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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 has been thoroughly answered in several previous questions, albeit not always under the exact question. Here is one link:math.stackexchange.com/questions/35991/logical-implication-help –  Asaf Karagila Sep 4 '11 at 9:22
You may also be interested in looking at this thread. –  t.b. Sep 4 '11 at 12:02

2 Answers 2

up vote 5 down vote accepted

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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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? –  Fahad Uddin Sep 4 '11 at 9:41
@fahad: In my answer, $Q$ is necessary for $P$, and $P$ is sufficient for $Q$. –  Asaf Karagila Sep 4 '11 at 9:46
@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 ($\neg p$). –  joriki Sep 4 '11 at 9:54
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. –  Doug Spoonwood Sep 4 '11 at 15:13
@LePressentiment: This is commonly known as a typo. –  Asaf Karagila Aug 28 '13 at 8:35

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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