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I am dissatisfied with logical implication. I understand the explanations that I've read. I understand the truth table. I understand how to use it. For example, I know the following statement is true.

if n is both even and odd then n=17

But, I am dissatisfied with why it's true. I'm looking for a good explanation of the logic of implication.

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closed as unclear what you're asking by David K, Pragabhava, user91500, naslundx, Behrouz Maleki Sep 23 '16 at 10:45

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ There are MANY such questions. $\endgroup$ – YoTengoUnLCD Sep 22 '16 at 21:06
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    $\begingroup$ Why shouldn't that implication be true? Exactly why do you find it dissatisfying? The reasons why you don't like the definition matter, because we have to explain why the mathematicians who developed logic thought the things they wanted implication to do were more important than your reasons for disliking their definition. $\endgroup$ – David K Sep 22 '16 at 21:16
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Showing that "if $n$ is even and odd, then $n=17$" is false (constructively) is exactly the same as exhibiting $n$ which is even and odd, but where $n \not = 17$. Can you find me such an $n$?

More abstractly, to disprove $A \Rightarrow B$ is precisely to show that there is a situation where $A$ is true but $B$ is false. "All cats are small" is "If it is a cat, then it is small"; to disprove that, it is necessary and sufficient to show that "not all cats are small", i.e. "there is a cat which is not small", i.e. to find a situation where there is a cat, but it is not small.

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