As Onar illustrates, the days when paradoxes are merely to be
disallowed, or 'explained away' must be gone (or quickly going).
there are now many approaches to these productive objects, hyperset
theory is one. There are now many different logics which allow for
partial inconsistency, self-reference, etc. The days of a restricted
(and some would argue, simply wrong) two-valued classical logic
having a monopoly are gone.
An approach that I like was illustrated by Louis Kauffman in his
lecture on "Vivid Logic" at the "Eistein meets Magritte" conference
(Quantum physics and logic stream). His is a mathematical approach,
in that he views statements like:
A = not A
as simply an equation, like x + 6 = 2. Whether or not it has any
solutions depends on the possible allowed 'values' in the logic (or
mathematical system) it is expressed in. The equation x+6 = 2 has no
solutions in the natural numbers - the classical greeks would have
denied solutions and meaning to it (after all if you have a number of
objects and add six more how _can_ you be left with 2?). Of course
in the different maths systems (neagtive numbers, reals, complex,
etc) the numbers involved are interpreted differently in relation to
the real world.
The same could be said to be true of statements like A = not A, it
simply does not have solutions in classical logic but *does* in many
three-valued logics (which have a single intermediatary value
between true and false, each with different intended
interpretations, e.g. 'unknown' (which would then be the solution to
the above equation)). This way of looking at the 'paradoxes' is
only a problem if you are philosophically wedded to a view of how
logic should be.
----------------------------------------------------------
Bruce Edmonds
Centre for Policy Modelling,
Manchester Metropolitan University, Aytoun Building,
Aytoun Street, Manchester, M1 3GH. UK.
Tel: +44 161 247 6479 Fax: +44 161 247 6802
http://bruce.edmonds.name/bme_home.html