Implies
"Implies" is the connective in propositional calculus which has the meaning "if 
is true, then 
is also true." In formal terminology, the term conditional
is often used to refer to this connective (Mendelson 1997, p. 13). The symbol
used to denote "implies" is 
, 
(Carnap
1958, p. 8; Mendelson 1997, p. 13), or 
. The Wolfram Language command Experimental`ImpliesRealQ[ineqs1,
ineqs2] can be used to determine if the system of real algebraic equations
and inequalities ineqs1 implies the system of real algebraic equations and
inequalities ineqs2.
In classical logic, 
is an abbreviation for 
, where
denotes NOT and 
denoted OR (though this is not the
case, for example, in intuitionistic logic).
is a binary operator that is implemented in
the Wolfram Language as Implies[A,
B], and can not be extended to more than two arguments.
has the following truth
table (Carnap 1958, p. 10; Mendelson 1997, p. 13).
 |  |  |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
If 
and 
(i.e., 
), then 
and 
are said to be
equivalent, a relationship which is written symbolically
as 
, 
, or
(Carnap 1958, p. 8).
Contact the MathWorld Team
p implies q