[FOM] Intuitionists and excluded-middle

Arnon Avron aa at tau.ac.il
Sun Oct 9 03:18:24 EDT 2005

On Thu, Oct 06, 2005 at 09:47:10AM +0200, Jeremy Clark wrote:
> 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!")

It is usually very unfruitful to argue with  declared intuitionists
(or declared relevantists, or ... ), because they pretend not to understand
the language we all speak. Still, I'll try to reply this time to
Jermey Clark, though I have no illusions that I might convince him
of anything. At best he might see why the gap between us is unbridgeable.

 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"). 
The proof of this last claim is very constructive: the overwhelming 
majority of mathematicians (and logicians) were not convinced. So
either the arguments were not convincing, or (constructive or!)
most of the logicians and mathematicians from Hilbert onwards
were and are hopelessly stupid. I leave to you the rest of the argument. 

 Second: when it comes to really basic principles, any "defense"
one can offer can be described as a "bluster" ("Because I say so").
What "defense" do the intuitionists have for the disjunctive syllogism
(which they accept,  but the relevantists reject)? Why do they accept 
that a=a is a logical truth, and take 0=1 as "absurd"? I wonder what 
explanations they can possibly give that will be any better than 
the explanation I am going to give below why the logical truth 
of excluded middle is forced on me as self-evident (Note: not why I "accept" 
excluded middle - such a terminology suggests that I have other choices.
I don't).

 So what is a "logical truth", you ask. Well, at least for me a proposition
is a basic logical truth (or logically valid) if its truth is necessarily 
seen by anyone who understands the meanings of the logical notions used
in that proposition. One might ask of course what is a 
"logical notion". This is not an easy question, but for understanding
excluded middle one need not be able to define "logical notion"
in general. It suffices to recognize "not" and "or"
as "logical notions", and to understand the meanings that normal
people (including even mathematicians and logicians) attach to these
notions. And to normal people "not A" is true precisely when 
A is not true, and "A or B" is true precisely when either A is true
or B is true (yes, perhaps the main characteristic of a "logical notion"
is that one cannot really explain it without using that very notion...).
Once this meaning of "or" and "not" is understood, one
sees/understands/feels/is forced to accept (make your choice) that 
excluded middle is self-evident.  So if you really cannot see this too, then 
you do have a big problem that I cannot solve for you.

 Now when intuitionists reject excluded middle (and "accept" other laws)
it is because they do not use the ordinary meanings of "not" and "or".
Actually, they are not denying this: their books usually explain in detail the
meanings they attach to the usual connectives. Thus "or" should be understood
constructively, while "not A" means that there is a procedure that transforms 
any proof of A to a proof of an absurdity (or maybe *the* absurdity?
it seems that for intuitionists all absurdities are equivalent. Why?
"because they say so"?).  Although I have some problems in understanding 
the intuitionists' explanations (how can a procedure
transform  one thing to another which by definition does not exist?),
I am somehow able to make (classical!) sense of them, and even 
see circumstances in which intuitionistic logic can be quite useful. 
Hence  I do not have a serious problem with the way intuitionists apply
their notions.  I do have some reservations about the fact that they 
insist on using words like "not" and "or" in a way which is very different 
from the ordinary meanings of these words, but if this would make
them happy, I am ready to leave these words to them, and use
"c-not", "c-or" etc for the classical notions (So many useless debates
could have been avoided if people would not insist on using certain 
specific words for describing their ideas and beliefs! Such insistence
is especially common in political battles. Thus in my country, Israel, almost 
every Jewish citizen, including myself, strongly support Zionism, but 
there are very hot debates what "Zionism" means...). 

  Needless to say, I know very well that even a willing of all classical
logicians and mathematicians to switch to a new terminology, leaving to
the intuitionists the use of the favorite words, would c-not make the
intuitionists happy. They would c-not agree that A c-or c-not A 
is a valid principle. Instead they would pretend that they do c-not
understand the meaning of "c-or" c-or "c-not" (in fact they would
claim that these words do c-not have meanings. Why they
do c-not have meanings? Because they say so!).

I declare that about this point I do c-not believe the intuitionists.
They understand and use "c-not" like all ordinary people. Take for
example their explanation of their "not". It involves the concept
of "absurdity". And what is an "absurdity" according to them?
Well, something that does c-not have a proof, (or perhaps can-c-not 
have a proof?).  They use here "c-not" rather than their own "not", because
had they define an absurdity to be a proposition for which there is a 
procedure that transforms any proof that this proposition has a proof
to a proof of absurdity, then *every* proposition would have counted
as an absurdity! Fortunately, I know for sure that the intuitionists
do c-not think that every proposition is absurd...

Arnon Avron

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