# Semantics for minimal logic

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.

• I find it boggling that one would throw in a gratuitous propositional constant without any axioms relating to it. I find it especially boggling that one would name the constant $\bot$ when it is not a bottom element. – user14972 Jan 28 '18 at 22:52
• Can I ask the reason of the recent downvote? – Taroccoesbrocco Apr 5 '18 at 19:04

## 1 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.

• Great reference, exactly what I was looking for. Thank you! – Taroccoesbrocco Jan 29 '18 at 0:41
• Do you know which is the first paper where a completeness theorem for (propositional and/or first-order) minimal logic has been proved? Maybe it is already in the paper by Johansson introducing minimal logic (1937), but it is in German and I don't speak German. – Taroccoesbrocco Jan 29 '18 at 8:24
• @Taroccoesbrocco Sorry, no I don't. – Peter Smith Jan 29 '18 at 8:55
• @Taroccoesbrocco - you can see also Almudena Colacito's Thesis: Minimal and Subminimal Logic of Negation with some historical references: maybe useful the ref to H.Rasiowa's algebraic semantics (1974). – Mauro ALLEGRANZA Jan 29 '18 at 9:17
• @MauroALLEGRANZA - Very useful, thank you! – Taroccoesbrocco Jan 29 '18 at 9:49