Minimal logic is a fragment of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet). Essentially, from proof-theoretical viewpoint, this means that in minimal logic the bottom $\bot$ is considered as a propositional variable, without any special inference rule involving it.
Question: Is there a semantics $\mathcal{S}$ for minimal logic such that a completeness theorem holds? By completeness theorem I mean a statement of the form (for both propositional and first-order languages):
For every formula $A$, $A$ is provable in minimal logic if and only if $A$ is valid in all $\mathcal{S}$-structures.
I guess such a semantics could be a generalization of intuitionistic Kripke models.
I would like also to have some references about completeness theorem in (propositional and/or first-order) minimal logic.
Best Answer
Yes, tweaking Kripke semantics based on that for intuitionistic logic does the trick. If I recall right, there are slightly different ways of doing this.
Here's a version for minimal logic and some variants, in a recent paper by Almudena Colacito, Dick de Jongh and Ana Lucia Vargas.