What does "imply" mean in a statement?

Question

Does imply means two directions are both true, i.e. "equivalent to," or just one direction, i.e. "shows?"

Answer 1

Just one direction. E.g. "$x$ is divisible by $9$" implies "$x^2$ is divisible by $9$," but not conversely (take $x=3$).

Note that implication as usually formalized has some non-intuitive features - e.g. a false statement implies any true statement. Questions about the material implication are treated elsewhere on this site.

Answer 2

For a given implication ($p\rightarrow q$), its contrapositive ($\lnot q \rightarrow\lnot p$) is always true but its converse and inverse need not to be always true. Consider the following two examples.

Conditional ($p\rightarrow q$): If a number is divisible by $6$, then it is divisible by $3$.

Converse ($q\rightarrow p$): If a number is divisible by $3$, then it is divisible by $6$.

Inverse ($\lnot p\rightarrow \lnot q$): If a number is not divisible by $6$, then it is not divisible by $3$.

Contrapositive ($\lnot q\rightarrow\lnot p$): If a number is not divisible by $3$, then it is not divisible by $6$.

It is easy to find counterexamples for the converse and inverse statements stated above but the contrapositive always hold.

Conditional ($p\rightarrow q$): If a triangle is right-angled, then $H^2=P^2+B^2$.

Converse ($q\rightarrow p$): If $H^2=P^2+B^2$, then triangle is right-angled.

Inverse ($\lnot p\rightarrow \lnot q$): If a triangle is not right-angled, then $H^2\neq P^2+B^2$.

Contrapositive ($\lnot q\rightarrow\lnot p$):If $H^2\neq P^2+B^2$, then triangle is not right angled.

With a bit of luck here, the converse and inverse statements are also true.

Answer 3

Think of the statement overall as a whole. Consider the statement

if X, then Y (that is, X implies Y).

You want to know if the implication is true. Call X the assumption and Y the conclusion. If your assumptions are false, then it does not matter, if the conclusion is true or false. Overall, the statement is true. It does not really tell you anything.

What matters more is when you have an assumption that is true. For X to imply Y, when X is true, Y must be true. Therefore, it is necessary for Y to be true, when X is true, for X to imply Y.

Example:

(0 > 1) implies (0 < 2). That is if 0 > 1, then 0 < 2.

The implication is (strange, but) true.

(0 > 1) implies (0 > 2). That is if 0 > 1, then 0 > 2.

The implication is (also strange, but ) true.

(0 < 1) implies (0 > 2). The implication is false. If it is necessary for 0 to be greater than 2, for 0 to be less than 1, this cannot be true.

(0 < 1) implies (0 < 2). The implication is true. Think about that statement. For 0 to be less than 1, it necessary for 0 to be less than 2, which makes sense.