[FOM] FOM: The Law of Excluded Middle

Richard Heck rgheck at brown.edu
Sun Oct 16 22:53:23 EDT 2005

Some remarks on Corcoran's questions, specifically with respect to how
the term "the Law of Excluded Middle" gets used in philosophy of logic.

I propose that we can categorize uses of this term using the following
two questions. (This is a proposal, not a statement.)

(i) Is the principle essentially meta-theoretic? or could it be stated
using propositional, substitutional, or some other such form of
quantification? Or again: Does the principle essentially involve a
notion of truth that is not simply disquotational? or can it be stated
using a purely disquotational truth-predicate?
(ii) Does the principle concern sentences or other forms of expressions?
or does it concern thoughts, propositions, or what have you?

These questions are stated in such a way as to presume that there are
only two relevant answers to each question. Perhaps that is not so. If
it is so, however, there will be four sorts of candidates for the title
of LEM.

(One of the possible meanings for "the Law of Excluded Middle" that
Corcoran mentioned is not included here, namely, "Every property either
belongs or does not belong to a given entity". I won't defend its
exclusion in detail, but I suggest that it will either reduce to one of
the other forms (if the notion of a "property" is sufficiently weak) or,
if it does not (which it won't if the notion of a "property" is strong),
then it isn't a form of LEM, anyway.)

Question (ii) should be clear enough for present purposes. So far as I
know, preferences as regards (ii) do not often surface in debates about
LEM, though they do from time to time. Kripke, for example, familiarly
denies that his theory of truth violates "bivalence" on the ground that,
while *sentences* may fail to be true or false, *propositions* may not.
Obviously, someone who prefers a sentential formulation (or, in a
language that is not context-independent, one in terms of utterances),
will want to insist that LEM applies only to certain sorts of sentences
(intuitively, ones that express propositions, though she will not wish
to put it that way).

A couple of words of explanation about (i), which could use to be
sharpened a fair bit. The intention here is to distinguish pairs like
this one:
(1) The disjunction of any sentence with its negation is true.
(2) Given any sentence, either it or its negation is true.
I think (1) can be regarded as equivalent, roughly speaking, to
something that could be stated using substitutional quantification. It
is precisely the sort of claim fans of disquotational truth-predicates
like to mention as, in effect, an infinite conjunction, which could be
stated using a purely disquotational truth-predicate. But (2) is not
intended to be understood that way. Given natural assumptions about
truth and how it interacts with sentential operators, it will imply the
first form, but it could be stated regarding an impoverished language
that did not contain disjunction. What is more important is that (2) is
not intended to follow from (1): That the logic of a theory is classical
implies (1), but that does not imply (2), as witness supervaluational
semantics and the like. (Of course, there is a question what
"disjunction" and "negation" are supposed to mean here, as well.)

Michael Dummett suggested that "the Law of Excluded Middle" be reserved
for something like (1) and "the Law of Bivalence" for something like
(2), /modulo/ the issue about question (ii) we've already discussed.
This suggestion has not universally been adopted, but I think a lot of
people have followed Dummett's lead, for lack of any better idea how to
use the terms.

Richard Heck

John Corcoran wrote:

>In the 1999 Cambridge Dictionary of Philosophy under "Laws of Thought" we read that the expression 'Law of Excluded Middle' is ambiguous. The article goes on to list several senses including: "Every proposition is either true or false", "Every property either belongs or does not belong to a given entity", "Given any proposition, either it or its negation is true".  The author explicitly disclaims giving an exhaustive list. I can
>think of others myself including several that are obviously false: "Every proposition is either known to be true or known to be false", "Every property is either known to belong or known not belong to a given entity", "Given any proposition, either it or its negation is known to be true".  Just starting here, given the notorious ambiguity of 'or', 'not', 'proposition' and 'known', we have an impressive array of choices to work though.
>Q1 Is there any generally accepted meaning of 'the law of excluded
>middle' or related expression as it occurs in foundational debates?
>Q2 Does any of the contributors to the FOM exchange have a specific
>preferred sense for 'the law of excluded middle' or related expression
>as it occurs in foundational debates?
>Q3 Is there a place in the literature that discusses the range of
>meanings that this expression has been made to carry over its long
>Q4. Do all of the contributors agree as to what this law is about:
>strings of characters, people's thoughts, people's assertions, abstract
>propositions in the sense of Alonzo Church 1956, "meaningful sentences"
>in the sense of Tarski 1956 or something else?
>Q5 Has any contributor to this debate prefaced his or her contribution
>with remarks sufficient to clarify the intended meaning of this
Richard G Heck, Jr
Professor of Philosophy
Brown University
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