What good is logic? -- Seventeen ways this book is different -- The two logics -- All of logic in two pages : an overview -- The three acts of the mind -- I. The first act of the mind : understanding -- Understanding : the thing that distinguishes man from both beast and computer -- Concepts, terms and words -- The problem of universals -- The comprehension and extension of terms -- II. Terms -- Classifying terms -- Categories -- (...) Predicables -- Division -- III. Material fallacies -- Fallacies of language -- Fallacies of diversion -- Fallacies of oversimplification -- Fallacies of argumentation -- Inductive fallacies -- Procedural fallacies -- Metaphysical fallacies -- Short story : love is a fallacy -- IV. Definition -- The nature of definition -- The rules of definition -- The kinds of definition -- The limits of definition -- V. Second act of the mind : judgment -- Judgments, propositions, and sentences -- What is truth? -- The four kinds of categorical propositions -- Logical form -- Euler's circles -- Tricky propositions -- The distribution of terms -- VI. Changing propositions -- Immediate inference -- Conversion -- Obversion -- Contraposition -- VII. Contradiction -- What is contradiction? -- The square of opposition -- Existential import -- Tricky propositions on the square -- Some practical uses of the square of opposition -- VIII. The third act of the mind : reasoning -- What does reason mean? -- The ultimate foundations of the syllogism -- How to detect arguments -- Arguments vs. explanations -- Truth and validity -- IX. Different kinds of arguments -- Three meanings of because -- The four causes -- A classification of arguments -- Simple argument maps -- Deductive and inductive arguments -- Combining deduction and induction : socratic method -- X. Syllogisms -- The structure and strategy of the syllogism -- The skeptics objection to the syllogism -- The empiricist's objection to the syllogism -- Demonstrative syllogisms -- How to construct convincing syllogisms -- XI. Checking syllogisms for validity -- By Euler's circles -- By Aristotle's six rules -- Barbara Celarent : mood and figure -- Venn diagrams -- XII. More difficult syllogisms -- Enthymemes : abbreviated syllogisms -- Sorites : chain syllogisms -- Epicheiremas : multiple syllogisms -- Complex argument maps -- XIII. Compound syllogisms -- Hypothetical syllogisms -- Reductio ad absurdum arguments -- The practical syllogism : arguing about means and ends -- Disjunctive syllogisms -- Conjunctive syllogisms -- Dilemmas -- XIV. Induction -- What is induction? -- Generalization -- Causal induction : Mill's methods -- Scientific hypotheses -- Statistical probability -- Arguments from analogy -- A fortiori and a minore arguments -- XV. Some practical applications of logic -- How to write a logical essay -- How to write a socratic dialogue -- How to have a socratic debate -- How to use socratic method on difficult people -- How to read a book socratically -- XVI. Some philosophical applications of logic -- Logic and theology -- Logic and metaphysics -- Logic and cosmology -- Logic and philosophical anthropology -- Logic and epistemology -- Logic and ethics. (shrink)

In this paper I examine Avicenna’s conception of the consequence relation. I will consider in particular his categorical and hypothetical logics. I will first analyse his definition of the implication and will show that this relation is not a consequence relation in his frame. Unlike the medieval logicians, he does not distinguish explicitly between material and formal consequences. The arguments discussed in al-Qiyās, where the conclusion is true only in some matters, and would seem close to a material consequence for (...) that reason, are rejected explicitly as not syllogistic. He also rejects the ‘enthymemes’ unless they are complemented by their missing premise and the superfluous premises which, according to him, should promptly be ruled out. It seems then that the consequence relation in his theory is formal. It can be characterized as being ‘productivity in all matters’ or ‘necessary truth preserving’. It is illustrated by some single premise arguments, and above all by all kinds of syllogisms which, in his theory, are more numerous and various than in Aristotle’s one. The syllogism may contain two or more premises, including disjunctive ones. When it is hypothetical, it may lead to several conclusions. The premises may be in conflict, but then, the conclusion is false. He thus rejects the principle according to which ‘anything follows from a contradiction’. But, unlike what some scholars say, he does not admit any connexive principle. In the compound syllogisms, the conclusion follows by steps, each step taking two premises at once. (shrink)

