Fred's semantics for McCall's syntactic presentation of Aristotle's assertoric and apodeictic syllogistic is altered to free it from Thom's objections that it is unAristotelian. The altered semantics rejects Baroco-XLL and Bocardo-LXL, which Thom says Aristotle should have accepted. Aristotle's proofs that use ecthesis are formalized by using singular sentences. With one exception the (acceptance) axioms for McCall's system L-X-M are derivable. Formal proofs are shown to be sound.

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)

In this paper, we provide a detailed critical review of current approaches to ecthesis in Aristotle’s Prior Analytics, with a view to motivate a new approach, which builds upon previous work by Marion & Rückert (2016) on the dictum de omni. This approach sets Aristotle’s work within the context of dialectic and uses Lorenzen’s dialogical logic, hereby reframed with use of Martin-Löf's constructive type theory as ‘immanent reasoning’. We then provide rules of syllogistic for the latter, and provide proofs (...) of e-conversion, Darapti and Bocardo and e-subalternation, while showing how close to Aristotle’s text these proofs remain. (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)

In this paper we first show that Robin Smith’s ecthetic system SE for Aristotle’s assertoric syllogistic is not complete, despite what is claimed by Smith. SE is then not adequate to establish that ecthesis allows one to dispense with indirect or per impossibile deductions in Aristotle’s assertoric logic. As an alternative to SE, we then present a stronger system EC which is adequate for this purpose. EC is a nonexplosive ecthetic system which is shown to be sound and complete (...) with respect to all valid syllogistic arguments with a consistent set of premises. (shrink)

There is very little information about the proving by Aristotle’s ecthesis method both in Aristotle’s and his commentators’ articles. Researches on ecthesis which were made by recent commentators are only on expository term. In our study, comments have been evaluated, points that are subject to contradiction have been determined, and opinions about ecthesis have been cited by giving proofs obtained by the ecthesis method.

Abstract“Natural syllogisms” are arguments formally identifiable with categorical syllogisms that have an implicit universal affirmative premise retrieved from semantic memory rather than explicitly stated. Previous studies with adult participants (Politzer, 2011) have shown that the rate of success is remarkably high. Because their resolution requires only the use of a simple strategy (known as ecthesis in classic logic) and an operational use of the concept of inclusion (the recognition that an element that belongs to a subset must belong to (...) the set but not vice versa), it was hypothesized that these syllogisms would be within the grasp of non‐adult participants, provided they have acquired the notion of deductive validity. Here, 11‐year‐old children were presented with natural syllogisms embedded in short dialogs. The first experiment showed that their performance was equivalent to adults' highest level of performance in standard experiments on syllogisms. The second experiment, while confirming children's proficiency in solving natural syllogisms, showed that they outperformed children who solved non‐natural matched syllogisms in the same experimental setting. The results are also in agreement with the argumentation theory of reasoning. (shrink)

We elaborate on the approach to syllogistic reasoning based on “case identification” (Stenning & Oberlander, 1995; Stenning & Yule, 1997). It is shown that this can be viewed as the formalisation of a method of proof that dates back to Aristotle, namely proof by exposition ( ecthesis ), and that there are traces of this method in the strategies described by a number of psychologists, from St rring (1908) to the present day. We hypothesised that by rendering individual cases (...) explicit in the premises, the chance that reasoners would engage in a proof by exposition would be enhanced, and thus performance improved. To do so, we used syllogisms with singular premises (e.g., this X is Y ). This resulted in a uniform increase in performance as compared to performance on the associated standard syllogisms. These results cannot be explained by the main theories of syllogistic reasoning in their current state. (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)

This book explores noteworthy approaches to modal syllogistic adopted by medieval logicians including Abélard, Albert the Great, Avicenna, Averröes, Jean Buridan, Richard Campsall, Robert Kilwardby, and William of Ockham. The book situates these approaches in relation to Aristotle's discussion in the Prior and Posterior Analytics, and other parts of the Organon, but also in relation to the thought of Alexander of Aphrodisias and Boethius on the one hand, and to modern interpretations of the modal syllogistic on the other. Problems explored (...) include: Aristotle's doctrine of modal conversion, the pure and mixed necessity-moods, modal ecthesis, the pure and mixed contingency-moods, and Aristotle's use of counter-examples. Medieval logicians brought various concepts to bear on these problems, including the distinction between per se and per accidens terms, the notion of essential predication, the distinction between ut nunc and simpliciter propositions, the distinction between de dicto and de re modals, and the notion of ampliation. All these are examined in this book. (shrink)

