[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 3

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| The Signs of Inclusion, Equality, Etc.
|
| I shall follow Boole in taking the sign of equality to signify identity.
| Thus, if v denotes the Vice-President of the United States, and p the
| President of the Senate of the United States,
|
| v = p
|
| means that every Vice-President of the United States is President of the
| Senate, and every President of the United States Senate is Vice-President.
| The sign "less than" is to be so taken that
|
| f < m
|
| means that every Frenchman is a man, but there are men besides Frenchmen.
| Drobisch has used this sign in the same sense.  It will follow from these
| significations of '=' and '<' that the sign '-<' (or '=<', "as small as")
| will mean "is".  Thus,
|
| f -< m
|
| means "every Frenchman is a man", without saying whether there are any
| other men or not.  So,
|
| 'm' -< 'l'
|
| will mean that every mother of anything is a lover of the same thing;
| although this interpretation in some degree anticipates a convention to
| be made further on.  These significations of '=' and '<' plainly conform
| to the indispensable conditions.  Upon the transitive character of these
| relations the syllogism depends, for by virtue of it, from
|
| f -< m
|
| and
|
| m -< a,
|
| we can infer that
|
| f -< a;
|
| that is, from every Frenchman being a man and every
| man being an animal, that every Frenchman is an animal.
|
| But not only do the significations of '=' and '<' here adopted fulfill all
| absolute requirements, but they have the supererogatory virtue of being very
| nearly the same as the common significations.  Equality is, in fact, nothing
| but the identity of two numbers;  numbers that are equal are those which are
| predicable of the same collections, just as terms that are identical are those
| which are predicable of the same classes.  So, to write 5 < 7 is to say that 5
| is part of 7, just as to write f < m is to say that Frenchmen are part of men.
| Indeed, if f < m, then the number of Frenchmen is less than the number of men,
| and if v = p, then the number of Vice-Presidents is equal to the number of
| Presidents of the Senate;  so that the numbers may always be substituted
| for the terms themselves, in case no signs of operation occur in the
| equations or inequalities.
|
| C.S. Peirce, CP 3.66
|
| 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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