"There are several different
conceptions of the nature of logic. Here I want to contrast an ontic
conception with an epistemic conception. On one ontic conception logic
investigates certain general aspects of 'reality', of 'being as such', in
itself and without regard to how (or even whether) it may be known by
thinking agents: in this connection logic has been called formal ontology.
On one epistemic conception, logic amounts to an investigation of deductive
reasoning per se without regard to what it is reasoning about; it
investigates what has been called formal reasoning. On this view, logic is
part of epistemology, viz. the part that studies the operational knowledge
known as deduction. It has been said that one of the main goals of
epistemically-oriented logic is to explicate the expression 'by logical
reasoning' as it occurs in sentences such as: a deduction shows how its conclusion can be obtained by
logical reasoning from its
premise-set.
Relevant to the axiomatic method there would be two branches of
epistemology: one to account for knowledge of the axioms and one to account
for how knowledge of the theorems is obtained from knowledge of the axioms,
in other words, one investigating induction and one investigating deduction.
The latter is logic according to the epistemic conception.
On the ontic view of logic, on the other hand, logic is an attempt to
gain knowledge of the truth of propositions expressible using only generic
nouns (individual, property, relation, etc.) and other 'logical'
expressions. In the framework of Principia Mathematica those are
propositions expressible using only variables and logical constants.
Principia Mathematica is an excellent example of an axiomatic
presentation of logic as formal ontology. Below are some typical laws of
formal ontology.
Excluded middle: Given any individual and any property either the
property belongs to the individual or the property does not belong tothe individual.
Noncontradiction: Given any individual and any property it is not the
case that the property both belongs to the individual and does not belong to
the individual.
Identity: Given any individual and any property, if the property belongs
to the individual then the individual has the property.
Dictum de omni: Every property A belonging to everything having
a given property B which in turn belongs to everything having another
property C likewise belongs to everything having that other
property C.
Dictum de nullo: Every property A belonging to nothing having a
given property B which in turn belongs to everything having another property
C likewise belongs to nothing having that other property C.
Commutation of Complementation with Conversion: Given any relation R the
complement of the converse of R is the converse of the complement of R.
From this sample of logic as ontic science we can see how the focus is on
ontology, or, as has been said by others, on the most general features of
reality itself and not on methods of gaining knowledge. According to Russell
Introduction to mathematical philosophy, 1919, 169, 'logic is concerned with the real world just as truly as zoology,
though with its more abstract and general features.' These six laws are
purely ontic in that they involve no concepts concerning a knowing agent or
concerning an epistemic faculty such as perception, judgement, or deduction.
This is not to deny that there is an epistemic dimension to logic as ontic
science but only to affirm that the focus if ontic. Every science in so far
as it is science has an epistemic dimension. The epistemic differs from the
ontic more as size differs from shape than as, say, animal differs from
plant.
Logic as ontic science was referred to above as formal ontology. Logic as
epistemic metascience may in like manner be called formal epistemology. It
is important and interesting to note that both are called formal logic but
for very different reasons. Some formal onticists justify the adjective
formal by reference to the fact that its propositions are expressed
exclusively in general logical terms without the use of names denoting
particular objects, particular properties, etc. cf.
Russell 1919, 197. Some formal epistemicists justify the adjective formal by
reference to the fact that the cogency of an argumentation is subject to a
principle of form and in particular to the following principles: (l) every
two argumentations in the same form are either both cogent or both non-cogent,
(2) every argumentation in the same form as a deduction is itself a
deduction. In fact, some formal epistemicists such as Boole claimed, with
some justification, that they were dealing with the forms of thought, i.e.
with the forms of cogent argumentations. For more on cogency of
argumentations and the principles of form see Corcoran 1989.
Formal onticists are often easy to recognize because of their tendency to
emphasize the fact that formal ontology does not study reasoning per se. In
fact, the formal onticists often think that the study of reasoning belongs
to psychology and not to logic. For example, Łukasiewicz in his famous book
on Aristotle's syllogistic makes the following two revealing remarks.
