Let us now consider reasoning from definition to definitum, and also the argument from enumeration. A defining proposition has a meaning. It is not, therefore, a merely identical proposition, but there is a difference between the definition and the definitum. According to the received doctrine, this difference consists wholly in the fact that the definition is distinct, while the definitum is confused. But I think that there is another difference. The definitum implies the character of being designated by a word, while the definition, previously to the formation of the word, does not. Thus, the definitum exceeds the definition in depth, although only verbally. In the same way, any unanalyzed notion carries with it a feeling,—a constitutional word,—which its analysis does not. If this be so, the definition is the predicate and the definitum the subject, of the defining proposition, and this last cannot be simply converted. In fact, the defining proposition affirms that whatever a certain name is applied to is supposed to have such and such characters; but it does not strictly follow from this, that whatever has such and such characters is actually called by that name, although it certainly might be so called. Hence, in reasoning from definition to definitum, there is a verbal increase of depth, and an actual increase of extensive distinctness (which is analogous to breadth). The increase of depth being merely verbal, there is no possibility of error in this procedure. Nevertheless, it seems to me proper, rather to consider this argument as a special modification of hypothesis than as a deduction, such as is reasoning from definitum to definition. A similar line of thought would show that, in the argument from enumeration, there is a verbal increase of breadth, and an actual increase of depth, or rather of comprehensive distinctness, and that therefore it is proper to consider this (as most logicians have done) as a kind of infallible induction. These species of hypothesis and induction are, in fact, merely hypotheses and inductions from the essential parts to the essential whole; this sort of reasoning from parts to whole being demonstrative. On the other hand, reasoning from the substantial parts to the substantial whole is not even a probable argument. No ultimate part of matter fills space, but it does not follow that no matter fills space.
1. This is quoted from Baynes (Port-Royal Logic, 2d ed., p. xxxiii), who says that he is indebted to Sir William Hamilton for the information.
2. Porphyry appears to refer to the doctrine as an ancient one.
3. The author of De Generibus et Speciebus opposes the integral and diffinitive wholes. John of Salisbury refers to the distinction of comprehension and extension, as something “quod fere in omnium ore celebre est, aliud scilicet esse quod appellativa significant, et aliud esse quod nominant. Nominantur singularia, sed universalia sig-nificantur.” (Metalogicus, lib. 2, cap. 20. Ed. of 1610, p. 111.)
Vincentius Bellovacensis (Speculum Doctrinale, Lib. Ill, cap. xi.) has the following: “Si vero quseritur utrum hoc universale ‘homo’ sit in quolibet homine secundum se totum an secundum partem, dicendum est quod secundum se totum, id est secundum quamlibet sui partem diffinitivam.… Non autem secundum quamlibet partem subjectivam.” William of Auvergne (PrantFs Geschichte, Vol. Ill, p. 77) speaks of “totalitatem istam, quae est ex partibus rationis seu diffinitionis, et hae partes sunt genus et differentiae; alio modo partes speciei individua sunt, quoniam ipsam speciem, cum de eis praedicatur, sibi invicem quodammodo partiuntur.” If we were to go to later authors, the examples would be endless. See any commentary Physics, Lib. I.
4. Part I, chap. ix.
5. Principia, Part I, §45 et seq.
6. Eighth Letter to Burnet.
7. Cf. Morin, Dictionnaire, Tome I, p. 685; Chauvin, Lexicon, both editions; Eustachius, Summa, Part I, Tr. I, qu. 6.
8. Prantl, Geschichte, Vol. Ill, p. 364.
9. Ibid. p. 134. Scotus also uses the term. Quodliheta, question 13, article 4.
10. Summa Theologica, Part I, question 53.
11. Part I, chap. X. (Ed. of 1488, fol. 6, c.)
12. Fol. 23. d. See also Tartaretus’ Expositio in Summulas Petri Hispani towards the end. Ed. of 1509, fol. 91, b.
13. Logic, p. 100. In the Summa Logices attributed to Aquinas, we read: “Omnis forma sub se habens multa, idest quod universaliter sumitur, habet quandam latitudi-nem; nam invenitur in pluribus, et dicitur de pluribus.” (Tr. 1, c. 3.)
14. I adopt the admirable distinction of Scotus between actual, habitual, and virtual cognition.
15. Logik, 2te Aufl., §54.
16. Formal Logic, p. 234. His doctrine is different in the Syllabus.
17. Laws of Thought, 4th ed., §§52, 80.
18. Logic, Part I, chap, ii, §5.
19. For the distinction of extensive and comprehensive distinctness, see Scotus, i, dist. 2, qu. 3.
20. That is, of whatever things it is applicable to.
21. See, for example, De Generibus et Speciebus, p. 548.
Notes
MS 152: November–December 1868
Note 1
The proof offered that
is fallacious. For the identities
cannot be obtained by simply substituting 1 and 0 for x in the identity
in as much as i and j have not been proved to be independent of x. Boole’s proof is the same.
It can be proved, however, that i and j are independent of x. For let x be logically multiplied by any factor m1 and then increased by any term a1 and we have
Now when the same pair of processes is performed upon an expression of this form the result is another expression of the same form for
Now to multiply by 1 or increase by 0 is the same as omitting to multiply or add at all; and therefore any series of multiplications and additions may by the intercalation of a multiplication by 1 between every pair of successive additions and of an increase by 0 between every pair of successive multiplications, be reduced to a series of alternate additions and multiplications and, therefore, the result of any operation composed only of additions and multiplications is of the form
But such an expression as ax does not come under this formula.
It could easily be shown by (11) and (14) that the difference or quotient of two quantities of the form