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I am trying to understand what “$p$ implies $q$” means. I read that $p$ is a sufficient condition for $q$, and $q$ is a necessary condition for $p$. Further from Wikipedia,

A necessary condition of a statement must be satisfied for the statement to be true. Formally, a statement $P$ is a necessary condition of a statement $Q$ if $Q$ implies $P,\quad (Q \Rightarrow P)$.

A sufficient condition is one that, if satisfied, assures the statement's truth. Formally, a statement $P$ is a sufficient condition of a statement $Q$ if $P$ implies $Q,\quad (P \Rightarrow Q)$.

Now what I am stuck with is that if $P$ is not satisfied will the condition still always be true?

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This is a simple matter answered by the truth table of $\Rightarrow$:

$$\begin{array}{ c | c || c | } P & Q & P\Rightarrow Q \\ \hline \text T & \text T & \text T \\ \text T & \text F & \text F \\ \text F & \text T & \text T \\ \text F & \text F & \text T \end{array}$$

This shows that when $P$ is false, the implication is true. Note that this is the definition of the table, there is no need to prove it. This is how $\Rightarrow$ is defined to work.

As an example, here is one:

$$\textbf{If it is raining then there are clouds in the sky}$$

In this case $P=$It is raining, and $Q=$There are clouds in the sky. Note that $P$ is sufficient to conclude $Q$, and $Q$ is necessary for $P$. There is no rain without clouds, and if there are no clouds then there cannot be any rain.

However, note that $P$ is not necessary for $Q$. There could be light clouds without any rain, and there could be clouds of snow and blizzard (which is technically not rain).

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  • $\begingroup$ Thanks for your answer. In your answer the statement P is false, but as I read that p is not neccessary but sufficient for q then shouldn't it be true as q is true? $\endgroup$ – user2857 Sep 4 '11 at 9:41
  • $\begingroup$ @fahad: In my answer, $Q$ is necessary for $P$, and $P$ is sufficient for $Q$. $\endgroup$ – Asaf Karagila Sep 4 '11 at 9:46
  • $\begingroup$ @fahad: No; that $p$ is not necessary for $q$ means that $p$ need not be true for $q$ to be true. It is a sufficient but not necessary condition for you to get rich that you win the lottery. It may well be true that you get rich ($q$) even though you don't win the lottery ($\neg p$). $\endgroup$ – joriki Sep 4 '11 at 9:54
  • $\begingroup$ Note that there exist 15 other possibilities for the last column where each entry belongs to {T, F}. None of the other ones fit with the meaning of implication as well as this one. $\endgroup$ – Doug Spoonwood Sep 4 '11 at 15:13
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    $\begingroup$ @Zephyr: No. Sufficient means that it is enough to know that P is true in order to know that Q is true. And that's that. Nothing more, and nothing less. We say that P is necessary if the falsehood of P implies the falsehood of Q. $\endgroup$ – Asaf Karagila Sep 24 '17 at 12:10
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Once I read a classic example which I would like to share with you. A politician said......... p(If I Will win) q(I will cut down taxes by half) People will feel cheated only in the case when the politician actually won but taxes remained the same. In this case, q is FALSE but p is true so p->q become false. We are okay with the rest of cases/situation, so rest of combination of p and q this statement is TRUE.

Lets examine "p is not necessary but sufficient for q"................

There can be a case when this politician lose but taxes get reduced so for the reduction of taxes p is not necessary but sufficient...........

regarding "q is a necessary condition for p" we can say reduction in taxes is necessary after the victory of the politician else the entire statement of the politician will be false or we can say p->q will be false.

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One way implications like this don't put any restrictions on what the antecedent can be (P), only what the consequent (Q) can be. P can always be whatever, but Q NEEDS to be true IF P is true (Otherwise the statement "If P is true, then Q is true will be false). However, if P is false, the Q can be whatever.

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