Corcoran, John. 1974. Aristotelian Syllogisms: Valid arguments or true generalized conditionals?, Mind 83, 278–81. MR0532928 (58 #27178) This tightly-written and self-contained four-page paper must be studied and not just skimmed. It meticulously analyses quotations from Aristotle and Lukasiewicz to establish that Aristotle was using indirect deductions—as required by the natural-deduction interpretation—and not indirect proofs—as required by the axiomatic interpretation. Lukasiewicz was explicit and clear about the subtle fact that Aristotle’s practice could not be construed as correctly performed indirect (...) proof. Lukasiewicz evidence is presented fully; it is irrefutable. But, instead of considering the possibility that Aristotle’s discourses were not intended to express indirect proofs of universalized conditions presupposing axiomatic premises, Lukasiewicz came to the amazing conclusion that Aristotle did not understand indirect proof. This paper builds on the admirable Lukasiewicz scholarship to establish a conclusion diametrically opposed to the one Lukasiewicz asserted. This paper points out that if Aristotle had not been establishing an underlying logic but he was instead axiomatizing a theory of terms, Aristotelians could not claim for Aristotle the title Founder of Logic. People who take Euclid, Peano, and Zermelo to have first axiomatized geometry, arithmetic, and set theory, respectively, do not think that this qualifies them for the titles Founder of Geometry, Founder of Arithmetic, and Founder of Set Theory, respectively. Such people do think that this would qualify the three for the titles Founder of Axiomatic Geometry, Founder of Axiomatic Arithmetic, and Founder of Axiomatic Set Theory, respectively: titles that carry no honors in logic. Being the Founder of Axiomatic Term Theory would likewise carry no honors in the field of logic. CORRECTION: The paper inadvertently implied that Euclid was the first to axiomatize geometry. In order to understand Aristotle’s Analytics as a treatise on demonstration it helps to realize that axiomatized geometry was studied in Plato’s Academy when Aristotle was a student. Euclid was the author of the only ancient axiomatized geometry now available. (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.

We present a coherence-based probability semantics for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions of the existential import which is required, for the validity of the syllogisms. Based on a generalization of de Finetti's fundamental theorem to conditional probability, we investigate the coherent probability (...) propagation rules of argument forms of the syllogistic Figures I, II, and III, respectively. These results allow to show, for all three Figures, that each traditionally valid syllogism is also valid in our coherence-based probability semantics. Moreover, we interpret the basic syllogistic sentence types by suitable defaults and negated defaults. Thereby, we build a knowledge bridge from our probability semantics of Aristotelian syllogisms to nonmonotonic reasoning. Finally, we show how the proposed semantics can be used to analyze syllogisms involving generalized quantifiers. (shrink)

This paper continues our work on a coherence-based probability semantics for Aristotelian syllogisms (Gilio, Pfeifer, and Sanfilippo, 2016; Pfeifer and Sanfilippo, 2018) by studying Figure III under coherence. We interpret the syllogistic sentence types by suitable conditional probability assessments. Since the probabilistic inference of P|S from the premise set {.

Aristotle characterizes a syllogism as “discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so.” This characterization of the syllogism does not require us to include as one of its constituent propositions the conclusion of a syllogism. When what are now called the premisses of a syllogism are stated, “something other than what is stated follows of necessity,” but what necessarily follows need not be a proposition in a syllogism on (...) a par with its premisses. The view that an Aristotelian syllogism contains only its premisses as its constituent propositions I call the premiss account of the Aristotelian syllogism. The more customary accounts of the Aristotelian syllogism are the inference account, according to which the syllogism is an inference whose premisses make necessary its conclusion, and the statement account, according to which it is a conditional statement whose conjunctive antecedent makes necessary its consequent. In this paper I shall not explain my reasons for adopting the premiss account and rejecting the inference and statement accounts of the Aristotelian syllogism, since these reasons are enumerated in another place. Instead, I shall discuss a complication which is involved in the premiss account but not involved in the inference and statement accounts. My explanation of this complication must await a more careful statement of the premiss account. (shrink)

