Peirce's 1870 Logic Of Relatives • Part 3
Author: Jon Awbrey
Demonstration of the sort called mathematical is founded on suppositions of particular cases. The geometrician draws a figure; the algebraist assumes a letter to signify a single quantity fulfilling the required conditions. But while the mathematician supposes an individual case, his hypothesis is yet perfectly general, because he considers no characters of the individual case but those which must belong to every such case. The advantage of his procedure lies in the fact that the logical laws of individual terms are simpler than those which relate to general terms, because individuals are either identical or mutually exclusive, and cannot intersect or be subordinated to one another as classes can. Mathematical demonstration is not, therefore, more restricted to matters of intuition than any other kind of reasoning. Indeed, logical algebra conclusively proves that mathematics extends over the whole realm of formal logic; and any theory of cognition which cannot be adjusted to this fact must be abandoned. We may reap all the advantages which the mathematician is supposed to derive from intuition by simply making general suppositions of individual cases.
(Peirce, CP 3.92).
In reference to the doctrine of individuals, two distinctions should be borne in mind. The logical atom, or term not capable of logical division, must be one of which every predicate may be universally affirmed or denied. For, let be such a term. Then, if it is neither true that all is nor that no is it must be true that some is and some is not and therefore may be divided into that is and that is not which is contrary to its nature as a logical atom.
Such a term can be realized neither in thought nor in sense.
Not in sense, because our organs of sense are special — the eye, for example, not immediately informing us of taste, so that an image on the retina is indeterminate in respect to sweetness and non-sweetness. When I see a thing, I do not see that it is not sweet, nor do I see that it is sweet; and therefore what I see is capable of logical division into the sweet and the not sweet. It is customary to assume that visual images are absolutely determinate in respect to color, but even this may be doubted. I know no facts which prove that there is never the least vagueness in the immediate sensation.
In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates.
A logical atom, then, like a point in space, would involve for its precise determination an endless process. We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate. Such a term as “the second Philip of Macedon” is still capable of logical division — into Philip drunk and Philip sober, for example; but we call it individual because that which is denoted by it is in only one place at one time. It is a term not absolutely indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them. Such differences we habitually disregard in the logical division of substances. In the division of relations, etc., we do not, of course, disregard these differences, but we disregard some others. There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse, and if be a term which in consequence of such neglect becomes indivisible in that discourse, we have in that discourse,
This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words individual (τὸ ἄτομον) and singular (τὸ καθ᾿ ἕκαστον); but as those who have used the word individual have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense.
(Peirce, CP 3.93).
Note. On the square bracket notation used above: Peirce explains this notation at CP 3.65.
I propose to denote the number of a logical term by enclosing the term in square brackets, thus,
The number of an absolute term, as in the case of is defined as the number of individuals it denotes.