Łukasiewicz 1957 pages 12 and 73, respectively. 'Logic has no more to do
with thinking than mathematics. "[Aristotle's] system is not a theory of the
forms of thought nor is it dependent on psychology; it is similar to a
mathematical theory...'
There are significant differences among formal onticists. For example,
even among those that emphasize the truth-preserving character of deduction
some accept the view that it is consequences-conservative as well and some
reject this view. For example, Łukasiewicz 1929, 16 explicitly rejects the
view that deduction is a process of information extraction. He says that in
deductive inference '...we may obtain quite new results, not contained in
the premises'." pp. 17-19
From: John Corcoran: The founding of logic. Modern
interpretations of Aristotle's logic - Ancient Philosophy, 14, 1994 pp.
9-24
INTRODUCTORY AND GENERAL READINGS ON ARISTOTLE'S LOGIC
Formale und nicht-formale Logik bei Aristoteles. Edited by Menne Albert and Öffenberger Niels. Hildesheim: Georg Olms 1985.
Zur Modernen Deutung der aristotelischen Logik (Band 2)
Modallogik und Mehrwertigkeit. Edited by Menne Albert and Öffenberger Niels. Hildesheim: Georg Olms 1988.
Zur Modernen Deutung der aristotelischen Logik (Band 3)
Logic, dialectic and science in Aristotle. Ancient Philosophy 14 1994.
Special issue edited by Robert Bolton and Robin Smith.
Contents: Introduction by the Editors 1; John Corcoran: The founding of logic 9; Timothy Smiley: Aristotle's completeness proof 25; Gisela Striker: Modal vs. assertoric syllogistic 39; James G. Lennox: Aristotelian problems 53; Michael Ferejohn: The immediate premises of Aristotelian demostration 79; Robert Bolton: The problem of dialectical reasoning in Aristotle 99; Robin Smith: Dialectic and the syllogism 133-151
Beiträge zum Satz vom Widerspruch und zur Aristotelischen Prädikationstheorie. Edited by Öffenberger Niels and Skarica Mirko. Hildesheim: Georg Olms 2000.
Zur Modernen Deutung der aristotelischen Logik (Band 8)
Logique et métaphysique dans l'Organon d'Aristote. Edited by Bastit Michel and Follon Jacques. Louvain: Peeters 2001.
Actes du colloque de Dijon
Rumänische Beiträge zur modernen Deutung der Aristotelischen Logik. Edited by Öffenberger Niels and Surdu Alexandru. Hildesheim: Georg Olms 2004.
Zur Modernen Deutung der aristotelischen Logik (Band 9)
Über den Folgerungsbegriff in der aristotelischen Logik. Edited by Menne Albert and Öffenberger Niels. Hildesheim: Georg Olms 1982.
Zur Modernen Deutung der aristotelischen Logik (Band 1)
Allen James, "The development of Aristotle's logic: part of an account in outline," Boston Area Colloquium in Ancient Philosophy 11: 177-205 (1995).
Barnes Jonathan. Grammar On Aristotle's terms. In Rationality in Greek thought. Edited by Frede Michael and Striker Gisela. New York: Oxford University Press 1996. pp. 175-202
"However that may be, Aristotelian syllogistic concerned itself exclusively with monadic predicates. Hence it could not begin to investigate multiple quantification. And that is why it never got very far. None the less, the underlying grammar of Aristotle's logic did not in itself block the path to polyadicity. The later Peripatetics were conservative creatures and they lacked logical imagination. Moreover, Aristotle himself had assured them that his syllogistic was adequate for all serious scientific needs. As for Aristotle, his service to logic is nonpareil, and it would be grotesque to chide him for lack of inventiveness. It is true that, in logical grammar, he did not climb above the level which he attained in the de Interpretatione. But the Analytics does not represent a fatal, or even a new, grammatical excursion. And the story of Aristotle's fall, like the story of the fall of Adam, is a myth." pp. 201-202
Berg Jan. Aristotle's theory of definition. In Atti del convegno internazionale di storia della logica. Organizzato dalla Società italiana di logica e filosofia delle scienze (SILFS), San Gimignano, 4-8 dicembre 1982. Edited by Abrusci Michele, Casari Ettore, and Mugnai Massimo. Bologna: CLUEB 1983. pp. 19-30
Buddensiek Fridemann. Die Modallogik des Aristoteles in den Analytica Priora A. Hildesheim: Georg Olms 1994.