In order to provide a consistent explanation for Aristotelian modal syllogistic, this paper reveals the reductions between the Aristotelian modal syllogism ◇I□A◇I-3 and the other valid modal syllogisms. Specifically, on the basis of formalizing Aristotelian modal syllogisms, this paper proves the validity of ◇I□A◇I-3 by means of the truth value definition of (modal) categorical propositions. Then in line with classical propositional logic and modal logic, generalized quantifier theory and set theory, this paper deduces the other (...) 47 valid Aristotelian modal syllogisms from the modal syllogism ◇I□A◇I-3. This study shows that the reasons why these syllogisms are reducible are: 1) any of Aristotelian quantifier can be defined by the other three Aristotelian quantifiers; 2) the Aristotelian quantifiers some and no have symmetry; 3) the possible modality ◇ and necessary modality □ can be mutually defined. This formal study of Aristotelian modal syllogistic not only conforms to the needs of formalization transformation of various information in the era of artificial intelligence, but also provides a unified mathematical research paradigm for other kinds of syllogistic. (shrink)

One semantic and two syntactic decision procedures are given for determining the validity of Aristotelian assertoric and apodeictic syllogisms. Results are obtained by using the Aristotelian deductions that necessarily have an even number of premises.

A coherence-based probability semantics for categorical syllogisms of Figure I, which have transitive structures, has been proposed recently (Gilio, Pfeifer, & Sanfilippo [15]). We extend this work by studying Figure II under coherence. Camestres is an example of a Figure II syllogism: from Every P is M and No S is M infer No S is P. We interpret these sentences by suitable conditional probability assessments. Since the probabilistic inference of ~????|???? from the premise set {????|????, ~????|????} is not (...) informative, we add ????(????|(????∨????))>0 as a probabilistic constraint (i.e., an “existential import assumption”) to obtain probabilistic informativeness. We show how to propagate the assigned (precise or interval-valued) probabilities to the sequence of conditional events (????|????,~????|????,????|(????∨????)) to the conclusion ~????|???? . Thereby, we give a probabilistic meaning to the other syllogisms of Figure II. Moreover, our semantics also allows for generalizing the traditional syllogisms to new ones involving generalized quantifiers (like Most S are P) and syllogisms in terms of defaults and negated defaults. (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)

Relational inferences are a well-known problem for Aristotelian logic. This book charts the development of thinking about this problem by logicians writing in Arabic from the ninth to the nineteenth century. It shows that that the development of Arabic logic did not - as is often supposed - come to an end in the fourteenth century.

Discussion of the Aristotelian syllogistic over the last sixty years has arguably centered on the question whether syllogisms are inferences or implications. But the significance of this debate at times has been taken to concern whether the syllogistic is a logic or a theory, and how it ought to be represented by modern systems. Largely missing from this discussion has been a study of the few passages in the Prior Analytics where Aristotle provides explicit guidance on how to (...) individuate syllogisms. I argue that syllogisms are individuated by unordered premise pairs and so may be fruitfully thought of as multi-succedent sequents. I conclude by drawing a few consequences for the representation of the syllogistic. (shrink)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows (...) from the premises: there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

In a brilliant article published in a past issue of the Transactions, Jorge A. Flórez examines Peirce’s theory of the origin of abduction in Aristotle. In the article Flórez makes two substantial points. In the first place, he argues that Peirce’s theory of the origin of abduction in the 25th chapter of the second book of the Prior Analytics is mistaken, because in that chapter Aristotle discusses first-figure syllogisms with a dialectic or contingent minor premise, and not, as Peirce (...) thought, second-figure syllogisms. The second substantial point that Flórez makes is that something having the form of a Peircean abduction is rather to be found in the 13th chapter of the first book of the Posterior Analytics... (shrink)

Drawing on the theory of ‘mental models’, I have previously shown that the valid syllogisms in the Aristotelian logical system, including all of its figures and moods, are very easy for the human mind. Indeed, they can even be used to predict inferences that people can make with quantified sentences. In this paper, I further argue that, if mental models theory is correct, then also the Aristotelian conversion rules are not hard for the human mind. My account (...) here again focuses on the distinction made by the mental models theory between canonical and noncanonical models. (shrink)

