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 AristotelianLogic. Nevertheless, even in the context of AristotelianLogic, 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 Aristotelianlogic, 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 Aristotelianlogic 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)
I want to pull together some well-known facts which, when taken together, provide us with a plausible, and I think persuasive, argument that Aristotle's logic is inconsistent. We cannot, of course, hope to show that it is formally inconsistent since he does not present us with a fully worked-out formal system. On the other hand, we do have Lukasiewicz's formal version of Aristotelianlogic which he proves consistent. (edited).
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 Aristotelianlogic (...) 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-Aristotelianlogic that are frequently neglected (such as Peripatetic hypothetical syllogistic, the Stoic axiomatic system of propositional logic and various later ancient developments).
. In the XIXth century there was a persistent opposition to Aristotelianlogic. 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-Aristotelianlogic (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)
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)
Since Freges terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this questionBegriffes syntaxhighercorresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcorans theory via predicate logic.
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)
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)
The author discusses what he deems an oversight in prior's article on lukasiewicz's book "aristotle's syllogistic". He thinks prior missed lukasiewicz's exposure of the "symbolic logicians' fairy tale" which is the attempt to fit aristotle's logic into the boolean and russellian systems by "the lopping and stretching of inconvenient limbs." he concludes that lukasiewicz has "broken the ice that had begun to form" on traditional logic and that logic did not begin in the nineteenth century. (staff).
'Aristotelianlogic', 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)