Formal Grammar by Annie Foret Glyn Morrill Reinhard Muskens Rainer Osswald & Sylvain Pogodalla

Author:Annie Foret, Glyn Morrill, Reinhard Muskens, Rainer Osswald & Sylvain Pogodalla
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

These are the standard rules of L / L1 (roughly as in [13]). We have rules to introduce either slash and ‘’ both on the right hand side of and on the left hand side of . We will now add two additional connectives, which are well-known from structural logics, namely and . These are not present in L / L1, have however been considered as extensions as early as in [14], and have been subsequently studied by [11].

This gives us the logic . Note that this slightly deviates from standard terminology, because usually, has an additional constant 0 (not to be confused with !). In our formulation, 0 and 1 coincide. In order to have logical counterparts for the bounded lattice elements and , we introduce two logical constants, which are denoted by the same symbol.2

() ()

This gives us the calculus . From a logical point of view, all these extensions of L are quite well-behaved: they are conservative, and also allow us to preserve the important result of [13], namely admissibility of the cut-rule.

We say that a sequent is derivable in a calculus, if it can be derived by the axiom and the rules of inference; we then write , , etc., depending on which calculus we use.

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