The main idea animating the present paper is that the general aim of debates, such as the one involving the notorious case of the Master Argument, is the ponderation of logical principles by confronting them with some set of assertions and other endorsed principles on the meaning explanation of connectives, quantifiers and modality. As suggested by Seel (2017), the point of the specific case of the MA is about examining Aristotle’s notion of possibility – as implemented by the Possibility Principle (...) (i.e., If it is necessary that if A then B, then B is possible if A is) and the Possibility Rule (i.e., If A is possible, and if B follows from the assumption of A, then B is is possible) in the context of a dialectical debate where it has been asserted that a given proposition is possible, even it is neither true now nor it will ever be, and where at the same time the irrevocability of the past has been endorsed. This suggests to study the MA debate in a setting based on Marion’s (2020) reconstruction of Aristotle’s Dialectical Bouts, adapted to propositions affected by modalities. The setting of Dialectical Bouts for the MA proposed includes•Propositions affected by Reichenbach’s (1948, § 5) three-folded temporal reference, namely: the point of speech S, the point of reference R and the point of the event E. This constitutes our take on Vuillemin’s (1977, 1979, 1984, 1996) suggestion of a “double indexation”.•The dialogical meaning explanation for modalities understood as involving hypotheticals. •The dialogical rules that settle the commitments undertaken when endorsing Aristotle’s Possibility Principle. In fact, in relation to the notion of possibility at work in Vuillemin’s (1996) reconstruction we explored different formulations of the Possibility Rule. Still, there remains a dearth of researches confronting the different formulations of the Possibility Rule with the texts studied by Crubellier (2010) and Rosen and Malink (2012). More generally and coming back to my remarks above, one of the most fruitful insights that can be gained from Vuillemin’s writings on the MA, is that such kind of debates provide an instance of a general pattern for reasoning that I shall call Dialectical Bouts of Material Admissibility (BMA). The rationale behind such kind of argumentation-pattern is to examine the incorporation of new (epistemological, ontological or logical) principles against the background of the commitments undertaken by endorsing a specific set of assertions and meaning explanations. They display the kind of abductive procedure stressed by Crubellier’s (2014) study of syllogism. Under this perspective Ebbinghaus’s (1964) casting of the inferential moves that reduce imperfect syllogisms to perfect ones as implementing an admissibility procedure, represents a logical generalization of the material procedures at stake in the Dialectical Bouts of Material Admissibility. Furthermore if, as stressed by Duthil-Novaes (2005) Medieval Obligations represent a development of Aristotelian Bouts aimed at the ludic exercise of consistency maintenance, DMA should be rather linked to the Islamic tradition of dialectical theory [jadal, munāẓara, ādāb al-baḥth], aimed at the epistemological task of what Young (2017) calls the forging of principles, including legal and deontic ones. In a more contemporary setting, DMA can be related to Brandom’s (2000, pp. 66-77) considerations on Non-Conservative Extensions, where the point of dialectical procedures is put under public Socratic scrutiny the discovery and elucidation of concepts. This general point on Dialectical Bouts of Material Admissibility can be seen as underlying Vuillemin’s project of linking the study of the MA with the classification of philosophical systems.The present paper does not contain a fully developed study on such Dialectical Bouts, but it is rather is an invitation to pick up the gauntlet that Jules Vuillemin threw into the ring. (shrink)

We give a simple definition of validity for syllogisms involving necessary and assertoric premises which validates all and only the Aristotelian apodeictic syllogisms.

