ONT Toward A Functional Conception Of Quantificational Logic
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david, you have called back to mind a number of loose threads
on the subject of quant calc that i cannot elaborate too much
at the moment but only try to collect them in one place again,
and maybe tie them to a title that i can find again in future.
exhibit 1.
"A Simple Desultory Philippic"
| 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
| 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 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. (CP 3.93, CE 2.389-390).
|
| [ Note on the square bracket notation that was used above.
| | Peirce explains this notation at CP 3.65, also CE 2.366.
| |
| | | I propose to denote the number of a logical term by
| | | enclosing the term in square brackets, thus, ['t'].
| |
| | The "number" of an absolute term, as in the case of 'I',
| | is defined as the number of individuals that it denotes.
| ]
|
| 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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