Basic Ideas
To show constructively that a theory has quantifier elimination, it suffices to show that we can eliminate an existential quantifier applied to a conjunction of literals, that is, show that each formula of the form:
where each is a literal, is equivalent to a quantifier-free formula. Indeed, suppose we know how to eliminate quantifiers from conjunctions of formulae, then if is a quantifier-free formula, we can write it in disjunctive normal form
and use the fact that
is equivalent to
Finally, to eliminate a universal quantifier
where is quantifier-free, we transform into disjunctive normal form, and use the fact that is equivalent to
Read more about this topic: Quantifier Elimination
Famous quotes containing the words basic ideas, basic and/or ideas:
“It is a strange fact that freedom and equality, the two basic ideas of democracy, are to some extent contradictory. Logically considered, freedom and equality are mutually exclusive, just as society and the individual are mutually exclusive.”
—Thomas Mann (1875–1955)
“The basic tool for the manipulation of reality is the manipulation of words. If you can control the meaning of words, you can control the people who must use the words.”
—Philip K. Dick (1928–1982)
“Modesty and taste are questions of latitude and education; the more people know,—the more their ideas are expanded by travel, experience, and observation,—the less easily they are shocked. The narrowness and bigotry of women are the result of their circumscribed sphere of thought and action.”
—Elizabeth Cady Stanton (1815–1902)