- Quine's variable-free notation, and in particular his symbol 'Ref', can help us to form a unified picture of the way in which the concept of identity works.
- The first of three functions performed by variables: indication of scope.
- The second function, indication of the direction of a relational expression, can be combined with the first function, scope-indication, by the device of binding variables.
- An inversion operator reversing the direction of relational expressions can take over the second function of variables, leaving quantifiers shorn of their variables to take over the first function unaided.
- The third function of variables is to allow a quantifier to operate on more than one argument-place of a polyadic predicable. This is achieved by repetition of a variable in the matrix of the quantified formula. Such repetition still required in the case of n-place predicables for n>2.
- Quine's two inversion operators enable us to position all repeated variables in final position in open sentences. These operators, together with 'Ref', take care of the third of the functions normally performed by bound variables.
- Quine's apparatus can be applied to molecular as well as to atomic sentences.
- The variable-free notation can be used to detect the fallacy in the argument put forward at the beginning of Chapter III ("Williams (Christopher) - The Need For a Sign of Identity").
- Existential generalization in some cases needs to be split into two operations: reflection and derelativization. This allows us to reject the derivation of (3A) from (2A), on the grounds that 'Paul thinks that' is opaque with respect to reflection although transparent with respect to derelativization.
- English vernacular equivalents of 'Ref'.
- Comparisons with other languages, particularly languages which have a middle voice.
- If reflection is analogous to the middle, inversion is analogous to the passive voice.
- Many languages, including an earlier form of English, use just one morpheme to do the work done in modern English by the two morphemes 'same' and 'self'.
- 'Same' and 'self' express the same concept, although there is another concept of self, connected with self-consciousness. This will be studied in Chapter IX ("Williams (Christopher) - Personal Identity").
- We need a sign for the concept of identity (expressed by 'same' or 'self'), but not for that of difference (expressible by 'other' or 'else'). Wittgenstein was wrong about identity but right about difference.
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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