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Propositional formula
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== Inductive definition == The classical presentation of propositional logic (see [[Herbert Enderton|Enderton]] 2002) uses the connectives <math>\lnot, \land, \lor, \to, \leftrightarrow</math>. The set of formulas over a given set of propositional variables is [[inductive definition|inductively defined]] to be the smallest set of expressions such that: * Each propositional variable in the set is a formula, * <math>(\lnot \alpha)</math> is a formula whenever <math>\alpha</math> is, and * <math> (\alpha\,\Box\,\beta)</math> is a formula whenever <math>\alpha</math> and <math>\beta</math> are formulas and <math>\Box</math> is one of the binary connectives <math>\land, \lor, \to, \leftrightarrow</math>. This inductive definition can be easily extended to cover additional connectives. The inductive definition can also be rephrased in terms of a [[closure (mathematics)|closure]] operation (Enderton 2002). Let ''V'' denote a set of propositional variables and let ''X<sub>V</sub>'' denote the set of all strings from an alphabet including symbols in ''V'', left and right parentheses, and all the logical connectives under consideration. Each logical connective corresponds to a formula building operation, a function from ''XX<sub>V</sub>'' to ''XX<sub>V</sub>'': * Given a string ''z'', the operation <math>\mathcal{E}_\lnot(z)</math> returns <math>(\lnot z)</math>. * Given strings ''y'' and ''z'', the operation <math>\mathcal{E}_\land(y,z)</math> returns <math>(y\land z)</math>. There are similar operations <math>\mathcal{E}_\lor</math>, <math>\mathcal{E}_\to</math>, and <math>\mathcal{E}_\leftrightarrow</math> corresponding to the other binary connectives. The set of formulas over ''V'' is defined to be the smallest subset of ''XX<sub>V</sub>'' containing ''V'' and closed under all the formula building operations.
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