This is sometimes abbreviated as .
For example, for any integer , the statement “ is positive” is equivalent to “ is not negative and ”.
More generally, one says that a formula is a logical consequence of a set of formulas, written
if whenever every formula in is true, so is . If is a singleton consisting of formula , we also write
Using this, one sees that
To see this: if , then and are both true, which means that if is true so is and that if is true so is , or and . The argument can be reversed.
Remark. Some authors call the above notion semantical equivalence or tautological equivalence, rather than logical equivalence. In their view, logical equivalence is a syntactic notion: and are logically equivalent whenever is deducible from and is deducible from in some deductive system.
|Date of creation||2013-03-22 13:17:00|
|Last modified on||2013-03-22 13:17:00|
|Last modified by||sleske (997)|