Editing Universal quantification (section)

=== Rules of inference ===

A [[rule of inference]] is a rule justifying a logical step from hypothesis to conclusion.  There are several rules of inference which utilize the universal quantifier.

''[[Universal instantiation]]'' concludes that, if the propositional function is known to be universally true, then it must be true for any arbitrary element of the universe of discourse.  Symbolically, this is represented as

:<math> \forall{x}{\in}\mathbf{X}\, P(x) \to P(c)</math>

where ''c'' is a completely arbitrary element of the universe of discourse.

''[[Universal generalization]]'' concludes the propositional function must be universally true if it is true for any arbitrary element of the universe of discourse.  Symbolically, for an arbitrary ''c'',

:<math> P(c) \to\ \forall{x}{\in}\mathbf{X}\, P(x).</math>

The element&nbsp;''c'' must be completely arbitrary; else, the logic does not follow: if ''c'' is not arbitrary, and is instead a specific element of the universe of discourse, then P(''c'') only implies an existential quantification of the propositional function.
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