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  "body": "\n## Definition\n\nFix a finite set $C$ of connectives, each with an arity.\nA logical matrix for $C$ is a triple $(V, D, \\set{f_c}_{c \\in C})$ with these parts.\n\n1. A set $V$ of truth values with at least three elements.\n2. A nonempty proper subset $D \\subseteq V$ of designated values.\n3. For each $n$-ary connective $c$, a [[function|0EPAT51DXQ]] $f_c \\colon V^n \\to V$, its truth table.\n\nA valuation assigns a value in $V$ to each propositional variable and extends to all formulas through the functions $f_c$.\nA formula is a tautology of the matrix when it takes a value in $D$ under every valuation.\nThe many-valued logic of the matrix is the set of its tautologies.\n\n## Remarks\n\nWith $V = \\set{0, 1}$, $D = \\set{1}$ and the usual tables, the tautologies are those of classical [[propositional logic|09K9MWD2QQ]]; clause 1 excludes this two-valued case.\n\nExample: Łukasiewicz's three-valued logic $\\mathrm{L}_3$ has $V = \\set{0, \\tfrac12, 1}$ and $D = \\set{1}$.\nIts tables are $\\neg x = 1 - x$, $x \\to y = \\min(1, 1 - x + y)$, $x \\lor y = \\max(x, y)$ and $x \\land y = \\min(x, y)$.\nThe formula $p \\to p$ is a tautology, since $\\min(1, 1 - x + x) = 1$ for every $x$.\n\nNon-example: $p \\lor \\neg p$ is a classical [[tautology|05006AJ7DZ]] but not a tautology of $\\mathrm{L}_3$.\nThe valuation $p \\mapsto \\tfrac12$ gives $\\max(\\tfrac12, \\tfrac12) = \\tfrac12 \\notin D$.\n\nInfinite-valued logics use, for example, $V = [0, 1]$ with the same tables.\n\n## Explanations\n\n### L1 human\n\nA many-valued logic replaces the two truth values by a larger set and gives a table for each connective on that set.\nA formula is a law when it always lands among the designated values.\nSome classical laws survive, and others, such as excluded middle in Łukasiewicz logic, do not.\n",
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