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| ==Selection 11== | | ==Selection 11== |
| | | |
− | <pre> | + | <blockquote> |
− | | The Signs for Multiplication (concl.)
| + | <p>'''The Signs for Multiplication''' (concl.)</p> |
− | |
| + | |
− | | The conception of multiplication we have adopted is that of
| + | <p>The conception of multiplication we have adopted is that of the application of one relation to another. So, a quaternion being the relation of one vector to another, the multiplication of quaternions is the application of one such relation to a second.</p> |
− | | the application of one relation to another. So, a quaternion
| + | |
− | | being the relation of one vector to another, the multiplication
| + | <p>Even ordinary numerical multiplication involves the same idea, for 2 × 3 is a pair of triplets, and 3 × 2 is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives.</p> |
− | | of quaternions is the application of one such relation to a second.
| + | |
− | |
| + | <p>If we have an equation of the form:</p> |
− | | Even ordinary numerical multiplication involves the same idea, for
| + | |
− | | 2 x 3 is a pair of triplets, and 3 x 2 is a triplet of pairs, where
| + | : <p>''xy'' = ''z'' |
− | | "triplet of" and "pair of" are evidently relatives.
| + | |
− | |
| + | <p>and there are just as many x's per y as there are, ''per'' things, things of the universe, then we have also the arithmetical equation:</p> |
− | | If we have an equation of the form:
| + | |
− | |
| + | : <p>[''x''][''y''] = [''z''].</p> |
− | | xy = z
| + | |
− | |
| + | <p>For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then:</p> |
− | | and there are just as many x's per y as there are,
| + | |
− | | 'per' things, things of the universe, then we have
| + | : <p>['t'][f] = ['t'f]</p> |
− | | also the arithmetical equation:
| + | |
− | |
| + | <p>holds arithmetically.</p> |
− | | [x][y] = [z].
| + | |
− | |
| + | <p>So if men are just as apt to be black as things in general:</p> |
− | | For instance, if our universe is perfect men, and there
| + | |
− | | are as many teeth to a Frenchman (perfect understood)
| + | : <p>[m,][b] = [m,b]</p> |
− | | as there are to any one of the universe, then:
| + | |
− | |
| + | <p>where the difference between [m] and [m,] must not be overlooked.</p> |
− | | ['t'][f] = ['t'f]
| + | |
− | |
| + | <p>It is to be observed that:</p> |
− | | holds arithmetically.
| + | |
− | |
| + | : <p>[!1!] = `1`.</p> |
− | | So if men are just as apt to be black as things in general:
| + | |
− | |
| + | <p>Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of ''expectation''.</p> |
− | | [m,][b] = [m,b]
| + | |
− | |
| + | <p>Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. (Peirce, CP 3.76).</p> |
− | | where the difference between [m] and [m,] must not be overlooked.
| + | </blockquote> |
− | |
| |
− | | It is to be observed that:
| |
− | |
| |
− | | [!1!] = `1`.
| |
− | |
| |
− | | Boole was the first to show this connection between logic and
| |
− | | probabilities. He was restricted, however, to absolute terms.
| |
− | | I do not remember having seen any extension of probability to
| |
− | | relatives, except the ordinary theory of 'expectation'.
| |
− | |
| |
− | | Our logical multiplication, then, satisfies the essential conditions
| |
− | | of multiplication, has a unity, has a conception similar to that of
| |
− | | admitted multiplications, and contains numerical multiplication as
| |
− | | a case under it.
| |
− | |
| |
− | | C.S. Peirce, CP 3.76
| |
− | |
| |
− | | 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).
| |
− | </pre> | |
| | | |
| ===Commentary Note 11.1=== | | ===Commentary Note 11.1=== |