Zur Modernen Deutung der aristotelischen Logik (Band 6)
Calogero Guido. I fondamenti della logica aristotelica. Firenze: Le Monnier 1927.
Second edition with appendixes by Gabriele Giannantoni and Giovanna Sillitti - Firenze, La Nuova Italia, 1968.
Charles David. Aristotle on meaning and essence. New York: Oxford University Press 2000.
Code Alan. Metaphysics and logic. In Aristotle today: essays on Aristotle's ideal of science. Edited by Matten Mohan. Edmonton: Academic Printing and Publishing 1987. pp. 127-149
de Rijk Lambertus Marie. Aristotle: semantics and ontology. Volume I: General introduction. The works on logic. Leiden: Brill 2002.
From the Preface: "In this book I intend to show that the ascription of many shortcomings or obscurities to Aristotle resulted from persistent misinterpretation of key notions in his work. The idea underlying this study is that commentators have wrongfully attributed anachronistic perceptions of `predication', and statement-making in general to Aristotle. In Volume I, what I consider to be the genuine semantics underlying Aristotle's expositions of his philosophy are culled from the Organon. Determining what the basic components of Aristotle's semantics are is extremely important for our understanding of his view of the task of logic -- his strategy of argument in particular.
In chapter 1, after some preliminary considerations I argue that when analyzed at deep structure level, Aristotelian statement-making does not allow for the dyadic 'S is P' formula. An examination of the basic function of `be' and its cognates in Aristotle's philosophical investigations shows that in his analysis statement-making is copula-less. Following traditional linguistics I take the `existential' or hyparctic use of `be' to be the central one in Greek (pace Kahn), on the understanding that in Aristotle hyparxis is found not only in the stronger form of `actual occurrence' but also in a weaker form of what I term `connotative (or intensional) be' (1.3-1.6). Since Aristotle's `semantic behaviour', in spite of his skilful manipulation of the diverse semantic levels of expressions, is in fact not explicitly organized in a well-thought-out system of formal semantics, I have, in order to fill this void, formulated some semantic rules of thumb (1.7).
In chapter 2 I provide ample evidence for my exegesis of Aristotle's statement-making, in which the opposition between `assertible' and `assertion' is predominant and in which `is' functions as an assertoric operator rather than as a copula (2.1-2.2). Next, I demonstrate that Aristotle's doctrine of the categories fits in well with his view of copula-less statement-making, arguing that the ten categories are `appellations' ('nominations') rather than sentence predicates featuring in an `S is P' formation (2.3-2.4). Finally, categorization is assessed in the wider context of Aristotle's general strategy of argument (2.5-2.7).
In the remaining chapters of the first volume (3-6) I present more evidence for my previous findings concerning Aristotle's `semantic behaviour' by enquiring into the role of his semantic views as we find them in the several tracts of the Organon, in particular the Categories De interpretatione and Posterior Analytics. These tracts are dealt with in extenso, in order to avoid the temptation to quote selectively to suit my purposes."
de Rijk Lambertus Marie. Aristotle: semantics and ontology. Volume II: The Metaphysics, semantics in Aristotle's strategy of argument. Leiden: Brill 2002.
