[FOM] Paradox and cognitive mechanism

[laureano luna]

In the remarks to my argument about paradox and
cognitive mechanism I affirmed that the class CLA of
all correct cognitive mechanical behaviors is not well
defined. I overlooked the possibility of CLA being
empty.

The argument behind my assertion was the following.
Consider the family of sentences (GX):

'This sentence is not a conclusion of a correct proof
by X' 

where 'X' stands for the complete description of an
algorithm.

It is undoubtedly well defined for each X whether (GX)
is a conclusion of a proof by X. Now suppose it is
always well defined whether a mechanical proof is a
correct one; then it would be well defined whether
(GX) is a conclusion of a correct proof by X; then
(GX) would have a truth value; and, since it cannot be
false, (GX) would be true for each X.

Consider a Turing machine M provided with a list L of
all Turing machines and with the instruction (I):

'For each X and each (GX), if X is in L, then prove
(GX)'

If it is always well defined whether a mechanical
proof is correct, then the instruction (I) would be
correct; but then M would correctly prove (GM)
rendering it false ipso facto.

So, it is not always well defined whether a mechanical
proof is a correct one. So, CLA is not well defined.

Unless there is no correct cognitive mechanical
behavior. If there is none, (I) would not be correct
when performed by a Turing machine and every (GX)
would be true.

According to consistent bivalent logic (CBL), if CLA
is well defined, then it is empty. But for the class
CL of all correct cognitive behaviors the
corresponding conditional does not hold. According to
cognitive mechanism (CM), CLA and CL are the same. So,
CBL and CM are mutually inconsistent.


Consider now the sentence:

'no correct cognitive mechanical behavior proves this
sentence'

or the sentence:

‘ ‘appended to its own quotation yields no sentence
some correct cognitive mechanical behavior proves’
appended to its own quotation yields no sentence some
correct cognitive mechanical behavior proves’

I considered them paradoxical i. e. having at least
one non propositional case. But if CLA is empty, both
sentences are true. 

Regards,

Laureano Luna


		
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