Canonical extensions and relational completeness of some substructural logics

Abstract

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

Relational semantics for full linear logic

“…Hence, the first step in obtaining relational semantics for substructural logics, using the method depicted in Figure 1, is to define canonical extensions for posets. This is worked out in Section 2 of [DGP05] where one can find a careful and clear explanation of this theory. We quickly recap the relevant definitions below.…”

“…Thereafter their ideas have been developed further, which has led to a smooth theory of canonical extensions applicable in a broad setting [GH01,GJ94]. In [DGP05] canonical extensions of partially ordered algebras are defined to obtain relational semantics for the implication-fusion fragment of various substructural logics.…”

“…In [DGP05] relational semantics for the basic implication-fusion fragment of linear logic was obtained. In this paper we extend this approach to derive relational semantics for full linear logic.…”

“…We focus on the parts of this general theory that are important for the remainder of our paper and refer the reader to [DGP05,Geh06] for more details. In Section 4 this method is applied to obtain relational semantics for the multiplicative additive fragment of linear logic (MALL).…”

“…In [DGP05] it is shown that the theory of relational semantics is also available in the more general setting of substructural logic, at least in an algebraic guise. Building on these ideas, in [Geh06] a type of frames is described which generalise Kripke frames and provide semantics for substructural logics in a purely relational form.…”