Abstract
It has been said that there is no scholarly consensus as to why Aristotle’s logics of proof and refutation would have borne the title Analytics. But if we consulted Tarski’s (Introduction to logic and the methodology of deductive sciences, Oxford University Press, New York, 1941) graduate-level primer, we would have the perfect title for them: Introduction to logic and to the methodology of deductive sciences. There are two strings to Aristotle’s bow. The methodological string is the founding work on the epistemology of science, and the logical string sets down conditions on the proofs that bring this knowledge about. The logic of proof presents a difficulty whose solution exceeds its theoretical reach. The logic of refutation takes the problem on board, and advances a solution whose execution is framed by fallacy-avoidance at the beginning and fallacy-adoption at the end. But with a difference: the avoidance-fallacies are of Aristotle’s own conception, whereas the adoption-fallacies, so judged in the modern tradition, aren’t fallacies at all for Aristotle. The avoidance-fallacies are begging the question and ignoratio elenchi, and the adoption-fallacies, fallacies in name only, are the ad hominem and ad ignorantiam, an inductive turning in the first instance, and an abductive finish in the second.
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Notes
By John Corcoran, for example, in his Logic School lectures on Aristotle’s demonstrative logic in conjunction with UNILOG 2015, the 2015 conference on Universal Logic meeting at the University of Istanbul. In these same lectures, Corcoran reported that the first known axiomatization of plane geometry was made by Thales in the 6th century BC.
John Corcoran (2003, p. 261).
Corcoran (2009). See also Angioni and Zuppolini (2019).
It could be that, irrespective of its subject-matter, Aristotle took a science to be axiomatizable if it has well-developed proof methods and/or some widely-shared unproved postulates framed in a way that gives rise to lawlike generalizations. In a word, if it is mature.
To accommodate formalist preferences, for example. For more on these differences in how axioms have been conceived of, see Blanchette (2019).
Contrary to Łukasiewicz (195). Ebbinghaus, Corcoran and Smiley frame the syllogistic in modern natural deduction terms. Clearly an improvement, Aristotle sees logic as philosophy, and philosophy in turn not as a science, but rather as a Speculative Art.
In Aristotle’s case, it is taken as given that all relevant terms are clear without express definition or have been well-defined in the run-up to axiomatic remodelling. Frege, whose Grundgesetze bears a striking structural similarity to Aristotle’s framework, gave definitions an express (and creative) role in the axiomatization process. But we needn’t take that further here.
The scholarly consensus is that the word ‘logic’ was first applied to Aristotle’s writings by Alexander of Aphrodisias (2nd-3rd century), in Wallies (1881–1883).
Notably, the concept of syllogism.
Since it is in no sense possible for a logical truth S* to be false, it is in no sense possible for a proposition S to be true and S* not true. On the biconditional definition, this difficulty cannot be avoided.
By parity of argument, if it is in no sense possible for ‘S and not-S’ to be true, there is no sense in which arbitrary S′ is true and “S and not-S” is not true.
It is true that Aristotle provides selective proofs of his own, as for example, the proof of the reducibility of syllogistic schemata in all figures to one or other of the first-figure schemata. For example, at Prior Analytics, 27a 9–12, Aristotle reduces a second figure syllogism by propositional conversion, and at 27a 36–27b he reduces a third figure syllogism “through the impossible.” I will come back to this in Sect. 1.4.
Harman (1970). Harman’s main objective was to blow the whistle on the inadequacy of the rules of the probability calculus for inductive inference. The deductive detour was his amuse-bouche to get the critical juices flowing.
Consider writing the advance manual that would regulate the proof of the binomial theorem and also as the unsolvability proof for Hilbert’s tenth problem.
The angles flank an ordered pair of elements. The curly brackets enclose the arguments premiss(es). The rightmost element is the argument’s conclusion.
It is even more promiscuous than that. From the commutativity of addition we have it that addition is commutative or beautiful Auntie Bea went swimming today in Monaco or the Blue Jays whipped the Orioles on Wednesday or the real numbers are uncountably many or … or …. We should note that principle R gives the truth conditions for inclusive alternation. That’s why we can simplify the discussion by switching to “or”.
The quotation marks are advisory. In no literal sense do propositions do anything. Propositions lack agency.
This, the arbitrary promissory expansion of valid arguments is disallowed.
To be a theorem of a given science, a proposition’s ancestral line must originate with an axiom. But no axiom is itself a theorem since no axiom is its own ancestor.
The fewer the axioms that get the job done, the less likely that an axiom will turn out to have been provable, hence no axiom at all.
