[FOM] Excluded Middle

[Lawrence Stout]

On October 10 Arnon Arvon answered Jeremy Clark's question
>
> On Oct 5, 2005, at 11:36 pm, Dmytro Taranovsky wrote:
>>
>> The law of the excluded middle is a logical truth.
>
> Please could somebody explain to me what a "logical truth" is and why
> the law of excluded middle merits status as such. This seems to me
> like another defense of classical logic by bluster. ("Because I say
> so!")
>
> Jeremy Clark
>
>

by attacking intuitionists (rather than the ideas of intuitionism) and 
saying that classical logic needs no defense.  I think that the issue 
of the law of excluded middle is more nuanced than his attack would 
suggest.

Classical logic is based on a theory of truth which is quite crisp and 
applies, on the face of it, only to propositions for which that theory 
of truth is appropriate.    In this theory of truth statements are 
either (completely) true or (completely) false.  The negation of a true 
statement is a false statement and other connectives are determined by 
the usual truth tables.  The two-valued-ness of the logic is built in 
from the start and conclusion that the law of the excluded middle holds 
almost seems circular:  because there are only two possibilities for 
truth values, there is not a third.

The application of classical logic to most reasoning in mathematics 
reflects the view of most mathematicians that mathematics is about a 
particular abstract reality in which meaningful statements all have 
crisp truth values.  (Since there do exist intuitionist mathematicians, 
pluralists, constructivists, formalists, structuralists, and other 
kinds of mathematicians, this view is not monolithic, though it is 
predominant.)

Many kinds of things can be reasoned about productively using classical 
logic, but not all things we want to reason about are so crisp.  
Intuitionists like Brouwer concern themselves with reasoning about 
mathematical knowledge, something much more limited than mathematical 
truth even for a committed mathematical realist.  Their use of 
"intuitionistic" propositional logic comes from an analysis of what it 
takes to know a compound statement using knowledge of the component 
parts.  Because there are mathematical statements whose truth values we 
do not know (favorite examples seem to involve long sequences appearing 
in the decimal expansion of pi) there will be situations in which the 
reasoning which is appropriate crisp truth does not apply.  
Intuitionists use a different logic because of the nature of the 
questions they want their logic to answer.  They accept classical logic 
for reasoning about finite things because they view finite things as 
being graspable in their entirety and thus having a crisp knowledge 
about their crisp truths.

Constructivists also use a logic in which  the law of the excluded 
middle is not used.  Their claims are not for the truth of statements 
but rather for the existence of a construction establishing that truth. 
  We might take the position that reasoning about arithmetic statements 
for a constructivist asks for a decision procedure.  Since we know that 
not all facts about arithmetic are decidable there will need to be 
situations in which we cannot decide which of p or not p holds, having 
no construction giving either, and we will consequently reject the 
excluded middle for constructible truth.

Situations in which the law of the excluded middle does not hold also 
arise in the internal logic of topoi, where they are simply the fact of 
the matter, not the result of a particular philosophical position on 
the nature of mathematics.  For example, in the topos Fin^2 of pairs of 
finite sets with a function between then (A---->B,  with A giving "now" 
and B giving "later")  the object of truth values has this shape:
      { 1, .5      --------------------->   {true,
	0}          --------------------->     false}
where we may interpret  1 as "true now and continuing to be true 
later", .5 as "false now, but true later", and 0 as "false forever".  
Notice that in this example everything is finite, but we still need to 
reject the law of the excluded middle.   Interestingly, the law of the 
excluded middle gets the "truth value"
	{.5} ------------> {true}
a middle value!

Application of the law of the excluded middle where the nature of truth 
in the domain of discourse is not crisp can lead to significant error:  
consider the case of "if you are not my friend, then you are my enemy"  
which allows no neutrality nor disinterested parties.  Not even third 
grade playgrounds operate that way.

Lawrence Neff Stout
Professor of Mathematics
Illinois Wesleyan University
Bloomington, IL 61702-2900
http://www.iwu.edu/~lstout