[FOM] EXCLUDED MIDDLE AS LOGICAL LAW
Laureano Luna
laureanoluna at yahoo.es
Wed Oct 12 17:14:59 EDT 2005
After reading recent Avron's post I think I should add a short reflection.
How can one argue for a basic logical principle? One should rather argue
from it... But in this case I feel that something can be achieved about the
question.
Let me first introduce the distinction between the ontological principle of
excluded middle and the logical principle of bivalence. The former asserts
that for any well defined situation or state-of-affairs we have either that
it is the case or that it is not; the latter affirms that every proposition
is either true or false. If I think that the latter can be argued for it's
only because I think that it rests on the former and the former is accepted
by intuitionists too.
Let me define a proposition as the description of a well defined situation.
If the described situation is the case, then the proposition is true;
otherwise it is false. Take this as a definition too.
Consider Goldbach's conjecture. It depicts a situation that either is the
case or is not; so it has to be either true or false. But intuitionists
interpret (so Heyting) Goldbach's conjecture as stating that Godbach's
conjecture has been proved and its negation as asserting that it has been
proved that a proof for Goldbach's conjecture would imply an absurd.
The point in two final remarks: first, it's evident that intuitionists do
not interpret the logical negation in the same way classical logic does,
which as Avron argues is precisely the way in which they interpret it in
ordinary everyday language; second, they do not interpret a proposition
such as Goldbach's conjecture according to ordinary linguistic rules. So,
when they use ordinary language to argue, in both cases they argue against
the very position which they argue from.
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