A logic puzzle that use a dummy variable to probe and whose value is never discovered

[repoman]

This is a specific version of the "knight and knave" logic puzzle. It uses "ja" and "da" which mean yes and no, but you never find out which is which and don't need to.

Also, for now ignore the Random character.

here is a good statement of the puzzle and a solution

https://puzzling.stackexchange.com/questions/2665/knights-and-knaves-in-a-foreign-language

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.

I did a full chart of the four possible formulations of the questions and how they would be answered by T, True, and F, False. I will show it in bullet points here

A) If I asked you "are you a liar?" would you say no?
B) If I asked you "are you a liar?" would you say yes?
C) If I asked you "are you a truth teller?" would you say no?
D) If I asked you "are you a truth teller?" would you say yes?

I am not sure what the mathematical jargon is, but I am will say positive or negative "valence". I am sure someone knows the correct term here.

Question A) has minus times minus = plus valence. Because the "liar" is a negative in this logic and no is negative.
Question B) has minus times plus = minus valence.
Question C) has plus times minus = minus valence.
Question D) has plus times plus = plus valence.


Now the "True" character has a plus valence and "False" character has a minus valence. Multiply the character valence times the question valence to get the answer valence.
Example,

Question A) asked to False: If I asked you "are you a liar?" would you say no?
False would initially say "no" self referentially, then would say "no" because he had said "no".

So, True's answers to A,B,C and D will be Yes, No, No and Yes.
False's answer to A,B,C and D will be No, Yes, Yes and No.

So how are Ja and Da used as dummy probes for Yes and No?

For example, if Ja means Yes then if you ask "are you a liar?" you are asking Question B:
If I asked you "are you a liar?" would you say yes?
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Let
Ja =Yes, Da = No

For T: If I asked you "are you a liar?" would you say yes? ----> If I asked you "are you a liar?" would you say Ja?
Answer: "No, I would not say I am a liar" -----> "Da , I would not say I am a liar". The italicized part is implied, but not said.

For F: If I asked you "are you a liar?" would you say Ja (yes)?
Answer: Yes ------> Ja

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Let
Ja = No, Da = Yes


For T: If I asked you "are you a liar?" would you say no (because ja means no now)? ----> If I asked you "are you a liar?" would you say Ja?
Answer: "Yes I would not say I am a liar" -----> "Da , I would not say I am a liar".

For F: If I asked you "are you a liar?" would you say Ja (no)?
Answer: "No, I would not say I am a liar" -----> "Ja , I would not say I am a liar".

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So, no matter the meaning of Ja and Da, for the question:

'If I asked you "are you a liar" would you say Ja?'

the answer from "True" is Da and the answer from "False" is Ja.

But Ja and Da are not known.

 

[skepticalbip]

That seems to be a take-off of a really, really old logic problem.

A hiker in the forest comes to a fork in the path. At the fork are two people, one who is known to always lie and the other known to always tell the truth. Without knowing which is which, what one question can the hiker ask to determine the proper path that will take him out of the forest?

 

[Swammerdami]

That seems to be a take-off of a really, really old logic problem.

A hiker in the forest comes to a fork in the path. At the fork are two people, one who is known to always lie and the other known to always tell the truth. Without knowing which is which, what one question can the hiker ask to determine the proper path that will take him out of the forest?

I saw this one long ago on Usenet. The solution was

Say "Did you know there giving away free beer at the forest edge?"
Ignore what they say, but follow them.​