We had a question about an island full of truth tellers and liars. There were a lot of questions, however there was one I couldn't wrap my head around.
It went as follows:
A random islander approaches you and says the following 2 declarations:
I like cookies (p)
Then right afterwards he says
If I like cookies (p), then I like cake (q)
Now from these 2 statements we had to conclude whether the islander was a:
-Truth teller
-Liar
-Unable to determined
The answer to this question is: He must be a truth teller. The explanation for this was as follows: Lets say the islander was a liar. His first declaration p would be false. This would mean that in the second declaration (p→q) the first part p must also be false. In the truth table for implication, if the first part is false, then the whole implication is ALWAYS true, regardless of q in this case. This would be the liar spoke the truth on the second declaration which is not possible, hence he must be a truth teller.
This was the explanation given by our teacher, however there is one thing I don't understand. It was made pretty clear to us that implication is NOT the equivalent of if…then statements in the natural language (which have causal relationships).
However in this question, the second declaration is clearly an if..then statement with a causal relationship.
My original answer was that it cannot be determined. I thought this was the case because if the islander doesn't like cookies, then there is NO way to determine whether the second declaration is true or false. We simply cannot know since he doesn't like cookies, so the second statement can either be the truth, or a lie.
In logic the "logic" behind implication seems to be true until proven false, which I sort of understand now. However in the example given it is pretty clearly a real life example, which holds a causal relationship. Hence why I think that you cannot apply implication to the second statement.
Best Answer
There is no causal relationship here. If $p$ then $q$ does not tell us that not $p$ implies not $q$. Regardless of what your intuition about everyday conversations says. A lot of logic puzzles work with the premise that those involved are master logicians (not always), in which case a lot of times even when 'plane English' is spoken, we're meant to interpret the statements as statements of pure logic.
I do understand what you're getting at. If a childs mom says "If you do your chores, then you can go outside". In this case she obviously means "Only if you do your chores, then you can go outside." In logic though, even in logic puzzles where the language seems plane, never assume "only if", when you're told "if".
Also, the example is not clearly a real life example. Whenever are you going to be on an island with only truth tellers and liars? You aren't. So if we're going to assume that scenario, then it's not too much more of a stretch to say they're also using if in the logic sense.