The notions of (1) two statements’ being compatible and (2) one statement’s being implied by the other are closely related. Here’s how:
X and Y are compatible if and only if both of the following are true:
1. X does not imply the falsity of Y.
2. Y does not imply the falsity of X.
1 and 2 are equivalent—they are either both true or both false—so we can also say this:
X and Y are compatible if and only if neither implies the falsity of the other.