I don't understand the truth table for logical consequence $(a \rightarrow b)$

Question

When I take a look at the truth table for implification, I don't get the logic:

a     b       a --> b
----------------------
1     1         1
1     0         0
0     1         1
0     0         1

I understand implication $a \rightarrow b$ means that $a$ becomes $b$ or I understand this wrong now? Because if I compare what I said with the table, it seems contradicting... My understanding works for all lines except for the last one where we have 0 0 1..

Anyone could please explain me?

Yes, you understand it wrong. The correct wording is "if $a$, then $b$". — , Oct 18 '16 at 20:31
$a\to b$ is just a typographic variant of $a\Rightarrow b$, meaning "if $a$ then $b$". There is no becoming. — hmakholm left over Monica, Oct 18 '16 at 20:31
Ok but how would you explain this "if $a$ then $b$" with that table? I don't get why if $a=0$, $b=0$ we have that $a\rightarrow b$ is $1$ — cnmesr, Oct 18 '16 at 20:32
For instance, it is arguably true that, if the moon were made of cheese, famine would not be a severe issue. $\ddot\smile$ — , Oct 18 '16 at 20:34
@cnmesr: See In classical logic, why is $p\to q$ True if both $p$ and $q$ are False? — hmakholm left over Monica, Oct 18 '16 at 20:36
Alright this is more complicated than I expected : / But thank you a lot for the link. I guess I have to live with the easy reasoning mentioned in your link: "the short answer is that if $a=0$, then it doesn't matter what $b$ is" — cnmesr, Oct 18 '16 at 20:42

score 6 · Accepted Answer · answered Oct 18 '16 at 20:36

The symbol $\rightarrow$ is read "implies", not "becomes". Yes, logical implication is an interesting one. The source of confusion is case 2), below.

Think of an insurance policy. Let $a$ mean "an accident has occurred", and $b$ "the insurance company has paid out." Then $(a \rightarrow b)$ means "the insurance company has kept within its contract." Let's examine the possibilities:

1) $a = F$ (no accident occurred), $b = F$ (no payout made). The company has kept with the contract, so $(a \rightarrow b) = T$.

2) $a = F$ (no accident occurred), $b = T$ (payout made). The company would be crazy to do this (pay when it doesn't have to), but would still be within the contract, so $(a \rightarrow b) = T$.

3) $a = T$ (accident occurred), $b = T$ (payout made). The company has met its contract: $(a \rightarrow b) = T$.

4) $a = T$ (accident occurred), $b = F$ (no payout made). Contract violated: $(a \rightarrow b) = F$.

Thank you a lot I had to read your answer 3 times and then I understood :P Still this is very mean thing to understand! — cnmesr, Oct 18 '16 at 20:52

score 2 · Answer 2 · answered Oct 18 '16 at 20:32

2

Rather than "$a$ becomes $b$", it should be read as "$a$ implies $b$".

The only "disproof" that $a$ implies $b$ is if $a$ is true, but $b$ is not. That's why only the second row in your table has a false value for $a\to b$.

answered Oct 18 '16 at 20:32

Nick Peterson

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I don't understand the truth table for logical consequence $(a \rightarrow b)$

2 Answers2