Lorenzo Valla's rhetorical reform of logic resulted in important changes in sixteenth-century mathematical sciences, and not only in mathematical education and in the use of mathematics in other sciences, but also in mathematical theory itself. Logic came to be identified with dialectic, syllogisms with enthymemes and necessary truth with the limit case of probable truth. Two main ancient authorities mediated between logical and mathematical concerns: Cicero and Proclus. Cicero's 'common notions' were identified with Euclid's axioms, so that mathematics could (...) be viewed as core knowledge shared by all humankind. Proclus' interpretation of Euclid's axioms gave rise to the idea of a universal human natural light of reasoning and of a mathesis universalis as a basic mathematics common to both arithmetic and geometry and as an art of thinking interpretable as algebra. (shrink)

Identifying the missing or unstated premisses of arguments is important, because their logical quality depends on them. Textbook authors regard enthymematic syllogisms (e.g., "Elvis is a man, so Elvis is mortal") as having an unstated premiss - the major premiss (e.g., "All men are mortal"). They are said to be such because these syllogisms become formally valid when the major premiss is added (i.e., it is a gap-filler). I argue that unstated major premises are not gap-fillers: they support a part (...) of the argument that is already given implicitly - the inference claim ("If the premisses are true then the conclusion is true"). As such, their logical status is the same as that of an unstated proposition that supports the minor premiss, which is not part of the argument. Therefore, the so-called "major premiss" is not an unstated premiss of the enthymeme. (shrink)

Flaubert himself, in an early and now famous letter, identifies in "bêtise" the effect of an inordinate desire to conclude: "Oui, la bêtise," he writes, "consiste à vouloir conclure. Nous sommes un fil et nous voulons savoir la trame" . This is to say stupidity, to Flaubert, is less a given content of discourse than a particular order of that discourse itself.1 It is the sign of an hasty and elliptical intervention into thought of a series of preconceived conclusions, the (...) source of which may be situated in the doxa and in the rhetoric of verisimilitude that sustains the persuasive power of the doxa. Stupidity, as the project of the Dictionnaire demonstrates, is an endless fabric of maxims and probable syllogisms the function of which is to determine the particular and the specific, the singular and the different, as paradigmatic exempla of the larger discourse of encyclopaedic universality expressed in the verisimilitude of received ideas. It is in this sense that one can see in Flaubert's notion of "bêtise" the denunciation by the writer of an especially vulgarised form of the Aristotelian concept of verisimilitude, which, built around the rhetorical figures of the probable syllogism—the enthymeme—and the exemplum , is directed towards winning adhesion to a particular thesis by appealing to generalities and probabilities, and which constructs its arguments from material drawn from the doxa.2 It is this rhetoric of persuasion by verisimilitude that Flaubert, in the various discourses of the lover, the dreamer, and the politician, will throw into ironic relief in Madame Bovary and L'Éducation sentimentale. · 1. Cf. Valéry, Oeuvres, 1:1452.· 2. The concept of verisimilitude is a difficult one and one which had received much critical attention in recent years. I have taken the term here to refer to the complex network of constraints by which the mimetic novelist, like the rhetorician, is able to engage his audience in a contract of mutual recognition and to persuade them of the "sense of reality" of his narrative, that his is a plausible interpretation of reality, worthy of belief . It is here that Aristotle's elaboration of mimesis and of the art of rhetoric is decisive. Both in the Poetics and in the Rhetoric, Aristotle distinguishes two concepts with regards to the manner in which the artist or the rhetorician solicits from his audience the belief in the justness of his reconstruction of reality. The first concept is that of pithanon, the plausible or the persuasive. This corresponds to the speculative consideration of what strategy will be most forceful in any given case. Rhetoric is indeed defined as "the faculty of observing in any given case the available means of persuasion " . As such, pithanon is the sign of a desire to convince, a decision on the part of an individual in a particular situation. For this desire to convince to become fully operative in the context of an audience, it needs to be recast not as a plausibility, but as a probability, as eikos. Aristotle defines eikos as "a thing that usually happens: not . . . anything whatever that usually happens, but only if it belongs to the class of the 'contingent' or 'variable.' It bears the same relation to that in respect of which it is probable as the universal bears to the particular" . Eikos is on one level a collection of contents, of topoi. But it is more than this. For otherwise this would mean that works deriving from different historical contexts would become unintelligible to the uninitiated reader. Eikos is a patterning of discourse, a rhetorical syntax, based upon the integration of the singular in the universal, and translated in the text by the enthymeme and the exemplum. The homogeneity of the mimetic novel derives from the way in which the desire to convince is mediated and dissimulated by a totalising, "natural" eikos, when, in other words, the narrator is "objective." It is when these two dimensions are dissociated, as in Bouvard et Pécuchet, that all manner of disturbance is generated. . Leslie Hill, fellow in Clare College, Cambridge University, is presently doing research on Flaubert and on general aspects of the modern French novel. This is his first publication. (shrink)