Which is the whole point of a theory of scientific knowledge.
Hans Hansen points out in correspondence, it is difficult to square this dismissal with Aristotle’s non-cause as cause fallacy. See here John Woods and Hans V. Hansen, “The subtleties of Aristotle on Non-Cause”, Logique et Analyse, 44 (2001), 325–415.
Pr. An. 25a 15–17; 28a 24–26; 28b 14–15; 28b 20–21; and 30a 9–10.
Corcoran writes: “In his recent book Aristotle’s Earlier Logic, John Woods 2001 presents further evidence that Aristotle addressed these issues in earlier works. (p. 263, n. 5)”.
Pr. An. I 4–6, but slightly adjusted upward in I7. We need not determine the exact number here. The most famous schema is Barbara: 〈{“Every A is B”, “Every B is C’}, “Every A is C”〉.
According to one modern critic, Frege’s approach to axioms − hence Aristotle’s too −was “a dinosaur” Blanchette (2014, p. 201.)
For a further discussion of why a first principle must itself be an unprovable truth, readers could consult Woods (2019; p. 31.) The Corcoran reference is to that book’s 2001 edition, which contains material on the turn to dialectic (Chap. 8) which, as I now regret, wasn’t retained in the second edition. The translation is Tredennick’s in Aristotle (1960).
Pasch, (1882).
Frege (1980), p. 37, The remark cited here is from the Frege-Hilbert correspondence in the interval 1895–1903.
Aristotle’s axiomatics preceded Euclid’s Elements by two generations.
Philosophical Papers, p. 2. Aristotle took pains with question-begging. For details, readers could consult Woods and Walton (1982).
Locke (1962). Locke’s purpose was to direct them to others so as “to prevail on their assent, or at least so awe them as to silence their opposition. For Aristotle, see Soph. Ref. 22 178b 17: 10, 170a 13, 17–18, 20;20. 177b 33–38; 33 183a 22, 24, 8; Top. 161a.
It is worth noting that a perfectly made syllogism can be a failed type of argument. This necessitates a distinction between a syllogism-as such and a syllogism-in use. Arguments satisfying the definition of syllogisms are, just so, syllogism-as such. Further conditions bind syllogism-in use, and these vary with the argument’s objectives and the procedural means of bringing them to heel. Discussion of the distinction pervades Woods (2014).
Named so for Locke (1962), but not as a fallacy.
Still, it must be noted, that substantial efforts by scholars of high standing have been made to bring the notions of Pr. An. 2.25 into fuller theoretical bloom. It might well be that, in due course, those efforts will achieve fruitful landing in the dialectical logic of refutation. But for the present, I think that we must wait to see. See for example Magnani (2017), especially Chap. 5, and Bellucci (2019), and a reply by Woods at pp. 565–570. Closer to the mark among recent writings, I see Park (2021).
References
Alexander of Aphrodisias (1881–1883). In Aristotelis topicorum libros octo commentaria, and In Aristotelis analyticorum priorum librum I commentarium, ed. M. Wallies, Commentaria in Aristotelum Graeca, Berlin: Reimer.
Aristotle. 1984. The Complete Works of Aristotle: the revised English translation, two volumes, ed. Jonathan Barnes, Princeton: Princeton University Press.
Aristotle. 1960. Aristotle, Posterior Analytics, Topics, edited and translated in the first instance by Hugh Tredennick and, in the second, by E. S. Forster, Cambridge, MA: Harvard University Press.
Bellucci, Francesco. 2019. Abduction in Aristotle. In D. M. Gabbay, Lorenzo Magnani, Woosuk Park, and Ahti-Veikko Pietarinen editors, Natural Arguments, volume. 40 of Tributes, pp. 551–564, London: College Publications.
Blanchette, Patricia A. 2019. Axioms in Frege. In Essays on Frege’s basic laws of arithmetic, eds. Philip A. Ebert, and Marcus Rossberg, pp. 31–56. Oxford: Oxford University Press.
Caret, Colin R., and Ole T. Hjortland. 2015. Foundations of Logical Consequence. Oxford: Oxford University Press.
Corcoran, John. 1972. Completeness of an ancient logic. Journal of Symbolic Logic 37: 696–702.
Corcoran, John. 2003. Aristotle’s prior analytics and Boole’s. Laws of Thought History and Philosophy of Logic 24: 261–288.
Corcoran, John. 2009. Aristotle’s demonstrative logic. History and Philosophy of Logic 30: 1–20.
