Truth, Reference and Realism

Absolute Identity and Absolute Generality

timothy WilliAmsoN absolute identity and absolute Generality The aim of this chapter is to tighten our grip on some issues about quantification by analogy with corresponding issues about identity on which our grip is tighter. We start with the issues about identity. i in conversations between native speakers, words such as ‘same’ and ‘identical’ do not usually cause much difficulty. We take it for granted that others use them with the same sense as we do. if it is unclear whether numerical or qualitative identity is intended, a brief gloss such as “one thing not two” for the former or “exactly alike” for the latter removes the unclarity. in this paper, numerical identity is intended. a particularly conscientious and logically aware speaker might explain what ‘identical’ means in her mouth by saying: “everything is identical with itself. if something is identical with something, then whatever applies to the former also applies to the latter.” it seems perverse to continue doubting whether ‘identical’ in her mouth means identical (in our sense). Yet other interpretations are conceivable. for instance, she might have been speaking an odd idiolect in which ‘identical’ means in love, under the misapprehension that everything is in love with itself and with nothing else (narcissism as a universal theory). let us stick to interpretations on which she spoke truly. let us also assume for the time being that we can interpret her use of the other words homophonically. We will make no assumption at this stage as to whether ‘everything’ and ‘something’ are restricted to a domain of contextually relevant objects. We can argue that ‘identical’ in her mouth is coextensive with ‘identical’ in ours. for suppose that an i4 Truth.indb 177 2011.08.15. 8:57 178 Truth, reference and realism object x is identical in her sense with an object y. By our interpretative hypotheses, if something is identical in her sense with something, then whatever applies to the former also applies to the latter. Thus whatever applies to x also applies to y. By the logic of identity in our sense (in particular, reflexivity), everything is identical in our sense with itself, so x is identical in our sense with x. Thus being such that x is identical in our sense with it applies to x. consequently, being such that x is identical in our sense with it applies to y. Therefore, x is identical in our sense with y. Generalizing: whatever things are identical in her sense are identical in ours. conversely, suppose that x is identical in our sense with y. By the logic of identity in our sense (in particular, leibniz’s law), if something is identical in our sense with something, then whatever applies to the former also applies to the latter. Thus whatever applies to x also applies to y. By our interpretative hypotheses , everything is identical in her sense with itself, so x is identical in her sense with x. Thus being such that x is identical in her sense with it applies to x. consequently, being such that x is identical in her sense with it applies to y. Therefore, x is identical in her sense with y. Generalizing: whatever things are identical in our sense are identical in hers. conclusion: identity in her sense is coextensive with identity in our sense.1 Of course, coextensiveness does not imply synonymy or even necessary coextensiveness. Thus we have not yet ruled out finer-grained differences in meaning between her use of ‘identical’ and ours. if we can interpret her explanation as consisting of logical truths, then, given that the principles that the argument invoked about identity in our sense (reflexivity and leibniz’s law) are also logical truths, we can show that the universally quantified biconditional linking identity in her sense with identity in ours is a logical truth, so that coextensiveness is logically guaranteed.2 if the relevant kind of logical truth is closed under the rule of necessitation from modal logic, then the necessitated universally quantified biconditional too is a logical truth, so that necessary coextensiveness is also logically guaranteed. But not even a logical 1 The argument goes back to Quine (1961); see the reprinted version in Quine 1966 (178). 2 The logic of indexicals is arguably not closed under the rule of necessitation (Kaplan 1989). such problems do not seem to arise for (1) and (2). i4 Truth.indb 178 2011.08.15. 8:57 [18.117.183.49] Project MUSE (2024-04-26 04...