The paper examines J. Hintikka’s thesis that Euclid’s procedure of geometrical proof had been «paradigm» or «model» for Kant’s notion of the mathematical method. The detailed re-construction of the researcher’s arguments allows to reveal Hintikka’s main thesis, namely, that écthesis as a structural element of Euclidean proposition allows explanation of Kant’s notion of construction. However, in-depth analysis of Euclidean proof structure, compared also with Proclus’ and Th. Heath’s comments, shows that terminologically and functionally this element does not perform its supposed (...) role of construction and is not fundamental to the struc-ture of proposition. Also Hintikka’s thesis that intuition is just a representation of the individu-al has weak explanatory potential because Kant’s explication of construction is always correla-tive with intuition a priori, that is, with imagination of a geometric concept in space as a form of pure intuition. In conclusion, the author mentions the three sources of Hintikka’s interpreta-tion that could be links between Kant and ancient origins of the mathematical method: notion of construction by Ch. A. Hausen, translation of the «Elements» by J. F. Lorenz and J. Ch. Schwab’s considerations concerning the structure of geometric proof. (shrink)

SYLLOGISTIC WITH NEGATIVE TERMS IN THE SEMANTICALLY TRANSPARENT FORMULATION S u m m a r y The paper refers to the classic works of Polish authors (Kukasiewicz, SleszyNski, Zawirski) and comprises the idea of proof by ecthesis (originally formulated by Borkowski) with the use of list notation to show that the five diagrams by Euler, which provide a starting point for SleszyNski’s analysis, adequately formulate the system of syllogistic with negative terms (IwanuX and Wedberg’s axiomatization).

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. (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.

This thesis is a critique of Jaakko Hintikka’s reconstruction of Kantian intuition in logical and mathematical reasoning. I argue that Hintikka’s reconstruction of Kantian intuition in particular and his reconstruction of Kant's philosophy of mathematics in general fails to be successful in two ways: First, the logical formula which contains an instantiated term (henceforth, instantial term) that is introduced by the rule of existential instantiation in the ecthesis part of a proof of an argument is not even a proper (...) singular proposition whose relation to its object is supposed to be immediate. It is not a proper singular proposition because its truth conditions are general, i.e., it makes a general statement about a class of individuals of the sort instantiated- a statement whose analysis is based on quantifiers. Second, I show that certain proofs in mathematics- those in the form of reductio ad absurdum- are not captured by Hintikka’s reconstruction of Kant’s philosophy of mathematics either. (shrink)

Aristotle assesses as valid three first figure syllogisms, each of which contains at least one premiss expressing a de re contingency. In fact, all three of these moods (namely, Barbara-QQQ, Barbara-XQM, and Barbara-LQM) are invalid. Utilizing the concept of ampliation, this paper shows how the mood Barbara-QQQ must be refined if it is to be deemed valid. It can then become clear as to how Barbara-XQM and Barbara-LQM can be disambiguated and ultimately validated. In treating all three moods, some theses (...) from S4 will be exploited in the context of distinguishing de dicto and de re modes of attributing possibility and necessity. Various Aristotelian propositional forms and rules of inference, including argumentation by ecthesis, will shape the presentation. The viability of Aristotle’s views on the convertibility of universal negative apodeictic propositions will emerge as decisive in evaluating the success of his modal syllogistic. (shrink)

Aristotle assesses as valid three first figure syllogisms, each of which contains at least one premiss expressing a de re contingency. In fact, all three of these moods (namely, Barbara-QQQ, Barbara-XQM, and Barbara-LQM) are invalid. Utilizing the concept of ampliation, this paper shows how the mood Barbara-QQQ must be refined if it is to be deemed valid. It can then become clear as to how Barbara-XQM and Barbara-LQM can be disambiguated and ultimately validated. In treating all three moods, some theses (...) from S4 will be exploited in the context of distinguishing de dicto and de re modes of attributing possibility and necessity. Various Aristotelian propositional forms and rules of inference, including argumentation by ecthesis, will shape the presentation. The viability of Aristotle’s views on the convertibility of universal negative apodeictic propositions will emerge as decisive in evaluating the success of his modal syllogistic. (shrink)