Online research in philosophy

This book examines the medieval tradition of Aristotelian logic from two perspectives.

Stewart Shapiro and Alan Weir have argued that a crucial part of the demonstration of Frege's Theorem (specifically, that Hume's Principle implies that there are infinitely many objects) fails if the Neo-logicist cannot assume the existence of the empty property, i.e., is restricted to so-called Aristotelian Logic. Nevertheless, even in the context of Aristotelian Logic, Hume's Principle implies much of the content of Peano Arithmetic. In addition, their results do not constitute an objection to Neo-logicism so (...) much as a clarification regarding the view of logic that the Neo-logicist must take. (shrink)

This paper addresses the question of what existential assumptions are needed for the Aristotelian interpretation of the relationships between the four categorical propositions. The particular relationships in question are those unique to the Aristotelian logic, namely, contrariety, subcontrariety, subaltemation, conversion by limitation, and contraposition by limitation. The views of several recent authors of logic textbooks are surveyed. While most construe the Aristotelian logic as capable of being preserved by assuming that the subject class has (...) a member, Irving Copi construes that logic as requiring that four assumptions about class membership be made. These are that the subject, predicate, complement of subject, and complement of predicate classes all have members. It is argued that only three assumptions about class membership are needed, viz., that subject, predicate, and complement of predicate classes have members. (shrink)

This comment calls attention to the nature of the Aristotelian and classical logics, and the difficulty of representing their judgments and inferences by means of Venn diagrams. The meaning of ‘all’ in the different calculi produces problems. A second problem is that the specification of existence in Venn diagrams for statements and arguments cannot be restricted to a single class, overlooked by Wiebe. This problem is further complicated by his adoption of classical (Renaissance) syllogistic, which is inconsistent. Aristotle’s term (...) logic is consistent. So also is the medieval extension, though the inclusion of singular premisses renders it less perspicuous though more flexible. (shrink)

This is part two of a complete exposition of Logic, in which there is a radically new synthesis of Aristotelian-Scholastic Logic with modern Logic. Part II is the presentation of the theory of propositions. Simple, composite, atomic, compound, modal, and tensed propositions are all examined. Valid consequences and propositional logical identities are rigorously proven. Modal logic is rigorously defined and proven. This is the first work of Logic known to unite Aristotelian logic (...) and modern logic using scholastic logic as the instrument. (shrink)

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with a focus on issues that may be of interest to contemporary logicians and covering important topics in Post-Aristotelian logic that are frequently neglected (such as Peripatetic hypothetical syllogistic, the Stoic axiomatic system of propositional logic and various later ancient developments).

The specialized essays in this collection study whether non-Aristotelian traditions of ancient logic had a role for medieval logicians. Special attention is given to Stoic logic and semantics, and to Neoplatonism.

Machine generated contents note: ARISTOTELIAN AND CARTESIAN LOGIC AT HARVARD -- by Rick Kennedy -- I. Introduction --II. Religiously-Oriented, Dogmatically-Inclined Humanistic Logics from the Renaissance to the Seventeenth Century -- A. Melanchthon and Aristotelianism 01 -- B. Richardson and Ramism 16 -- C. Aristotelianism, Ramism, and Schematic Thinking 25 -- D. Puritan Favoritism From Ramus to Descartes 32 -- E. Cartesian Logic and Christian Skepticism 37 -- F. The Religious and Dogmatic Orientation of The Port-'Royalfogic 42 -- (...) G. Cartesian Logic in British Textbooks 52 -- III. Charles Morton and c A; logick System -- A. Charles Morton 62 -- B. Morton's cAfogick System 78 -- IV. William Brattle and the Compendium of logick -- A. Intellectual Reform in the Puritans' Collapsing World 91 -- B. The Compendium ofJogick 93 -- c. Brattle: Tutor and Unofficial Professor of Divinity 108 -- V. Epilogue: Later Constituencies of Religious Logics and 133 -- The Separation of Logic and Divinity at Harvard. (shrink)

This book shows otherwise. John Martin rehabilitates Neoplatonism, founded by Plotinus and brought into Christianity by St. Augustine.

