XPost: comp.theory, sci.logic, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 17/11/2025 22:59, olcott wrote:   
   > On 11/17/2025 4:45 PM, Tristan Wibberley wrote:   
   >> On 17/11/2025 22:15, Alan Mackenzie wrote:   
   >>   
   >>> There is no proper academic conversation to be had over 2 + 2 = 4.    
   >>> It is   
   >>> firm, unassailable knowledge, unchallengeable.  The Halting Theorem   
   >>> is of   
   >>> the same status, proven using the same methodology from the same   
   >>> fundamentals.   
   >>   
   >>   
   >> It's a completely different league from 2 + 2 = 4.   
   >> It's closer to x = 1/2 + x/2 but it's still conceptually /much/ harder   
   >> than that.   
   >> It's more like the problem of whether a fixed point exists or not, but   
   >> it's for the fixed point of a limit of a particular, conceptually weird,   
   >> sequence of functions.   
   >>   
   >> It really is quite peculiar.   
   >>   
   >   
   > Ultimately it is essentially the Liar Paradox in disguise.   
   >   
   > The Liar Paradox formalized in the Prolog Programming language   
   >   
   > This sentence is not true.   
   > It is not true about what?   
   > It is not true about being not true.   
   > It is not true about being not true about what?   
   > It is not true about being not true about being not true.   
   > Oh I see you are stuck in a loop!   
   >   
   > This is formalized in the Prolog programming language below.   
   >   
   > ?- LP = not(true(LP)).   
   > LP = not(true(LP)).   
   > ?- unify_with_occurs_check(LP, not(true(LP))).   
   > false.   
      
   true/0   
   use \+/1 rather than not/1   
      
      
   > Failing an occurs check seems to mean that the   
   > resolution of an expression remains stuck in an   
     ^^^^^^^^^^   
   > infinite loop.   
      
   You mean "judgement" ?   
      
   (1) I can see how a judgement looping with a negation in the middle   
   should be rejected as contradiction.   
      
   (2) But looping with double negation is merely nondetermining isn't it?   
      
      
   Examples   
      
   (1) A = \+(A).   
   (2) A = \+(\+(A)).   
      
   yet both fail to unify with an occurs check in prolog. I think you need   
   a deeper theory.   
      
   Before you mention intuitionistic double negation vs classical:   
      
   ?- unify_with_occurs_check(\+A, \+(\+(\+A))).   
   false.   
      
   of course.   
      
   I should probably functionalise negation and see what's what.   
      
   --   
   Tristan Wibberley   
      
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