ARISTOTLE S PROOFS BY ECTHESIS

Transcription

1 Ewa Żarnecka Bia ly ARISTOTLE S PROOFS BY ECTHESIS 1. The proofs by ecthesis (εχϑεσιζ translated also as the English exposition and the German Heraushebung) are mentioned in a few fragments of Aristotle s Prior Analytics, without however a detailed explanation of the nature of this kind of proof. At first, the ecthetic proofs are suggested to be an alternative way of proving assertoric III Figure moods: Darapti, Datisi, Disamis and Bocardo; later, for two modal moods Baroco II and Bocardo (with apodeictic premisses) this way of proving is recommended as the only one (reductio ad impossibile has been found to be inapplicable in those cases). There were some discussions and quarrels among Aristotle s early commentators about his proofs by ecthesis and the problem perhaps still remains intriguing. 2. Let SYL be a reconstruction of Aristotle s assertoric syllogistic described in a natural deduction style 1 as follows: Language. a) name symbols: S, P, M,..., and operators: a, e, i, o used as in traditional reconstructions. b) individual terms x*, y*..., to be used (together with symbol) in the course of proving only, and with restrictions as usually applied in natural deduction systems with the rule of existential instantiating. 1. Basic Rules. R Barbara MaP R Celarent MeP SaM SaM SeP 1 Cf. Lear s interpretation in [2]. 40

2 2. Conversion Rules. R e SeP R a R i SiP PeS PiS PiS 3. Ecthetic Rules. RE /a ( a Elimination) x S & x P RE /i ( i Elimination) RE ( o Elimination) SiP x S & x P x S & x P (on the supposition that x did not appear earlier in the proof). RE /& ( & Elimination) α & β α α & β β (α, β are atomic ecthetic formulas like x S, x S) RI /i ( i Introduction) RI /o ( o Introduction) x S, x P SiP x S, x P R Barbara R Camestres R Celarent x S x P SeP x S x P x S x P 41

3 An Example. The ecthetic proof in SYL for Darapti. 1. MaP Darapti s major premiss 2. MaS Darapti s minor premiss 3. x M & x P 1, RE /a 4. x M 3, RE /& 5. x S 2,4, R Barbara 6. x P 3, RE /& 7. SiP 5,6, RI /i As for other places, where ecthesis intervenes in the proofs of other assertoric syllogistic moods in the Prior Analytics, the following R rules can be used: Moode, Figure Baroco II: Darapti III: Datisi III: Disamis III: Bocardo III: R rules R Camestres, RE /o, RI /o R Barbara, RE /a, RI /i R Barbara, RE /i, RI /i R Barbara, RE /i, RI /i R Barbara, RE /o, RI /o All the R proofs are here coherent with the Aristotelian comments on ecthesis; it is to be noticed, however, that the motivation for using R rules for moods with i conclusion seems more convincing than that for Baroco and Bocardo. 3. Now let us take SYL as another reconstruction of Aristotle s syllogistic with ecthesis: SYL differs from SYL in that it has a new ecthetic rule (RE /o) instead of all the R rules. The RE /o rule introduces the new syllogistic term of the same syntactic category as traditional syllogistic terms. 2 2 Cf. Mostowski s set theoretical interpretation of Aristotle s syllogistic in [4] and his definition of a relation. 42

4 RE /o X as & X ep The ecthetic proofs for Baroco and Bocardo rewritten in SYL are also in agreement with Aristotle s ecthetic instruction and, what more, also with the Galenian interpretation perhaps preserving Aristotle s tradition. On the other hand an attempt to follow this R style of ecthesis for Darapti would result in circularity ( Lukasiewicz and Patzig had avoided it, but alas! none of them obeyed Aristotle s instruction or intuitions, so that they did not solve the question of how the ARISTOTELIAN concept of ecthesis is to be understood. 4. The comparison of results obtainable within SYL and SYL supports the hypothesis that there were two ways of dealing with ecthesis, both of them present in Prior Analytics: (i) the updated one elaborated as essential for the above mentioned modal Baroco and Bocardo ; and (ii) an earlier one as suggested for Darapti, quite probably rooted in some (forgotten or suppressed) practices of visualization [Venn style PRA DIAGRAMS?] of syllogistic relations. 5. Suppose that for SYL not only R rules are allowed, but also indirect proofs (per impossibile): then the basic rules (R1 Barbara and Celarent) would be no longer neeeded, and the list of R rules could be limited to the following four: RE /i, RE /o, R Barbara and R Celarent. This would work against Aristotle s style of grouping syllogisms into three figures, with the perfect first figure moods. Still, just at the very beginning of constructing his system, Aristotle justifies the operation of e conversion, using indirect argumentation and AS IF using ecthesis at the same time (the word did not happen to be used in this context). It seems to count as an extra argument for the more archaic origin of ecthesis, later on re elaborated by Aristotle under radically different circumstances. One can find that in Prior Analytics Aristotle sometimes uses individ- 43

5 ual terms (e.g. proper names Socrates, Kalias... ); sometimes he also uses his variable like letters as referring to the individuals (instead of common names). But Aristotle had no motivation to pay more attention to the individual terms and in some contexts perhaps he could treat analogs of our x X, {x } ax, and of our X ax as working in the same way. Thus perhaps he would be a bit surprised if somebody taught him about so many ecthetic rules as enlisted in our SYL : surprised, but not disgusted. References [1] Aristotle s Prior and Posterior Analytics, W.D. Ross, Oxford, [2] J. Lear, Aristotle and Logical Theory, Cambridge University Press, [3] J. Lukasiewicz, Aristotle s Syllogistic from the Standpoint of Modern Formal Logic, Oxford, [4] A. Mostowski, Logika matematyczna, Warszawa Wroc law, [5] G. Patzig, Aristotle s theory of the Syllogism. A Logico Philological Study of Book A of the Prior Analytics, D. Reidel, Dordrecht, [6] E. Żarnecka Bia ly, Premonition of Mathematical Logic in Aristotle s Prior Analytics, Logic Counts (E. Żarnecka Bia ly, ed.), Dordrecht, Department of Logic Jagiellonian University Grodzka Kraków Poland 44