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First of all I took discrete course, I know all about truth tables and stuff so I am not uninformed.I have confusion about following:

All sources says $p\implies q$ means basically $p$ implies $q$ which is if $p$ is true then $q$ is true.It sounds ok at first but why do they assume $p\implies q$ in the first place? $p$ can take the true value and $q$ can take the false value? Why they don't consider this and say if $p$ is true $q$ is true.

I am sure there must be logical explanation and other must have been thought it before me.

Kevin Long
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aintnosunshinewhenyouaregone
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    $p \implies q$ is not always a true statement. Why it is true or not true depends on each specific instance. – wgrenard Oct 20 '16 at 01:09
  • suppose p⟹q is false and if p is true q must be false basically. But they are defining or explaining p⟹q as if p is true then q is true.I know it can change in specific instance why all textbooks,wolframalpha,wikipedia doesn't see this notion. – aintnosunshinewhenyouaregone Oct 20 '16 at 01:11
  • If $p\implies q$ is false, then $p$ must be true and $q$ must be false. – Kevin Long Oct 20 '16 at 01:13
  • Yes as I say so please search p implies q and read what is written.Is it common misleading due to English language or a semantic thing? – aintnosunshinewhenyouaregone Oct 20 '16 at 01:21
  • Assuming that you are using $\Rightarrow$ for the conditional connective, it is not correct to say that it means "$p$ implies $q$". It is more correct to say that it means "if $p$, then $q$". – Mauro ALLEGRANZA Oct 20 '16 at 06:25

4 Answers4

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You have to consider the truth of the implication as a whole. P $\implies$ Q is true, if

  1. P is true and Q is true
  2. P is false (Q can be either true or false)

The only time the implication as an overall statement is false is when P is true and Q is false

See this answer:

What does "imply" mean in a statement?

Community
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RJM
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  • I doesn't mean that.When you search P ⟹ Q It is told or defined ,whatever, as if p is true then q is true.Isn't it faulty p can be true and q can be false overally the statement would be false.Why they act like as it is a premise? – aintnosunshinewhenyouaregone Oct 20 '16 at 01:15
  • You are thinking about the assumption (P) and conclusion (Q) separately. The implication is a logical construct in it's own right. It shows a relationship between the truth of P and Q. (Oranges are apples) $\implies$ (the sky is yellow) is logically true. As is (0 > 1) $\implies$ (1 < 2). The implication is equivalent logically to $\lnot$ P $\lor$ Q. – RJM Oct 20 '16 at 01:22
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The question is not very clear, but I'm going to make a guess at the source of your confusion and try to answer that; I apologize if my guess is wrong. The condition "if $p$ is true then $q$ is true" is not the definition of the sequence of symbols $p\implies q$. Rather, it is the definition of under what circumstances $p\implies q$ counts as true. So the condition "if $p$ is true then $q$ is true" might well be false; it is not assumed as a premise. It might well be the case that $p$ is true and $q$ is false, but in that situation $p\implies q$ would also be false.

Andreas Blass
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  • Yes exactly Andreas so as my intituive understanding P ⟹ Q is statement whether takes truth values 1 or 0 and p implies q means P ⟹ Q≡1; am I right? – aintnosunshinewhenyouaregone Oct 20 '16 at 01:28
  • And when we say If I swim then I will be drown; can is it mean that we are assuming the whole statement true and when I swim(1) so drown(1). Also saying if p then q for P ⟹ Q is a way to understand truth table or remembering it it is not the actual meaning of implication for sake of holding 1⟹1 will be 1 .Am I right ,I hope you will answer despite I wrote very long.Thanks – aintnosunshinewhenyouaregone Oct 20 '16 at 01:32
  • @aintnosunshinewhenyouaregone Your intuitive understanding that $p \implies q$ is a statement that can either have truth values $1$ or $0$ is correct, and this is exactly how you should think about this. Your statement that "$p$ implies $q$ means $p \implies q \equiv 1$" is not correct. "$p$ implies $q$" is just the way that we verbally say $p \implies q$. Therefore "$p$ implies $q$" is the exact same statement as $p \implies q$. – wgrenard Oct 20 '16 at 01:42
  • wgrenard let suppose you are right saying p⟹q same with verbally p implies q. – aintnosunshinewhenyouaregone Oct 20 '16 at 01:48
  • If I say you If Trump wins then(implies) Middle East will be destroyed and Trump wins will you expect to Middle east to be destroyed(1) or the possiblity of 0.I mean If trump wins can middle east not be destroyed? – aintnosunshinewhenyouaregone Oct 20 '16 at 01:49
  • If you answer is middle east will be destroyed you are holding the whole statement as 1(true) or we can say premise. – aintnosunshinewhenyouaregone Oct 20 '16 at 01:49
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    If you say "If Trump wins then Middle East will be destroyed" and then Trump wins I will not expect the Middle East to be destroyed because I do not know if the statement you told me (the whole bit in quotation marks) is true or not. You can make up any crazy implication that you want but there is no reason to expect it to be true. In other words you cannot just assume that some statement "$p$ implies $q$" is true. You either have to prove that it is true or be given that it is true. – wgrenard Oct 20 '16 at 01:58
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I'm not sure this question has any informational content; you're confused about a semantic issue (I think?). The statement has to start somewhere. Why shouldn't P imply Q? Suppose $x=2$

P: x is an even number

Q: x is a multiple of 2

P implies Q because this is the way we define even numbers. I can't say the same if

Q: x is a dog

Then, of course, P does not imply Q. There is a requirement that Q does, in fact, follow from P before you can use the statement.

