Paraconsistent logic

Remarks

Example: Priest's logic of paradox $\mathrm{LP}$ has the truth values $t$, $b$ ("both") and $f$, ordered $f < b < t$. Negation swaps $t$ and $f$ and fixes $b$; conjunction is the minimum and disjunction the maximum. The designated values are $t$ and $b$, and $\Gamma \vdash \psi$ means that every valuation giving all of $\Gamma$ designated values gives $\psi$ one too. The valuation $p \mapsto b$, $q \mapsto f$ makes $p$ and $\neg p$ designated and $q$ not, so $\set{p, \neg p} \nvdash q$.

Non-example: classical propositional logic is explosive. No valuation makes both $\varphi$ and $\neg\varphi$ true, so every $\psi$ follows from them. Intuitionistic logic is explosive too, through the axiom $\bot \to \psi$.