dition the various functions of judgment. Reality and Negation are not, of course, like Substantiality and the other Categories of Relation, valid as forms of actual existence. They only denote a relation between our thoughts and actual existence. This, however, only justifies the attack upon the Kantian table of Categories, not Knauer's own doctrine. On the other hand, the axiom of Knauer's is correct (in which he recognises the master of Stagira' as his ally, but which is not a new doctrine, even in the sense that it had been lost sight of since Aristotle, and was first brought to light again by Knauer), that necessary contradiction exists only between affirmation and negation of the same thing, and not between judgments whose predicates are opposed contradictorily. See §§ 77-80. [Sir W. Hamilton, with Mansel and Thompson, refuse to recognise the modality of judgments as any part of their logical treatment. The mode, they say, belongs to the matter, and must be determined by a consideration of the matter, and therefore is extralogical.'] § 70. QUANTITY is the extent in which the predicate is affirmed or denied in the sphere of the subject-notion. Some logicians divide judgments according to Quantity into Universal, Particular, and Singular. Singular judgments are to be subsumed under the other two classes: under the first when the subject is definite and individually designated (e.g. Caesar, or this man); under the second when the subject is indefinite and designated only by a general notion (e.g. a man, or a great general). For in the first case the predicate is affirmed or denied of the whole sphere of the subject (which in this case is reduced to an individual), and in the other case of an indefinite part of the sphere of the subject-notion. [ Cf. Hamilton's Lect. on Log. i. 257; Mansel's Aldrich's Rudimenta, 4th ed. p. 46 n.; Proleg. Log. 2nd ed. Note II.] Aristotle distinguishes Universal, Particular, and Indefinite Judgments: πρότασις—ἢ καθόλου, ἢ ἐν μέρει, ἢ ἀδιόριστος. The Judgment Indefinite according to quality, which Aristotle makes co-ordinate with the Universal and Particular, is not properly a third kind, but an incomplete, or incompletely expressed, judgment.2 Kant recognised three kinds—Singular, Particular or Plurative, and Universal Judgments—and traces them to the three Categories of Quantity-Unity, Plurality, and Universality. He teaches that singular judgments belong to the same class as the universal.3 Herbart says, that individual judgments are only to be reckoned along with universal ones when they have a distinct subject. When the meaning of a general expression is limited by the indefinite article to any individual not more definitely designated, those judgments are to be reckoned with the particular. This manner of reduction shows itself to be the correct one, partly in itself, because it does not depend upon the absolute number of subject-individuals, but on the relation of this number to the number of individuals falling under the subjectnotion generally; partly in its application to the forms of inference.5 The subject of the particular judgment is any part of the sphere of the subject-notion, and at least any single individual falling under this notion. Its limits may be enlarged up to coincidence with the whole sphere, so that the particular judgment does not exclude, but comprehends, the possibility of the universal. The rule that the judgment, indesignate in reference to quantity, is universal if affirmative, and particular if negative, is more grammatical than logical, and not unconditionally valid. 1 Anal. Pri. i. 1. 2 [Cf. Hamilton's Lect. on Log. i. 243.] 3 Krit. d. r. Vern. §§ 9-11; Proleg. § 20; Logik, § 21. 5 Cf. below, § 107. § 71. By combination of the divisions of judgments according to QUALITY and QUANTITY four kinds arise: 1. Universal Affirmative of the form-All S are P. Logicians have been accustomed to denote these forms by the letters a, e, i, o (of which a and i are taken from affirmo, e and o from nego). It will be seen from a comparison of spheres, that in every universal judgment the subject is posited universally, and particularly in every particular judgment; but the predicate is posited particularly in every affirmative judgment, or, if universally, only by accident (for, according to the form of the judgment, both in a and i its sphere can lie partly outside of the subject), and universally in every negative judgment (for in e the sum total of S, and in o the part of S concerned, must always be thought as separated from the whole sphere of the predicate). The judgments of the form a (Sa P-All S are P) can be represented in a scheme by the combination of the two following figures: a, 1. SP e, 2. S P The following scheme is for judgments of the form e (Se P -No S is P): Judgments of the form i (S i P—At least a part of S is P) require the combination of the four following figures (of which 1 and 2 are peculiar to the form i, but 3 and 4 repeat the schema of the form a): Judgments of the form o (So P-At least one or some S are not P) are to be represented by the combination of the three following figures (of which 1 and 2 are peculiar to the form o, while 3 repeats the schema of the form e: If the definite be denoted by a continuous, and the indefinite by a dotted line, the symbol of judgments of the form a may be reduced to the one figure: SP The Symbol for the judgments of the form i under the same presupposition: The use of these Schemata is not confined to that apprehension of the judgment which finds it to be only a subsumption of the lower subject-notion or conception under the higher predicate-notion, and which, therefore, requires that the predicate-notion be made substantive in cases where this is actually unsuitable. If the predicate-notion is the proper genus-notion of the subject, it is quite natural to take it for substantive, but not when it denotes a property or action. This last case does not require to be reduced to the first for the sake of a comparison of spheres. It is not necessary (although in many cases very convenient) to attach such a meaning to the circle P as to make it embrace the objects which fall under the substantive predicate-notion. The sphere of an adjective or verbal conception can be also understood by the sphere P. It may mean the sum total of the cases in which the corresponding property or action occurs, while S may denote the sphere of a substantive conceptionthe sum total of the objects in which the corresponding property or action occurs. On this presupposition the coincidence of the circles or parts of the circles is not to be taken to be the symbol of the identity of objects, but as the symbol of the co-existence of what subsists and what inheres. Cf. § 105. In a, 1 all S are only a part of P, but in a, 2 all S are all P ; in i, 1 some S are some P, &c. The Quantification of the Predicate consists in paying attention to these relations. It |