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  "body": "\n## Definition\n\nA Kripke frame is a pair $(W, R)$ of a nonempty set $W$ of worlds and a [[binary relation|05AVARPXA2]] $R$ on $W$, the accessibility relation.\nA Kripke model is a triple $\\mathfrak{M} = (W, R, V)$ where $V$ maps each propositional variable to a subset of $W$, an element of the [[power set|00E0YZCERH]].\nTruth of a [[modal formula|C3RZY20ETK]] at a world $w \\in W$ is defined by induction.\n\n1. $\\mathfrak{M}, w \\models p$ if and only if $w \\in V(p)$, and $\\mathfrak{M}, w \\not\\models \\bot$.\n2. The connectives $\\neg$, $\\land$, $\\lor$ and $\\to$ are evaluated at $w$ as in classical logic.\n3. $\\mathfrak{M}, w \\models \\square\\varphi$ if and only if $\\mathfrak{M}, v \\models \\varphi$ for every $v \\in W$ with $w R v$.\n\nA formula is valid on the frame $(W, R)$ when it is true at every world of every model $(W, R, V)$.\n\n## Remarks\n\nThen $\\mathfrak{M}, w \\models \\lozenge\\varphi$ if and only if some $v$ with $w R v$ has $\\mathfrak{M}, v \\models \\varphi$.\n\nExample: on any frame, the axiom $\\square(p \\to q) \\to (\\square p \\to \\square q)$ is valid.\nAt a world $w$ where $\\square(p \\to q)$ and $\\square p$ hold, every $v$ with $w R v$ satisfies $p \\to q$ and $p$, hence $q$.\n\nNon-example: $\\square p \\to p$ is not valid on the frame $(\\set{w}, \\emptyset)$.\nWith $V(p) = \\emptyset$, the formula $\\square p$ holds at $w$ because no world is accessible, while $p$ fails at $w$.\n\nThe formulas valid on all frames form the logic $\\mathrm{K}$.\nThe formulas valid on all reflexive frames form $\\mathrm{T}$, and those valid on all reflexive and transitive frames form $\\mathrm{S4}$.\n\nFor intuitionistic logic, the relation is a preorder, $V(p)$ is closed upward, and $\\varphi \\to \\psi$ holds at $w$ when $\\psi$ holds at every $v \\ge w$ where $\\varphi$ holds.\n\n## Explanations\n\n### L1 human\n\nKripke semantics reads \"necessarily\" as \"in every world I can see\".\nDifferent conditions on the arrows between worlds give different laws of necessity.\nFor instance, if every world sees itself, then what is necessary is true.\n",
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