I don't understand the truth table for logical consequence $(a \rightarrow b)$ 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?
 A: 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: $(a \rightarrow b) = F$.
A: 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 $a\to b$.
