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  "body": "\n## Definition\n\nLet $\\vdash$ be a [[consequence relation|1HK0VYWTV7]] between sets of formulas and formulas of a language with a [[negation|029JBBF693]] $\\neg$.\nThe relation $\\vdash$ is explosive when $\\set{\\varphi, \\neg\\varphi} \\vdash \\psi$ for all formulas $\\varphi$ and $\\psi$.\nA logic is paraconsistent when its consequence relation is not explosive.\nThat is, there are formulas $\\varphi$ and $\\psi$ with $\\set{\\varphi, \\neg\\varphi} \\nvdash \\psi$.\n\n## Remarks\n\nExample: Priest's logic of paradox $\\mathrm{LP}$ has the truth values $t$, $b$ (\"both\") and $f$, ordered $f \u003c b \u003c t$.\nNegation swaps $t$ and $f$ and fixes $b$; conjunction is the minimum and disjunction the maximum.\nThe designated values are $t$ and $b$, and $\\Gamma \\vdash \\psi$ means that every valuation giving all of $\\Gamma$ designated values gives $\\psi$ one too.\nThe valuation $p \\mapsto b$, $q \\mapsto f$ makes $p$ and $\\neg p$ designated and $q$ not, so $\\set{p, \\neg p} \\nvdash q$.\n\nNon-example: classical [[propositional logic|09K9MWD2QQ]] is explosive.\nNo valuation makes both $\\varphi$ and $\\neg\\varphi$ true, so every $\\psi$ follows from them.\nIntuitionistic logic is explosive too, through the axiom $\\bot \\to \\psi$.\n\n## Explanations\n\n### L1 human\n\nA paraconsistent logic is any logic in which some contradiction fails to prove some statement.\nSuch logics weaken negation, or the notion of consequence, so that a contradiction can be true in a model without making everything true.\n",
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