A general metaphysical account of logic, meaning, and reference that developed from the Greeks through the medievals and up into modem times can be called Aristotelian. “Copernican” claims (Kant, Frege), radically to replace this paradigm as quasi-“Ptolemaic,” actually participated in the prolonged decline of scholasticism, after Aquinas in particular. We need to recognize, or to remember, thepriority of being to truth and not to conflate them. We need to explicate the origin of thinking (abstraction) as at one remove (...) from immediate sense-experience. Syllogistic logic then emerges as a true causal account of reasoning in general; it is not some primitive attempt to outline a formal logical system. An account of suppositio as controlling the analogous uses of our finite store of words in reference to an infinite reality itself shaped by criss-cross patterns of likenesses, governs the general picture supplied here. (shrink)

What is the function of logic in al-Kind's theory of categories as it was presented in his epistle On the Number of Aristotle's Books (F treats the Categories as a logical book, but in a manner different from that of the classical Aristotelian tradition. He ascribes a special status to the categories Quantity (kammiyya) and Quality (kayfiyya), whereas the rest of the categories are thought to be no more than different combinations of these two categories with the category (...) Substance. The discussion will pay special attention to the function of the categories of Quantity and Quality as mediators between logic and mathematics. (shrink)

This article is currently available as a free download on ingentaconnect.

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

Gödel’s incompleteness applies to any system with recursively enumerable axioms and rules of inference. Chaitin’s approach to Gödel’s incompleteness relates the incompleteness to the amount of information contained in the axioms. Zurek’s quantum Darwinism attempts the physical description of the universe using information as one of its major components. The capacity of quantum Darwinism to describe quantum measurement in great detail without requiring ad-hoc non-unitary evolution makes it a good candidate for describing the transition from quantum to classical. A baby-universe (...) diffusion model of cosmic inflation is analyzed using quantum Darwinism. In this model cosmic inflation can be approximated as Brownian motion of a quantum field, and quantum Darwinism implies that molecular interaction during Brownian motion will make the quantum field decohere. The quantum Darwinism approach to decoherence in the baby-universe cosmic-inflation model yields the decoherence times of the baby-universes. The result is the equation relating the baby-universe’s decoherence time with the Hubble parameter, and that the decoherence time is considerably shorter than the cosmic inflation period. (shrink)

Logic as a discipline starts with the transition from the more or less unreflective use of logical methods and argument patterns to the reflection on and inquiry into these and their elements, including the syntax and semantics of sentences. In Greek and Roman antiquity, discussions of some elements of logic and a focus on methods of inference can be traced back to the late 5th century BCE. The Sophists, and later Plato (early 4th c.) displayed an interest in (...) sentence analysis, truth, and fallacies, and Eubulides of Miletus (mid-4th c.) is on record as the inventor of both the Liar and the Sorites paradox. But logic as a fully systematic discipline begins with Aristotle, who systematized much of the logical inquiry of his predecessors. His main achievements were his theory of the logical interrelation of affirmative and negative existential and universal statements and, based on this theory, his syllogistic, which can be interpreted as a system of deductive inference. Aristotle's logic is known as term-logic, since it is concerned with the logical relations between terms, such as ‘human being’, ‘animal’, ‘white’. It shares elements with both set theory and predicate logic. Aristotle's successors in his school, the Peripatos, notably Theophrastus and Eudemus, widened the scope of deductive inference and improved some aspects of Aristotle's logic. (shrink)

ABSTRACT: A very brief summary presentation of western ancient logic for the non-specialized reader, from the beginnings to Boethius. For a much more detailed presentation see my "Ancient Logic" in the Stanford Encyclopedia of Philosopy (also on PhilPapers).

The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward L?wenheim-Skolem theorem to proper classes is used to argue that (...) a sound and complete axiomatization of the logic of unrestricted universal quantification results from adding all sentences of the form 'There are at least n individuals' as axioms to a standard axiomatization of the first-order predicate calculus. (shrink)

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

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

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

ABSTRACT: English translation of the 2nd/3rd century Peripatetic Philosopher's Alexander of Aphrodisias commentary on Aristotle's non-modal syllogistic, i.e. on one of the most influential logical texts of all times. -/- Volume includes introduction on Alexander of Aphrodisias and the early commentators, translation with notes and comments, appendices with a new translation of Aristotle's text, a summary of Aristotle's non-modal syllogistic and textual notes.

