I am reading introductory texts on logic and am having a hard time understanding intuitively logical implication.
Specifically, I am wondering if logical implication can always be determined by two statements, P and Q alone? I'll try to give an example to clarify my question.
For example if the statement P is $x = 2$, and the statement Q is $x^2 < 6$, then determining the truth of P $\Rightarrow$ Q is fairly straightforward:
If $x$ does equal 2, then $4<6$ and Q is true, so $P \Rightarrow Q$ is true. And if $x$ does not equal 2, $P \Rightarrow Q$ is vacuously true.
In this example, with two statements P and Q, I can evaluate the truth/falsity of the statement P implies Q. However, what if the statement Q is changed such that it is now the statement $y = 5$?
So now I am trying to prove from that $$x = 2 \Rightarrow y = 5$$
Here is where things get murky for me. Say $x$ is not equal to 2, then $P \Rightarrow Q$ is vacuously true. But what about when $x$ does equal 2, now the statement Q could be true, but it could also be false, depending on some 'context'.
For example, if I said at the start: "Here are two statements, P and Q. Know that x = 2 and y = 4." Given this 'context', $P \Rightarrow Q$ is false. But if I say "Here are two statements, P and Q. Know that x = 2 and y = 5", then now $P \Rightarrow Q$ is true.
My discomfort here is that I assumed with two statements P and Q, I can determine the truth/falsity of $P \Rightarrow Q$ with just this information. Is this the wrong assumption?
As a secondary question, how can the truthfulness of the statement $P \Rightarrow Q$ be variable, depending on context? I have read on some other questions that implication can be interpreted as a promise. So how could it be that depending on some context the promise is sometimes true but sometimes false? Wouldn't this actually mean the promise is not always consistent and is therefore always false?
Some further information in case it is relevant, I recognise that in the first example, the statement Q is 'dependent' in some way on the statement P, since the object under examination in both statements is $x$, whereas in the second example there are two different objects which I am trying to make a logical connection between, which I am thinking may not be valid for some reason. I also recognise that there is still 'context' in the first example, in that we need information on whether $x$ is equal to 2 or not, however in that case it doesn't matter, since $P \Rightarrow Q$ is always true anyway.
Thanks very much in advance.