Among Tolstoy's absolute statements are those that exhibit characteristics of both biblical commands and proverbs—and of other types of absolute statements as well. He also draws, for example, on logical propositions, mathematical deductions, laws of nature and human nature, dictionary definitions, and metaphysical assertions. The language of all these forms is timeless, anonymous, and above all categorical. Their stylistic features imply that they are not falsifiable and that they are not open to qualification: they characteristically include words like "all," "each," (...) "every," "only," and "certainly" and phrases like "there neither is nor can be," "the human mind cannot grasp," and "it is impossible that." Even in sentences that omit such phrases, the very refusal to use a qualifier of any kind can assert unqualifiability. When Tolstoy's absolute statements take the form of syllogisms, the use of the word "therefore" or some explicit or implicit equivalent carries the force of logical inevitability. It carries the same force with Tolstoy's enthymemes, which omit the major premise for the reader to reconstruct. [An example] from The Death of Ivan Ilysch,1 cited above, for instance, contains a minor premise and a conclusion of a syllogism; the reader himself must supply the major premise, which would be: "The simpler and more ordinary a life is, the more terrible it is."· 1. "Ivan Ilysch's life was the most simple and the most ordinary, and therefore the most terrible."Leo Tolstoy, The Death of Ivan Ilysch, ed. John Bayley, trans. Louise and Aylmer Maude , p.225.Gary Saul Morson is an associate professor of Russian literature at the University of Pennsylvania, and the author of The Boundaries of Genre: Dostoevsky's Diary of a Writer and the Traditions of Literary Utopia, and The Broken Frame: The Anti-tradition of Russian Literature. The present article is from a theoretical study of literary creativity and the biography of authors. (shrink)

Mediaeval logicians inherited from Boethius an account of conditional propositions and the syllogisms which may be constructed using them. In the following paper it is shown that there are considerable difficulties with Boethius' account which arise from his failure to understand the nature of compound propositions and in particular to provide for their negation. Boethius suggests that there are two different conditions which may be imposed for the truth of a conditional proposition but he really gives no adequate account of (...) how such propositions may be obtained. The true greatness of Peter Abaelard as a philosophical logician is revealed in what he is able to do with the material which he found in Boethius. It is shown that he developed a precise theory of conditionals giving an account of how true conditionals may be obtained and principles which may be used to reject others as false. Unlike Boethius Abaelard properly appreciates that conjunctions must be treated as logical units. Even he, however, falls victim to difficulties which arise when this connective is brought into contact with negation and the conditions which he lays down for the truth of a conditional. (shrink)

In Enthymemes and Topoi in Dialogue, Ellen Breitholtz presents a novel and precise account of reasoning from an interactional perspective. The account draws on the concepts of enthymemes and topoi, originating in Aristotelian rhetoric and dialectic, and integrates these in a formal dialogue semantic account using TTR, a type theory with records. Argumentation analysis and formal approaches to reasoning often focus the logical validity of arguments on inferences made in discourse from a god's-eye perspective. In contrast, Breitholtz's account (...) emphasises the individual perspectives of interlocutors and the function and acceptability of their reasoning in context. This provides an analysis of interactions where interlocutors have access to different topoi and therefore make different inferences. (shrink)