For more than twenty years, introductory logic students have relied on this text to provide clear lessons as well as practical applications of the discipline. Robert Baum emphasizes formal logic and utilizes such elements of popular culture as cartoons and advertisements to illustrate technical concepts. Logic, 4/e addresses all the basic concepts, including informal analysis of statements, arguments, Aristotelian logic, propositional logic, quantificational logic, enumerative induction, the scientific method, probability, informal fallacies, definitions, and (...) applied logic. As with previous editions, Logic, 4/e is extremely flexible--most of the chapters can be included or excluded from a particular course depending on the goals of the course and the time available. This fourth edition features hundreds of additional exercises throughout. (shrink)

According to the received view (Boche?ski, Kneale), from the end of the fourteenth to the second half of nineteenth century, logic enters a period of decadence. If one looks at this period, the richness of the topics and the complexity of the discussions that characterized medieval logic seem to belong to a completely different world: a simplified theory of the syllogism is the only surviving relic of a glorious past. Even though this negative appraisal is grounded on good (...) reasons, it overlooks, however, a remarkable innovation that imposes itself at the beginning of the sixteenth century: the attempt to connect the two previously separated disciplines of logic and mathematics. This happens along two opposite directions: the one aiming to base mathematical proofs on traditional (Aristotelian) logic; the other attempting to reduce logic to a mathematical (algebraical) calculus. This second trend was reinforced by the claim, mainly propagated by Hobbes, that the activity of thinking was the same as that of performing an arithmetical calculus. Thus, in the period of what Boche?ski characterizes as ?classical logic?, one may find the seeds of a process which was completed by Boole and Frege and opened the door to the contemporary, mathematical form of logic. (shrink)

This study is the first modern account of the development of philosophy during the Carolingian Renaissance. In the late eighth century, Dr Marenbon argues, theologians were led by their enthusiasm for logic to pose themselves truly philosophical questions. The central themes of ninth-century philosophy - essence, the Aristotelian Categories, the problem of Universals - were to preoccupy thinkers throughout the Middle Ages. The earliest period of medieval philosophy was thus a formative one. This work is based on a (...) fresh study of the manuscript sources. The thoughts of scholars such as Alcuin, Candidus, Fredegisus, Ratramnus of Corbie, John Scottus Eriugena and Heiric of Auxerre is examined in detail and compared with their sources; and a wide variety of evidence is used to throw light on the milieu in which these thinkers flourished. Full critical editions of an important body of early medieval philosophical material, much of it never before published, are included. (shrink)

We consider the history of logic in pre-Petrine. Petrine. and immediate post-Pctrine Russia (from the 15th to the mid-18th centuries) and especially of the Petrine era from the late 17th to early 18th century. Throughout much of this time, the clergy evinced strong hostility towards logic. Nevertheless, a small number of academics and clerics such as Stefan Iavorskii and Fcofan Prokopovich kept Aristotelian logic alive during this period and provided the foundation for its development in the (...) modern era. (shrink)

This article is about the history of logic in Australia. Douglas Gasking (1911?1994) undertook to translate the logical terminology of John Anderson (1893?1962) into that of Ludwig Wittgenstein's (1921) Tractatus. At the time Gilbert Ryle (1900?1976), and more recently David Armstrong, recommended the result to students; but it is reasonable to have misgivings about Gasking as a guide to either Anderson or Wittgenstein. The historical interest of the debate Gasking initiated is that it yielded surprisingly little information about Anderson's (...) traditional (syllogistic or Aristotelian) logic and its relation to classical (first-order predicate or Russellian) logic, the ostensible topic; but the materials now exist to interpret Anderson's logic in classical logic, possibly as an algebra of classes. This would be of little interest to contemporary logicians, but it might shed some light on Anderson's philosophy. (shrink)

The Consolations of Philosophy by Boethius, whose English translators include King Alfred, Geoffrey Chaucer, and Queen Elizabeth I, ranks among the most remarkable books to be written by a prisoner awaiting the execution of a tyrannical death sentence. Its interpretation is bound up with his other writings on mathematics and music, on Aristotelian and propositional logic, and on central themes of Christian dogma. -/- Chadwick begins by tracing the career of Boethius, a Roman rising to high office under (...) the Gothic King Theoderic the Great, and suggests that his death may be seen as a cruel by-product of Byzantine ambitions to restore Roman imperial rule after its elimination in the West in AD 476. Subsequent chapters examine in detail his educational programme in the liberal arts designed to avert a threatened collapse of culture and his ambition to translate into Latin everything he could find on Plato and Aristotle. -/- Boethius has been called `last of the Romans, first of the scholastics'. This book is the first major study in English of a writer who was of critical importance in the history of thought. (shrink)

