If both $P$ and $Q$ are true , how can I tell that $P$ implies $Q$? I am trying to understand the fundamentals of mathematical logic in order to be able to study discrete mathematics and computer science soon.

I have a big problem understanding Implication. I understand the idea intuitively very well; for example:

$$ x &gt;0  \implies 2x&gt;0 .$$

Or

$$ x \text{ is a prime number} \implies x \geq 2 .$$

I notice that there is always a (connection) between the hypothesis and the conclusion; they seem to be related and fall in the same context.

What I don't understand is: according to the truth table, any two propositions can be linked through an implication even if they are not related at all, or they belong to different contexts.

For example :

$$ \text{A day is 24 hours} \implies \text{A cat has four legs and a tail} .$$

which is logically or mathematically TRUE, because both statements are true and according to the truth table when when both inputs are true for any two statements then the implication is true.

How can that be true?

Another example:

$$ 2 \text{ is a prime number} \implies \text{An hour is 60 minutes}.$$

again, which is logically or mathematically TRUE, because both statements are true and according to the truth table when when both inputs are true for any two statements, then the implication is true.

How can that be true? That's my first question.

The second question could be the same:


  How can we use truth tables with implication anyway?


What I understand is truth tables are used to list the probabilities of the output based on the logical values of the inputs. So the value of the output of any line in the truth table depends (only) on the logical values of the inputs in that line and has nothing to do with the conditional connection between the two inputs. 

How can we display a conditional statement in a truth table the same way we display a logic gate or so? In other words, how can I tell only from the values of $P$ and $Q$ that $P \implies Q$?
Here is the truth table for an implication:

$$ \begin{array}{ccccc}  
   P &amp; &amp; Q &amp; &amp; P \to Q \\
   T &amp; &amp; T &amp; &amp; T \\
   T &amp; &amp; F &amp; &amp; F \\
   F &amp; &amp; T &amp; &amp; T \\
   F &amp; &amp; F &amp; &amp; T \end{array} $$

You can think of an implication as a conditional promise. If you keep the promise, it's true. If you break the promise, it's false. 

If I tell my kids, "I'll give you a cookie if you clean up." Then they clean up. I better give them a cookie. If I don't, I've lied. However, if they don't clean up, I can either give them a cookie or not. I didn't promise either if they didn't keep up their end of the bargain.

So in other words, an implication is false only if the hypothesis is true and conclusion is false.

Logically $P \to Q$ is equivalent to $\neg P \vee Q$

I had a computer science professor who was fond of promising his kids things given a false premise. This way he wasn't compelled to follow through. Example: "If the moon is made of green cheese, I'll give you an x-box."
Our common thinking of implication ($P \implies Q$) is that P allows us to say wether Q is true. But if we already know that Q is true, this way of thinking does not make sense anymore. 

Your confusion stems from the fact that you think P would have any implication on Q, which cannot be the case because Q is already given. The statement is true nevertheless, see here.

Take the example given in an other answer, where "clean up" implies "get cookie". I think it helps understanding a special case, but it is not a good one, because we can assume "clean up" would lead to "get cookie", if "get cookie" were false. But the statement holds true even for things that will never lead to "get cookie". In other words, if it is really decided, that "get cookie" is true, you can say anything in your if-sentence and the statement still holds:

If you crash the car, I will give you a cookie.


Our common understanding of this sentence is, that the cookie is the reward for crashing the car. But this does not make sense in this situation, so nobody would normally formulate it this way, but rather:

You can do anything, I will give you a cookie.