logically equivalent
Two formulas and are said to be logically equivalent (typically shortened to equivalent) when is true if and only if is true (that is, implies and implies ):
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.
Title | logically equivalent |
Canonical name | LogicallyEquivalent |
Date of creation | 2013-03-22 13:17:00 |
Last modified on | 2013-03-22 13:17:00 |
Owner | sleske (997) |
Last modified by | sleske (997) |
Numerical id | 10 |
Author | sleske (997) |
Entry type | Definition |
Classification | msc 03B05 |
Synonym | tautologically equivalent |
Synonym | semantically equivalent |
Synonym | tautological equivalence |
Synonym | semantical equivalence |
Synonym | tautological consequence |
Synonym | semantical consequence |
Related topic | Biconditional |
Defines | logical equivalence |
Defines | logical consequence |