[FOM] Intuitionists and excluded-middle

Paul Hollander paul at paulhollander.com
Sun Oct 16 15:26:44 EDT 2005

I'd like to comment.

When discussing the attractions (or otherwise) of different systems 
of logic, I think it's important to keep in mind several 
distinctions.  I believe that paying heed to these distinctions goes 
a long way towards improving the quality of criticism and debate.

First, I think there is a distinction to be made between the 
MOTIVATION for developing a non-classical system of logic, on one 
hand, and on the other hand the CONSEQUENCES of developing such a 
system.  Sometimes the motivation and the consequences stand or fall 
with relative independence of one another.  If what one objects to is 
the motivation for developing intuitionist logic, that's one thing. 
If what one objects to is the consequences of such a development as 
it has actually occurred, that's often a different issue.  One might 
have sympathy for the motivation behind intuitionist logic, while 
disagreeing with the consequences of advocating it.

In this respect I find it interesting that Lukasiewicz seems 
regularly to escape the ire of those who rail against intuitionist 
logic.  He also was motivated to call into question Excluded Middle, 
but his approach led to different consequences because he did so by 
the semantic device of using more than two truth values, as opposed 
to the syntactic approach that characterizes the development of 
intuitionist logic.  If this is acceptable, then I think the critics 
of intuitionist logic must ask themselves whether they object to the 
motivation for questioning Excluded Middle, or simply to the 
consequences of doing so in that manner.

Second, I think there is a three-fold distinction to be made 
regarding the function of logic.  Logic has a DESCRIPTIVE function: 
it serves to describe how we actually reason.  (I realize I leave 
myself open to charges of psychologism by writing this, but I also 
believe that this debate has moved so far beyond Frege's criticism of 
the associationist psychology of the British empiricists that its 
relevance today is questionable.)  Logic also has a NORMATIVE 
function:  it identifies examples of reasoning well, and examples of 
reasoning poorly.  Finally, logic has a PRESCRIPTIVE function:  it 
tells us how to do a better job of reasoning.

Also, a logic might have a RESTRICTED function:  it might seek to 
describe only certain restricted types of reasoning, or to evaluate 
or prescribe only in certain restricted types of discourse.

So, when I read accusations leveled at intuitionists such as, "they 
pretend not to understand the language we all speak," I can't help 
but suspect that the descriptive function of intuitionist logic is 
being confused with its normative and prescriptive functions, and 
that the consequences of having developed intuitionist logic are 
being confused with the motivation behind developing it.  I also 
wonder if intuitionist logic is being taken to apply without 
restriction, as opposed to relegating it to a restricted domain of 
discourse.  Intuitionist logic might fail at describing how we 
actually do reason (and it might not), but that does not mean that it 
automatically fails at its normative or prescriptive functions of 
telling us how we ought to reason, or that it applies without 
restriction to all domains of discourse.

Finally, regarding whether to accept the Law of Excluded Middle, I 
get the impression that many logicians overlook clues that the 
history of philosophy provide for how to approach the issue.  Both 
Plato (e.g., in the Theaetetus) and Aristotle (e.g., in the 
Metaphysics) address this.  They accept Excluded Middle, but they do 
so for DIALECTICAL reasons.  Such reasons are irremediably ad 
hominem.  The general approach is to point out that language loses 
its meaning if we don't accept Excluded Middle.  But the approach is 
brought home in the context of some proposition that the disputant 
herself cares deeply about.  THAT proposition, presumably, DOES abide 
by Excluded Middle.  Then, once one proposition is seen to abide by 
Excluded Middle, induction will presumably lead the disputant to see 
that more, perhaps all, propositions abide by Excluded Middle.  I 
think this is where the motivation for adopting a logical principle, 
and the consequences of doing so, join up.

As I assume we all know, Aristotle himself had doubts about Excluded 
Middle when it came to future contingents in On Interpretation.  And, 
the fact that his modalities in the Analytics are not duals 
('Necessarily P' is not logically equivalent to 'Not possibly not P') 
might also be construed as a rejection of Excluded Middle in that 
context.  I'm not saying that I necessarily agree with Aristotle, but 
I do think that he serves as an example of how an advocate of 
Excluded Middle himself can have doubts about its unrestricted 

So, for my money, I think a better way to convince an intuitionist to 
accept Excluded Middle is to engage in a dialectical discussion about 
some proposition whose truth value she cares deeply about.  Not 
necessarily about a theoretical proposition, but possibly about a 
homely, practical issue.  It doesn't really matter so long as the 
disputant cares about the meaning of the proposition in question.  If 
Plato and Aristotle are correct, THIS is the fulcrum point from which 
to elevate one to an acceptance of Excluded Middle.   And, this 
approach requires a bit more tact and finesse than simply accusing 
intuitionists of pretending "not to understand the language we all 

At least this is how I see it.


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

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