logically equivalent

Two formulasMathworldPlanetmathPlanetmath A and B are said to be logically equivalent (typically shortened to equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath) when A is true if and only if B is true (that is, A implies B and B implies A):


This is sometimes abbreviated as AB.

For example, for any integer z, the statement “z is positive” is equivalent to “z is not negative and z0”.

More generally, one says that a formula A is a logical consequence of a set Γ of formulas, written


if whenever every formula in Γ is true, so is A. If Γ is a singleton consisting of formula B, we also write


Using this, one sees that

AB  iff  AB and BA.

To see this: if AB, then AB and BA are both true, which means that if A is true so is B and that if B is true so is A, or AB and BA. 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: A and B are logically equivalent whenever A is deducibleMathworldPlanetmath from B and B is deducible from A 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 BiconditionalMathworldPlanetmathPlanetmath
Defines logical equivalence
Defines logical consequence