From the Preface to the first volume: "The lion's part of volume two (chapters 7-11) is taken up by a discussion of the introductory books of the Metaphysics (A-E) and a thorough analysis of its central books (Z-H-O). I emphasize the significance of Aristotle's semantic views for his metaphysical investigations, particularly for his search for the true ousia. By focusing on Aristotle's semantic strategy I hope to offer a clearer and more coherent view of his philosophical position, in particular in those passages which are often deemed obscure or downright ambiguous.
In chapter 12 1 show that a keen awareness of Aristotle's semantic modus operandi is not merely useful for the interpretation of his metaphysics, but is equally helpful in gaining a clearer insight into many other areas of the Stagirite's sublunar ontology (such as his teaching about Time and Prime matter in Physics). In the Epilogue (chapter 13), the balance is drawn up. The unity of Aristotelian thought is argued for and the basic semantic tools of localization and categorization are pinpointed as the backbone of Aristotle's strategy of philosophic argument.
My working method is to expound Aristotle's semantic views by presenting a running commentary on the main lines found in the Organon with the aid of quotation and paraphrase. My findings are first tested (mainly in Volume II) by looking at the way these views are applied in Aristotle's presentation of his ontology of the sublunar world as set out in the Metaphysics, particularly in the central books (ZHO). As for the remaining works, I have dealt with them in a rather selective manner, only to illustrate that they display a similar way of philosophizing and a similar strategy of argument. In the second volume, too, the exposition is in the form of quotation and paraphrase modelled of Aristotle's own comprehensive manner of treating doctrinally related subjects: he seldom discussed isolated problems in the way modern philosophers in their academic papers, like to deal with special issues tailored to their own contemporary philosophic interest."
Deslauriers Marguerite. Aristotle on definition. Leiden: Brill 2007.
Detel Wolfgang. Aristotle's logic and theory of science. In A Companion to Ancient philosophy. Edited by Gill Mary Louise and Pellegrin Pierre. Malden: Blackwell 2006. pp. 245-269
Gourinat Jean-Baptiste. Principe de contradiction, principe du tiers-exclu et principe de bivalence: philosophie première ou organon? In Logique et métaphysique dans l'Organon d'Aristote. Edited by Bastit Michel and Follon Jacques. Louvain: Peeters 2001. pp. 63-91
Hintikka Jaakko, "Commentary on James Allen The development of Aristotle's logic: part of an account in outline," Boston Area Colloquium in Ancient Philosophy 11: 206-215 (1995).
Lear Jonathan. Aristotle and logical theory. Cambridge: Cambridge University Press 1980.
Leszl Walter. Logic and metaphysics in Aristotle. Aristotle's treatment of types of equivocity and its relevance to his metaphysical theories. Padova: Antenore 1970.
Leszl Walter, "Aristotle's logical works and his conception of logic," Topoi.An Internationale Review of Philosophy 23: 71-100 (2004).
"I provide a survey of the contents of the works belonging to Aristotle's Organon in order to define their nature, in the light of his declared intentions and of other indications (mainly internal ones) about his purposes. No unifying conception of logic can be found in them, such as the traditional one, suggested by the very title Organon, of logic as a methodology of demonstration. Logic for him can also be formal logic (represented in the main by the De Interpretatione), axiomatized syllogistic (represented in the main by the Prior Analytics) and a methodology of dialectical and rhetorical discussion. The consequent lack of unity presented by those works does not exclude that both the set of works called Analytics and the set of works concerning dialectic (Topics and Sophistici Elenchi) form a unity, and that a certain priority is attributed to the analytics with respect to dialectic."
Łukasiewicz Jan. Elements of mathematical logic. Warsaw: Warsaw University 1929.
English translation by Olgierd Wojtasiewicz edited with footnotes by Jerzy Slupecki, New York, Macmillan, 1963.
Łukasiewicz Jan. Über den Satz des Widerspruchs bei Aristoteles. Hildesheim: Georg Olms 1993.