Which properties are characteristic of the enthymeme in Aristotle's Rhetoric? There is no consensus on this point. The present discussion centres on three properties. 1. Is there always an implicit premise? (Answer: Above all, a pragmatic level and a logical level must be distinguished.) 2. Do the premises consist by definition of probabilities and signs? (Answer: No.). 3. Are all enthymemes reducible to a syllogistic form? (Answer: The literature pertaining to this question is dominated by a false dilemma: an (...) enthymeme does not have either a topical or a syllogistic structure). In general, Aristotle's approach to the enthymeme in the Rhetoric appears to shift from argumentation theory to logic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Rhetoric 34.2 (2001) 93-112 [Access article in PDF] Enthymemes, Common Knowledge, and Plausible Inference Douglas Walton The study of enthymemes has always been regarded as important in logic, critical thinking, and rhetoric, but too often it is the formal or mechanistic aspect of it that has been in the forefront. This investigation will show that there is a kind of plausibilistic script-based reasoning, of a (...) kind that has mainly been studied in artificial intelligence, that should have a much more important role to play in the study of enthymemes. But then curiously, as will also be shown, this plausibilistic type of reasoning was familiar in the ancient world, to the Sophists, as well as to leading philosophers such as Plato and Aristotle. By linking this ancient notion of plausibility to the modern notion used in computer science, this investigation reveals an important basis for the enthymeme that has a type of logical structure in its own right, but also has an informal aspect.An enthymeme, in current usage, is an argument that has one or more premises, or possibly a conclusion, not explicitly stated in the text, but that needs to have these propositions explicitly stated to extract the complete argument from the text. Sometimes enthymemes are described as arguments with "missing premises." That vocabulary is awkward, however, because the nonexplicit statement that needs to be added can be a conclusion, at least in a minority of cases. To make the exposition below smoother, the term nonexplicit assumption will be used to cover either the case of a nonexplicit premise or that of a nonexplicit conclusion. The problem with enthymemes is that if the nonexplicit assumptions in an argument are supposed to be propositions used by the arguer (as opposed to just the propositions needed to make the argument structurally correct, according to some standard), reasonable people can have differences of opinion on what the nonexplicit assumptions are supposed to be. The problem is that filling in the missing parts of enthymemes depends on interpreting the natural language [End Page 93] in which the argument was put forward to try to determine what the speaker meant to say.The solution to the problem comes through the recognition that enthymemes rest not only on formal (structural) criteria, but also on informal criteria. One of the most important of these informal criteria is something often called "common knowledge" (Govier 1992; Freeman 1995). But as shown below in a set of selected case studies of enthymemes, "common knowledge" is not really a kind of knowledge at all. It is really plausibility, or eikos, something well known in the ancient world and often misleadingly translated as "probability." Curiously, Aristotle's original doctrine of the enthymeme was based on this notion of plausibility. This historical fact has often been a source of puzzlement and confusion, and sometimes it has even been taken to indicate a defect or contradiction in Aristotle's treatment of the enthymeme. However, the goal of this investigation is not primarily historical. It is to work out one of the most important required steps toward a solution to the problem of enthymemes. But to do this, it is necessary to come back to the ancient notion of plausible inference. 1. The problem with enthymemes Once you have any formal theory of inference, like syllogistic, for example, you will be confronted by cases of arguments in everyday discourse that meet all the requirements for a structurally correct inference except that some part or parts, usually a premise or a conclusion, is missing. By "missing" is meant that the proposition in question has not been explicitly stated in the text of discourse, even though it may be clear enough that the speaker (writer) was relying on it, or including it, as part of the argument. The classic example is the following inference: All men are mortal; therefore, Socrates is mortal. To evaluate such an argument, surely some account of the "hidden" or "missing" assumption needs to be taken. The problem with enthymemes (Burke 1985; Gough and Tindale 1985; Hitchcock 1985) is that, if given carte... (shrink)

This essay argues that the affective component inherent in the enthymeme is the essence of aristotle's concept of the enthymeme as practical reasoning. 'affective component' refers to emotions and feelings. The three proofs of the thesis are the etymology of 'enthymeme', Aristotle's works on human action and practical wisdom, And aristotle's rhetoric. These sources show the inherent relation between enthymemes and phronesis, Or practical reasoning, Not nous, Or abstract intellect.

