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  "body": "\n## Definition\n\nThe modal formulas are built from propositional variables $p, q, \\dots$ and the constant $\\bot$ with the connectives $\\neg$, $\\land$, $\\lor$, $\\to$ and the unary operator $\\square$.\nThe formula $\\lozenge\\varphi$ abbreviates $\\neg\\square\\neg\\varphi$.\nA normal modal logic is a set $\\Lambda$ of modal formulas with these properties.\n\n1. $\\Lambda$ contains every substitution instance of a [[tautology|05006AJ7DZ]] of [[propositional logic|09K9MWD2QQ]].\n2. $\\Lambda$ contains every formula $\\square(\\varphi \\to \\psi) \\to (\\square\\varphi \\to \\square\\psi)$, the axiom $\\mathrm{K}$.\n3. If $\\varphi \\in \\Lambda$ and $\\varphi \\to \\psi \\in \\Lambda$, then $\\psi \\in \\Lambda$, the rule of [[modus ponens|0Y0MQYST0M]].\n4. If $\\varphi \\in \\Lambda$, then $\\square\\varphi \\in \\Lambda$, the rule of necessitation.\n5. If $\\varphi \\in \\Lambda$, then each formula obtained from $\\varphi$ by substituting formulas for its variables is in $\\Lambda$.\n\nThe smallest normal modal logic is called $\\mathrm{K}$.\n\n## Remarks\n\nExample: the logic $\\mathrm{S4}$ is the smallest normal modal logic containing $\\square p \\to p$ and $\\square p \\to \\square\\square p$.\nIt satisfies the five clauses by construction, as the intersection of all normal modal logics that contain these two formulas.\n\nNon-example: the set of all tautologies of propositional logic, read as modal formulas, is not a normal modal logic.\nIt fails clause 2, since $\\square(p \\to q) \\to (\\square p \\to \\square q)$ is not an instance of a tautology.\n\nOther standard normal modal logics are $\\mathrm{T}$, given by $\\square p \\to p$, and $\\mathrm{S5}$, given by $\\square p \\to p$ and $\\lozenge p \\to \\square\\lozenge p$.\nThe word \"modal logic\" also names the whole field that studies such systems, including non-normal ones.\n\n## Explanations\n\n### L1 human\n\nA modal logic is a list of the modal formulas taken as laws.\nEvery normal modal logic keeps all laws of classical logic and adds the rule that what is proved is necessary.\nExtra axioms, such as \"what is necessary is true\", select the logic that fits a reading of $\\square$.\n",
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