Relational proof system for linear and other substructural logics

In this paper we give relational semantics and an accompanying relational proof system for a variety of intuitionistic substructural logics, including (intuitionistic) linear logic with exponentials. Starting with the (Kripke-style) semantics for FL as discussed in [13], we developed, in [11], a relational semantics and a relational proof system for full Lambek calculus. Here, we take this as a base and extend the results to deal with the various structural rules of exchange, contraction, weakening and expansion, and also to deal with an involution operator and with the operators ! and ? of linear logic. To accomplish this, for each extension X of FL we develop a Kripke-style semantics, RelKripkeX semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with distinguished elements, ternary relations and a list of conditions on the relations. For each extension X, RelKripkeX semantics is accompanied by a Kripke-style valuation system analogous to that in [13]. Soundness and completeness theorems with respect to FLX hold for RelKripkeX -models. Then, in the spirit of the work of Orlowska [16], [17], and Buszkowski & Orlowska [4], we develop relational logic RFLX for each extension X. The adjective relational is used to emphasize the fact that RFLX has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ→α in FLX is provable if, a translation, t(γ1 [sdot ]…[sdot ]γn ⊃α)<εvu, has a cut-complete proof tree which is fundamental. This result is constructive: that is, if a cut-complete proof tree for t(γ1 [sdot ]…[sdot ]:γn ⊃α)εvu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 [sdot ]…γn ⊃α) and from this, build a RelKripkeX countermodel for γ1 [sdot ]…[sdot ]γn ⊃α.