Re: The Smullyan paradox

On Feb 16, 6:33 pm, "LordBeotian" <pokips...@xxxxxxxx> wrote:
Smullyan formulated the following paradox:

--
let A and B be two numbers such that A=2B or B=2A
than we can prove that

(*) The difference A-B in case A>B is strictly lower than the difference B-A
in case B>A.

Proof: let's fix x:=A
in case A>B we have B=x/2 and A-B=x/2
in case A<B we have B=2x and B-A=2x
and obviously 2x>x/2 QED.

This is clearly paradoxical because in the same way we can prove the converse
statement.
---

Now my question is: is (*) a meaningful mathematical statement? It seems to
be so, but if it is so how can it be formulated in (for example) the language
of PA or any other formal language?


This paradox only arises if we insist that there is absolute truth,
independent of our theories. To see this, first consider a consistent
theory T1 of arithmetic such that:

1. T1 proves the existence of a positive number A with an explicit
"construction" for A (e.g. A=100). i.e., T1 "fixes" A.

2. T1 proves the existence of a positive number B such that:

B=2A v B=A/2 (1)

But T1 does not "fix" B, i.e., T1 does not provide an explicit
construction for B.

It is obvioius that T1 proves the statement (*) noted above, namely,

(*) The difference A-B in case A>B is strictly lower than the
difference B-A in case B>A.

Next consider a consistent theory T2 of arithmetic such that

3. T2 proves the existence of a positive number B with an explicit
"construction" for B (e.g. B=50). i.e., T2 "fixes" B.

4. T2 proves the existence of a positive number A such that:

A=2B v A=B/2 (2)

But T2 does not "fix" A, i.e., T2 does not provide an explicit
construction for A.

It is obvioius that T2 proves the "converse" of the statement (*)
noted above, namely,

(**) The difference B-A in case B>A is strictly lower than the
difference A-B in case A>B.

A paradox arises if we insist that the constants A and B must have an
existence independently of our theories T1 and T2. For then we have to
conclude that either (*) or (**) is true, but not both, since these
are contradictory assertions if both A and B are "fixed".

But if we simply take the view that there is no absolute truth, but
only truth with respect to theories, we could possibly take (*) as
true with respect to T1 and (**) as true with respect to T2, and the
paradox would seemingly be resolved. However, then the law of the
exluded middle as applied to the statements (1) and (2) above is
problematic, for it seems to assert the existence of the constants B
and A independently of our theories T1 and T2 respectively, and that
is precisely what we are trying to deny.

In the logic NAFL, this paradox is resolved by noting that both T1 and
T2 are inconsistent theories (by the NAFL yardstick). For if a
consistent NAFL theory proves a disjunction, it must necessarily prove
one of the disjuncts.

Indeed, NAFL requires that each of the disjuncts in (1) or (2) can
only be "true" via an axiomatic declaration in the human mind, i.e.,
truth=provability and truth is only with respect to theories. In which
case (1) and (2) would be equivalent to the absurdity that the human
mind has axiomatically asserted one of the disjuncts, but it does not
know which (or equivalently, the absurdity that T1 (T2) proves one of
the disjuncts in (1) ((2)), but we do not know which).

Hence this paradox will not arise in NAFL.

Regards, RS
.

Relevant Pages

  • Re: The Smullyan paradox
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