Aristotle · Prior Analytics §1.2.8

Refutation of Premises by Conversion in the First Figure

Passage 59 of 123 · Greek

Summary

Aristotle defines refutation through "conversion" (contraposition) and examines how premises are refuted when the contradictory or contrary of the conclusion is assumed in first-figure universal and particular syllogisms.

§1.2.8Τὸ δʼ ἀντιστρέφειν ἐστὶ τὸ μετατιθέντα τό συμπέρασμα ποιεῖν τὸν συλλογισμόν ὅτι ἢ τὸ ἄκρον τῷ μέσῳ οὐχ ὑπάρξει ἢ τοῦτο τῷ τελευταίῳ.

To convert is to transpose the conclusion and produce a syllogism to the effect that either the extreme does not belong to the middle or the middle to the last extreme.

ἀνάγκη γὰρ τοῦ συμπεράσματος ἀντιστραφέντος καὶ τῆς ἑτέρας μενούσης προτάσεως ἀναιρεῖσθαι τὴν λοιπήν· εἰ γὰρ ἔσται, καὶ τὸ συμπέρασμα ἔσται.

For it is necessary, when the conclusion is converted and the other premise remains, that the remaining premise is refuted; for if it is true, the conclusion will also be true.

διαφέρει δὲ τὸ ἀντικειμένως ἢ ἐναντίως ἀντιστρέφειν τὸ συμπέρασμα· οὐ γὰρ ὁ αὐτὸς γίνεται συλλογισμός ἑκατέρως ἀντιστραφέντος·

But converting the conclusion contradictorily differs from converting it contrarily; for the same syllogism is not produced in both ways of conversion.

δῆλον δὲ τοῦτʼ ἔσται διὰ τῶν ἑπομένων.

This will be clear from what follows.

λέγω δʼ ἀντικεῖσθαι μὲν τὸ παντὶ τῷ οὐ παντὶ καὶ τὸ τινὶ τῷ οὐδενί, ἐναντίως δὲ τὸ παντὶ τῷ οὐδενὶ καὶ τὸ τινὶ τῷ οὐ τινὶ ὑπάρχειν.

I say that to be opposed contradictorily is for "belonging to all" to be opposed to "not belonging to all", and "belonging to some" to "belonging to none", while to be opposed contrarily is for "belonging to all" to be opposed to "belonging to none", and "belonging to some" to "not belonging to some".

ἔστω γὰρ δεδειγμένον τὸ Α κατὰ τοῦ Γ διὰ μέσου τοῦ Β. εἰ δὴ τὸ Α ληφθείη μηδενὶ τῷ Γ ὑπάρχειν, τῷ δὲ Β παντί, οὐδενὶ τῷ Γ ὑπάρξει τὸ Β. καὶ εἰ τὸ μὲν Α μηδενὶ τῷ Γ, τὸ δὲ Β παντὶ τῷ Γ, τὸ Α οὐ παντὶ τῷ Β καὶ οὐχ ἀπλῶς οὐδενί· οὐ γὰρ ἐδείκνυτο τὸ καθόλου διὰ τοῦ ἐσχάτου σχήματος.

For let A be proved of C through the middle term B. If, then, A is assumed to belong to no C, but to all B, B will belong to no C. And if A belongs to no C, and B to all C, A will not belong to all B, and not simply to none; for the universal was not proved through the last figure.

ὅλως δὲ τὴν πρὸς τῷ μείζονι ἄκρῳ πρότασιν οὐκ ἔστιν ἀνασκευάσαι καθόλου διὰ τῆς ἀντιστροφῆς· ἀεὶ γὰρ ἀναιρεῖται διὰ τοῦ τρίτου σχήματος· ἀνάγκη γὰρ πρὸς τὸ ἔσχατον ἄκρον ἀμφοτέρας λαβεῖν τὰς προτάσεις.

In general, it is not possible to refute the premise containing the major extreme universally through conversion; for it is always refuted through the third figure, since it is necessary to take both premises in relation to the last extreme.

καὶ εἰ στερητικὸς ὁ συλλογισμός, ὡσαύτως.

And if the syllogism is negative, likewise.

δεδείχθω γὰρ τὸ μηδενὶ τῷ Γ ὑπάρχον διὰ τοῦ Β. οὐκοῦν ἂν ληφθῇ τὸ τῷ Γ παντὶ ὑπάρχειν, τῷ δὲ Β μηδενί, οὐδενὶ τῷ Γ τὸ Β ὑπάρξει.

For let A be proved to belong to no C through B. Therefore, if A is assumed to belong to all C, and to no B, B will belong to no C.

καὶ εἰ τὸ Α καὶ τὸ Β παντὶ τῷ Γ, τὸ τινὶ τῷ Β· ἀλλʼ οὐδενὶ ὑπῆρχεν.

And if A and B belong to all C, A will belong to some B; but it belonged to none.

Ἐὰν δʼ ἀντικειμένως ἀντιστραφῇ τὸ συμπέρασμα, καὶ· οἱ συλλογισμοὶ ἀντικείμενοι καὶ οὐ καθόλου ἔσονται.

But if the conclusion is converted contradictorily, the syllogisms will also be contradictory and not universal.

γίνεται γὰρ ἡ ἑτέρα πρότασις ἐν μέρει, ὥστε καὶ τὸ συμπέρασμα ἔσται κατά μέρος.

For the other premise becomes particular, so that the conclusion will also be particular.

ἔστω γὰρ κατηγορικὸς ὁ συλλογισμός, καὶ ἀντιστρεφέσθω οὕτως.

For let the syllogism be affirmative, and let it be converted in this way.

