Gelfand quantales are complete unital quantales with an involution, *, satisfying the property that for any element a, if a b a for all b, then a a* a = a. A Hilbert-style axiom system is given for a propositional logic, called Gelfand Logic, which is sound and complete with respect to Gelfand quantales. A Kripke semantics is presented for which the soundness and completeness of Gelfand logic is shown. The completeness theorem relies on a Stone style representation theorem for (...) complete lattices. A Rasiowa/Sikorski style semantic tableau system is also presented with the property that if all branches of a tableau are closed, then the formula in question is a theorem of Gelfand Logic. An open branch in a completed tableaux guarantees the existence of an Kripke model in which the formula is not valid; hence it is not a theorem of Gelfand Logic. (shrink)

We present a general framework for proof systems for relational theories. We discuss principles of the construction of deduction rules and correspondences reflecting relationships between semantics of relational logics and the rules of the respective proof systems. We illustrate the methods developed in the paper with examples relevant for the Lambek calculus and some of its extensions.

In this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system (...) analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t( $\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ , has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for $t(\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ is not fundamental, we can use the failed proof search to build a relational countermodel for $t(\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha)\varepsilon \upsilon u$ and from this, build a RelKripke countermodel for $\gamma_1 \bullet \cdots \bullet \gamma_n \supset \alpha$ . These results allow us to add FL, the basic substructural logic, to the list of those logies of importance in computer science with a relational proof theory. (shrink)