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)

The rediscovery of Aristotle’s works on syllogisms in the Latin world, especially the Sophistici Elenchi and then the Prior Analytics, gave rise to sophisticated views on the nature of syllogistic form and syllogistic matter in the thirteenth century. It led to debates on the ontology of the syllogism as studied in the Prior Analytics, i.e. the syllogism made of letters and the four logical constants a/e/i/o, with deep consequences on the definition of logic as a universal method for all (...) sciences and as a science itself. (shrink)

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

Behaviorism is consistent with the assumptions of perfect competition, with the homo economicus model, and with a form of ethics that enshrines market-based notions of utility, justice, and rights and encourages rational maximizing. Economics and business courses foster this deficient form of ethics, assuming an overriding desire for money, which, according to MacIntyre and Aristotle, crowds out the associative virtues. These beliefs, often associated with Taylor and Friedman, lead to such practices as incentive compensation, which would be effective only if (...) employees were homines economici and thus rational in the sense of knowing how to satisfy whatever desires they have. Against Davidson’s view that psychology cannot be a natural science, Aristotle believes that there can be a teleological, if not exact, science of behavior. Aristotle believes that emotions and objectives may be rational or irrational. But rationality—phronesis, practical wisdom—is not a matter of applying algorithms. Doris questions whether character generates the first premises of an agent’s practical syllogisms, while Haidt has argued that we often act on emotions rather than on the basis of the reasons that we claim motivate us. Aristotle agrees that many people are not virtuous and that weakness of the will is a common occurrence, and he agrees that emotion drives actions. But if you are virtuous, your emotions and desires support your rationality. Aristotle gives us reason to reject the notion that people are homines economici and to treat them as rational deliberators, in part by creating an appropriate culture. The claim that it is in the nature of human beings to deliberate rationally is in part aspirational, but also in part scientific—in Aristotle’s modest sense. (shrink)

The rediscovery of Aristotle’s works on syllogisms in the Latin world, especially the Sophistici Elenchi and then the Prior Analytics, gave rise to sophisticated views on the nature of syllogistic form and syllogistic matter in the thirteenth century. It led to debates on the ontology of the syllogism as studied in the Prior Analytics, i.e. the syllogism made of letters and the four logical constants a/e/i/o, with deep consequences on the definition of logic as a universal method for all (...) sciences and as a science itself. (shrink)

n this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods of Barbara and Datisi. By adding first-degree propositional negation to this system, we prove that the square of opposition holds without using many of the other (...) rules of classical logic (including double negation elimination). We then show that the Propositional Substructural Logic SLe is the best logic to study Aristotelian Syllogisms. Also, based on the IFLe square of opposition, the rules of conversation and the rules of negation are completely proved in Muzaffar's logic. For this purpose, we used the monadic first-order logic with the same standard deductive apparatus of quantifiers in classical logic, plus the axioms of "some a is an a" and "some not-a is a not-a". Finally, to show that there is no existential commitment to general terms in categorical logic, the Strong Four-Valued Relevant-classical Logic KR4 was used. With the same existential interpretation of the quantifiers and the standard translation of the quarter quantified. (shrink)

Aristotle’s aitiai are middle terms in Aristotelian syllogisms. I argue that stating the aitia of a thing therefore amounts to re-describing this same thing in an alternative and illuminating way. This, in turn, means that a thing and its aitiai really are one and the same thing under different descriptions. The purpose of this paper is to show that this view is implied by Aristotle’s account of explanation, and that it makes more sense than one might expect.

The psychological study of reasoning with quantifiers has predominantly focused on inference patterns studied by Aristotle about two millennia ago. Modern logic has shown a wealth of inference patterns involving quantifiers that are far beyond the expressive power of Aristotelian syllogisms, and whose psychology should be explored. We bring to light a novel class of fallacious inference patterns, some of which are so attractive that they are tantamount to cognitive illusions. In tandem with recent insights from linguistics that (...) quantifiers like “some” are treated as wh-questions, these illusory inferences are predicted by the erotetic theory of reasoning, which postulates that a process akin to question asking and answering is behind human inference making. (shrink)

Philosophers since the ancient Greeks have investigated the nature of different kinds of inference. Although deductive inference in the form of Aristotelian syllogisms and Fregean formal logic has predominated, much attention has also been paid to induction, inference where the conclusion does not follow necessarily from the premises.

