[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 4

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| The Signs for Addition
|
| The sign of addition is taken by Boole so that
|
| x + y
|
| denotes everything denoted by x, and, 'besides',
| everything denoted by y.
|
| Thus
|
| m + w
|
| denotes all men, and, besides, all women.
|
| This signification for this sign is needed for
| connecting the notation of logic with that of the
| theory of probabilities.  But if there is anything
| which is denoted by both terms of the sum, the latter
| no longer stands for any logical term on account of
| its implying that the objects denoted by one term
| are to be taken 'besides' the objects denoted by
| the other.
|
| For example,
|
| f + u
|
| means all Frenchmen besides all violinists, and,
| therefore, considered as a logical term, implies
| that all French violinists are 'besides themselves'.
|
| For this reason alone, in a paper which is published
| in the Proceedings of the Academy for March 17, 1867,
| I preferred to take as the regular addition of logic
| a non-invertible process, such that
|
| m +, b
|
| stands for all men and black things, without any implication that
| the black things are to be taken besides the men;  and the study of
| the logic of relatives has supplied me with other weighty reasons for
| the same determination.
|
| Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in
| a tract called 'Pure Logic, or the Logic of Quality' [1864], had anticipated me in
| substituting the same operation for Boole's addition, although he rejects Boole's
| operation entirely and writes the new one with a '+' sign while withholding from
| it the name of addition.
|
| It is plain that both the regular non-invertible addition
| and the invertible addition satisfy the absolute conditions.
| But the notation has other recommendations.  The conception
| of 'taking together' involved in these processes is strongly
| analogous to that of summation, the sum of 2 and 5, for example,
| being the number of a collection which consists of a collection of
| two and a collection of five.  Any logical equation or inequality
| in which no operation but addition is involved may be converted
| into a numerical equation or inequality by substituting the
| numbers of the several terms for the terms themselves --
| provided all the terms summed are mutually exclusive.
|
| Addition being taken in this sense,
| 'nothing' is to be denoted by 'zero',
| for then
|
| x +, 0 = x,
|
| whatever is denoted by x;  and this is the definition
| of 'zero'.  This interpretation is given by Boole, and
| is very neat, on account of the resemblance between the
| ordinary conception of 'zero' and that of nothing, and
| because we shall thus have
|
| [0] = 0.
|
| C.S. Peirce, CP 3.67
|
| 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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