[FOM] FOM: The Law of Excluded Middle

[Paul Hollander]

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.

My answer also fails to address the descriptive question of whether 
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