I am taking a proof course and the implication is really causing me trouble. Every definition I've looked up for sufficient and necessary says something along the lines
"If P suffices for Q, this causes P to guarantee the result Q"
"If Q is necessary for P, P cannot be true without Q being true"
I can understand these definitions and the terminology of sufficient and necessary only if we assume P->Q is already true.
Otherwise I don't see how P can be sufficient. I look at the line on the truth table when P is true, but Q is false.
We know P is true, yet Q is false. How could P guarantee for Q in that case?
Thank you.
Here's the definition of sufficient where I am getting hung up on.
A condition A is said to be sufficient for a condition B, if (and only if) the truth (/existence /occurrence) [as the case may be] of A guarantees (or brings about) the truth (/existence /occurrence) of B... This is the definition I get stuck on.
If there's a possibility that A is true, but B ends up false. How can A always be sufficient for B?
These are just phrases that we use to express logical statements. The English sentence "P suffices for Q" (or "P implies Q", or "if P then Q") translates to the formal statement "P $\implies$ Q". It's a declaration. Similarly "Q is necessary for P" (or "P only if Q") translates to "$\neg$Q $\implies$ $\neg$P" (which is actually logically equivalent to "P $\implies$ Q".)
You're right, this is not a tautology. There are lines in the truth table where "P $\implies$ Q" evaluates to be false. "Necessary" and "sufficient" are just terms we use to describe some possible relationships between boolean variables.