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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 (pā†’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.

Clement Yung
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Silver
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3 Answers3

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Let's assume he was a liar , which means the statement 'i like cookies' is false. But if in $p\rightarrow q $ is true when p is false then that would mean 'If i like cookies then i like cake ' is true but the liar never speaks truth.

aryan bansal
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  • Well yeah that was the original answer to the question. However my whole point of this post is that the second statement cannot be an implication since it's an if...then statement which implies a causal relationship. And a causal relationship is NOT the equivalent of implication, hence why I think this is not an implication and cannot be answered. ā€“ Silver Feb 12 '20 at 10:46
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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.

Melody
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    I guess this makes sense, it's just that our teacher told us not to try and understand implication using natural language, but then proceeds to give us an example like this. It's just a bit weird for me to take stuff by definition, or interpret stuff in one way when it can be interpreted in another way in a course called literal "logic". The logic given isn't really that logical to me, since stuff has to be assumed which defeats the purpose of it.

    Also what I meant with real world example is an example with actual language instead of just p -> q. Not a realistic example.

     ā€“ Silver Feb 12 '20 at 10:55
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"If I like cookies (p), then I like cake (q)."

If he doesn't like cookies, then this statement has no information. A statement that makes no claim is true by default.

Mark
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