- Rules of passage (logic)
In

mathematical logic , the**rules of passage**govern howquantifier s distribute over the basiclogical connective s offirst-order logic . The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent formula inprenex normal form , and vice versa.**The rules**See Quine (1982: 119, chpt. 23). Let "Q" and "Q" 'denote ∀ and ∃ or vice versa. β denotes a closed formula in which "x" does not appear. The rules of passage then include the following sentences, whose main connective is the

biconditional :*$Qx\; [lnotalpha\; (x)]\; leftrightarrow\; lnot\; Q\text{'}x\; [alpha\; (x)]\; .$

*$Qx\; [eta\; or\; alpha\; (x)]\; leftrightarrow\; (eta\; or\; Qx\; alpha\; (x)).$

*$exist\; x\; [alpha\; (x)\; or\; gamma\; (x)]\; leftrightarrow\; (exist\; x\; alpha\; (x)\; or\; exist\; x\; gamma\; (x)).$

*$Qx\; [eta\; and\; alpha\; (x)]\; leftrightarrow\; (eta\; and\; Qx\; alpha\; (x)).$*$forall\; x\; ,\; [alpha(x)\; land\; gamma(x)]\; leftrightarrow\; (forall\; x\; ,\; alpha(x)\; land\; forall\; x\; ,\; gamma(x)\; ).$

The following conditional sentences can also be taken as rules of passage:

*$exist\; x\; [alpha\; (x)\; and\; gamma\; (x)]\; ightarrow\; (exist\; x\; alpha\; (x)\; and\; exist\; x\; gamma\; (x)).$

*$(forall\; x\; ,\; alpha(x)\; or\; forall\; x\; ,\; gamma(x))\; ightarrow\; forall\; x\; ,\; [alpha(x)\; or\; gamma(x)]\; .$

*$(exists\; x\; ,\; alpha(x)\; and\; forall\; x\; ,\; gamma(x))\; ightarrow\; exists\; x\; ,\; [alpha(x)\; and\; gamma(x)]\; .$"Rules of passage" first appeared in French, in the writings of

Jacques Herbrand . Quine employed the English translation of the phrase in each edition of his "Methods of Logic", starting in 1950.**ee also***

First-order logic

*Prenex normal form

*Quantifier **References***

Willard Quine , 1982. "Methods of Logic", 4th ed. Harvard Univ. Press.

*Jean Van Heijenoort , 1967. "From Frege to Godel: A Source Book on Mathematical Logic". Harvard Univ. Press.**External links***

Stanford Encyclopedia of Philosophy : " [*http://plato.stanford.edu/entries/logic-classical/ Classical Logic*] -- by Stewart Shapiro.

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