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)

Positions in dialogic dispute are presented enthymematically. It is important to explore the position the disputant holds. A model is offered which relies on the presentation of a counter-example to an inferred missing premiss. The example may be: [A+J embraced as falling under the rule; [A-] rejected as basically changing the position; or, [R] rejected as changing the proffered missing premiss. In each case the offered model indicates the next appropriate action. The focus of the model is on uncovering the (...) position actually held by the disputant as opposed to identifying the "logically correct" enthymematic premiss. (shrink)

Examines the concept of the enthymeme in ancient Greek rhetoric, arguing that it is a technique of storytelling aimed at eliciting from the audience an inference about a narrative.

본 논문의 목적은 아리스토텔레스로부터 시작하여 초기 페리파토스학파를 거쳐 고대후기에 이르기까지 ‘아리스토텔레스 논리학’의 특정 요소가 발전해 가는 과정을 추적하는 것이다. 고대 후기에는 가언삼단논법으로 분류된 4개의 논증형식이 있었다. 긍정적 긍정식(modus ponendo ponens), 부정적 부정식(modus tollendo tollens), 긍정적 부정식(modus ponendo tollens), 부정적 긍정식(modus tollendo ponens)이 그것들이다. 우리가 다루게 될 문제는 다음과 같은 것들이다. 앞에서 제시되었던 4개의 논증형식은 정확하게 어떤 상황에서 생겨났는가? 이 4개의 논증형식이 왜 ‘가언’ 삼단논법이라고 명명되었는가? 어떤 근거에서 이 4개의 논증형식이 타당하게 되는가? 논의를 진행하는 가운데 우리는 이 4개의 논증형식이 『분석론 전서』와 (...) 『토피카』에 보존되어 있는 아리스토텔레스 논리학의 특정 요소들이 오랜 기간에 걸쳐 점진적으로 발전한 결과임을 논증할 것이다.BR 우리의 논의결과는 이렇다. 첫째, 고대 후기에 오늘날의 조건명제와 선언명제가 가언명제로 분류된 이유는 단순히 조건명제와 선언명제에 조건적 요소가 포함되어 있기 때문이었다. 둘째, 고대 후기에 4가지 유형의 가언삼단논법이 ‘가언’ 삼단논법이라고 명명된 이유는 이 4가지 유형의 논증에 귀결관계와 비정합성관계의 조합으로부터 획득되는 가언명제가 적어도 한 개는 포함되어 있기 때문이었다. 셋째, 고대 후기에 4가지 유형의 가언삼단논법이 타당한 것으로 간주된 이유는 그것들 모두가 ‘모두스 포넨스’(modus ponens)유형의 형식을 갖고 있기 때문이었다. (shrink)

Traditionally, an enthymeme is an incomplete argument, made so by the absence of one or more of its constituent statements. An enthymeme resolution strategy is a set of procedures for finding those missing elements, thus reconstructing the enthymemes and restoring its meaning. It is widely held that a condition on the adequacy of such procedures is that statements restored to an enthymeme produce an argument that is good in some given respect in relation to which the enthymeme itself is (...) bad. In previous work, we emphasized the role of parsimony in enthymeme resolution strategies and concomitantly downplayed the role of charity . In the present paper, we take the analysis of enthymemes a step further. We will propose that if the pragmatic features that attend the phenomenon of enthymematic communication are duly heeded, the very idea of reconstructing enthymemes loses much of its rationale, and their interpretation comes to be conceived in a new light. (shrink)

In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic. We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition.

This paper argues for a reinterpretation of Aristotle's concept of an enthymeme and also his wider informal logic in terms of arguments that are defeasible. They are represented by forms of argument that are called argumentation schemes, considered to be similar to forms of argument found in deductive logic, but different from the foregoing in virtue of their being defeasible. Indeed, the most interesting schemes have been put forward as a helpful way of characterizing structures of human reasoning that have (...) proved troublesome to model deductively. The paper sheds new light on Aristotle's topics and how to dene `enthymeme'. If the traditional denition of an enthymeme in logic accepted for over two thousand years is a misnomer, the question is raised whether we ought to redene it as a defeasible argumentations scheme or leave things as they are. (shrink)

The Legacy of Aristotelian Enthymeme provides a historical-logical analysis of Aristotle's rhetorical syllogism, the enthymeme, through its Medieval and Renaissance interpretations. Bringing together notions of credibility and proof, an international team of scholars highlight the fierce debates around this form of argumentation during two key periods for Aristotle's beliefs.Reflecting on medieval and humanist thinkers, philosophers, poets and theologians, this volume joins up dialectical and rhetorical argumentation as key to the enthymeme's interpretation and shows how the enthymeme was the source of (...) a major interpretive conflict. As a method for achieving the standards for proof and credibility that persist across diverse fields of study today including the law, politics, medicine and morality, this book takes in Latin and Persian interpretations of the enthymeme and casts contemporary argumentation in a new historical light. (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.

