# 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, 2018 at 22:52
• Can I ask the reason of the recent downvote? Apr 5, 2018 at 19:04

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! Jan 29, 2018 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. Jan 29, 2018 at 8:24
• @Taroccoesbrocco Sorry, no I don't. Jan 29, 2018 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• @Taroccoesbrocco - you can see also Almudena Colacito's Thesis: Minimal and Subminimal Logic of Negation with some historical references: maybe useful the ref to H.Rasiowa's algebraic semantics (1974). Jan 29, 2018 at 9:17
• @MauroALLEGRANZA - Very useful, thank you! Jan 29, 2018 at 9:49

Update 2023-12-06: The proof was incomplete. I didn't cover the case where the $$\bot$$ pseudovariable was assigned the interpretation $$\varnothing$$.

You can tweak the topological semantics for S4 and IPC to get a semantics for minimal logic.

I'll assume the following two results.

1. The standard topology on $$\mathbb{R}$$ (call it $$\tau^R$$) with $$\mathbb{R}$$ as the sole distinguished truth value is a sound and complete semantics for IPC.
2. Minimal logic is exactly equivalent to IPC with $$\bot$$ interpreted as an extra propositional variable.

Given those two results, we can define the following semantics for minimal logic.

Let each unary and binary connective be defined on $$\tau^R$$ as for IPC. $$\land$$ and $$\lor$$ are $$\cap$$ and $$\cup$$ respectively. $$\to$$ is $$i(\cdot^c \cup \cdot)$$ where $$i$$ is the interior and $${}^c$$ is the complement. Let $$\lnot \cdot$$ abbreviate $$\cdot \to \bot$$ as usual.

Let $$\bot$$ be interpreted as $$(-1, 1)$$ (for concreteness, any fixed nonempty open set will do).

Here's a proof of the soundness and completeness of this semantics.

Let $$v$$ be a mapping from variable symbols including the $$\bot$$ pseudo-variable to $$\tau^R$$.

Suppose the interpretation of $$\bot$$ is not $$\varnothing$$. Let $$k$$ be an affine function sending the interpretation of $$\bot$$ to $$(-1, 1)$$. Let $$v''$$ be $$v$$ with $$k$$ applied to the interpretation of each function.

Suppose the interpretation of $$\bot$$ is $$\varnothing$$. Then define a function $$F$$ in the following way: $$F(0) = \{-1, 1\}$$, $$F(\alpha)$$ is $$1 + \alpha$$ when $$\alpha > 0$$, $$F(\alpha)$$ is $$-1 + \alpha$$ when $$\alpha < 0$$.

Let $$F(X)$$ be defined as $$(-1, 1) \cup \bigcup_{x \in X} F(x)$$.

Intuitively, $$F$$ duplicates the point $$0$$ and then parts the waters between the positive and negative reals. We then throw back in $$(-1, 1)$$.

I claim that $$v \models \varphi$$ if and only if $$v'' \models \varphi$$. As proof, observe that the affine functions $$k$$ and our line-splitter function $$F$$ commute with each operation $$\cap, \cup, {}^c, i$$. And that $$kX$$ is $$\mathbb{R}$$ if and only if $$X$$ is $$\mathbb{R}$$ and likewise for $$F$$.

Therefore this topological semantics is a sound and complete semantics for IPC with $$\bot$$ interpreted as an extra variable.

Therefore this topological semantics is a sound and complete semantics for minimal logic.