In his posthumous book from 1914, "New foundations of logic, arithmetic and set theory", Julius Konig develops his philosophy of mathematics. In a previous contribution, we attracted attention on the positive part (his truth and falsehood predicates being excluded) of his "pure logic": his "isology" being assimilated to mutual implication, it constitutes a genuine formalization of positive intuitionistic logic. Konig's intention was to rebuild logic in such a way that the excluded third's principle could no longer (...) be logical. However, his treatment of truth and falsehood (boiling down to negation) is purely classical. We explain here this discrepancy by the choice of the alleged more primitive notions to which the questioned notions of truth and falsehood have been reduced. Finaly, it turns out that the disjunctive and conjunctive forms of the principles of the excluded third and of contradiction have effectively been excluded, but none of their implicative forms. (shrink)

Abstract. As a general theory of reasoning—and as a general theory of what holds true under every possible circumstance—logic is supposed to be ontologically neutral. It ought to have nothing to do with questions concerning what there is, or whether there is anything at all. It is for this reason that traditional Aristotelian logic, with its tacit existential presuppositions, was eventually deemed inadequate as a canon of pure logic. And it is for this reason that modern (...) quantification theory, too, with its residue of existentially loaded theorems and patterns of inference, has been claimed to suffer from a defect of logical purity. The law of non-contradiction rules out certain circumstances as impossible—circumstances in which a statement is both true and false, or perhaps circumstances where something both is and is not the case. Is this to be regarded as a further ontological bias? (shrink)

“Formal logic”, an expression created by Kant to characterize Aristotelian logic, has also been used as a name for modern logic, originated by Boole and Frege, which in many aspects differs radically from traditional logic. We shed light on this paradox by distinguishing in this paper five different meanings of the expression “formal logic”: (1) Formal reasoning according to the Aristotelian dichotomy of form and content, (2) Formal logic as a formal science (...) by opposition to an empirical science, (3) Formal systems in the sense of Hilbert, Curry and the formalist school, (4) Symbolic logic, a science using symbols, such as Venn diagrams, (5) Mathematical logic, a mathematical approach to reasoning. We argue that these five meanings are independent and that the meaning (5) is the one which better characterized modern logic, which should therefore not be called “formal logic”. (shrink)

Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, “The logic and meaning of plurals, II”, aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of (...) plural constructions that result from the view that plural constructions are, by and large, devices for talking about many things (as such). The account of logic presented in the papers surpasses contemporary Fregean accounts in its scope. This extension of the scope of logic results from extending the range of languages that logic can directly relate to. Underlying the view of language that makes room for this is a perspective on reality that locates in the world what plural constructions can relate to. The papers suggest that reflections on plural constructions point to a broader framework for understanding logic, language, and reality that can replace the contemporary Fregean framework as this has replaced its Aristotelian ancestor. (shrink)

. In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What reasons forced him to reassess the status of particular propositions and (...) to replace the square of opposition by the triangle of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the new ‘imaginary’ logic, as he called it? (shrink)

In his Introduction to Logical Theory, Strawson argues that Aristotelian logic can be given a successful interpretation into ordinary English, but not into the symbolism of Principia Mathematica, on the grounds that Aristotelian logic and ordinary English share something absent in PM, namely, the doctrine of presupposition. It is argued that Strawson is mistaken. PM does justice to the logical rules of Aristotelian logic and also has a fully articulated doctrine of presupposition.

