A treatment of vagueness that serves to dualize the semantics of supervaluationism. Both supervaluationism and subvaluationism evaluate formulae by means of collections $S$ of classical valuations, or precisifications. Call a formula subtrue with respect to a set $S$ of precisifications if there exists a $v\in S$ such that $v$ evaluates that formula as true (a formula is subfalse if there exists a valuation in $S$ evaluating it as false). Subvaluationism defines validity as the preservation of subtruth from a set of premisses $\text{Γ}$ to a set of conclusions $\Delta$, i.e.,

• • $\Delta$ is a consequence of $\text{Γ}$ if for every set of classical valuations $S$, if for each $\phi \in \text{Γ}$ there exists a $v\in S$ evaluating $\phi$ as true, then there exists a $\psi \in \Delta$ and a $v^{\prime }\in S$ such that $v^{\prime }$ evaluates $\psi$ as true.

Notably, the semantics of subvaluationism are in a sense paraconsistent, as the inconsistency of a set of premisses $\text{Γ}$ does not entail that there exists a subtrue formula in an arbitrary $\Delta$, e.g., in general, one cannot assume that $\left\{\psi \right\}$ is a consequence of $\left\{\phi ,\neg \phi \right\}$. However, the semantics does validate the inference to arbitrary sets $\Delta$ from, e.g., a set $\text{Γ}$ including a contradiction $\phi \wedge \neg \phi$; while $\phi$ and $\neg \phi$ can simultaneously be subtrue, there is no classical valuation according to which $\phi \wedge \neg \phi$ is true, whereby $\phi \wedge \neg \phi$ can never be subtrue.