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Universal Truth Machine


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Question

Someone claims to have invented a Universal Truth Machine (UTM), a machine that takes a proposition as input, and returns "true", "false", or "undecidable" as output. Example:

 

Input                                   Output

1+3 = 4                               true

1+2= 4                                false

this proposition is false      undecidable

  

Devise a true proposition that the UTM will claim to be false, thereby disproving the inventor's claim.

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(You will answer me with" False" to 1+1=2)


The UTM will respond with " False "..which means that my expectation was true, and it responds with false  .
There is no possiblity to respond with " True ", because 1+1=2
Edited by wolfgang
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Is this an accurate paraphrase of the OP?

 

The machine takes a proposition and responds with one of  {"true", "false", "undecidable"}

Given its responses to these three particular inputs, ...

devise a true statement to which the machine will respond "false".

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This UTM seems pretty foolproof...

This proposition is decidedly undecidable... or false.

:P

I know this isn't exactly what the OP is asking for, but it seems that's as close as I can get it.

 

You're right, I think; the OP asks for a true statement.

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I am baffled.

 

If any statement like that existed, why couldn't the UTM output "true" or "undecidable" instead? The "false" output is already wrong. The other two outputs can't be any more wrong, can they? They're just as good as the "false" answer.

 

On the other hand, I can easily think of an answer that fits bonanova's criteria: "The reader of this proposition is human."

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How about slightly modifying bonanova's criteria.

 

"If the statement were posed to the UTM, the UTM would answer False. If any rational third answerer (who you can call observer X if you like, and may be human, machine, or flying spaghetti monster) were told that the statement was being posed to the UTM, then observer X would say that the statement is True."

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Suppose the machine outputs "true" to every input except "1+2= 4" and "this proposition is false". Then no true proposition will be reported as false. The only proposition which will be reported as false is "1+2= 4", which is false. So there must be some additional assumption you make about the machine in order for a solution to be reached.

Another way to look at it: consider the proposition you had in mind for the answer. Why would the machine be incapable of answering "true" or "undecidable" to this proposition?

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"This statement is true and the Universal Truth Machine will report it as false."

The UTM would have to declare it as false. It certainly couldn't call it true. And because calling it false would not pose a paradox that I can see (because it's an AND statement it would be ok to call it false if the left-hand side is false), I don't think the UTM would be forced to call it unanswerable.

 

However there's no reason a third party couldn't call the statement true.

 

But this is sort of dangerously skirting on the issue of whether a self-referential statement can be considered to be true or false simply on the grounds that such an answer would be logically consistent. By such logic, the statement "This statement is true" might then be called either true or false as you see fit. A third party could call it true, but I guess they wouldn't be forced to do so.

Edited by plasmid
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How about slightly modifying bonanova's criteria.

 

"If the statement were posed to the UTM, the UTM would answer False. If any rational third answerer (who you can call observer X if you like, and may be human, machine, or flying spaghetti monster) were told that the statement was being posed to the UTM, then observer X would say that the statement is True."

 

 

Yes.

 

Are you saying plasmid's post is the correct answer?

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How about slightly modifying bonanova's criteria.

 

"If the statement were posed to the UTM, the UTM would answer False. If any rational third answerer (who you can call observer X if you like, and may be human, machine, or flying spaghetti monster) were told that the statement was being posed to the UTM, then observer X would say that the statement is True."

 

 

Yes.

 

Are you saying plasmid's post is the correct answer?

 

I am saying his rephrasing of what you attempted to clarify is an appropriate rephrasing.

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