οὐκοῦν εἰ τὸ Α οὐ παντὶ τῷ Γ, τῷ δὲ Β παντί, τὸ Β οὐ παντὶ τῷ Γ· καὶ εἰ τὸ μὲν Α μὴ παντὶ τῷ Γ, τὸ δὲ Β παντί, τὸ Α οὐ παντὶ τῷ Β. ὁμοίως δὲ καὶ εἰ στερητικὸς ὁ συλλογισμός.

Therefore, if A does not belong to all C, but to all B, B will not belong to all C; and if A does not belong to all C, and B to all C, A will not belong to all B. And likewise if the syllogism is negative.

εἰ γὰρ τὸ Α τινὶ τῷ Γ ὑπάρχει, τῷ δὲ Β μηδενί, τὸ Β τινὶ τῷ Γ οὐχ ὑπάρξει, οὐχ ἁπλῶς οὐδενί· καὶ εἰ τὸ μὲν Α τῷ Γ τινί, τὸ δὲ Β παντί, ὥσπερ ἐν ἀρχῇ ἐλήφθη, τὸ τινὶ τῷ Β ὑπάρξει.

For if A belongs to some C, and to no B, B will not belong to some C, and not simply to none; and if A belongs to some C, and B to all, as was assumed at the beginning, A will belong to some B.

Ἐπὶ δὲ τῶν ἐν μέρει συλλογισμῶν ὅταν μὲν ἀντικειμένως ἀντιστρέφηται τὸ συμπέρασμα, ἀναιροῦνται ἀμφότεραι αἱ προτάσεις, ὅταν δʼ ἐναντίως, οὐδετέρα.

But in particular syllogisms, when the conclusion is converted contradictorily, both premises are refuted, but when contrarily, neither.

οὐ γὰρ ἔτι συμβαίνει, καθάπερ ἐν τοῖς καθόλου, ἀναιρεῖν ἐλλείποντος τοῦ συμπεράσματος κατὰ τὴν ἀντιστροφήν, ἀλλʼ οὐδʼ ὅλως ἀναιρεῖν.

For it no longer happens, as in universal syllogisms, that refutation occurs because the conclusion falls short in the conversion, but there is no refutation at all.

δεδείχθω γὰρ τὸ Α κατὰ τινὸς τοῦ Γ. οὐκοῦν ἂν ληφθῇ τὸ Α μηδενὶ τῷ Γ ὑπάρχειν, τὸ δὲ Β τινί, τὸ Α τῷ Β τινὶ οὐχ ὑπάρξει· καὶ εἰ τὸ Α μηδενὶ τῷ Γ, τῷ δὲ Β παντί, οὐδενὶ τῷ Γ τὸ Β. ὥστʼ ἀναιροῦνται ἀμφότεραι.

For let A be proved of some C. Therefore, if A is assumed to belong to no C, and B to some C, A will not belong to some B; and if A belongs to no C, and to all B, B will belong to no C. So both are refuted.

ἐὰν δʼ ἐναντίως ἀντιστραφῇ, οὐδετέρα.

But if it is converted contrarily, neither is refuted.

εἰ γὰρ τὸ τινὶ τῷ Γ μὴ ὑπάρχει, τῷ δὲ Β παντί, τὸ Β τινὶ τῷ Γ οὐχ ὑπάρξει, ἀλλʼ οὔπω ἀναιρεῖται τὸ ἐξ ἀρχῆς· ἐνδέχεται γὰρ τινὶ ὑπάρχειν καὶ τινὶ μὴ ὑπάρχειν.

For if A does not belong to some C, but B belongs to all, B will not belong to some C, but the original premise is not yet refuted; for it is possible to belong to some and not to belong to some.

τῆς δὲ καθόλου, τῆς Β, ὅλως οὐδὲ γίνεται συλλογισμός· εἰ γὰρ τὸ μὲν Α τινὶ τῷ Γ μὴ ὑπάρχει, τὸ δὲ Β τινὶ ὑπάρχει, οὐδετέρα καθόλου τῶν προτάσεων.

And for the universal premise B, no syllogism is produced at all; for if A does not belong to some C, and B belongs to some, neither of the premises is universal.

ὁμοίως δὲ καὶ εἰ στερητικὸς ὁ συλλογισμός· εἰ μὲν γὰρ ληφθείη τὸ Α παντὶ τῷ Γ ὑπάρχειν, ἀναιροῦνται ἀμφότεραι, εἰ δὲ τινί, οὐδετέρα.

And likewise if the syllogism is negative; for if A is assumed to belong to all C, both are refuted, but if to some, neither.

ἀπόδειξις δʼ ἡ αὐτή.

And the proof is the same.

Notes

1.2.8τῷ δὲ Β παντί — Meaning "and to all B", where the subject `τὸ Α` and the infinitive `ὑπάρχειν` are understood from the context ("A belongs to all B"). The dative `τῷ Β` is governed by `ὑπάρχειν`.
1.2.8ὥσπερ ἐν ἀρχῇ ἐλήφθη — The adverbial phrase "as was assumed at the beginning" modifies the immediately preceding clause `τὸ δὲ Β παντί` (B belongs to all C), indicating that this is identical to the original premise (the minor premise of the first figure).
p.87ἐλλείποντος τοῦ συμπεράσματος κατὰ τὴν ἀντιστροφήν — A genitive absolute construction. It explains that in the process of conversion, the resulting opposed conclusion falls short in logical strength (to refute the original premise) or is less than universal.

Cite this passage

Aristotle, Prior Analytics §1.2.8. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg0086.tlg001.humanitext-grc2:1.2.8

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