Lattice-like semantics for the propositional fragment of Minimal Logic

Question

It's been previously asked for references for the semantics of minimal logic. The answers mention different kinds of semantics, but I'm wondering if there are also semantics using lattices or order theory.

Answer 1

The semantics presented in the linked question should apply verbatim to the propositional fragment, but since you asked for a lattice-like semantics specifically, I'll show you how to derive one here. I'll also prove that a complete lattice-style semantics cannot be finite-valued.

The right notion of semantics is immediate from the description you linked:

in minimal logic the bottom ⊥ is considered as a propositional variable, without any special inference rule involving it

This means that you can start with any complete semantics for intuitionistic logic, forget that $\bot$ was supposed to have a special interpretation, interpret it like you would interpret any other propositional variable, and thus immediately obtain a complete semantics for minimal logic.

Take a Heyting algebra $(H,\cap,\cup,\Rightarrow,0,1)$. Recall that a valuation $v$ assigns elements of some $H$ to formulae of intuitionistic propositional logic so that $v(\top)=1, v(\bot)=0, v(x \wedge y) = v(x) \cap v(y), v(x \vee y) = v(x) \cup v(y), v(x \rightarrow y) = v(x) \Rightarrow v(y)$ hold for all formulae $x,y$. The soundness/completeness of the semantics states a formula $x$ is a tautology of intuitionistic propositional logic precisely if $v(x)=1$ holds for any valuation $v$ in any Heyting algebra.

So, one obtains a sound and complete semantics for minimal logic by removing the requirement $v(\bot)=0$ in the definition above, getting the notion of a "minimal valuation": a formula $x$ is a tautology of propositional minimal logic precisely if $v(x)=1$ under any minimal valuation $v$.

The proof of soundness/completeness is immediate from the proof-theoretic translation from minimal logic to intuitionistic logic which replaces $\bot$ with a fresh propositional variable that does not otherwise occur in the formula $x$.

Note that there is no way to have a single fixed finite $n$-valued semantics (like the $0/1$-valued Boolean semantics of classical logic) for minimal logic, for essentially the same reason you cannot have one for intuitionistic logic. The disjunction property holds for minimal logic just as well as it holds for intuitionistic logic. An $n$-valued semantics always assigns $1$ to some finite disjunction of atomic biimplications $(P_1 \leftrightarrow Q_1) \vee \dots \vee (P_m \leftrightarrow Q_m)$: if this was a tautology, then it follows from the disjunction property that minimal logic would have to prove one of the individual disjuncts $P_i \leftrightarrow Q_i$ where $P_i$ and $Q_i$ are two different propositional letters. This is impossible.