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  "body": "\n## Definition\n\nFix a language of [[predicate logic|086QKPF0A6]].\nA *sequent* is an expression $\\Gamma \\vdash \\Delta$, where $\\Gamma$ and $\\Delta$ are finite lists of formulas.\nThe *sequent calculus* LK has the following parts.\n\n1. Axioms: $A \\vdash A$ for each formula $A$.\n2. Structural rules: weakening, contraction and exchange on each side, and the cut rule. The cut rule infers $\\Gamma, \\Pi \\vdash \\Delta, \\Lambda$ from $\\Gamma \\vdash \\Delta, A$ and $A, \\Pi \\vdash \\Lambda$.\n3. Logical rules: for each connective and quantifier, rules that introduce it on the left and on the right.\n\nThe logical rules for [[conjunction|0M28ASWG8T]], [[disjunction|0H5GKDTB01]], [[implication|0ZNX6DQ3EJ]], [[negation|0XR7A7RSXB]] and the quantifiers are these.\n\n- $\\wedge$R: from $\\Gamma \\vdash \\Delta, A$ and $\\Gamma \\vdash \\Delta, B$ infer $\\Gamma \\vdash \\Delta, A \\wedge B$.\n- $\\wedge$L: from $A, \\Gamma \\vdash \\Delta$ infer $A \\wedge B, \\Gamma \\vdash \\Delta$ and $B \\wedge A, \\Gamma \\vdash \\Delta$.\n- $\\vee$L: from $A, \\Gamma \\vdash \\Delta$ and $B, \\Gamma \\vdash \\Delta$ infer $A \\vee B, \\Gamma \\vdash \\Delta$.\n- $\\vee$R: from $\\Gamma \\vdash \\Delta, A$ infer $\\Gamma \\vdash \\Delta, A \\vee B$ and $\\Gamma \\vdash \\Delta, B \\vee A$.\n- $\\to$R: from $A, \\Gamma \\vdash \\Delta, B$ infer $\\Gamma \\vdash \\Delta, A \\to B$.\n- $\\to$L: from $\\Gamma \\vdash \\Delta, A$ and $B, \\Pi \\vdash \\Lambda$ infer $A \\to B, \\Gamma, \\Pi \\vdash \\Delta, \\Lambda$.\n- $\\neg$R: from $A, \\Gamma \\vdash \\Delta$ infer $\\Gamma \\vdash \\Delta, \\neg A$.\n- $\\neg$L: from $\\Gamma \\vdash \\Delta, A$ infer $\\neg A, \\Gamma \\vdash \\Delta$.\n- $\\forall$R: from $\\Gamma \\vdash \\Delta, A(y)$ infer $\\Gamma \\vdash \\Delta, \\forall x\\, A(x)$, where the variable $y$ is not free in $\\Gamma, \\Delta, \\forall x\\, A(x)$.\n- $\\forall$L: from $A(t), \\Gamma \\vdash \\Delta$ infer $\\forall x\\, A(x), \\Gamma \\vdash \\Delta$, where $t$ is a term.\n- $\\exists$L: from $A(y), \\Gamma \\vdash \\Delta$ infer $\\exists x\\, A(x), \\Gamma \\vdash \\Delta$, where $y$ is not free in $\\Gamma, \\Delta, \\exists x\\, A(x)$.\n- $\\exists$R: from $\\Gamma \\vdash \\Delta, A(t)$ infer $\\Gamma \\vdash \\Delta, \\exists x\\, A(x)$, where $t$ is a term.\n\nA *derivation* is a finite tree of sequents.\nEach leaf is an axiom, and each other node follows from its children by a rule.\nThe calculus LJ for intuitionistic logic is LK with the restriction that each right side has at most one formula.\n\n## Remarks\n\nExample: the [[law of excluded middle|0RYYK8PG4S]] has an LK derivation.\nStart from $A \\vdash A$ and get $\\vdash A, \\neg A$ by $\\neg$R.\nThen $\\vee$R twice gives $\\vdash A \\vee \\neg A, A \\vee \\neg A$, and contraction gives $\\vdash A \\vee \\neg A$.\nNon-example: this is not an LJ derivation, because $\\vdash A, \\neg A$ has two formulas on the right.\nIn fact $\\vdash A \\vee \\neg A$ has no LJ derivation at all.\nA sequent $A_1, \\dots, A_m \\vdash B_1, \\dots, B_n$ means the same as the formula $(A_1 \\wedge \\dots \\wedge A_m) \\to (B_1 \\vee \\dots \\vee B_n)$.\nSome books use multisets or sets in place of lists, and then drop the exchange rule.\n",
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