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  "body": "\n## Definition\n\nThe formulas of propositional linear logic are built from propositional variables with these connectives and constants.\n\n1. Multiplicatives: the conjunction $A \\otimes B$ (tensor), the disjunction \"$A$ par $B$\", and the units $1$ and $\\bot$.\n2. Additives: the conjunction $A \\mathbin{\\\u0026} B$ (with), the disjunction $A \\oplus B$ (plus), and the units $\\top$ and $0$.\n3. Exponentials: $!A$ (of course) and $?A$ (why not).\n4. Linear implication: $A \\multimap B$.\n\nLinear logic is the set of sequents $\\Gamma \\vdash \\Delta$, with $\\Gamma$ and $\\Delta$ finite lists of formulas, derivable in Girard's [[sequent calculus|W72CA3R9MP]].\nThat calculus has the logical rules of each connective and the exchange rule.\nIt has weakening and contraction only for formulas $!A$ on the left and $?A$ on the right.\n\n## Remarks\n\nThe two conjunctions differ in how they treat the context.\nThe right rule of $\\otimes$ derives $\\Gamma, \\Gamma' \\vdash A \\otimes B, \\Delta, \\Delta'$ from $\\Gamma \\vdash A, \\Delta$ and $\\Gamma' \\vdash B, \\Delta'$, so the context is split.\nThe right rule of $\\\u0026$ derives $\\Gamma \\vdash A \\mathbin{\\\u0026} B, \\Delta$ from $\\Gamma \\vdash A, \\Delta$ and $\\Gamma \\vdash B, \\Delta$, so the context is shared.\n\nExample: $A \\vdash A \\mathbin{\\\u0026} A$ is derivable for a variable $A$: apply the $\\\u0026$ rule to two copies of the axiom $A \\vdash A$.\n\nNon-example: $A \\vdash A \\otimes A$ is not derivable for a variable $A$.\nBy cut elimination, a derivation can be taken cut-free, and then its last rule is the $\\otimes$ rule.\nThat rule would need $A$ for one premise and the empty context for the other, and $\\vdash A$ is not derivable.\nIn classical [[propositional logic|09K9MWD2QQ]], $A \\to A \\land A$ holds, which shows the role of contraction.\n\n## Explanations\n\n### L1 human\n\nLinear logic counts its hypotheses.\nThe sequent $A \\vdash B$ reads \"one copy of $A$ can be turned into one copy of $B$\".\nThe exponential $!A$ marks a hypothesis that may be copied or discarded freely, and so recovers ordinary reasoning when needed.\n",
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