Peircers,
Writing "in reference to the doctrine of individuals" in his
"Description of a Notation for the Logic of Relatives" (1870),
Peirce's approach is, in its basic principles, so far ahead of
his time that it overleaps the dustbin of Logical Atomism and
anticipates ideas about element-free set theory that will not
come into their own until the latter part of the 20th Century.
Here is a collection of excerpts that I gathered for
several previous occasions, here and elsewhere, when
the topic of Peirce's approach to individuals arose.
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DOI. Doctrine Of Individuals
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DOI. Note 1
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| 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 'A' be such a term. Then, if it is neither true that all 'A' is 'X'
| nor that no 'A' is 'X', it must be true that some 'A' is 'X' and some 'A' is
| not 'X'; and therefore 'A' may be divided into 'A' that is 'X' and 'A' that
| is not 'X', 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 sweetUess 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 of 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 'I' be a term which in consequence of such
| neglect becomes indivisible in that discourse, we have in
| that discourse,
|
| ['I'] = 1.
|
| 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' ('to atomon') and 'singular' ('to kath
| ekaston'); 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.
|
| C.S. Peirce, 'Collected Papers', CP 3.93
Peirce defines the "number" ['t'] of a logical term 't' as follows:
| I propose to assign to all logical terms, numbers; to an absolute term,
| the number of individuals it denotes; to a relative term, the average
| number of things so related to one individual. Thus in a universe of
| perfect men ('men'), the number of "tooth of" would be 32. The number
| of a relative with two correlates would be the average number of things
| so related to a pair of individuals; and so on for relatives of higher
| numbers of correlates. I propose to denote the number of a logical term
| by enclosing the term in square brackets, thus ['t'].
|
| C.S. Peirce, 'Collected Papers', CP 3.65
The "number" of an absolute term, as in the case of 'I',
is defined as the number of individuals that it denotes.
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DOI. Note 2
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| The old logics distinguish between 'individuum signatum' and 'individuum
vagum'.
| "Julius Caesar" is an example of the former; "a certain man", of the latter.
| The 'individuum vagum', in the days when such conceptions were exactly
| investigated, occasioned great difficulty from its having a certain
| generality, being capable, apparently, of logical division. If we
| include under the 'individuum vagum' such a term as "any individual
| man", these difficuluties appear in a strong light, for what is true
| of any individual man is true of all men. Such a term is in one sense
| not an individual term; for it represents every man. But it represents
| each man as capable of being denoted by a term which is individual; and
| so, though it is not itself an individual term, it stands for any one
| of a class of individual terms. If we call a thought about a thing
| in so far as it is denoted by a term, a 'second intention', we may
| say that such a term as "any individual man" is individual by
| second intention. The letters which the mathematician uses
| (whether in algebra or in geometry) are such individuals
| by second intention. Such individuals are one in number,
| for any individual man is one man; they may also be regarded
| as incapable of logical division, for any individual man, though
| he may either be a Frenchman or not, is yet altogether a Frenchman
| or altogether not, and not some one and some the other. Thus, all
| the formal logical laws relating to individuals will hold good
| of such individuals by second intention, and at the same
| time a universal proposition may at any moment be
| substituted for a proposition about such an
| individual, for nothing can be predicated
| of such an individual which cannot be
| predicated of the whole class.
|
| C.S. Peirce, 'Collected Papers', CP 3.94
|
| Charles Sanders Peirce,
|"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', Volume 9, pages 317-378, 26 January 1870,
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).
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DOI. Note 3
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| Individual
|
| (As a technical term of logic, 'individuum' first appears
| in Boethius, in a translation from Victorinus, no doubt
| of 'atomon', a word used by Plato ('Sophistes', 229 D)
| for an indivisible species, and by Aristotle, often in
| the same sense, but occasionally for an individual.
| Of course the physical and mathematical senses of
| the word were earlier. Aristotle's usual term for
| individuals is 'ta kath ekasta', Latin 'singularia',
| English 'singulars'.)
|
| Used in logic in two closely connected senses.
|
| (1) According to the more formal of these an individual is an object (or term)
| not only actually determinate in respect to having or wanting each general
| character and not both having and wanting any, but is necessitated by
| its mode of being to be so determinate. See Particular (in logic).