Zur Modernen Deutung der aristotelischen Logik (Band 5).
Translated from the Polish O zasadzie sprzecznosci u Arystotelesa (1910) by Jacek Barski; with a preface by Joseph Bochenski-
Translated in Italian as: Del principio di contraddizione in Aristotele - A cura di Gabriele Franci e Claudio Antonio Testi; presentazione di Maurizio Matteuzzi - Macerata, Quodlibet, 2003.
Translted in French as: Du principe de contradiction chez Aristote - Paris, Édition Éclat, 2000
Mariani Mauro, "Numerical identity and accidental predication in Aristotle," Topoi.An Internationale Review of Philosophy 19: 99-110 (2000).
Öffenberger Niels. Zur Vorgeschichte der mehrwertigen Logik in der Antike. Hildesheim: Georg Olms 1990.
Zur Modernen Deutung der aristotelischen Logik (Band 4)
Öffenberger Niels and Vigo Alejandor G. Südamerikanische Beiträge zur modernen Deutung der Aristotelischen Logik. Hildesheim: Georg Olms 1997.
Zur Modernen Deutung der aristotelischen Logik (Band 7)
Parry William and Hacker Edward. Aristotelian logic. New York: State University of New York Press 1991.
Pasquale Gianluigi. Aristotle and the principle of non-contradiction. Sankt Augustin: Academia Verlag 2006.
Traduzione italiana: Il principio di non-contraddizione in Aristotele - Torino, Bollati-Boringhieri, 2008
Perreiah Alan R., "Aristotle's axiomatic science: Peripatetic notation or pedagogical plan?," History and Philosophy of Logic 14: 87-99 (1993).
"To meet a dilemma between the axiomatic theory of demonstrative science in "Posterior analytics" and the non-axiomatic practice of demonstrative science in the physical treatises, Jonathan Barnes has proposed that the theory of demonstration was not meant to guide scientific research but rather scientific pedagogy. The present paper argues that far from contributing directly to oral instruction, the axiomatic account of demonstrative science is a model for the written expression of science. The paper shows how this interpretation accords with related theories in the "Organon", including the theories of dialectic in "Topics" and of deduction in "Prior analytics"."
Sainati Vittorio. Storia dell' "Organon" aristotelico. I: Dai "Topici" al "De Interpretatione". Firenze: Le Monnier 1968.
Sainati Vittorio. Storia dell' "Organon" aristotelico. II: L'analitica. Parte prima. La crisi epistemologica della Topica. Firenze: Le Monnier 1973.
Ristampato con il titolo: Dalla Topica all'Analitica in Teoria, 2, 1993 pp. 1-117
Sainati Vittorio, "Aristotele. Dalla Topica all'Analitica," Teoria.Rivista di Filosofia 2: 1-117 (1993).
Scritto nel 1973.
Sisson Edward, "The copula in Aristotle and afterwards," Philosophical Review 48: 57-64 (1939).
Solmsen Friedrich. Die Entwicklung der aristotelischen Logik und Rhetorik. Berlin: Weidmannsche Buchhandlung 1929.
Sorbi Luca. Aristotele: la logica comparativa. Firenze: Olschki 1999.
Due volumi: I (1999); II (2002).
Striker Gisela. Aristotle and the uses of logic. In Method in ancient philosophy. Edited by Gentzler Jyl. New York: Oxford University Press 1998. pp. 209-226
"Aristotle, as we all know, invented formal logic. Over the last fifty years or so, scholars have learned to recognize that what he presented in the first few chapters of the Prior Analytics (An. pr.) is the real thing -- a system of formal logic, whether or not the inspiration for the discovery of the syllogism had anything to do with Platonic division. We no longer hear about the magical force of the middle term or the alleged demonstrative power of first figure syllogisms as opposed to, say, the superficial subtleties of Stoic logic. Although Aristotle's syllogistic covers only a small part of' the field of modern mathematical logic, what he offered contained all the elements of a formal deductive system. He introduces the system of syllogistic moods by defining its technical terms, stating and justifying the primitive rules, and then providing formally correct proofs of the derivative rules. In other words, he developed a complete system of natural deduction, limited indeed by the assumption that all propositions must be simple subject-predicate sentences, but otherwise flawless. (1)
(...)
