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  "body": "\n## Definition\n\nLet $P$ be a set of propositional variables.\nLet $F$ be the set of formulas of classical [[propositional logic|09K9MWD2QQ]] over $P$ built with $\\wedge$, $\\vee$, $\\neg$, $\\top$ and $\\bot$.\nLet $T \\subseteq F$ be a theory.\nFor $\\varphi, \\psi \\in F$, put $\\varphi \\sim_T \\psi$ if $T \\vdash \\varphi \\leftrightarrow \\psi$.\nThis is an [[equivalence relation|0G8VEZ3WTK]] on $F$; write $[\\varphi]$ for the class of $\\varphi$.\nThe *Lindenbaum–Tarski algebra* of $T$ is the [[quotient set|0011MTQMR6]] $F / {\\sim_T}$ with the operations\n$$[\\varphi] \\wedge [\\psi] = [\\varphi \\wedge \\psi], \\quad [\\varphi] \\vee [\\psi] = [\\varphi \\vee \\psi], \\quad \\neg [\\varphi] = [\\neg \\varphi],$$\nand the constants $0 = [\\bot]$ and $1 = [\\top]$.\n\n## Remarks\n\nThe operations are well defined because provable equivalence is compatible with the connectives, and the result is a [[Boolean algebra|0XADJKM699]].\nIts order is given by $[\\varphi] \\le [\\psi]$ if and only if $T \\vdash \\varphi \\to \\psi$.\n\nExample: for $P = \\set{p}$ and $T = \\emptyset$, every formula is equivalent to one of $\\bot$, $p$, $\\neg p$, $\\top$, and no two of these are equivalent, as their truth tables differ.\nSo the algebra is the four-element Boolean algebra with atoms $[p]$ and $[\\neg p]$.\n\nNon-example: identify formulas only when they are equal as strings.\nThen the classes $[p \\wedge q]$ and $[q \\wedge p]$ differ, so $\\wedge$ is not commutative and the result is not a Boolean algebra.\n\nThe same construction for intuitionistic logic gives a Heyting algebra.\nA theory $T$ is consistent if and only if $0 \\ne 1$ in its algebra.\n",
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