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Aristotle · Prior Analytics §1.2.6

Circular Proof and Proving Negative Premises in the Second Figure

Passage 57 of 123 · Greek

Summary

Discusses the applicability of circular proof in the second figure, demonstrating that negative premises can be proved while affirmative ones cannot, and explains the conditions under which premises in particular syllogisms can be proved.

§1.2.6Ἐν δὲ τῷ δευτέρῳ σχήματι τὸ μὲν καταφατικὸν οὐκ ἔστι δεῖξαι διὰ τούτου τοῦ τρόπου, τὸ δὲ στερητικὸν ἔστιν.
In the second figure, it is not possible to prove the affirmative premise in this way, but it is possible to prove the negative one.
τὸ μὲν οὖν κατηγορικόν οὐ δείκνυται διὰ τὸ μὴ ἀμφοτέρας εἶναι τὰς προτάσεις καταφατικάς· τό γὰρ συμπέρασμα στερητικόν ἐστι, τὸ δὲ κατηγορικὸν ἐξ ἀμφοτέρων ἐδείκνυτο καταφατικῶν.
The affirmative premise is not proved because both premises do not become affirmative; for the conclusion is negative, and the affirmative premise was proved from both being affirmative.
τὸ δὲ στερητικὸν ὧδε δείκνυται.
But the negative premise is proved as follows.
ὑπαρχέτω τὸ Α παντὶ τῷ Β, τῷ δὲ Γ μηδενί· συμπέρασμα τὸ Β οὐδενὶ τῷ Γ. ἐὰν οὖν ληφθῇ τὸ Β παντὶ τῷ Α ὑπάρχον, ἀνάγκη τὸ Α μηδενὶ τῷ Γ ὑπάρχειν· γίνεται γὰρ τὸ δεύτερον σχῆμα·
Let A belong to every B, and to no C; the conclusion is that B belongs to no C. If, then, B is assumed to belong to every A, it is necessary for A to belong to no C; for the second figure is produced, and B is the middle term.
μέσον τὸ Β. εἰ δὲ τὸ Β στερητικὸν ἐλήφθη, θάτερον δὲ κατηγορικόν, τὸ πρῶτον ἔσται σχῆμα.
But if B was assumed as negative, and the other as affirmative, the first figure will be produced.
τὸ μὲν γὰρ Γ παντὶ τῷ Α, τὸ δὲ Β οὐδενὶ τῷ Γ, ὥστʼ οὐδενὶ τῷ τὸ Β· οὐδʼ ἄρα τὸ Α τῷ Β. διὰ μὲν οὖν τοῦ συμπεράσματος καὶ τῆς μιᾶς προτάσεως οὐ γίνεται συλλογισμός, προσληφθείσης δʼ ἑτέρας ἔσται.
For C belongs to every A, and B to no C, so that B belongs to no A; therefore neither does A belong to B. Therefore, a syllogism is not produced through the conclusion and the one premise, but there will be one if another is added.
εἰ δὲ μὴ καθόλου ὁ συλλογισμός, ἡ μὲν ἐν ὅλῳ πρότασις οὐ δείκνυται διὰ τὴν αὐτὴν αἰτίαν ἥνπερ εἴπομεν καὶ πρότερον, ἡ δʼ ἐν μέρει δείκνυται, ὅταν ᾖ τὸ καθόλου κατηγορικόν·
But if the syllogism is not universal, the universal premise is not proved for the same reason that we also said before, but the particular one is proved when the universal premise is affirmative.
ὑπαργέτω γὰρ τὸ Α παντὶ τῷ Β, τῷ δὲ Γ μὴ παντί· συμπέρασμα Β Γ ἐὰν οὖν ληφθῇ τὸ Β παντὶ τῷ Α, τῷ δὲ Γ οὐ παντί, τὸ Α τινὶ τῷ οὐχ ὑπάρξει·
For let A belong to every B, and not to every C; the conclusion is about B and C [i.e., that B does not belong to some C]. If, then, B is assumed to belong to every A, and not to every C, A will not belong to some C; the middle term is B.
μέσον Β. εἰ δʼ ἐστὶν ἡ καθόλου στερητική, οὐ δειχθήσεται ἡ Α Γ πρότασις ἀντιστραφέντος τοῦ Α Β· συμβαίνει γὰρ ἢ ἀμφοτέρας ἢ τὴν ἑτέραν πρότασιν γίνεσθαι ἀποφατικήν, ὥστ᾿ οὐκ ἔσται συλλογισμός.
But if the universal premise is negative, the AC premise will not be proved when AB is converted; for it happens that either both premises or one of them becomes negative, so that there will be no syllogism.
ἀλλʼ ὁμοίως δειχθήσεται ὡς καὶ ἐπὶ τῶν καθόλου, ἐὰν ληφθῇ, ὧ τὸ Β τινὶ μὴ ὑπάρχει, τὸ τινὶ ὑπάρχειν.
But it will be proved in the same way as in the case of universal syllogisms, if it is assumed that to whatever B does not belong to some, [A] belongs to some.

Notes

  1. 15διὰ τὸ μὴ ἀμφοτέρας εἶναι τὰς προτάσεις καταφατικάς — To prove an affirmative premise (τὸ κατηγορικόν), both premises of the new syllogism obtained in the process of circular proof must be affirmative (καταφατική), but this is impossible because the conclusion of the original second figure is negative.
  2. 25ὥστʼ οὐδενὶ τῷ τὸ Β — The term Α is omitted after the definite article τῷ. The context clearly indicates the conclusion of the first-figure Celarent: "B belongs to no A" (τὸ Β οὐδενὶ τῷ Α ὑπάρχειν).
  3. 30συμπέρασμα Β Γ — A highly condensed notation representing the particular negative conclusion "B does not belong to some C," where B is the subject and C is the predicate.
  4. 30τὸ Α τινὶ τῷ οὐχ ὑπάρξει — The term Γ is omitted after the definite article τῷ. This means "A will not belong to some C," which circularly derives one of the premises of the second-figure Baroco.

Cite this passage

Aristotle, Prior Analytics §1.2.6. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg0086.tlg001.humanitext-grc2:1.2.6

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