[233] συνακτικοὶ or περαντικοὶ, and ἀσυνακτικοὶ or ἀπέραντοι, or ἀσυλλόγιστοι. Sext. Pyrrh. ii. 137; Math. viii. 303 and 428; Diog. 77. [234] Syllogisms which are conclusive in point of fact, but wanting in precision of form, were called περαντικοί in the narrower sense; those complete also in form, συλλογιστικοί. Diog. 78; Ps. Galen, Εἰσαγ. διαλ. 58. [235] An inference is true (ἀληθὴς) when not only the illation is correct (ὑγιὴς), but when the individual propositions, the premisses as well as the conclusion, are materially true. The λόγοι συνακτικοὶ may therefore be divided into true and false. Sext. Pyrrh. ii. 138; Math. viii. 310 and 412; Diog. 79. [236] Sext. Pyrrh. ii. 140 and 135; Math. viii. 305; 313; and 411: True forms of inference are divided into ἀποδεικτικοὶ and οὐκ ἀποδεικτικοὶ. ἀποδεικτικοὶ = οἱ διὰ προδήλων ἄδηλόν τι συνάγοντες· οὐκ ἀποδεικτικοὶ when this is not the case, as in the inference: If it is day, it is light—It is day, ∴ It is light; for the conclusion, It is light, is known as well as it is known that It is day. The ἀποδεικτικοὶ may proceed either ἐφοδευτικῶς from the premisses to the conclusions, or ἐφοδευτικῶς ἅμα καὶ ἐκκαλυπτικῶς· ἐφοδευτικῶς when the premisses rest upon belief (πίστις and μνήμη); ἐκκαλυπτικῶς when they are based on a scientific necessity. [237] According to Diog. 79, Sext. Pyrrh. ii. 157, others added other forms of ἀναπόδεικτοι. Cic., in adding a sixth and seventh (Top. 14, 57), must have been following these authorities. [238] Consult, on these five ἀναπόδεικτοι of Chrysippus (which need not be given here more at length, and are absolutely identical with those of Theophrastus) Diog. 79–81 (on p. 79 we must read συλλογιστικῶν for συλλογισμὼν. See p. 118, 2); Sext. Pyrrh. ii. 156–159; 201; Math. viii. 223–227; Cic. Top. 13; Simpl. Phys. 123, b; Ps. Galen, Εἰσαγ. διαλ. 17; Prantl, 473, 182; on the πέμπτος ἀναπόδεικτος διὰ πλειόνων Sext. Pyrrh. i. 69; Cleomed. Meteora, pp. 41 and 47; Prantl, p. 475. [239] Two such cases are distinguished, one in which all three clauses, the other in which the conclusion and minor premiss are identical. The first class are called διφορούμενοι· If it is day, it is day; It is day, ∴ It is day. The second class, ἀδιαφόρως περαίνοντες· It is either day or night; It is day, ∴ It is day. The latter term is, however, applied to both kinds. See Alex. Anal. Pr. 7, a; 53, b; Top. 7; Schol. in Arist. 294, b, 25; Cic. Acad. ii. 30, 96; Prantl, 476, 185. [240] Cic. Top. 15, 57: ex his modis conclusiones innumerabiles nascuntur. Sext. Math. viii. 228, in which passage it is striking that ἀναπόδεικτοι should be divided into ἁπλοῖ and οὐχ ἁπλοῖ. It has been suggested that ἀποδεικτικῶν should be substituted for ἀναποδείκτων, but it is also possible that the latter word may be used in a narrow as well as in a wider sense.