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  "body": "\n## Definition\n\nLet $B$ be a complete [[Boolean algebra|0XADJKM699]], that is, one in which every subset has a supremum and an infimum.\nLet $\\mathcal{L}$ be a first-order language.\nA *$B$-valued model* $\\mathcal{M}$ of $\\mathcal{L}$ consists of a nonempty set $M$ and of the following data:\n\n1. For each $n$-ary relation symbol $R$ (including equality), a map $M^n \\to B$, written $(a_1, \\ldots, a_n) \\mapsto \\norm{R(a_1, \\ldots, a_n)}$.\n2. For each $n$-ary function symbol $f$, a map $f^{\\mathcal{M}} \\colon M^n \\to M$.\n\nThese data must satisfy, for all elements of $M$:\n\n3. $\\norm{a = a} = 1$ and $\\norm{a = b} = \\norm{b = a}$.\n4. $\\norm{a = b} \\wedge \\norm{b = c} \\le \\norm{a = c}$.\n5. $\\norm{a_1 = b_1} \\wedge \\cdots \\wedge \\norm{a_n = b_n} \\wedge \\norm{R(a_1, \\ldots, a_n)} \\le \\norm{R(b_1, \\ldots, b_n)}$ for each relation symbol $R$.\n6. $\\norm{a_1 = b_1} \\wedge \\cdots \\wedge \\norm{a_n = b_n} \\le \\norm{f^{\\mathcal{M}}(a_1, \\ldots, a_n) = f^{\\mathcal{M}}(b_1, \\ldots, b_n)}$ for each function symbol $f$.\n\nThe truth value of a formula with parameters from $M$ is defined by recursion: $\\norm{\\neg \\varphi} = \\neg \\norm{\\varphi}$, $\\norm{\\varphi \\wedge \\psi} = \\norm{\\varphi} \\wedge \\norm{\\psi}$, and\n$$\\norm{\\exists x \\, \\varphi(x)} = \\sup_{a \\in M} \\norm{\\varphi(a)}, \\qquad \\norm{\\forall x \\, \\varphi(x)} = \\inf_{a \\in M} \\norm{\\varphi(a)}.$$\n\n## Remarks\n\nExample: if $B = \\set{0, 1}$ and $\\norm{a = b} = 1$ exactly when $a = b$, then a $B$-valued model is an ordinary [[structure|983GZBQYT4]], where $R(a_1, \\ldots, a_n)$ holds when its value is $1$.\nNon-example: if $B$ is not complete, a supremum over $M$ may not exist, and then the value of $\\exists x \\, \\varphi(x)$ is not defined.\nThis is why the definition asks for a complete algebra.\nFor a complete Boolean algebra $B$, the Boolean-valued model $V^B$ of set theory gives every axiom of ZFC the value $1$.\nSome authors also give function symbols values in $B$, by treating them as relation symbols.\n",
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