Implications

A very large share of mathematical statements take the form "if p, then q", and understanding exactly what this claims — and what it does not claim — is essential.

Consider r: "If you are born in some country, then you are a citizen of that country," built from p: "you are born in some country" and q: "you are a citizen of that country." The sentence "if p then q" asserts only this: whenever p is true, q must also be true. It says nothing at all about what happens when p is false — if you were not born in that country, the statement places no demand whatsoever on q; the citizenship question is simply left open. Nor does "if p then q" claim that p ever actually happens.

Using r: "If a number is a multiple of 9, then it is a multiple of 3" (with p: "a number is a multiple of 9", q: "a number is a multiple of 3"), the same relationship between p and q can be phrased in several equivalent ways:

1. p implies q, written $p \Rightarrow q$ — being a multiple of 9 implies being a multiple of 3.
2. p is a sufficient condition for q — merely knowing a number is a multiple of 9 is enough to guarantee it is a multiple of 3.
3. p only if q — a number is a multiple of 9 only if it is a multiple of 3.
4. q is a necessary condition for p — being a multiple of 3 is required (necessary) for being a multiple of 9. …