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I was just learning the truth table of the propositional logic . I understand the truth table for the conjunction and disjunction because they make sense in the real life. The conjunction A∧B means "A and B", and it is intuitively correct that "A and B" is true iff both A is true and B is true. The disjunction A∨B means "A or B", and it makes sense also that unless both A and B are false, A∨B is true. But for the material implication, A implies B, I can't really get the truth table. For example:

Let A be the statement that "I washed my dishes today." Let B be the statement that "It rained yesterday."

Assumed that both statements are true, then according to the material implication, A implies B is also true, but there is no single connection between the chores and the weather. It sounds a little bit bizarre to say that "because I washed my dishes today, so it rained yesterday". That just sounds weird.

So why does the truth table of material implication be like that?

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It's called material implication, not causal implication for that reason.

$A \to B$ is the claim that "$B$ is found to be true whenever $A$ is found to be true."

So the statement "I washed my dishes $\to$ It rained yesterday" is not a statement that you will somehow cause rain to have happened (retrospectively) when you wash dishes the day after.   It's the statement that you only wash dishes on a day after it rains.

The truth table means the statement is only falsified by observing a day you wash dishes when it did not rain the day before.

$$\begin{array}{l | l | l} \text{I washed dishes today} & \text{It rained yesterday} & A\to B \\ \hline \text{No}&\text{Yes} & \checkmark \\ \hline \text{No}&\text{No} & \checkmark \\ \hline \text{Yes}&\text{Yes} & \checkmark \\ \hline \text{Yes}&\text{No} &\times \end{array}$$

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  • $\begingroup$ Can I interpret it this way, instead of thinking about "knowing the truth value of A and B, compute the truth value of A→B", I will think "knowing the truth value of A and 'A→B'compute the truth value of B"? For example, let A be "This triangle is the right triangle", and let B be "a^2+b^2=c^2". $\endgroup$ – MathnerdXjh Mar 11 '15 at 2:54
  • $\begingroup$ Assume A is true, and from trig we know that A infer B is true also, so B must be true. $\endgroup$ – MathnerdXjh Mar 11 '15 at 2:55
  • $\begingroup$ $A, A\to B \vdash B$ is the rule of inference called modus ponens: "If proposition $A$ is valid, and the implication $A\to B$ is justified, then we can prove proposition $B$ is valid." $\endgroup$ – Graham Kemp Mar 11 '15 at 3:17
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This is a very common problem for beginners. I can still recall asking my teacher about this very point when I was in high school, many years ago.

I like to use the example of the statement: "If it is raining, then it is cloudy."

It is tempting to think this suggests that cloudiness somehow causes rain, or that rain somehow causes cloudiness. Neither is the case, of course.

EDIT: We simply mean that, at the moment, it is not both raining and not cloudy. Looked at this way, the truth table makes perfect sense. The implication would be false only if it is raining and not cloudy. See my more recent blog posting on material implication. There, I develop, among other things, a rationale for the usual definition of material implication.

Using symbols: $Raining \implies Cloudy \equiv\neg[Raining \land \neg Cloudy]$

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