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Stephen Read presented harmonious inference rules for identity in classical predicate logic. I demonstrate here how this approach can be generalised to a setting where predicate logic has been extended with epistemic modals. In such a setting, identity has two uses. A rigid one, where the identity of two referents is preserved under epistemic possibility, and a non-rigid one where two identical referents may differ under epistemic modality. I give rules for both uses. Formally, I extend Quantified Epistemic Multilateral Logic (...) |
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The logic of identity contains riches not seen through the coarse lens of predicate logic. This is one of several lessons to draw from the subtle treatment of identity in Martin‐Löf type theory, to which the reader will be introduced in this article. After a brief general introduction we shall mainly be concerned with the distinction between identity propositions and identity judgements. These differ from each other both in logical form and in logical strength. Along the way, connections to philosophical (...) |
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The calculus of Natural Calculation is introduced as an extension of Natural Deduction by proper term rules. Such term rules provide the capacity of dealing directly with terms in the calculus instead of the usual reasoning based on equations, and therefore the capacity of a natural representation of informal mathematical calculations. Basic proof theoretic results are communicated, in particular completeness and soundness of the calculus; normalisation is briefly investigated. The philosophical impact on a proof theoretic account of the notion of (...) |
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We study natural deduction systems for a fragment of intuitionistic logic with propositional identity from the point of view of proof-theoretic semantics. We argue that the identity connective is a natural operator to be treated under the elimination rules as basic approach. |
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The status of the equality predicate as a logical constant is problematic. In the paper we look at the problem from the proof-theoretic standpoint and survey several ways of treating equality in formal systems of different sorts. In particular, we focus on the framework of sequent calculus and examine equality in the light of criteria of logicality proposed by Hacking and Došen. Both attempts were formulated in terms of sequent calculus rules, although in the case of Došen it has a (...) |
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The best-known syntactic account of the logical constants is inferentialism. Following Wittgenstein’s thought that meaning is use, inferentialists argue that meanings of expressions are given by introduction and elimination rules. This is especially plausible for the logical constants, where standard presentations divide inference rules in just this way. But not just any rules will do, as we’ve learnt from Prior’s famous example of tonk, and the usual extra constraint is harmony. Where does this leave identity? It’s usually taken as a (...) |
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Harmony and conservative extension are two criteria proposed to discern between acceptable and unacceptable rules. Despite some interesting works in this field, the exact relation between them is still not clear. In this article, some standard counterexamples to the equivalence between them are summarized, and a recent formulation of the notion of stability is used to express a more refined conjecture about their relation. Then Prawitz's proposal of a counterexample based on the truth predicate to this refined conjecture is shown (...) |
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