equiangular triangle has sides of different sizes). If it is proved that: No unequilateral triangle is equiangular, it follows by mere logical conversion, that: No equiangular triangle is unequilateral (every equiangular triangle is equilateral). No square has a diagonal commensurable with one of the sides; and no figure, with a diagonal commensurable with one of the sides, is a square. We may take the theory of parallels as an example of the conversion of the corresponding hypothetical judgment. The proposition may be proved (it may be known. without the aid of the eleventh axiom of Euclid): If two straight lines (in one plane) are so intersected by any third that the corresponding angles are equal to each other, or that the interior angles on the same side of the intersecting line are together equal to two right angles, these lines will never meet in any one point. It follows, by mere conversion, without the necessity of going back upon the mathematical construction: If two straight lines (in one plane) meet in any point, they are never so intersected by any third that the corresponding angles are equal to each other, or that the interior angles, on the same side of the intersecting line, are together equal to two right angles. (In other words: Two angles in any triangle are never together equal to two right angles. But it cannot be asserted in this way that these two angles together with the third make two right angles; nor that, when the intersected lines do not meet, the corresponding angles are equal to each other.)

Aristotle holds that the universal negative judgment of possibility does not admit of altogether simple conversion: Anal.. Pr. i. c. iii. : ὅσα δε τῷ ὡς ἐπὶ πολὺ καὶ τῷ πεφυκέναι λέγεται ἐνδέχεσθαι—ἡ μὲν καθόλου στερητικὴ πρότασις οὐκ ἀντιστρέφει, ἡ δ' ἐν μέρει ἀντιστρέφει cf. c. xiii. c. xvii.: ὅτι οὐκ ἀντιστρέφει τὸ ἐν τῷ ἐνδέχεσθαι στερητικόν. If the judgment is given: τὸ Α ἐνδέχεται μηδενὶ τῷ B, it does not necessarily follow that τὸ Β ἐνδέχεσθαι μηδενὶ τῷ Α. Aristotle understands the first proposition in this sense: Every B, each by itself, is in the state of possibility to have or not to have A for a predicate. He understands the second proposition similarly, in this sense:

Every A, each by itself, is in the state of possibility to have or not to have B for a predicate (cf. § 98). Now the case may occur, as Aristotle rightly remarks, where all B are in that double state of possibility, while some A are in the state of necessity, not to have в for a predicate. Hence the Schema is :

In cases of this kind the first judgment (τὸ ἡ ἐνδέχεται μηδενὶ τῷ B) is true, and the second (Tò B évdéxetai μndevì TO A) is false. Hence the second is not the necessary consequence of the first. In this sense Aristotle's doctrine is well founded. But it does not contradict our proposition (which Theophrastus and Eudemus had recognised'), that universal negative propositions of any modality, and consequently the problematical, are converted with Quantity, Quality, and Modality unchanged. The contradiction is not overcome by the circumstance that the Aristotelian vdéxeolar does not denote subjective uncertainty like the perhaps of the problematic judgment, but the objective possibility of Being and Not Being, more especially (in distinction from dúvao@ai) in the sense of there being nothing to hinder it. For the argument of Aristotle remains correct, if subjective uncertainty be substituted for the objective possibility. If it is uncertain of all B, whether they are or are not A, it does not follow that it must be uncertain of all A, whether they are or are not B. The certainty that they are not в may exist of some A. But this does not prejudice the above demonstration that from the proposition: Perhaps no в is A, the proposition follows: Perhaps no A is B. For this last propoa sition is not equivalent to that, which can not be deduced: It is uncertain of all A, and of each one by itself, whether they are or are not B. It is equivalent to the following: It is uncertain whether all A are not B, or whether there be at least

1 Cf. Prantl, Gesch. der Log. i. 364.

any one a which is B. And this proposition can very well exist along with the certainty that some A are not B. Similarly, from the proposition: It is (objectively) possible that no B is A, the proposition follows necessarily : It is (objectively) possible that no A is B (while it is also possible that at least some one a is B). Conversion in the Aristotelian way, according to which the possibility not to be B is adjudged to every individual ▲, holds good (as Aristotle himself shows') in two cases-(1) when by ἐνδέχεσθαι is understood what might be expressed by it όμω výμws: to be at least in the state of possibility, without exclusion of the necessity; and (2) where all necessity whatever is excluded, and with it necessity in the direction from A to B, so that no A are present which are in the state of necessity not to be B. The apparent contradiction between the doctrine enunciated in the text of this paragraph and the Aristotelian is solved in this way.2

§ 88. Nothing follows from the conversion of the particular negative judgment. The particular negative categorical judgment asserts, that some S have not the predicate P, without saying anything definite about the rest of S. Its Schema is accordingly the combination in the three figures:

or in the one figure, which, comprehending the three

1 Anal. Pr. i. c. iii.

2 Cf. Prantl, Gesch. der Log. i. 267, 364.

possible cases, denotes the definite by the continuous, and the indefinite by the dotted lines:

Р

According to this, it can happen that when some S are not P: (1) Some P are not S, and other P are S ;— (2) All P are S; and (3) No P are S. Nothing can be said universally of the relation of P to S in a judgment whose subject is P.

Similarly, the Schema of the particular negative hypothetical judgment: Sometimes when ▲ is, в is not, is the following:

It may happen that when B is, (1) A sometimes is and sometimes is not; (2) A always is, and (3) A never is. Hence the general relation of B to A is quite indefinite.

Examples of these different possible cases are the following:

Of the particular negative categorical judgments of the form 1: Some parallelograms are not regular figures. Of the form 2: Some parallelograms are not squares; or: Some rectilineal plane figures, which are divided by a diagonal into two coincident triangles, are not parallelograms. Of the form 3: (At least) some parallelograms are not trapezoids; or: (At least) some rectilineal plane figures, which are divided by a diagonal into two triangles not coincident, are not parallelograms.

Of the particular negative hypothetical judgment of the

form 1: Sometimes, when the accused has confessed himself to be guilty, the accusation is not established. Of the form 2: Sometimes, when unestablished accusations are raised, there is not calumny (only error). Of the form 3: At least sometimes, when the advocate of a higher ideal principle is condemned to death by the advocates of a principle which is less in accordance with reason, but has become an historical power, the right and wrong have not been shared equally by both parties.

§ 89. Contraposition is that change of form, according to which the parts of the judgment change places with reference to its relation, but at the same time one of them receives the negation, and the quality of the judgment changes. Contraposition in categorical judgment consists in this, that the contradictory opposite of the predicate notion becomes the subject, and in this transference the quality of the judgment passes over to its opposite. In the hypothetical judgment, it consists in this, that the contradictory opposite of the conditioned becomes the conditioning proposition, and there is, instead of an affirmative nexus between the two parts of the judgment, a negative one, and instead of a negative an affirmative one.

The internal correctness of Contraposition is to be decided by the same axioms as that of Conversion (cf. § 84).

The term 'conversio per contrapositionem,' used by Boëthius (§ 82), where contrapositio' means the change of one member into its contradictory opposite, is in itself unobjectionable, if the notion of conversion is sufficiently widely understood and defined. But then a term would be needed to designate the first kind of Conversion in the wider sense, or Conversion in the stricter sense. Boëthius (cf. § 82) calls it 'conversio