In the Organon Aristotle describes some deductive schemata in which inconsistencies do not entail the trivialization of the logical theory involved. This thesis is corroborated by three different theoretical topics by him discussed, which are presented in this paper. We analyse inference schema used by Aristotle in the Protrepticus and the method of indirect demonstration for categorical syllogisms. Both methods exemplify as Aristotle employs classical reductio ad absurdum strategies. Following, we discuss valid syllogisms from opposite premises (contrary and (...) contradictory) studied by the Stagerian in the Analytica Priora (B15). According to him, the following syllogisms are valid from opposite premises, in which small Latin letters stand for terms such as subject and predicate, and capital Latin letters stand for the categorical propositions such as in the traditional notation: (i) in the second figure, Eba,Aba ` Eaa (Cesare), Aba, Eba ` Eaa (Camestres), Eba, I ba ` Oaa (Festino), and Aba,Oba ` Oaa (Baroco); (ii) in the third one, Eab,Aab ` Oaa (Felapton), Oab,Aab ` Oaa (Bocardo) and Eab, Iab ` Oaa (Ferison). Finally, we discuss the passage from the Analytica Posteriora (A11) in which Aristotle states that the Principle of Non-Contradiction is not generally presupposed in all demonstrations (scientific syllogisms), but only in those in which the conclusion must be proved from the Principle; the Stagerian states that if a syllogism of the first figure has the major term consistent, the other terms of the demonstration can be each one separately inconsistent. These results allow us to propose an interpretation of his deductive theory as a broad sense paraconsistent theory. Firstly, we proceed to a hermeneutical analysis, evaluating its logical significance and the interplay of the results with some other points of Aristotle’s philosophy. Secondly, we point to a logical interpretation of the Aristotelian syllogisms from opposite premises in the antilogisms method proposed by Christine Ladd-Franklin in 1883, and we also present a logical treatment of the Aristotelian demonstration with inconsistent material in the paraconsistent logics Cn, 1 _ n _!, introduced by da Costa in 1963. These two issues seem having not yet been analysed in detail in the literature. (shrink)

There has long been a history of studies investigating how people (“ordinary people”) perform on tasks that involve deductive reasoning. The upshot of these studies is that people characteristically perform some deductive tasks well but others badly. For instance, studies show that people will typically perform MP (“modus ponens”: from ‘If A then B’ and ‘A’, infer ‘B’) and bi-conditional MP (from: ‘A if and only if B’ and ‘A’, infer ‘B’) correctly when invited to make the inference and additionally (...) can discover of their own accord when such inferences are appropriate. On the other hand, the same studies show that people typically perform MT (“modus tollens”: from ‘If A then B’ and ‘not-B’, infer ‘not-A’) and biconditional MT badly. They not only do not recognize when it is appropriate to draw such inferences, but also they will balk at doing them even when they are told that they can make it. Related to these shortcomings seems to be the inability of people to understand that contrapositives are equivalent (that ‘If A then B’ is equivalent to ‘If not-B then not-A’). [Studies of people’s deductive inference-drawing abilities have a long history, involving many studies in the early 20th century concerning Aristotelian syllogisms. But the current spate of studies draws much of its impetus from Wason (Wason, 1968; see also Wason & Johnson-Laird, 1972). ] The general conclusion seems to be that there are specific areas where “ordinary people” do not perform very logically. This conclusion will not come as a surprise to teachers of elementary logic, who have long thought that the majority of “ordinary people” are inherently illogical and need deep and forceful schooling in order to overcome this flaw. (shrink)

In this paper I will attempt a unified analysis of the various examples of the fallacy of accident given by Aristotle in the Sophistical Refutations. In many cases the examples underdetermine the fallacy and it is not trivial to identify the fallacy committed. To make this identification we have to find some error common to all the examples and to show that this error would still be committed even if those other fallacies that the examples exemplify were not. Aristotle says (...) that there is only one solution “against the argument” as opposed to “against the man”, and it is this solution the paper attempts to find. It is a characteristic mark of my analysis that some arguments that we might normally be inclined to say are fallacious turn out to be valid and that some arguments that we would normally be inclined to say are valid turn out to be fallacious. This is (in part) because what we call validity in modern logic is not the same as the apodicticity that Aristotelian syllogisms require in order to be used in science. The fallacies of accident, uniquely among the fallacies, are failures of apodicticity rather than failures of, in particular, semantic entailment. This makes sense in a tensed and token-based logic such as Aristotle’s. I conclude that the closest analogue to the fallacy of accident that we can point to is a fallacy in modal logic, viz., the fallacy of necessity. (shrink)

