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 Implies[p, q] can be used to represent the logical implication .

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).

REFERENCES:

Carnap, R. Introduction to Symbolic Logic and Its Applications. New York: Dover, p. 8, 1958.

Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.