How to be really contraction free

Studia Logica 52 (3):381 - 391 (1993)
Abstract
A logic is said to becontraction free if the rule fromA (A B) toA B is not truth preserving. It is well known that a logic has to be contraction free for it to support a non-trivial naïve theory of sets or of truth. What is not so well known is that if there isanother contracting implication expressible in the language, the logic still cannot support such a naïve theory. A logic is said to berobustly contraction free if there is no such operator expressible in its language. We show that a large class of finitely valued logics are each not robustly contraction free, and demonstrate that some other contraction free logics fail to be robustly contraction free. Finally, the sublogics of (with the standard connectives) are shown to be robustly contraction free.
Keywords proof theory   Curry
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DOI 10.1007/BF01057653
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David Ripley (2013). Paradoxes and Failures of Cut. Australasian Journal of Philosophy 91 (1):139 - 164.
Shawn Standefer (2015). On Artifacts and Truth-Preservation. Australasian Journal of Logic 12 (3):135-158.
Daniel Nolan (2016). Conditionals and Curry. Philosophical Studies 173 (10):2629-2647.

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