laureano luna laureanoluna at yahoo.es
Fri Jun 16 15:33:28 EDT 2006

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 iff not-p, has to lack any truth value. From the standpoint of some logic permitting a proposition to possess more than one truth value or some truth value different from true and false, the proof might not succeed. So, 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.
  Roughly speaking, the argument below is founded on the fact that the attempt of an informal gödelian diagonalization on a human being able to understand the try produces a paradoxical sentence while it cannot be so if the attempt is made on a robot.
  I kindly request you to examine the following argument.
   1. DEFINITION 1: let R be an algorithmic device that implements some logical capacity. 
  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. ASSUMPTION: R is able of performing the following:
  a. R deduces: 
  (1)             (G) is true if and only if R does not deduce that (G) is true
  b. R can correctly apply the following logical rules: BICONDITIONAL ELIMINATION, TRANSPOSITION, DOUBLE NEGATION, MODUS PONENS regarding (1) and its atomic components.
  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.
  4. LEMMA 1: (G) lacks any truth value and (G) has a definite truth value.
  5. PROOF: 
  a. (G) has no truth value:
  i. Assume (G) is true; then R does not deduce (G) is true; and R deduces it doesn’t (3. d.); so R deduces that (G) is true (3. a., 3. b.); then (G) is not true. CONTRADICTION.
  ii. Assume (G) is false; then R deduces (G) is true; and R deduces it does so (3. d.); thus R does not deduce (G) is true (3. a., 3. b., 3. c.); then (G) is true. CONTRADICTION.
  b. (G) has a definite truth value:
  (G) refers to the behavior of a formally definable algorithmic device; so that it states a well-defined state of affairs which has either to be or not to be the case; thus (G) has a definite truth value.
  6. LEMMA 2: there exists no algorithmic device R able to perform the required in 3.
  7. PROOF: the contradiction in lemma 1 forces to reject the assumption in 3.
  8. DEFINITION 3: let “classical bivalent logic” be the logic that includes that every proposition (but not necessarily every sentence) is either true or false (Bivalence) and that no proposition and its negation are simultaneously true (Contradiction).
  9. DEFINITION 4: let “Strong AI” be the claim that every human logical capacity can be reproduced by some algorithmic device.
  10. THEOREM: if there exists some human being H able of performing the required in 3. (substituting “H” for “R” in all of its occurrences, particularly in (G)) during some time period T and if classical bivalent logic is correct, then Strong AI is false.
  11. PROOF: directly from lemma 2 and definition 4. 
   The argument shows that the existence of some human H, as defined in 10., and classical logic are logically incompatible with Strong AI.
  Since the existence of H seems an empirical evidence, the theorem renders Strong AI, as defined, highly improbable from the standpoint of classical bivalent logic.
  As suggested above, the reason to make explicit the assumption of classical bivalent logic is the use of it in 5. a. in order to infer that (G) has no truth value, for it is a particular direct consequence of classical bivalent logic that paradoxical sentences have no truth value.
  Laureano Luna Cabañero


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