Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or contingent identity. The arguments seem to prove too much, but their conclusion is hard to resist if we want to keep Leibniz's Law. We present a novel approach to this issue, based on an appropriate modification of the notion of logical consequence.
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In general, it might be risky to apply conditional proof after the rule of necessitation. In the present case, however, there is no problem since necessitation is applied to something that can be derived independently of the premises discharged in the application of conditional proof.
Schechter (2011) calls a logic ‘weakly classical’ if it preserves all classically valid inferences (though not, perhaps, all classically valid metainferences). Our approach to truth and vagueness weakly classical in this sense although, strictly speaking, it yields classical logic for the classical vocabulary and an extension of classical logic for languages enjoying transparent truth or similarity predicates. See Ripley (2012) for proof-theoretic results on the theory of transparent truth.
In our 2012b paper, it is shown how to define analogues of the non-transitive ST logic for possible world semantics. We shall make some simplifying assumptions below, like that all atomic formulas are of the form x = y. We leave the study of more general options for future research.
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The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement n° 229 441-CCC and from the Ministerio de Economía y Competitividad, Government of Spain project “Borderlineness and Tolerance” ref. FFI2010-16984.
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Cobreros, P., Egré, P., Ripley, D. et al. Identity, Leibniz's Law and Non-transitive Reasoning. Int Ontology Metaphysics 14, 253–264 (2013). https://doi.org/10.1007/s12133-013-0125-2