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You want to visit your close friend in the city Averola. For the long travel you brought your dog with you. Unfortunately, the guards in front of the gate to the city won't let you through. One of the guards gives you a riddle to have a laugh with the other guard. In his opinion it is unsolvable.

The guards riddle:
I let you into the city if you tell me a statement that's true.
I let your dog into the city if you tell me a statement that's false.

Rule: You can't use a paradox.

What can you say to bring you both into the city?

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  • $\begingroup$ Now if one guard had spoken each rule, I could state "your rule doesn't mention my dog". $\endgroup$
    – aschepler
    Oct 8, 2019 at 0:33

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Not entirely sure if this counts as "using a paradox", but you might try

"You'll let my dog in."

If he does, you have told a true statement, so you get to go in, too.

If he doesn't, he is violating his own rules, since you have just told a false sentence. So he cannot do that.

This is all based on the observation that

the guard didn't say "I'll let you in only if.."

so letting both in isn't against the guard's rules as stated.

By similar logic,

"You'll stop me"

works as well; stopping you is now out of the question, which means the statement is false and doggo gets to go in, too.

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You could say

"This statement is true"

Reasoning

The statement can be consistently assigned as either "true" or "false" so the guard would have to admit both of you. Note that this isn't a paradox as it doesn't appear to contradict itself or cause a logical inconsistency.

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  • $\begingroup$ Good thought. But if the statement can be assigned, doesn't that mean the guard can assign it to a value his preference and just let one through? Either way at some point it will be assigned as true or false and the guard will let only one through according to the given value. $\endgroup$
    – Nati
    Oct 7, 2019 at 10:35
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My dog is in the city

Initially it will be outside so that will be false and they will let the dog in.
Then it will be true and you will be let in.

It does depend on assessing them in that order. I guess for the original order one could do:

I am outside the city with my dog (true)

I get let in

I am outside the city with my dog (now false)

My dog gets let in.

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  • $\begingroup$ Clever approach! $\endgroup$
    – Stilez
    Oct 7, 2019 at 20:27
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Your statement is:

"A dog owner may not enter the city, if their dog may not enter the city."

Because

If this statement is TRUE, then the guard will admit the person. But then, because the statement is true, they must also allow the dog in.

On the other hand,

If the statement is FALSE, then the guard will admit the dog. But then, there is no rule barring the guard from allowing the person to enter as well.

Comment

I'm fairly sure this can be refined using a basic truth table, with sufficient "and-not", or "if and only iff", to make it more robust....

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I have told you a false statement

It is false when you say it, but true after. Thus they let your pup in, then let you in.

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  • $\begingroup$ That's a paradox isn't it? Because it's never either true or false. $\endgroup$
    – Nati
    Oct 8, 2019 at 6:49

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