| 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:
“Let’s start with the three fundamental Rules of Robotics.... We have: one, a robot may not injure a human being, or, through inaction, allow a human being to come to harm. Two, a robot must obey the orders given it by human beings except where such orders would conflict with the First Law. And three, a robot must protect its own existence as long as such protection does not conflict with the First or Second Laws.”
—Isaac Asimov (1920–1992)
“The early Christian rules of life were not made to last, because the early Christians did not believe that the world itself was going to last.”
—George Bernard Shaw (1856–1950)
“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)