This article investigates the nature of Aristotelian syllogistics and shows that the categorical syllogism is fundamentally about showing the connection, in the premises of the syllogism, between the major and minor terms as stated in the conclusion. It discusses how this is important for the use of the syllogism in scientific demonstration. The article then examines modern deductive logic with an eye to they way in which it contrasts with Aristotelian syllogistics. It shows howmodern logic is about making (...) necessary connections between each proposition by means of external or second order rules. In the syllogism, on the other hand, the necessity between the premises as a whole unit and the conclusion is based on the internal middle term. The article concludes with a discussion of Günther Patzig’s claim that Aristotelian syllogisms are best thought of as tautological propositions. If this were the case, then the differences asserted to exist between syllogistic and modern logic would not hold. However, it is shown that Patzig’s assimilation of syllogistics to modern deductive logic is illegitimate. (shrink)

This article investigates the nature of Aristotelian syllogistics and shows that the categorical syllogism is fundamentally about showing the connection, in the premises of the syllogism, between the major and minor terms as stated in the conclusion. It discusses how this is important for the use of the syllogism in scientific demonstration. The article then examines modern deductive logic with an eye to they way in which it contrasts with Aristotelian syllogistics. It shows howmodern logic is about making (...) necessary connections between each proposition by means of external or second order rules. In the syllogism, on the other hand, the necessity between the premises as a whole unit and the conclusion is based on the internal middle term. The article concludes with a discussion of Günther Patzig’s claim that Aristotelian syllogisms are best thought of as tautological propositions. If this were the case, then the differences asserted to exist between syllogistic and modern logic would not hold. However, it is shown that Patzig’s assimilation of syllogistics to modern deductive logic is illegitimate. (shrink)

The categorical syllogism is the foundation of Aristotelian logic, and Aristotle's logic is the foundation of modern logic. Proper fallacies were rather errors in reasoning based on ambiguities, such as those Aristotle speaks of in Sophistical Refutations and on which Galen comments in De Captionibus (On Fallacies). Each categorical fallacy is a violation of a rule for the formation of valid syllogisms. Formal fallacies, according to Aldrich, include any reasoning that violates the law of identity, law of non‐contradiction, (...) or law of excluded middle. This chapter deals with one of the common fallacies in Western philosophy, the exclusive premises (EP). EP occurs when a syllogism has two negative premises, which are propositions that deny a connection between the subject and predicate. The quality or quantity of the premises of a syllogism can make them exclusive to one another. (shrink)

Jan Łukasiewicz’s analysis of Aristotle’s syllogism drew attention to the nature of syllogisms as conditionals rather than premise-conclusion arguments. His further idea that syllogisms should be understood as theorems of an axiom system seems a step too far for many logicians. But there is evidence to suggest that Aristotle’s syllogism was to regularise some of the steps made in ‘dialogue games.’ This way of seeing the syllogism is explored in the framework of modern formal dialogue systems. A modern (...) formal syllogistic game, DLSyll, is set out and analysed in use. (shrink)

