Abstract
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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Notes
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.
“Although it is better to be methodical in our investigations and to consider the economics of research, yet there is no positive sin against logic in trying any theory which may come into our heads, so long as it is adopted in such a sense as to permit the investigation to go on unimpeded and undiscouraged. On the other hand, to set up a philosophy which barricades the road of further advance toward the truth is the one unpardonable offence in reasoning, as it is also the one to which metaphysicians have in all ages shown themselves the most addicted” (Peirce 1931, I. 3. §4).
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Acknowledgments
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
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DOI: https://doi.org/10.1007/s12133-013-0125-2