Source: J. Symbolic Logic
Volume 72, Issue 3
Consider a general class of structural inference rules
such as exchange, weakening, contraction and their generalizations.
Among them, some are harmless but others do harm to cut elimination.
Hence it is natural to ask under which condition cut elimination is
preserved when a set of structural rules is added to a
structure-free logic. The aim of this work is to
give such a condition by using algebraic semantics.
We consider full Lambek calculus (FL), i.e., intuitionistic logic
without any structural rules, as our basic framework.
Residuated lattices are the algebraic structures corresponding to
FL. In this setting, we introduce a criterion,
called the propagation property, that can be stated
both in syntactic and algebraic terminologies.
We then show that, for any set ℛ of structural rules,
the cut elimination theorem holds for FL enriched with
if and only if ℛ satisfies the propagation property.
As an application, we show that any set ℛ of structural rules
can be “completed" into another set ℛ*, so that
the cut elimination theorem holds for FL enriched with
ℛ*, while the provability remains the same.
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