- The Theory of Descriptions itself has the consequence that every informative identity proposition will be representable as being of one or other of two forms which I label (F) and (J).
- In neither of these forms does the identity sign occur unnegated: i.e., we never have 'x is the same as y', only 'x is different from y'.
- Wittgenstein suggested an interpretation of quantificational formulae which allows the clauses having the sense 'x is different from y' to be deleted from (F) and (J) without altering their meaning. The truncated formulae which result I label (K) and (L).
- Wittgenstein's innovation consists, not in expressing identity of object by identity of sign, but in expressing difference of object by difference of sign. No relation is involved in 'No two things phi'. Expressions of the form 'x = x' are useless, and therefore meaningless.
- Roger White has pointed out the significance of the fact that in (F) and (J) the argument places of the identity predicable are occupied by bound variables, never by names. He shows the appropriateness of expressing the concept of identity, like that of existence, by the apparatus of quantification.
- 'The phier is the same as the phier' is not, on any of these views, the contradictory of 'The phier is different from the phier'.
- Wittgenstein's doctrine of identity is required by his theory of a necessary truth, but has independent justification.
- Hintikka provides a variety of rules for interpreting quantifiers in the manner suggested by Wittgenstein.
- Hintikka's 'exclusive' interpretations of quantification and the more familiar 'inclusive' interpretation correspond to different ways in which the ordinary-language analogues of quantifiers are used. Linguistic intuitions neither constrain us nor forbid us to adopt an exclusive interpretation of the quantifiers.
- Hintikka's claim that an exclusive interpretation of the quantifiers can solve Russell's Paradox and other paradoxes of set theory.
- Hintikka seems to be mistaken in his claim that, as well as its being possible to find a translation in a language with inclusively interpreted quantifiers together with identity for every formula expressible with exclusively interpreted quantifiers, translation is always possible in the opposite direction. This appears to contradict Wittgenstein's claim that in a correct Begriffsschrift there is no way of expressing such things as 'Everything is identical with itself or 'Something is the same as a'.
- Roger White claims that the full import of the doctrine that identity is not a relation lies in just this refusal to allow such sentences to be meaningful.
- Kripke, like Russell, regards propositions of the form 'x is the same as y', where 'x' and 'y' are replaced by proper names, as necessarily true. But Wittgenstein would not allow strings of words of this sort to be propositions at all, and neither he nor Russell would approve of including names and descriptions in a single category of 'rigid designators'.
Photocopy of complete Book filed in "Various - Papers on Identity Boxes: Vol 19 (W)".
Footnote 1: Taken from "Williams (Christopher) - What Is Identity?: Introduction and Analytical Table of Contents". The numbering corresponds to Williams’s section-numbering.
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