|
| This definition does not prevent two distinct individuals from being
| precisely similar, since they may be distinguished by their hecceities
| (or determinations not of a generalizable nature); so that Leibniz's
| principle of indiscernibles is not involved in this definition.
|
| Although the principles of contradiction and excluded middle may be regarded
| as together constituting the definition of the relation expressed by "not",
| yet they also imply that whatever exists consists of individuals. This,
| however, does not seem to be an identical proposition or necessity of
| thought; for Kant's Law of Specification ('Krit. d. reinen Vernunft',
| 1st ed., 656, 2nd ed., 684; but it is requisite to read the whole
| section to understand his meaning), which has been widely accepted,
| treats logical quantity as a continuum in Kant's sense, i.e., that
| every part of which is composed of parts. Though this law is only
| regulative, it is supposed to be demanded by reason, and its wide
| acceptance as so demanded is a strong argument in favour of the
| conceivability of a world without individuals in the sense of
| the definition now considered.
|
| Besides, since it is not in the nature of concepts adequately
| to define individuals, it would seem that a world from which
| they were eliminated would only be the more intelligible.
|
| A new discussion of the matter, on a level with
| modern mathematical thought and with exact logic,
| is a desideratum. A highly important contribution
| is contained in Schroeder's 'Logik', iii, Vorles. 10.
| What Scotus says ('Quaest. in Met.', VII 9, xiii & xv)
| is worth consideration.
|
| C.S. Peirce, 'Collected Papers', CP 3.611-612
|
|'Dictionary of Philosophy and Psychology',
| J.M. Baldwin (ed.), Macmillan, New York, NY,
| Volume 1, pp. 537-538, 2nd edition 1911.
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DOI. Note 4
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| Individual (cont.)
|
| (2) Another definition which avoids the above difficulties is that
| an individual is something which reacts. That is to say, it does
| react against some things, and it is of such a nature that it
| might react, or have reacted, against my will.
|
| This is the stoical definition of a reality; but since the Stoics were
| individualistic nominalists, this rather favours the satisfactoriness
| of the definition than otherwise.
|
| It may be objected that it is unintelligible; but in the sense
| in which this is true, it is a merit, since an individual is
| unintelligible in that sense. It is a brute fact that the
| moon exists, and all explanations suppose the existence
| of that same matter. That existence is unintelligible
| in the sense in which the definition is so. That is
| to say, a reaction may be experienced, but it cannot
| be conceived in its character of a reaction; for
| that element evaporates from every general idea.
|
| According to this definition, that which alone immediately presents itself as
| an individual is a reaction against the will. But everything whose identity
| consists in a continuity of reactions will be a single logical individual.
| Thus any portion of space, so far as it can be regarded as reacting, is
| for logic a single individual; its spatial extension is no objection.
|
| With this definition there is no difficulty about the truth that whatever
| exists is individual, since existence (not reality) and individuality are
| essentially the same thing; and whatever fulfills the present definition
| equally fulfills the former definition by virtue of the principles of
| contradiction and excluded middle, regarded as mere definitions of
| the relation expressed by "not".
|
| As for the principle of indiscernibles, if two individual things are
| exactly alike in all other respects, they must, according to this
| definition, differ in their spatial relations, since space is
| nothing but the intuitional presentation of the conditions of
| reaction, or of some of them. But there will be no logical
| hindrance to two things being exactly alike in all other
| respects; and if they are never so, that is a physical
| law, not a neccesity of logic. This second definition,
| therefore, seems to be the preferable one.
|
| C.S. Peirce, 'Collected Papers', CP 3.613
|
|'Dictionary of Philosophy and Psychology',
| J.M. Baldwin (ed.), Macmillan, New York, NY,
| Volume 1, pp. 537-538, 2nd edition 1911.
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DOI. Doctrine of Individuals -- 2002
00. http://suo.ieee.org/ontology/thrd24.html#04332
01. http://suo.ieee.org/ontology/msg04332.html
02. http://suo.ieee.org/ontology/msg04348.html
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DOI. Doctrine of Individuals -- 2003
Ontology List
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DOI. Doctrine of Individuals -- 2005
00. http://stderr.org/pipermail/inquiry/2005-January/thread.html#2320
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