laureano luna laureanoluna at yahoo.es
Wed Jun 21 11:08:47 EDT 2006

I'm afraid my previous post published on 06-19-2006 contains an error.
  Really if a human H believed 
  (1)   (G') is true if and only if H does not deduce (G')
  where the sentence (G') refers to H, and satisfied at the same the rest of the requirements on R's logical capability, then he/she would probably be inconsistent, for he/she would probably conclude that (G') lacks any truth value and thence (through (1)) that (G') is true.
  It appears to me that this can be satisfactorily rectified. I write the corrected argument below.
  Within a classical bivalent logic including the principles of Bivalence and Contradiction it is easy to prove that any paradoxical sentence p, such that p if and only if not-p, has to lack any truth value. From the standpoint of some logic permitting a proposition to possess no truth value, or more than one truth value, or some truth value different from true and false, the proof might not succeed. Since a sentence devoid of truth value is no proposition, the treatment of paradoxical sentences as non propositional expressions is a trait of classical bivalent logic not shared by all the proposed non classical logics.
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.), 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.
The theorem shows the inconsistence of these three claims when taken together: the existence of a human H as defined in 12., classical bivalent logic and Strong AI.The existence of a human H as defined in 12. is a very weak condition, especially because it refers to only two sentences, a few elemental logical rules, a very limited self-consciousness and is restricted to a time period T whatsoever; so, that existence seems empirically evident. Therefore the theorem renders highly improbable the simultaneous correctness of classical bivalent logic and Strong AI. 
  The reason to make explicit the assumption of classical bivalent logic is the use of it in 7. in order to infer that (G) has no truth value. As I wrote above, it is a particular direct consequence of classical bivalent logic that paradoxical sentences have no truth value.
  Best regards
  Laureano Luna Cabañero


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