# [FOM] FOM: The Law of Excluded Middle

Paul Hollander paul at paulhollander.com
Tue Oct 18 21:13:07 EDT 2005

```John Corcoran, 3:28 PM -0400 10/16/05:
>Q1 Is there any generally accepted meaning of 'the law of excluded
>middle' or related expression as it occurs in foundational debates?

I'd like to re-phrase this by asking, can we find such a meaning?
And I'd like to try answering it from the perspective of someone
trained primarily in philosophy, as opposed to mathematics.

I would like to suggest that we can find such a meaning of 'Law of
Excluded Middle'.  But getting there involves recognizing that the
word 'ambiguous' is ITSELF ambiguous, as between at least three
different meanings.

First, two expressions (tokens) are ambiguous when the extension of
one is a proper subset of the extension of the other.  Typical
examples (types) are 'Xerox', 'Kleenex', 'Kodak', etc.  In one sense
'Xerox' refers to photocopies produced only by Xerox brand copiers;
in another sense it refers to photocopies produced by any brand of
photocopier.

Second, two expressions are ambiguous when their extensions are
disjoint, with examples such as the word 'bank'.  In one sense 'bank'
refers to buildings containing certain financial institutions; in
another sense it refers to the sides of rivers, and in another sense
it refers to turns made by airplanes.  These are disjoint sets.

Third, two expressions are ambiguous when their extensions are
disjoint but analogically related.  The traditional example here,
drawn from the history of philosophy, is the word 'being', which
presumably has a different but related sense when applied to, say, a
substance as opposed to an attribute (to use examples drawn from
Aristotle).

Given two expressions X and Y that are ambiguous in the final sense
listed above (their extensions are disjoint but analogically
related), often we find there is a third ambiguous expression, Z,
related to X and Y according to the first type of ambiguity listed
above.  That is, the extension of X, and of Y, are each proper
subsets of the extension of Z.  A typical example here, also borrowed
from Aristotle, is the word 'healthy':  in one sense it refers to
food; in another sense it refers to activities; but in both senses it
refers to things that are conducive to health.

Assuming the above account is correct, then the search for an
inclusive meaning of 'Law of Excluded Middle' can be described as the
search for a meaning of the expression that brings together the
disjoint but analogically related extensions of it into one single
extension.  Again, I think Aristotle's discussion in the Metaphysics
offers a clue, because that discussion is cast in terms of existence
versus non-existence (being versus non-being).  So on this approach,
all instances of Excluded Middle boil down to the matter of whether a
thing is, and of whether it isn't, and of how the two relate.

What I offer here is a rough-and-ready sketch of an answer.  The
devil is in the details, of course.  I'm sure the word 'analogical'
raises lots of skeptical eyebrows, and I'm sure we can find ambiguous
and/or fatally vague examples of it.  Those who oppose this answer
will naturally argue that there is no analogical relation between all
the different meanings, or at least none sufficient for the task.
The same goes for 'thing', 'is', etc.

or not such a meaning of 'Law of Excluded Middle' is generally
accepted in foundational debates.  But as I pointed out in my
previous post, there is a difference between descriptive claims (what
mathematicians actually do), on one hand, and normative and
prescriptive claims (what mathematicians ought to do).  I'm
addressing the normative and prescriptive issues, not the descriptive
one.

-paul

Paul J. Hollander
Visiting Lecturer
Dept. of Communications and Humanities
Corning Community College
Corning, NY
```