The necessary condition in the implication "$p\Rightarrow q$"

Question

My understanding is that in the implication $$p {\implies} q,$$ $p$ is sufficient to conclude $q$, and $q$ is necessary condition for $p$.

Let $p$ be "it is raining," and $q$ be "the sidewalk is wet."

The sufficient condition statement "If it is raining then it is sufficient to conclude that the sidewalk is wet" seems correct to me.

But the necessary condition statement "For the sidewalk to be wet, it is necessary that it is raining" does not seem correct, because the sidewalk can be wet for other reasons; for instance, the water-sprinkler trucks can also wet the sidewalk.

Please let me know if my necessary conditional statement is correct or not, and also please share the reason.

It is necessary for the sidewalk to be we if it is raining but that doesn't mean that the sidewalk cannot be wet if it is not raining. — John Douma, Sep 18 '24 at 04:26
Your necessary condition $q$ is "the sidewalk are going to be wet". — peterwhy, Sep 18 '24 at 04:28
You misapplied the principle: should be "wet sidewalk is a necessary condition for it to have been raining". — user2661923, Sep 18 '24 at 04:30
@JohnDouma, what's difference between your statement and mine. — novice programmer, Sep 18 '24 at 06:32
@noviceprogrammer -- you may replace the topics: genteel men do shave, but not every shaved man is genteel, is he? Don't take this example as a political comment, I only like to show, some conclusions are not like flipping coins (two sided coins). — m-stgt, Sep 18 '24 at 07:37
@MauroALLEGRANZA, I have read the post but still not able to accurately relate to my post. — novice programmer, Sep 18 '24 at 15:23
Voted to reopen. The question at the link deals with biconditionals, asking which conditional is assumed to be for the sufficient and which for the necessary aspect. — Dan Christensen, Sep 19 '24 at 19:46
Also: Need an intuitive example for how "P is necessary for Q" means "Q$\implies$P"? — MJD, Sep 19 '24 at 19:59

score 3 · Accepted Answer · answered Sep 18 '24 at 08:52

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As pointed out in the comments, the principle you state at the beginning is correct, but you misapplied it in your example.

The statement is $p\implies q$, with $p=$"It is raining" and $q=$"The sidewalk is wet". When you say that $q$ is necessary for $p$, the phrase you should form is "A wet sidewalk is necessary for it to be raining", or, using the same structure you propose, "For it to be raining, it is necessary for the sidewalk to be wet". If you think about it, these statements agree with common sense, since, as you notice, the sidewalk could be wet for other reasons, so it's not enough (i.e., it's not sufficient) to conclude that it is raining.

answered Sep 18 '24 at 08:52

Carlos Santi Toledo

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So my necessary condition statement is correct but understanding is not correct ?That’s what you want to say ? – novice programmer Sep 18 '24 at 15:12
@noviceprogrammer no, your necessary condition statement is backwards. Rain is sufficient for wetness, so wetness is necessary for rain (i.e. it can't rain without being wet). Note that necessity is not the same as causality. – Karl Sep 18 '24 at 18:55
@karl, Thanks for your comment. One more last question, do necessary condition guarantees the outcome ? – novice programmer Sep 19 '24 at 14:25

ryang · Answer 2 · 2024-11-25T16:42:32.937

Let $P$ be "it is raining," and $Q$ be "the sidewalk is wet."$$P {\implies} Q$$

The sufficient condition statement
"If it is raining, then it is sufficient to conclude that the sidewalk is wet"

the necessary condition statement
"For the sidewalk to be wet, it is necessary that it is raining"
because the sidewalk can be wet for other reasons; the water-sprinkler trucks can also wet the sidewalk.

You're mixing up $P$ and $Q:$ the sufficient condition refers to statement $P$ while the necessary condition refers to statement $Q$ (and note that $Q$ being necessary is relative to $P$ being sufficient, and vice versa).

Moreover, you are misunderstanding the structure and meaning of those "sufficient"/"necessary" assertions. Do you see that all the following assertions are equivalent to one another?

If it is raining, then the sidewalk is wet
$P\implies Q$
For $\boldsymbol Q$ to be true, it is sufficient that $\boldsymbol P$ is true
For $P$ to be true, $Q$ being true is necessary
For $P$ to be true, $Q$ cannot be false
If $Q$ is false, then $P$ must be false
$\lnot Q\implies \lnot P$
If the sidewalk is dry, then it isn't raining
For it to be raining, the sidewalk cannot be dry
For it to be raining, it is necessary that the sidewalk is wet.

The last one is actually looks the necessary condition. But my question still unanswered which is "How side walk being wet is actually necessary condition for it to be raining? A sidewalk can also be wet via water-sprinkler truck" As far as I know. A necessary condition must be met to reach the conclusion which is not happening in this case. — novice programmer, Sep 18 '24 at 15:09
Thanks for the suggestion. are necessary consequence and necessary condition same ? — novice programmer, Sep 18 '24 at 17:00
@noviceprogrammer In the context of P⟹Q, 'necessary condition' and 'necessary consequence' synonymously refer to Q; the former emphsises logical requirement, while the latter emphasises the act of concluding. — ryang, Nov 25 '24 at 16:49

5xum · Answer 3 · 2024-09-19T04:30:39.593

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The necessary condition statement is not what you wrote, but rather "It can't be raining if the sidewalk isn't wet". In either case, the way you wrote it,

For the sidewalk to be wet, it is necessary that it is raining

you wrote it as if "it is raining" is necessary for "the sidewalk is wet", when in fact, it's the other way around.

edited Sep 19 '24 at 04:30

answered Sep 18 '24 at 10:05

5xum

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You just wrote contrapositive of the of the necessary conditional statement $q\implies p$ which is equivalent to mine. I am not sure what wrong you found . – novice programmer Sep 18 '24 at 14:52
@noviceprogrammer I added more to my answer. – 5xum Sep 19 '24 at 04:30

score 0 · Answer 4 · answered Sep 19 '24 at 04:12

In the implication $p \implies q $ (where $ p $ is "it is raining" and $ q $ is "the sidewalk is wet"):

Sufficient Condition: $ p $ being true is sufficient to conclude $ q $. In other words, if it is raining, then the sidewalk is wet. This is the forward direction of the implication, and it seems fine in your example.
Necessary Condition: For $p \implies q $, $ q $ must be a necessary condition for $ p $. This means that for the sidewalk to be wet $ q $, it does not necessarily have to be raining $p $. The sidewalk can be wet for reasons other than rain, such as sprinklers or other sources of water. Thus, $ q $ (the sidewalk being wet) does not require $p $ (it raining).

Therefore, your necessary conditional statement is correct: "For the sidewalk to be wet, it is not necessary that it is raining," because there can be other reasons for the sidewalk being wet. In other words, rain is a sufficient condition but not a necessary condition for a wet sidewalk.

The necessary condition in the implication "$p\Rightarrow q$"

4 Answers4