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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?

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    $\begingroup$ I don't get what you're asking. "P is sufficient for Q" translated into formal language is $P\to Q.$ So if you don't assume it's true, you aren't assuming $P$ is sufficient for $Q.$ The line on the truth table when $P$ is true and $Q$ is false says $P\to Q$ is false, i.e. $P$ does not guarantee $Q$. So $P$ doesn't guarantee $Q$ in that case. $\endgroup$ Feb 2 '17 at 5:34
  • $\begingroup$ 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? $\endgroup$
    – dfk3
    Feb 2 '17 at 5:48
  • $\begingroup$ To see that they mean the same thing via truth tables, note that if $P$ and $P\to Q$ are both true, then $Q$ must be true en.wikipedia.org/wiki/… $\endgroup$ Feb 2 '17 at 5:48
  • $\begingroup$ If there's a possibility that $A$ is true but $B$ ends up false, then $A$ is is not sufficient for $B.$ $\endgroup$ Feb 2 '17 at 5:54
  • $\begingroup$ So before we can say that A is sufficient for B. We have to assume A->B is true. And the same before saying B is necessary for A? $\endgroup$
    – dfk3
    Feb 2 '17 at 5:56
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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.

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  • $\begingroup$ Ok, I think I am understanding it now. So when you say possible relationships, we still leave the possibility open that P's existence does not guarantee the existence of Q? $\endgroup$
    – dfk3
    Feb 2 '17 at 5:51
  • $\begingroup$ A is not always sufficient for B, for undefined statements A and B. But in a specific case, it can be true, and it's nice to have the word "sufficient" to describe that scenario. For example, let A = "My name is Bob", and B = "There exists a man named Bob". Then A is sufficient for B. $\endgroup$
    – nkm
    Feb 2 '17 at 5:55
  • $\begingroup$ So before we can say A is sufficient for B, we have to assume that A->B is true. Does truth of the implication matter when applying the relationship terms (sufficient, necessary)? $\endgroup$
    – dfk3
    Feb 2 '17 at 5:59
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$P \implies Q$ does not mean that $\neg P \implies \neg Q$.

Moreover, $(P \implies Q) \land (\neg P \implies \neg Q) \equiv (P \implies Q) \land (Q \implies P) \equiv (P \iff Q)$.

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