418 The Science of Logic
For example, there are the laws or principles of conversion. The principle of
E-conversion states that
A holds of no B if and only if B holds of no A.
Aristotle states the principle, and offers a concise argument in its favour.
Commentators were puzzled by the argument, and critics found much to
criticize. Alexander has a long discussion of the problem, in the course of
which he says this:
He shows it [i.e. E-conversion] by way of items already shown and established—I
mean ‘of every’ and ‘of no’ and ‘in as in a whole’ and ‘in no’. For it is by using these
items that he shows that non-modal universal negatives convert. For it being laid
down that A holds of no B, it follows—he says—that B follows no A. For if B holds
of some A (that is the opposite of what is laid down, and one or the other of them
must be true), let it hold of C—let C be something of A of which B holds. Then C
will be in B as in a whole and will be something of it, and B is said of every C (for
‘in as in a whole’ and ‘of every’ are the same). But C was something of A, and if it is
in it as in a whole, A will be said of all of it. But C was something of B. Therefore A
too will be predicated of something of B. But it was laid down that A is said of no B;
and to be said of no B is for there to be nothing of B of which A will be predicated.
(in APr 32.8–21)⁶⁵
According to this passage, the principle of E-conversion depends upon—and
can be shown on the basis of—the dictum de nullo.
Alexander’s presentation is convoluted. That is in part because Alexander
is an essentially convoluted chap and in part because he is commenting on
Aristotle’s text: his aim is not simply, nor even primarily, to indicate that
E-conversion can be based on the dictum—rather, it is to show that Aristotle
so bases it. The convolutions in Alexander’s text are there because he is trying
to stick close to the Analytics while insisting on the importance of the dictum.
In fact, Alexander’s interpretation is quixotic: there is no sniff of the dictum
in the pertinent text of the Analytics—andthereismorethanasniffof
⁶⁵ δείκνυσι δὲ διὰ τῶν ἐφθακότων δεδεῖχθαί τε καὶ κεῖσθαι· ἔστι δὲ ταῦτα τό τε κατὰ
παντὸς καὶ τὸ κατὰ μηδενὸς καὶ ἐν ὅλῳ καὶ ἐν μηδενί· τούτοις γὰρ προσχρώμενος δείκνυσι
τὴν τῆς καθόλου ἀποφατικῆς ὑπαρχούσης ἀντιστροφήν. κειμένου γὰρ τοῦ Α μηδενὶ τῷ Β
φησὶν ἕπεσθαι τούτῳ τὸ καὶ τὸ Β μηδενὶ τῷ Α· εἰ γὰρ τὸ Β τινὶ τῷ Α ὑπάρχει (τοῦτο γάρ
ἐστι τὸ ἀντικείμενον τῷ κειμένῳ, καὶ δεῖ τὸ ἕτερον αὐτῶν ἀληθὲς εἶναι), ὑπαρχέτω τῷ Γ·
ἔστω γὰρ τοῦτο τὶ τοῦ Α ᾧ ὑπάρχει τὸ Β. ἔσται δὴ τὸ Γ ἐν ὅλῳ τῷ Β καὶ τὶ αὐτοῦ, καὶ τὸ
Β κατὰ παντὸς τοῦ Γ· ταὐτὸν γὰρ τὸ ἐν ὅλῳ καὶ τὸ κατὰ παντός. ἀλλ᾿ ἦν τὸ Γ τὶ τοῦ Α· ἐν
ὅλῳ ἄρα καὶ τῷ Α τὸ Γ ἐστίν· εἰ δὲ ἐν ὅλῳ, κατὰ παντὸς αὐτοῦ ῥηθήσεται τὸ Α. ἦν δὲ τὸ Γ
τὶ τοῦ Β· καὶ τὸ Α ἄρα κατὰ τινὸς τοῦ Β κατηγορηθήσεται· ἀλλ᾿ ἔκειτο κατὰ μηδενὸς τὸ Α
τοῦ Β· ἦν δὲ κατὰ μηδενὸς τὸ μηδὲν εἶναι τοῦ Β καθ᾿ οὗ τὸ Α κατηγορηθήσεται.