ABSTRACT: This paper traces the earliest development of the most basic principle of deduction, i.e. modus ponens (or Law of Detachment). ‘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: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with an emphasis on topics which may be of interest to contemporary logicians. Content: 1. Pre-Aristotelian Logic 1.1 Syntax and Semantics 1.2 Argument Patterns and Valid Inference 2. Aristotle 2.1 Dialectics 2.2 Sub-sentential Classifications 2.3 Syntax and Semantics of Sentences 2.4 Non-modal Syllogistic 2.5 Modal Logic 3. The early Peripatetics: Theophrastus and Eudemus 3.1 Improvements and Modifications of Aristotle's Logic 3.2 Prosleptic Syllogisms (...) 3.3 Forerunners of Modus Ponens and Modus Tollens 3.4 Wholly Hypothetical Syllogisms 4. Diodorus Cronus and Philo the Logician 5. The Stoics 5.1 Logical Achievements Besides Propositional Logic 5.2 Syntax and Semantics of Complex Propositions 5.3 Arguments 5.4 Stoic Syllogistic 5.5 Logical Paradoxes 6. Epicurus and the Epicureans 7. Later Antiquity. (shrink)

The Aristotelian origins of formal systems are outlined, together with Aristotle's use of causal terms in describing syllogisms. The precision and exactness of a formalism, based on the projection of logical forms into perceptive signs, is contrasted with foundational, abstract concepts, independent of any formalism, which are presupposed for the understanding of a formal language. The definition of a formal system by means of a Turing machine is put in the context of Wittgenstein's general considerations of a machine (...) understood as a sign. A modification of Łukasiewicz's logic Ł3 with the inclusion of justification terms is proposed in order to formally analyse some features of formalistic reasoning as a mechanical, causal affair within a wider context of "indeterminacy". (shrink)

In this paper, we discuss the syllogisms from both fronts : Aristotle and Boethius. We mainly focus on the differences with respect to categorical and hypothetical syllogisms in Aristotle and Boethius. Regarding Aristotle’s works on logic, it is not unusual to claim that Aristotle extensively worked on categorical syllogisms. In Prior Analytics, Aristotle gave proofs for many valid moods. However we cannot find a similar treatment for hypothetical syllogism in his works. Thus, it might be a reason (...) for his students to improve Aristotle’s theory of syllogisms by filling the gaps in hypothetical syllogisms. As Boethius already emphasized in the Book I of De Hypotheticis Syllogismis, the Peripatetic School only initiated the start. The remaining and detailed treatise was due to Boethius himself. The impression we got from Boethius’s and Aristotle’s works on syllogisms is that the categorical syllogism is almost entirely developed by Aristotle himself, whereas the extensive and the detailed study of the hypothetical syllogism was carried out by Boethius, although he was not the first figure in the history who gave concise expositions on hypothetical syllogism. Boethius’s translations of syllogisms in medieval era were the primer resource and textbook for Aristotelian logic during the early Medieval ages. Therefore, the scholars of those times had no other chance but to work from Boethius’s commentaries. So, the evident properties of Aristotelian figures were not then visible. Boethius wrote several textbooks in logic. A detailed investigation might yield significant information on Boethius’s logic compared with the influence of Aristotle on Boethius. (shrink)

Aristotle's modal syllogistic has been an object of study ever since the time of Theophrastus; but these studies have been somewhat desultory. Remarkably, in the 1990s several new lines of research have appeared, with series of original publications by Fred Johnson, Richard Patterson and Ulrich Nortmann. Johnson presented for the first time a formal semantics adequate to a de re reading of the apodeictic syllogistic; this was based on a simple intuition linking the modal syllogistic to Aristotelian metaphysics. Nortmann (...) developed an ingenious de dicto analysis. Patterson articulated the links between the modal syllogistic and the metaphysics, using an analysis which strictly speaking is neither de re nor de dicto. My own studies in this field date from 1976, when my colleague Peter Roeper and I jointly wrote a paper "Aristotle's apodeictic syllogisms" for the XXIInd History of Logic Conference in Krakow. This paper contained the disjunctive reading of particular affirmative apodeictic propositions, which I still favour. Nonetheless, I did not consider that paper's results decisive or comprehensive enough to publish, and my 1981 book The Syllogism contained no treatment of the modal syllogism. The paper's ideas lay dormant till 1989, when I read Johnson's and Patterson's initial articles. I began publishing on the topic in 1991. Gradually my thoughts acquired a certain comprehensiveness and systematicity, till in 1993 I was able to take a semester's sabbatical to write up a draft of this book. (shrink)