Aristotle was interested both in logic as a theory and in its more humdrum uses in philosophical, or indeed everyday, argument, and more than half of the text of the Prior Analytics is concerned with the uses of logic in argument, rather than with either the exposition of a formal system or what we would calf logical theory. This is what one should expect, since Aristotle invented formal logic for the purposes of his general theory of argument, not just as a formal theory of deductive proof or an 'underlying logic' for demonstrative science. (5) In order to show how the perspective of a general theory of argument differs from that of logical theory, I will argue that although syllogistic can be shown to be complete in the modern logician's sense, it was not considered by its author to be complete in the sense relevant to his project. A deduction system is complete in the modern sense if it allows one to deduce all (and only) the valid formulae.
What Aristotle has in mind when he set out to show that 'every deductive argument (sullogismos) is one of the (syllogistic) figures' (A23 40b20-22) was the claim that every valid deductive argument can be formulated as one or more syllogisms in the narrow sense. This, as Aristotle recognized, is not the case (A 44. 50b2-3). However, I will also argue that he thought syllogistic captured at least a necessary component of every valid deductive argument, and perhaps that it was indeed sufficient as an account of the logical form of scientific demonstration. Finally, I will illustrate the role of formal syllogistic in the theory of argument by a few examples from the second half of book A and from book B." pp. 210-211
(1) This summarizes the conclusion of J. Corcoran, 'Aristotle's Natural Deduction System', in idem (ed.), Ancient Logic and its Modern Interpretations (Dordrecht: Reidel. 1974), 122-3.
(5) Corcoran 'Aristotle's Natural Deduction System'. 98.
Surdu Alexandru. Aristotelian theory of prejudicative forms. Hildesheim: Georg Olms 2006.
Zur Modernen Deutung der aristotelischen Logik (Band 10)
Theron Stephen, "The interdependence of semantics, logic, and metaphysics as exemplified in the Aristotelian tradition," International Philosophical Quarterly 42: 63-91 (2002).
"We need to recognize, or to remember, the priority 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 crisscross patterns of likenesses, governs the general picture supplied here."
Viano Carlo Augusto. La logica di Aristotele. Torino: Taylor 1955.
Vuillemin Jules. De la Logique à la théologie. Cinq études sur Aristote. Paris: Flammarion 1967.
Nouvelle version remaniée et augmentée par l'auteur editée et prefacée par Thomas Benatouil - Louvain-La-Neuve, Peeters, 2008.
Wedin Michael, "Aristotle on the existential import of singular sentences," Phronesis 23: 179-196 (1978).
Wedin Michael, "Negation and quantification in Aristotle," History and Philosophy of Logic 19: 131-150 (1990).
"Two main claims are defended. The first is that negative categorical statements are not to be accorded existential import insofar as they figure in the square of opposition. Against Kneale and others, it is argued that Aristotle formulates his O statements, for example, precisely to avoid existential commitment. This frees Aristotle's square from a recent charge of inconsistency. The second claim is that the logic proper provides much thinner evidence than has been supposed for what appears to be the received view, that is, for the view that insofar as they occur in syllogistic negative categoricals have existential import. At most there is a single piece of evidence in favor of the view -- a special case of echthesis or the setting out of a case in proof."
Weidemann Hermann, "In defence of Aristotle's theory of predication," Phronesis.A Journal for Ancient Philosophy 25: 76-87 (1980).
Weidemann Hermann, "Aristotle on inferences from signs (Rhetoric I 2, 1357 b 1-25)," Phronesis 34: 343-351 (1989).
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