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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.

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    $\begingroup$ 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. $\endgroup$
    – user14972
    Jan 28, 2018 at 22:52
  • $\begingroup$ Can I ask the reason of the recent downvote? $\endgroup$
    – Taroccoesbrocco
    Apr 5, 2018 at 19:04

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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.

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  • $\begingroup$ Great reference, exactly what I was looking for. Thank you! $\endgroup$
    – Taroccoesbrocco
    Jan 29, 2018 at 0:41
  • $\begingroup$ 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. $\endgroup$
    – Taroccoesbrocco
    Jan 29, 2018 at 8:24
  • $\begingroup$ @Taroccoesbrocco Sorry, no I don't. $\endgroup$
    – Peter Smith
    Jan 29, 2018 at 8:55
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    $\begingroup$ @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). $\endgroup$
    – Mauro ALLEGRANZA
    Jan 29, 2018 at 9:17
  • $\begingroup$ @MauroALLEGRANZA - Very useful, thank you! $\endgroup$
    – Taroccoesbrocco
    Jan 29, 2018 at 9:49

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