How to be really contraction free
It is well known that the inference of contraction (from \(A\to (A\to B)\) to derive \(A\to B\)) is necessary for the deduction of absurdity from paradoxical statements like ``if this is true, then absurdity follows''. If we call this kind of contraction ``2-1 contraction'' (from two uses of the antecedent \(A\) to one use), then the picture generalises. For a logic to resist absurd conclusions from paradoxical statements like these, it must not only be ``2-1 contraction'' free, but also ``\(n-m\) contraction'' free where \(n>m\). This paper shows that no finitely valued logic is \(n-m\) contraction free for each \(n\) and \(m\), and it gives examples of other logics which are \(2-1\) contraction free but not \(n-m\) contraction free.
scientific article; zbMATH DE number 488343