| Transformation rules |
|---|
| Propositional calculus |
|
Rules of inference Rules of replacement Commutativity Distributivity Double negation De Morgan's laws Transposition Material implication Exportation Tautology |
| Predicate logic |
| Universal generalization Universal instantiation Existential generalization Existential instantiation |
A rule of inference is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the existential quantifier.
Existential introduction (∃I) concludes that, if the propositional function is known to be true for a particular element of the domain of discourse, then it must be true that there exists an element for which the proposition function is true. Symbolically,
The reasoning behind existential elimination (∃E) is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is necessarily true, as long as it does not contain the name. Symbolically, for an arbitrary c and for a proposition Q in which c does not appear:
must be true for all values of c over the same domain X; else, the logic does not follow: If c is not arbitrary, and is instead a specific element of the domain of discourse, then stating P(c) might unjustifiably give more information about that object.
Read more about this topic: Existential Quantification, Properties
Famous quotes containing the words rules of, rules and/or inference:
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—Henry David Thoreau (1817–1862)
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—Louise Bourgeois (b. 1911)
“The inference is, that God has restated the superiority of the West. God always does like that when a thousand white people surround one dark one. Dark people are always “bad” when they do not admit the Divine Plan like that. A certain Javanese man who sticks up for Indonesian Independence is very lowdown by the papers, and suspected of being a Japanese puppet.”
—Zora Neale Hurston (1891–1960)