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  "body": "\n## Definition\n\nConsider formulas of [[propositional logic|09K9MWD2QQ]] built from variables with [[negation|029JBBF693]] $\\neg$ and [[implication|0NMG7W7Q11]] $\\to$.\nThe standard *Hilbert system* for this language has the following axioms, for all formulas $\\varphi$, $\\psi$, $\\chi$:\n\n1. $\\varphi \\to (\\psi \\to \\varphi)$;\n2. $(\\varphi \\to (\\psi \\to \\chi)) \\to ((\\varphi \\to \\psi) \\to (\\varphi \\to \\chi))$;\n3. $(\\neg \\varphi \\to \\neg \\psi) \\to (\\psi \\to \\varphi)$.\n\nIts only rule of inference is modus ponens: from $\\varphi$ and $\\varphi \\to \\psi$, infer $\\psi$.\nLet $\\Gamma$ be a set of formulas.\nA *derivation* of $\\varphi$ from $\\Gamma$ is a finite sequence of formulas that ends with $\\varphi$.\nEach formula in it is an axiom, a member of $\\Gamma$, or follows from two earlier formulas by modus ponens.\nOne writes $\\Gamma \\vdash \\varphi$ when such a derivation exists.\n\n## Remarks\n\nHere is a derivation of $\\varphi \\to \\varphi$ from no hypotheses.\n\n1. $(\\varphi \\to ((\\varphi \\to \\varphi) \\to \\varphi)) \\to ((\\varphi \\to (\\varphi \\to \\varphi)) \\to (\\varphi \\to \\varphi))$, axiom 2.\n2. $\\varphi \\to ((\\varphi \\to \\varphi) \\to \\varphi)$, axiom 1.\n3. $(\\varphi \\to (\\varphi \\to \\varphi)) \\to (\\varphi \\to \\varphi)$, modus ponens from lines 2 and 1.\n4. $\\varphi \\to (\\varphi \\to \\varphi)$, axiom 1.\n5. $\\varphi \\to \\varphi$, modus ponens from lines 4 and 3.\n\nThe sequence $\\varphi, \\; \\varphi \\to \\psi$ is not a derivation of $\\varphi \\to \\psi$ from no hypotheses: its first line is neither an axiom nor obtained by a rule.\nMore generally, a *Hilbert system* is any proof system of this shape: axiom schemas, few rules, and proofs as lists.\nFor [[predicate logic|086QKPF0A6]] one adds axioms for the quantifiers and the rule of generalization.\n",
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