[FOM] INCOMPATIBILITY OF STRONG AI. A CORRECTION

[laureano luna]

My previous post on this topic contained two errors.
   
  The first (less serious) is essentially that I had assumed the Church-Turing thesis without making it explicit.
   
  The second (definitive) is that my requirements in point 5. are inconsistent.
   
  I apologize and I thank Martin Davis for his kindness and tolerance.
   
  I reproduce my previous argument below with my comments in capitals in the flawed step 7. b. and I'll try to work out my idea more carefully.
   
  Best regards.
   
  Laureano Luna Cabañero.
   
  1. DEFINITION 1: let R be an algorithmic device that implements some cognitive capability. 
   
  2. DEFINITION 2: let be (G) the following sentence:
   
  (G)                      R does not deduce that this __expression is true
   
  where “R” stands for a complete description of R.
   
  3. LEMMA 1: (G) has a definite truth value
   
  4. PROOF: (G) refers to the behavior of a formally definable algorithmic device, so that it states a well-defined state of affairs that has either to be or not to be the case.
   
  5. ASSUMPTION: R is capable of performing the following:
   
  a. R deduces: 
   
  (1)   if (G) has a truth value, then (G) is true if and only if R does not deduce (G) is true
   
  b. R can correctly apply to (G) and its components the following logical rules: MODUS PONENS, BICONDITIONAL ELIMINATION, TRANSPOSITION, DOUBLE NEGATION.
   
  c. R is consistent regarding the truth value of (G).
   
  d. If R deduces that (G) is true, then R deduces that R does so; if R does not deduce that (G) is true, then R deduces that R doesn’t.
   
  e. R deduces lemma 1. 
   
  6. LEMMA 2: under the assumption in 5. (G) has no truth value 
   
  7. PROOF: 
   
  a. Assume (G) that is true and that therefore has a truth value; then R does not deduce (G) is true; and R deduces it doesn’t (5. d.); then, since R deduces that (G) has a truth value (5. e.), R deduces that (G) is true (5. a., 5. b.); then, since (G) has a truth value, (G) is not true. CONTRADICTION.
   
  b. Assume (G) that is false and that therefore has a truth value; then R deduces (G) is true; and R deduces it does so (5. d.); then, since R deduces that (G) has a truth value (5. e.)  NO. 5. e. IS NOT SUFFICIENT FOR THIS, FOR LEMMA 1 IS UNDER THE CONDITION THAT R IS AN ALGORITHM. SO, R SHOULD RECOGNIZE THE CONCRETE (G) REFERRING TO R AS HAVING A TRUTH VALUE AND FOR THIS IT MUST RECOGNIZE ITS OWN SPECIFICATION AS THE SPECIFICATION OF AN ALGORITHM. THIS REQUIRES THE SET OF ALL ALGORITHMS TO BE ALGORITHMICALLY ENUMERABLE. BUT WHEN 5. e. IS SO MODIFIED IT IS INCONSISTENT WITH THE REST OF 5. SO THAT NO CONSISTENT HUMAN H ABLE TO ACCOMPLISH THE REQUIRED IN 5. CAN EXIST.  then R does not deduce (G) is true (5. a., 5. b., 5. c.); then, since (G) has a truth value, (G) is true. CONTRADICTION.
   
  8. LEMMA 3: there exists no algorithmic device R able to perform the required in 5.
   
  9. PROOF: the contradiction between lemma 1 and lemma 2 forces to reject the assumption in 5.
   
  10. DEFINITION 3: let “classical bivalent logic” be the logic that includes that every proposition (but not necessarily every sentence) is either true or false and has no other truth value (Bivalence), and that no proposition is both at the same time (Contradiction).
   
  11. DEFINITION 4: let “Strong AI” be the claim that every human cognitive capability can be reproduced by some algorithmic device.
   
  12. THEOREM: if there exists some human being H able of performing the required in 5. (where “H” is to be substituted for “R” everywhere while sentences (G) and (1) remain unaltered) during a time period T whatsoever and if classical bivalent logic is correct, then there is no algorithmic device R reproducing the cognitive capability of H at T, and therefore Strong AI is false.
   
  13. PROOF: directly from lemma 3 and definition 4.
   
   

 		
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