Ebbinghaus, Kurt. 1964. Ein formales Modell der Syllogistik des Aristotles. Gottingen: Vanderhoek and Ruprecht.
Etchemendy, John. 1990. The Concept of Logical Consequence. Cambridge, MA: Harvard University Press.
Field, Hartry. 2008. Saving Truth from Paradox. Oxford: Oxford University Press.
Frege, Gottlob. 1979. On Euclidean geometry (1899–1906), in Frege, Posthumous Writings, Hans Hermes, Friedrich Kambartel and Friedrich Kaulbach, editors, translated by Peter Long and Roger White, pages 167–196, Oxford: Basil Blackwell.
Frege, Gottlob. 1980. Philosophical and Mathematical Correspondence, abridged from the German by Brian MacGuinness and translated by Hans Kaal, pages 31–52, Oxford: Basil Blackwell.
Harman, Gilbert. 1970. Induction: a discussion of the relevance of knowledge to the theory of induction. In Induction, Acceptance and Rational Belief, ed. Marshall Swain, 83–99. Dordrecht: Reidel.
Locke, John. 1961. An essay concerning Human understanding, edited and with an introduction by John W. Yolton, in two volumes. London: Dent. First published in 1690.
Łukasiewicz, Jan. 1957. Aristotle’s Syllogistic From the Standpoint of Modern Logic, 2nd edition, Oxford: Clarendon Press.
Magnani, Lorenzo. 2017. The Abductive structure of Scientific Creativity: An Essay on the Ecology of Cognition. Cham: Springer.
Moore, Robert C. 1983. Semantical considerations on non-monotonic logic. In Proceedings of the IJCAI, pp. 272–279, Karlsruhe: Morgan Kaufmann.
Moore, Robert C. 1988. Autoepistemic logic. In Non-Standard Logics for Automated Reasoning, ed. Philip Smets, E. H. Mandani, Didier Dubois and Henri Prade, pp. 105–136. New York: Academic Press.
Park, Woosuk. 2021. On abducing the axioms of mathematics. In Abduction in Cognition and Action: Scientific Inquiry and Social Practice, ed. John R. Shook, and Sami Paavola, pp. 161–175. Cham: Springer.
Pasch, Moritz. 1882. Vorlesungen über neuere Geometrie. Leipzig: Teubner.
Ramsey, F. P. 1990. Philosophical Papers, ed. D. H. Mellor, Cambridge: Cambridge University Press.
Smiley, Timothy J. 1973. “What is a syllogism?”. Journal of Philosophical Logic 2: 136–154.
Tarski, Alfred. 1941. Introduction to Logic and the Methodology of Deductive Sciences. New York: Oxford University Press.
Woods, John. 2001. Aristotle’s earlier logic. Oxford: Hermes Science.
Woods, John. 2021. What did Frege take Russell to have proved?. Synthese 198: 3949–3977. published online 22 July 2019.
Woods, John, and V. Hansen Hans. 2001. The subtleties of Aristotle on non-cause. Logique et Analyse 44: 325–415.
Woods, John. 2022. How abduction fares in mathematical space. In Handbook of Abductive Cognition, ed. Lorenzo Magnani, Cham: Springer.
Woods, John, and Douglas Walton. 1982. “The petitio: Aristotle’s five ways”. Canadian Journal of Philosophy 12: 77–100.
Woods, John. 2014. Aristotle’s Earlier Logic, 2nd edition revised and expanded, vol. 53 of Studies in Logic, London: College Publications.
Woods, John. 2023. Deduction’s Sovereign Spaces, and How They Matter for Proof.
Acknowledgements
For instructive example and generous assistance on matters discussed in this essay, I warmly thank in alphabetical order Francesco Bellucci, George Boger, Luca Castagnoli, John Corcoran, Paolo Fait, Hans V. Hansen, David Hitchcock, Jaakko Hintikka, Erik Krabbe, Lorenzo Magnani, Woosuk Park, Christos Pechlivanides, Larry Powers and Alirio Rosales. I am especially indebted to an anonymous referee for scrupulous and helpful assessment, and to Hans V. Hansen again, guest-editor of the present special issue, for his rich scholarship and constructive forbearance.
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In Memoriam: I dedicate this essay to the memory of John Corcoran, March 8th 1937–January 8th, 2021.
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Woods, J. Fallacies and Their Place in the Foundations of Science. Argumentation 37, 181–199 (2023). https://doi.org/10.1007/s10503-023-09609-6
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DOI: https://doi.org/10.1007/s10503-023-09609-6