16. Identity

Harold R. (Hal) Foster’s Prince Valiant

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Identity

The propositional function "x is identical with y" is expressed by
                      x = y.

This will be defined (cf. *13^.01), but, owing to certain difficult points involved in the definition, we shall here omit it (cf. Chapter II). We have, of course,

       ├ . x = x (the law of identity),

       ├ : x = y . º . y = x,

       ├ : x = y . y = z . É . x = z.

The first of these expresses the reflexive property of identity: a relation is called reflexive when it holds between a term and itself, either universally, or whenever it holds between that term and some term. The second of the above propositions expresses that identity is a symmetrical relation: a relation is called symmetrical if, whenever it holds between x and y, it also holds between y and x. The third proposition expresses that identity is a transitive relation: a relation is called transitive if, whenever it holds between x and y and between y and z, it holds also between x and z.

We shall find that no new definition of the sign of equality is required in mathematics: all mathematical equations in which the sign of equality is used in the ordinary way express some identity, and thus use the sign of equality in the above sense.

If x and y are identical, either can replace the other in any proposition without altering the truth-value of the proposition; thus we have

├ : x = y . É . φx º φy.

This is a fundamental property of identity, from which the remaining properties mostly follow.

It might be thought that identity would not have much importance, since it can only hold between x and y if x and y are different symbols for the same object. This view, however, does not apply to what we shall call "descriptive phrases," i.e. "the so-and-so." It is in regard to such phrases that identity is important, as we shall shortly explain. A proposition such as "Scott was the author of Waverley" expresses an identity in which there is a descriptive phrase (namely " the author of Waverley"); this illustrates how, in such cases, the assertion of identity may be important. It is essentially the same case when the newspapers say "the identity of the criminal has not transpired." In such a case, the criminal is known by a descriptive phrase, namely "the man who did the deed," and we wish to find an x of whom it is true that "x = the man who did the deed." When such an x has been found, the identity of the criminal has transpired.

Script: E. Nelson Bridwell  Pencils and Inks: Alex Toth  Super Friends © Respective copyright/trademark holders

Identity is important to all people, most especially adolescents trying desperately to break the bonds of childhood and become a certain kind of adult. In relation to our subject here, identity is important to understand in constructing propositional equations containing complex or ambiguous elements. The main importance is to know for sure what we are dealing with, and keeping tightly to the path of our perception. As long as we are clear in our minds what we are planning to convey in our work, this clarity will show through in the equations themselves. If we lose sight of identity in this sense, all our work is in vain.

This work is one of metaphysics, having very much less to do with mathematics than how to think about them, and speak/write clearly about them, or any other subject we chose to think, speak or write about. However a word of warning; for I have found as my thought and reasoning has transformed my perception and expression, this clarity puts an obstacle of separation from most of society in my day to day life.     

Story and Art: John Stanley  Little Lulu © Respective copyright/trademark holders.

Harold R. (Hal) Foster’s Prince Valiant

© Respective copyright/trademark holders.