# How to be*really* contraction free

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DOI: 10.1007/BF01057653

- Cite this article as:
- Restall, G. Stud Logica (1993) 52: 381. doi:10.1007/BF01057653

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## Abstract

A logic is said to be*contraction free* if the rule from*A* → (*A* →*B*) to*A* →*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 is*another* contracting implication expressible in the language, the logic still cannot support such a naïve theory. A logic is said to be*robustly* 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 Ł_{ω} (with the standard connectives) are shown to be robustly contraction free.