Writings of Charles S. Peirce: A Chronological Edition, Volume 8: 1890–1892

34. Examination of the Copula of Inclusion

217 34 Examination of the Copula of Inclusion Spring 1891 Houghton Library Definition of this copula. A. We are justified in writing X R Y, provided that assuming X to be true, we find ourselves forced to admit Y is true. A. We are obliged to take X R Y, if we cannot assume X to be true. B. If X R Y, and we assume X to be true, we are bound to admit that Y is true. General form with one copula X R Y This becomes necessarily true in the special case (1) X R X This is the principle of identity. Proof. Assume X to be true, then we are forced to admit X is true. Hence, by A, the formula holds X R X. General forms with two copulas Form I is true in two special cases: (2) X R (Y R Y) For assuming X to be true, we are forced by (1) to admit Y R Y. Hence, by A, X R (Y R Y). (3) X R (Y R X) For assume X is true, then we must admit Y R X is true. For assume Y is true we must admit X is true, according to the former assumption. Hence by A the formula holds. Or (3) follows from I. X R (Y R Z) II. (X R Y) R Z Writings of C. S. Peirce 1890–1892 218 (3) X R (Y R Z)  Y R (X R Z). Form II gives no necessary formula. General forms with three copulas Form I gives the necessary formulae mere cases under (2) and (3). Form II gives the necessary formulae Form III gives the necessary formulae (4) [(X R X) R Y] R Y Proof. Assume (X R X) R Y is true. Then we are bound to admit Y is true. For by (1) X R X must be assumed true and thus by B, we have to admit that Y is true. So by A the formula holds. (5) [(X R Y) R X] R X Proof. For assume (X R Y) R X. Then if we are forced to admit X is true, the formula holds by A. But by B if we assume X R Y we are bound to admit X is true. And if we assume X is not true by A we are to take X R Y. Form IV gives two necessary formulae, of which one, X R [(Y R Z) R X] is a case of (3) and the other is (6) X R [(X R Y) R Y]. For by A we have only to prove that if we assume X is true we must admit (X R Y) R X. But to show this by A is true we have only to show that assuming further X R Y we must admit Y is true. Now by B, if we I. X R [Y R (Z R W)] II. (X R Y) R (Z R W) III. [(X R Y) R Z] R W IV. X R [(Y R Z) R W] V. [X R (Y R Z)] R W X R [Y R (Z R Z)] X R [Y R (Z R Y)] by (3) X R [Y R (Z R X)] by (3) (X R Y) R (X R Y) case of (1) (X R Y) R (Z R Z) case of (2) 34. Examination of Copula, 1891 219 assume X is true, as we do, we must assume Y is true. Or this follows at once from (3). Form V yields no necessary formula. General forms with four copulas Form I gives the evident formulae Form II gives the evident formulae Form III gives the evident formulae These last follow at once from the principle of contraposition that if X R Y then (Y R Z) R (X R Z). Form IV gives the evident formulae I. U R {V R [X R (Y R Z)]} II. (U R V) R [X R (Y R Z)] III. [(U R V) R X] R (Y R Z) IV. U R [(V R X) R (Y R Z)] V. [U R (V R X)] R (Y R Z) VI. U R {V R [(X R Y) R Z]} VII. (U R V) R [(X R Y) R Z] VIII. [(U R V) R (X R Y...