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

The Ikhwan al-Safa (Brethren of Purity), the anonymous adepts of a tenth-century esoteric fraternity based in Basra and Baghdad, hold an eminent position in the history of science and philosophy in Islam due to the wide reception and assimilation of their monumental encyclopaedia, the Rasa'il Ikhwan al-Safa ( Epistles of the Brethren of Purity ). This compendium contains fifty-two epistles offering synoptic accounts of the classical sciences and philosophies of the age; divided into four classificatory parts, it treats themes in (...) mathematics, logic, natural philosophy, psychology, metaphysics, and theology, in addition to didactic fables. The Rasa'il constitutes a paradigmatic legacy in the canonization of philosophy and the sciences in mediaeval Islamic civilization, as well as having shown a permeating influence in Western culture. The present volume is the second of this definitive series, consisting of the very first critical edition of the Rasa'il in its original Arabic, complete with the first fully annotated English translation. Prepared by Professor Carmela Baffioni, Epistles 10-14 comprise the foundations of logic, which remained a fundamental component in pedagogy until the twentieth century. The Ikhwan treat the Isagoge and the larger part of the Organon , both of which were circulating through the Islamic world at that time, as they set about detailing the ten categories of existents, the five predicables, and other such commonplaces of Aristotelian logic, including his seminal method of syllogistic inference. With the claim that logic is the noblest of man's arts, and man the noblest of creatures, the Ikhwan cast Aristotelian tropes in a spiritual light. (shrink)

The Ikhwan al-Safa (Brethren of Purity), the anonymous adepts of a tenth-century esoteric fraternity based in Basra and Baghdad, hold an eminent position in the history of science and philosophy in Islam due to the wide reception and assimilation of their monumental encyclopaedia, the Rasa'il Ikhwan al-Safa' (Epistles of the Brethren of Purity). This compendium contains fifty-two epistles offering synoptic accounts of the classical sciences and philosophies of the age; divided into four classificatory parts, it treats themes in mathematics, (...) class='Hi'>logic, natural philosophy, psychology, metaphysics, and theology, in addition to didactic fables. The Rasa'il constitutes a paradigmatic legacy in the canonization of philosophy and the sciences in mediaeval Islamic civilization, as well as having shown a permeating influence in Western culture. -/- The present volume is the second of this definitive series, consisting of the very first critical edition of the Rasa'il in its original Arabic, complete with the first fully annotated English translation. Prepared by Professor Carmela Baffioni, Epistles 10-14 comprise the foundations of logic, which remained a fundamental component in pedagogy until the twentieth century. The Ikhwan treat the Isagoge and the larger part of the Organon, both of which were circulating through the Islamic world at that time, as they set about detailing the ten categories of existents, the five predicables, and other such commonplaces of Aristotelian logic, including his seminal method of syllogistic inference. With the claim that logic is the noblest of man's arts, and man the noblest of creatures, the Ikhwan cast Aristotelian tropes in a spiritual light. (shrink)

For more than twenty years, introductory logic students have relied on this text to provide clear lessons as well as practical applications of the discipline. Robert Baum emphasizes formal logic and utilizes such elements of popular culture as cartoons and advertisements to illustrate technical concepts. Logic, 4/e addresses all the basic concepts, including informal analysis of statements, arguments, Aristotelian logic, propositional logic, quantificational logic, enumerative induction, the scientific method, probability, informal fallacies, definitions, and (...) applied logic. As with previous editions, Logic, 4/e is extremely flexible--most of the chapters can be included or excluded from a particular course depending on the goals of the course and the time available. This fourth edition features hundreds of additional exercises throughout. (shrink)

Due to the current availability of the English translation of almost all of Lesniewski's works it is now possible to give a clear and detailed picture of his ideas. Lesniewski's system of the foundation of mathematics is discussed. In abrief ouüine of his three systems Mereology, Ontology and Protothetics his positions conceming the problems of the forms of expression, proper names, synonymity, analytic and synthetic propositions, existential propositions, the concept of logic, and his views of theory of science and (...) metaphysics are sketched. The influence of Mill, Lukasiewicz, Austrian philosophy and especially Petrazycki on his thinking is evaluated and an interpretation is suggested setting him squarely in a tradition of classical Aristotelian logic. (shrink)

Although Kant envisaged a prominent role for logic in the argumentative structure of his Critique of pure reason, logicians and philosophers have generally judged Kant's logic negatively. What Kant called `general' or `formal' logic has been dismissed as a fairly arbitrary subsystem of first order logic, and what he called `transcendental logic' is considered to be not a logic at all: no syntax, no semantics, no definition of validity. Against this, we argue that Kant's (...) `transcendental logic' is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first order logic. The main technical application of the formalism developed here is a formal proof that Kant's Table of Judgements in §9 of the Critique of pure reason, is indeed, as Kant claimed, complete for the kind of semantics he had in mind. This result implies that Kant's 'general' logic is after all a distinguished subsystem of first order logic, namely what is known as geometric logic. (shrink)

Introduction to my 'Contemporary Aristotelian Metaphysics' volume.

