[FOM] Unknowability of AI?

[laureano luna]

  I think it's established that no human mathematician, while correctly reasoning for the solution of instances of the halting problem, uses an algorithm he/she can know is sound.
   
  The condition "he/she can know is sound" is usually considered the weal link in the use of the preceding as an argument against Artificial Intelligence (AI). I've tried to ascertain how much can be proven without that condition and by reasoning about the truth value of seemingly paradoxical sentences.
   
  As fas as I know this technique is new, though I may easily be wrong about this. I'm aware that this kind of reasoning will look suspect to most readers.
   
  Anyway my result (a result of unknowability) is the following. I'd be grateful for any comments.
   

   
  DEFINITION 1: let “AI” name the claim that any possible cognitive behavior can be reproduced by an algorithm.
   
  PROPOSITION 1: if AI holds, then any possible correct cognitive behavior can be reproduced by an algorithm.
   
  PROPOSITION 2: no correct cognitive behavior C able to understand the reasoning below from 1. to 8., able to apply Modus Ponens to 8. and able to deduce (G’) from “(G’) is true”, deduces AI.
   
  
  PROOF?
   
  1. Let (G’) be the following sentence:
   
                  no correct cognitive behavior deduces this sentence
   
  2. If any correct cognitive behavior C deduces (G’), then (G’) is not true.  So, no C correctly deduces (G’). 
   
  3. No C deduces (G’) (from C’s correctness and 2.). 
   
  4. If (G’) has a truth value, then (G’) is true (from 3.).
   
  5. Let’s assume AI. Then, for any C, there exists an algorithm AC that exactly reproduces C (from definition 1 and proposition 1) and there is a sentence SC asserting that AC does not deduce (G’).
   
  6. Each SC has a truth value since it refers to the behavior of a mechanical device and therefore to a well-defined state of affairs.
   
  7. (G’) is logically equivalent to the proposition asserting the correctness of the theory TH that contains all statements SC and only them. Since each SC has a truth value, TH exists as a theory whose correctnes is a well-defined state of affairs that has either to be or not to be the case; so, (G’) has a truth value.
   
  8. If AI holds, then (G’) is true (from 4. and 5.-7.). 
   
  9. Let C* be a correct cognitive behavior able to understand the reasoning above from 1. to 8., able to apply Modus Ponens to 8. and able to deduce (G’) from “(G’) is true”.
   
  10. If C* deduces AI, then C* deduces that (G’) is true (from 8.) and then C* deduces (G’). 
   
  11. C* does not deduce AI (from 3. and 10.).  Q.E.D.
   
  PROPOSITION 3: no correct cognitive behavior C deduces AI.
   
  PROOF: C* can have any strength compatible with correctness.
   
  REMARK
   
  In some steps of the argument the correctness of the following (as in proposition 2) or of the preceding (as in 9. and proposition 3) is assumed, but no step is based on the assumption of its own correctness
   
  Regards
   
  Laureano Luna Cabañero
   

 		
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