Anthony P
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  • Anthony you are getting to my point thanks so I think p implies q is not the same thing with P ⟹ Q – aintnosunshinewhenyouaregone Oct 20 '16 at 01:23
  • P ⟹ Q is statement whether takes truth values 1 or 0 and p implies q means P ⟹ Q≡1 – aintnosunshinewhenyouaregone Oct 20 '16 at 01:26
  • Anthony, Am i right? – aintnosunshinewhenyouaregone Oct 20 '16 at 01:27
  • I don't understand what you're asking ... P $\rightarrow$ Q has to be verified before it can be used ... I can't just make any statement $P$ and say it implies $Q$ just because they're both true. For example, I can't say P: The sun is bright Q:My dog is yellow, and somehow draw an implication here. – Anthony P Oct 20 '16 at 01:27
  • x is not really defined here, though. Let x=3, then P(x) $\implies$ Q(x) is true in both cases. – RJM Oct 20 '16 at 01:29
  • Sorry, I meant to assume P is true. Assume x is, in fact, an even number – Anthony P Oct 20 '16 at 01:30
  • So in a specific instance Sun is bright=1(true);My dog is yellow=1(true) – aintnosunshinewhenyouaregone Oct 20 '16 at 01:35
  • Sun is bright⟹My dog is yellow≡1(truth value for definition and tables) but it does not implies.Implying(english context and mathematical implication is not the same thing.Am I right? – aintnosunshinewhenyouaregone Oct 20 '16 at 01:36
  • Then, the sun is bright logically implies your dog is yellow. Logically, but has no helpful meaning. – RJM Oct 20 '16 at 01:37
  • That is correct. Mathematical implication does not always agree with the intuition of "implies" in English. – RJM Oct 20 '16 at 01:39
  • I think, to get to the root of the confusion, you first have to come up with a reason for saying $P \rightarrow Q$ ...

    If P is true, and Q follows from P, then you can use the various axioms developed to accomplish what your goal is.

     – Anthony P Oct 20 '16 at 01:41
  • Sorry I couldn't understand what are the axioms? – aintnosunshinewhenyouaregone Oct 20 '16 at 01:42
  • @Anthony Paonessa ,@ Robert J McGinness And when we say If I swim then I will be drown; can is it mean that we are assuming the whole statement true and when I swim(1) so drown(1). Also saying if p then q for P ⟹ Q is a way to understand truth table or remembering it it is not the actual meaning of implication for sake of holding 1⟹1 will be 1 – aintnosunshinewhenyouaregone Oct 20 '16 at 01:44
  • You're probably going to have to find the answer in your native language; I simply can't understand what you're trying to say. – Anthony P Oct 20 '16 at 01:46
  • That logical statement would only be true if you drown swimming. I am swimming = true. If you drown, then the implication is true. If you do not drown, the implication is false. But, logically if you are not swimming, the implication is true no matter what. There is no causality necessary in a mathematically logical statement. – RJM Oct 20 '16 at 01:50
  • @AnthonyPaonessa If I say you If Trump wins then(implies) Middle East will be destroyed and Trump wins will you expect to Middle east to be destroyed(1) or the possiblity of 0.I mean If trump wins can middle east not be destroyed?
    If you answer is middle east will be destroyed you are holding the whole statement as 1(true) or we can say premise.     – aintnosunshinewhenyouaregone Oct 20 '16 at 01:50
  • If you are not swimming and drown in the bath, the logical implication is true. – RJM Oct 20 '16 at 01:51
  • To summarize I am convinced p implies q is a way for getting involve in p⟹q. When you say p implies q and p is true then q is true.But if you show as p⟹q and p is true q can be also false.Implying is a word for expressing ourself it is not actually mathematical . – aintnosunshinewhenyouaregone Oct 20 '16 at 01:55
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Suppose you give a statement here in MSE "If my friend knows some Mathematics then I will join MSE". Here

$p$: My friend knows Mathematics.

$q$: I join MSE.

Then what you said is $p\rightarrow q$. Now, suppose one of the old member of MSE files a case against you and imagine that you are in the court for hearing. See the following possibilities:

$1$. Your friend knows Mathematics ($T$) and you join MSE ($T$), then you are innocent ($T$) (as you stated nothing wrong in your statement) i.e. $T\rightarrow T\equiv T$.

$2$.Your friend knows Mathematics ($T$) and you dn't join MSE ($F$), then you are not innocent ($F$) (as you contradicted your statement) i.e. $T\rightarrow F\equiv F$.

$3$.Your friend doesn't know Mathematics ($F$) and you join MSE ($T$), then you are innocent ($T$) (as you never stated anything about his unawareness of Mathematics) i.e. $F\rightarrow T\equiv T$.

$4$.Your friend doesn't know Mathematics ($F$) and you do not join MSE ($F$), then you are innocent ($T$) (as you never stated anything about his unawareness of Mathematics) i.e. $F\rightarrow F\equiv T$.

Note: You cannot be punished for what you never claimed or stated. Consider another innocent statement, "If I am the creator of this world then elephants fly in the sky".

Nitin Uniyal
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  • I know thanks it is a good example given your example if my friend knows maths can you specify whether I join MSE or not.As you say and I think we cannot.If ı don't join i am fraud(not innocent,F) ; if I join it is 1(innoncent).But is it mathematically like this? Suppose If x is even then x^2 is even.What do you expect when x is even can x^2 be odd as mathematically it must be yes but in social,semantic or in a court it is not. – aintnosunshinewhenyouaregone Oct 20 '16 at 02:00