As a theory of reasoning, logic has—or ought to have—nothing to do with metaphysics. It ought to have nothing to do with questions concerning what there is, or whether there is anything at all. It is precisely because of its metaphysical commitments that Aristotelian syllogistics, for example, was eventually deemed inadequate as a canon of pure logical reasoning. The inference from an A-form statement such as (1) All humans are mortal to the corresponding I-form statement, (2) Some humans (...) are mortal, is syllogistically valid. But it depends on the existence of humans beings and should not, therefore, count as valid as a matter of pure logic. (It depends on the existence of human beings because, in a world with no such beings, (2) would be false whereas (1) would be true, although vacuously.) Likewise, modern quantification theory1 has been found inadequate insofar as it sanctions as valid the inference from a universal statement such as (3) Everything is mortal to the corresponding existential statement (4) Something is mortal, whose truth-conditions, unlike those of (3), appear to clash with the metaphysical possibility that there is nothing at all. It also sanctions as valid the inference from (3) to any of its substitution instances, such as.. (shrink)

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

This chapter begins with a discussion of Kant's theory of judgment-forms. It argues that it is not true in Kant's logic that assertoric or apodeictic judgments imply problematic ones, in the manner in which necessity and truth imply possibility in even the weakest systems of modern modal logic. The chapter then discusses theories of judgment-form after Kant, the theory of quantification, Frege's Begriffsschrift, C. I. Lewis and the beginnings of modern modal logic, the proof-theoretic approach to modal (...) logic, possible world semantics, correspondence theory, and modality and quantification. (shrink)

This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; ...

In the present paper we propose a system of propositional logic for reasoning about justification, truthmaking, and the connection between justifiers and truthmakers. The logic of justification and truthmaking is developed according to the fundamental ideas introduced by Artemov. Justifiers and truthmakers are treated in a similar way, exploiting the intuition that justifiers provide epistemic grounds for propositions to be considered true, while truthmakers provide ontological grounds for propositions to be true. This system of logic is then (...) applied both for interpreting the notorious definition of knowledge as justified true belief and for advancing a new solution to Gettier counterexamples to this standard definition. (shrink)

Rabern and Rabern (Analysis 68:105–112 2 ) and Uzquiano (Analysis 70:39–44 4 ) have each presented increasingly harder versions of ‘the hardest logic puzzle ever’ (Boolos The Harvard Review of Philosophy 6:62–65 1 ), and each has provided a two-question solution to his predecessor’s puzzle. But Uzquiano’s puzzle is different from the original and different from Rabern and Rabern’s in at least one important respect: it cannot be solved in less than three questions. In this paper we solve Uzquiano’s (...) puzzle in three questions and show why there is no solution in two. Finally, to cement a tradition, we introduce a puzzle of our own. (shrink)

Even among those philosophers who hold particular aspects of Hegel's philosophy in high regard, there have been few since the 19th century who have found Hegel's "metaphysics" plausible, and just as few not sceptical about the coherency of the "logical" project on which it is meant to be based. Indeed, against the type of work characteristic of the late nineteenth-century logical revolution which issued in modern analytic philosophy, it is often difficult to see exactly how Hegel's "logical" writings can be (...) read as a contribution to logic at all. Furthermore, any tendency toward skepticism here can only have been reinforced by the well-known views of Bertrand Russell about the logical inadequacy of the "Hegelian" approach of his predecessors. (shrink)

Friends, welcome to the first page of Logic in India. It is for Indian students prepared for first paper entitled Principles of Logic in Diploma-in-Reasoning course of Department of Philosophy, Kurukshetra University, Kurukshetra, where I taught four years. It is also beneficial for graduate students who have elementary logic course in their syllabus. Basically I used both printed books and internet sources to prepare it. You can find the course syllabus in my post “Philosophy is Nothing without (...) Logic” at The Positive Philosophy page and also in the side links of this page. This is only a draft, kindly send your suggestions and ideas to dr.sirswal@gmail.com or niyamak.drs@gmail.com, I shall be highly thankful to you. A short list of reference books are mentioned below of the Table of Contents and reference sites are linked with this page. This page introduces the basic conceptions of formal logic, informal logic and also Symbolic logic. (shrink)

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