this passage does not bring a complete proof of the Principle of Non-Contradiction, it corresponds to its first step, which consists in determining the conditions of meaning necessary for discourse. That passage encloses in nuce the reasons which underlie Aristotelian conviction con- cerning the conventional nature of names and also brings to light the way this conven- (...) tional nature of names must be understood according to Aristotle. (shrink)

"However that may be, Aristotelian syllogistic concerned itself exclusively with monadic predicates. Hence it could not begin to investigate multiple quantification. And that is why it never got very far. None the less, the underlying grammar of Aristotle's logic did not in itself..

The dissertation is an investigation into the structure of Aristotle's modal propositions through careful attention to the text of the Prior Analytics. I take account not only of recent attempts to formalize Aristotle's modal syllogistic but also of the discussion that Aristotle himself provides about modal statements. I provide evidence that his modal propositions are to be construed in a de re manner and then go on to investigate the problems raised by a de re analysis, particularly those problems concerned (...) with Aristotle's modal conversion principles. A large part of my project is to show that these can be given a valid de re analysis that sits well with the results Aristotle sketches. Terms in modal syllogistic premises are shown to be restricted in ways that reflect Aristotle's early metaphysics and notions of predication. (shrink)

Natural syllogisms are expressed in terms of classes and properties of the real world. They exploit a categorisation present in semantic memory that provides a class inclusion structure. they are enthymematic and typically occur within a dialogue. Their form is identical to a formal syllogism once the minor premise is made explicit. It is claimed that reasoners routinely execute natural_syllogisms in an effortless manner based on ecthesis, which is primed by the class inclusion structure kept in long term memory.

Aristotle's General Definition of the Syllogism may be taken as consisting of two parts: the Inferential Conditions and the Final Clause. Although this distinction is well known, traditional interpretations neglect the Final Clause and its influence on syllogistic. Instead, the aforementioned tradition focuses on the Inferential Conditions only. We intend to show that this neglect has severe consequences not just on syllogistic but on the whole exegesis of Aristotle's Prior Analytics I. Due to these consequences, our objective is to analyse (...) the General Definition's Final Clause and its consequences on syllogistic. We propose a reading of the Final Clause as an additional criterion for distinguishing some arguments as properly syllogistic ones and as a main theme which connects all parts of the Prior Analytics I into one coherent piece of work. (shrink)

Aristotle's theory of demonstration, developed in the Posterior Analytics, is not restricted to determining the formal requirements for formulating probative arguments that establish properly the results of scientific investigation. To the probative aspect of demonstration it shall be added its primarily explanatory character, orientated by theses of strong ontological and metaphysical content and involving notions like substance, essence and causation. We shall analyze the relation between those two ranges of Aristotle's philosophy of science and investigate how the formal features of (...) demonstration maintain an affinity with the metaphysical background to which scientific activity is conditioned. (shrink)

We can now recognize Aristotle's many accomplishments in logical theory, not the least of which is treating the deduction process itself as a subject matter and thus establishing the science of logic. Aristotle took logic to be that part of epistemolo gy used to establish knowledge of logical consequence. Prior Analytics is a metalogical treatise on his syllogistic system in which Aristotle modelled his deduction system to demonstrate certain logical relationships among its rules. Aristotle's n otion of substitution distinguishes logical (...) syntax from semantics and enabled him to distinguish validity from deducibility sufficiently to note the completeness of his logic. (shrink)

Aristotle studies syllogistic argumentation in Sophistical Refutations and Prior Analytics. In the latter he focuses on the formal and syntactic character of arguments and treats the sullogismoi and non-sullogismoi as argument patterns with valid or invalid instances. In the former Aristotle focuses on semantics and rhetoric to study apparent sullogismoi as object language arguments. Interpreters usually take Sophistical Refutations as considerably less mature than Prior Analytics. Our interpretation holds that the two works are more of a piece than previously believed (...) and, indeed, that Aristotle's treatment of fallacious reasoning presupposes the results of the formal theory. (shrink)

While the nature of warrants is unclear in both Toulmin’s Uses of Argument and in textbook pedagogy based on it, the theory of non-monotonic reasoning could clarify and enhance our understanding of warrants.

This presentation traces an historical root of the reductio ad absurdum mode of argumentation in Greek philosophy. I propose a new understanding of the liar paradox as an instance of this mode of argumentation. I show that the paradox was crea ted as part of a refutational argument in the controversy over the justification of realism and the realists concepts of truth and certainty. The paradox was part of the dialectical style of Greek scepticism, which was characterized, inter alia, by (...) the u se of the reductio ad absurdum. The paradox turns out to be a metaphysical and epistemological argument. (shrink)