1 Discussion and Related Work Proposition 4 If`lke(c 0 ) A, Then a Is Veriied in All C 0 -frames

A generalization of analytic deduction via labelled deductive systems.Part I: Basic substructural logics. 39 In this way, Plotkin and Stirling generate intuitionistic modal logics in which modal operators are interpreted similarly to the intuitionistic quantiiers 8 and 9, and the two modal operators are not interdeenable. An extension of the work presented in this paper to full substructural modal logics (including negation) will allow better comparisons with PS86] and others. In future papers we also plan to extend our research in the following directions: accommodating Girard's modalities (the \exponentials" of Linear Logic) into the framework outlined in the present paper; investigating the decision problem for modal substructural logics, on the basis of the LKE approach; this involves introducing a suitable notion of completed or saturated LKE tree, and adapting existing constraint satisfaction methods for solving the systems of inequalities generated when the applications of the labelled PB rule are carried out with variable labels (see Example 10 in Section 8); extending the present approach to cover modal many-valued logics; investigating the relation between our methods and the methods based on Belnap's Display Logic ((Bel82, Bel90, Wan94]). Abr91] V.M. Abrusci. Phase semantics and sequent calculus for pure non-commutative classical linear propositional logic. 38 in particular D^ os88, D^ os89], but also Avr88]. The Kripke-style \seman-tics" of implication described below, and which can be found in a number of papers, is nothing but a generalization of Urquhart's semantics for relevant implication described in Urq72]. For more semantic investigations into sub-structural logics which can be related to the contents of Section 3 see also Wan93], Ono93], Sam93], AD93], Abr91], Mac96]. Previous work on the speciic topic of this paper (combining modal and substructural logics) has been concentrating on the intuitionistic modal logics. The implication fragments of these logics belong to the wide family of substructural modal systems presented in this paper (i.e. those satisfying all the structural rules of Table 2, and are captured by the corresponding LKE systems, i.e. the LKEC systems obtained by including in the set C all the structural constraints. We consider in particular the works of Bo zi c and Do sen BD84], and Plotkin and Stirling PS86] (see also Sym93] for an overview). The main problem in deening modal connectives on Kripke-style intuitionistic structures is to guarantee the intuitionistic hereditary property for modal formulae. This can be achieved either by deening approriate satissaction clauses for the modal …