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From: Peter Olcott <OCR4Screen>
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Subject: Re: The Halting Problem is based on an ill-formed Question
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On 6/9/2012 8:39 PM, Joshua Cranmer wrote:
> On 6/9/2012 6:58 PM, cplxphil wrote:
>> Perhaps you've been over this before in your lengthy discussions, but
>> let me see if I have this straight.
>>
>> You are saying that the question, "Does machine M halt on input I?"
>> may, for certain M and I, be impossible to answer either yes or no?
>
> I think Peter believes that this is not the proper way to phrase the 
> question really being asked. Although, when he was pushed into a 
> corner, I think there was a tacit admission that the answer of such a 
> question depends on who you asking it of.
>
>> Also, before this continues too long:  It sounds like you are quite
>> confident that you're right about this.  I am quite confident that you
>> are not.  What would it take to convince you that you are wrong?
>
> Again, I think the answer is that he has no issue with proof, per se. 
> His umbrage is with the interpretation of the result: he believes that 
> it is possible to make a Halt decider that is "essentially" correct 
> but fails in some cases which are "necessary" (use of quotation marks 
> to indicate that the terms contained within are slippery and 
Neccessity and Possibility form the fundamental basis of Model Logic:
http://en.wikipedia.org/wiki/Modal_logic

> ill-defined). Lots of people have attempted to illustrate several 
> alternate derivations to show that the incorrectness is not so 
> well-contained, but they have all been ignored because either:
> a) it happens to fall under the "necessary" failures,
> b) it relies on a similar "incorrect" result (the uncountability of 
> real numbers and Godel's incompleteness theorem have also been 
> explicitly cited as incorrect proofs due to being similar [1]), or
> c) he doesn't understand it, so it has to be either case a or case b 
> because he is OBVIOUSLY right.
>
> Trying to come up with a simple, alternate proof of the Halting 
> problem that is understandable and skirts any other proofs that have 
> anything smacking of diagonalization or self-reference is indeed hard, 
> but I doubt he'd accept anything less. I also doubt that he'd accept 
> even that much, though...
>
> [1] This just makes it seem to me that he has a really hard time 
> accepting that a proof by contradiction is indeed a valid proof.
>
Ultimately to understand what I am saying requires an understanding of 
the mathematics of the meaning of words.
Since the last time that we spoke I have done extensive reading and 
found that the field called Formal Semantics in Linguistics provides the 
most complete specification of all knowledge of mathematics of the 
meaning of words.

This single source describes the basis for essentially all of this work:
Formal Semantics: An Introduction (Cambridge Textbooks in Linguistics) 
Ronnie Cann