[FOM] Intuitionists and excluded-middle

Nik Weaver nweaver at dax.wustl.edu
Wed Oct 12 12:45:22 EDT 2005

Arnon Avron wrote:

> So first of all: Classical logic needs no defense. These are its
> attackers who need to justify their rejection of some of its laws.
> Despite of almost 100 years of desperate attempts, they have failed to
> provide any convincing argument for rejecting excluded middle (except
> "Because I say so").

Many messages on this list appear to have strong emotive content,
which I find odd given the subject matter.  Intuitionistic logic
is necessary when one is reasoning about indefinite domains.  If
one is reasoning about a family of entities which is only partially
defined and could, in any conceivable circumstance, always be
enlarged, then "P or not-P" is generally not acceptable.

Intuitionists regard the natural numbers in this way, a view that
I do not share, though I would not dismiss it so contemptuously.
I, and I believe most practicing mathematicians, feel that we have
a convincing mental picture of omega which justifies treating it
as a definite entity and accepting the law of excluded middle for
all arithmetical statements.

On the other hand, I do regard the universe of sets in this way,
as a necessarily incomplete entity which is always capable of
extension.  In fact I find arguments to the contrary nonsensical.
This is connected to the philosophical incoherence of the "iterative"
conception of the universe which I discussed in a previous message
in relation to power sets.  If one does not regard the universe of
sets as a well-defined complete entity then not only is the power
set axiom unjustified, as I argued previously, but also one must
use intuitionistic logic.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu

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