"implies" in this case is shorthand for the Logical Implication operator (→) in Propositional Logic. (See

http://en.wikipedia.org/wiki/Propositional_calculus.)

Given propositions *p* and *q*, *p* → *q* is defined to be True in all cases except True → False, which is False.

The confusing thing is that the result of the operator is about the validity of the **statement** "if *p* then *q*", not about the validity of *q*.

"If cows have 4 legs, then sheep have 6 legs." This is T → F = F. The statement is False because sheep don't have 6 legs.

"If cows have 6 legs, then ...." This is F → X^{1} = T. Whatever comes after the "then" doesn't matter since cows don't have 6 legs. However worthless the statement may be, it is True.

I don't remember the formal usage of implication in Propositional Logic; that was nearly 40 years ago in school; imagine an engineer in an "Introduction to Logic" course taught by the Philosophy department!

--Mark

^{1} X means "don't care" in equations and truth tables.