Some problems cannot be solved in the paradigms where they first appear, which is why we keep recurring to them without quite freeing ourselves from the loops in which they ensnare us. Questions about the supposed uniqueness of supposed individuals and the dyadic relation of identity are as old as the ship of Theseus and the morning and evening star(s) we steer by.
Peirce, of course, took another course …
As fortune has it, I find myself waylaid between bouts of travel, with promises to keep when it comes to Peirce’s information formula, so let me leave this for now with a link to one of the most critical passages in all of Peirce’s explorations:
Selection from C.S. Peirce, “Logic Of Relatives” (1870), CP 3.45–149
93. 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 explains his use of the square bracket 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.
- Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870. Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429. Online (1) (2) (3).
- Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
- Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.
- Peirce’s 1870 Logic Of Relatives
- C.S. Peirce • Doctrine Of Individuals
- Mathematical Demonstration and the Doctrine of Individuals • (1) • (2)