## Ambiguities in Logic

You visit an island of three towns: Trueton, Lieberg and Alterborough. Folks living in Trueton always tell the truth. Those who live in Lieberg, always lie. People from Alterborough alternate strictly between truth and lie. You meet an islander who says:

Two plus two is five. Also, three plus three is six.

Can you determine which town he is from?

It should be easy. He made two statements: the first one is false, the second is true. So he must be from Alterborough.

But what about “also”? How should we interpret this transition? There are many ways to interpret this “also.” On one hand it could mean: In addition to the previous statement I am making another statement. On the other hand it could mean: The previous pause shouldn’t be considered as the end of the statement; the whole thing should be interpreted as one statement. Besides this person was speaking not writing. Are we sure that the first period was not meant to be a comma or a semi-colon? If we assume that the quote is one statement, then the speaker might be either a liar or an alternator.

Here is a puzzle for you from the same island:

One night a call came into 911: “Fire, help!” The operator couldn’t ID the phone number, so he asked, “Where are you calling from?” “Lieberg.” Assuming no one had overnight guests from another town, is there an emergency? If so, where should help be sent? And was it a fire?

Now find the ambiguities.

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1. #### L33tminion:

No one from Lieberg will say they’re from Lieberg (because that would be true). No one from Trueton will say they’re from Lieberg (because that would be a lie). Therefore the caller is from Alterborough, and their answer to the question “Where are you calling from?” is a lie. Therefore, their previous statement is true.

The question is what sort of unit these rules about truth-telling and lying apply to. Presumably, only “statements” (some sort of unit of speech with a truth-value). (So no need to worry about the truth of the operator’s question, probably.) So is “fire, help!” a single unit of speech or two? And if it’s two, does the second part have a truth-value or not (is it an implicit statement of fact or just some sort of imperative without a truth-value). Personally, I think that it makes sense to interpret “fire, help” as a single statement, analogous to “there is a fire and I need help because of the fire”. But this is a logic puzzle, so all sorts of arbitrarily silly things could be the case.

2. #### Sergei:

Given that the caller is from Alterborough and that they lied about Lieberg, the previous statement has to be true. Now, there’s some ambiguity in whether or not the previous statement includes “Fire” or not, but either way the “help” part is true, so the policeman should probably send help.

3. #### Philip Petrov:

Good problem.

If you do not mind, I did rewrite your problem with real Bulgarian villages names here:
http://www.cphpvb.net/fun/9295-emergency-phone/

4. #### Philip Petrov:

P.S. The Alterborough are not required to be alternating. They can be random too ðŸ™‚