Relational Syllogisms and the History of Arabic Logic, 900–1900. By Khaled El-Rouayheb. Islamic Philosophy, Theology and Science, vol. 80. Leiden: Brill, 2010. Pp. viii + 295. $168.

This article examines two syllogistic arguments contrasted in an ancient Chinese book, the Mozi, which expounds doctrines of the Mohist school of philosophers. While the arguments seem to have the same form, one of them is valid but the other is not. To explain this difference, the article uses English plural constructions to formulate the arguments. Then it shows that the one-horse argument is valid because it has a valid argument form, the plural cousin of a standard form of valid (...) categorical syllogisms, and argues that the two-horse argument involves equivocal uses of a key predicate that has the distributive/nondistributive ambiguity. In doing so, the article discusses linguistic differences between Chinese and English and explains why the logic of plural constructions is applicable to Chinese arguments that involve no plural constructions. (shrink)

This paper discusses the poetic art of cultivating the sacred by connecting the thought of two unlikely figures, twentieth-century anthropologist and systems theorist Gregory Bateson and nineteenth-century poet-philosopher Friedrich Hölderlin. Gregory Bateson’s theory of mental process within the ecology of mind, characterized in terms of metaphor and simile, or poetry and prose thinking, is illustrated with the aid of two syllogisms, the syllogism in Barbara and the syllogism in Grass. In light of these syllogisms, Friedrich Hölderlin’s views on the task (...) of the poet and the logic behind tragic song are then examined. Hölderlin’s poetic theory reveals how two elements of the song, tragic metaphor and caesura, invoke a living representation of the sacred, or the whole of ecology of mind. (shrink)

The enthymeme is essential because it reflects what humans do when they think. It provides an effective pedagogical approach to the analysis and synthesis of ideas in the classroom. In this volume, such an approach is applied to various areas of content for the purpose of helping students prepare themselves for the challenges of modern life.

Since Aristotle it is recognised that a valid syllogism cannot have two particular premises. However, that is not how a lay person sees it; at least as long as the premises read many, most etc, instead of a plain some. The lay people are right if one considers that these syllogisms do not have strict but approximate (Zadeh) validity. Typically there are only particular premises available in everyday life and one is dependent on such syllogisms. – Some rules on the (...) usage of particular premises are given below. (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)

This article is about self-referential arguments that are akin to the paradox of the liar. They make reference to their own validity, invalidity, soundness, unsoundness, or to the truth or falsity of their own premises or conclusions. Several very puzzling examples of such arguments are considered and their paradoxicality is defended against objections.

How to use truth tables to narrow down the number of possible candidates for missing premise. and, how to use philosophical analysis to pick the most plausible candidate from among those. this activity is a nice capstone to a course in logic for it combines formal and informal procedures.

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

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.

In the first half of this paper, we present a fragment of relational syllogisms named RELSYLL consisting of quantified statements with a special set of numerical quantifiers, and introduce a number of concepts that are useful for the later sections, including indirect reduction, quantifier transformations and equivalence of syllogisms. After determining the valid and invalid syllogisms in RELSYLL, we then introduce two Derivation Methods which can be used to derive valid relational syllogisms based on known valid simple syllogisms. We also (...) show that the two Methods are sound and complete for RELSYLL. In the second half of this paper, we discuss ways to extend the Derivation Methods, including the use of more valid syllogisms and the use of existential assumptions. In this way, we are able to derive more relational syllogisms that contain other types of non-classical quantifiers, including “only” and proportional quantifiers. Finally, we state and prove a proposition concerning the relationship between the two Methods. (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)