Sue asked:

What logical arguments have been developed for valid reasoning that avoids the excluded middle and
proof by contradiction? I have been training as a mathematician, which I am now applying to Buddhist
philosophy regarding the nature of reality. Starting from the single premise that "An agent of change
must not be non-existent to have an effect", I find that one can prove that there is no free will.
However, I am concerned because I easily fall into using excluded middle arguments, and in quantum
mechanics one ends up with atoms being neither existent in the way we usually think of existent, nor
non-existent. So I want to make arguments regarding phenomena that are in this grey area, between
inherent existence and nihilism. What has already been done in this regard?

I am aware that Erret Bishop, a constructivist mathematician, did mathematics without use of the
excluded middle, but that is probably deeper into the mathematical side than the philosophy I am
working on. Have there been philosophers who have also worked on this?

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Well if you've been training as a mathematician, and interested in this area, I'm surprised you haven't
heard of fuzzy logic, developed by Zadeh. It treats extensively of the continuum of probabilities
between true and false, and has in fact been applied to the construction of analog circuits in, for
example, elevator controls. One can also look up quite a bit of literature on many-valued logic, an
area I really have no expertise in... but three-valued is just a starter. As for whether "philosophers"
have worked in this area... I guess it depends on whether you regard logicians as philosophers. You
might also try Whitehead on "process" for another take on causality.

Which brings me to another point. Your sentence, "An agent of change must not be non-existent to
have an effect" seems to assume quite a bit of metaphysics. What is "change", how is it brought
about, what is an "effect", and so forth are questions which are still being debated. Whitehead, C.
Taylor, and others do not give conventional answers to these questions. As far as quantum
mechanics goes, that's no help for the free will problem, since you're just dealing with probabilities
there, and not different types of "existence", whatever that term means. You might also look at John
McCrone, Peter Carruthers, Daniel Wegner, Jaegwon Kim, and Palmquist for a few modern
treatments of free will.

In addition, just what do you mean by "free will"? After reading a bit on this subject, I find that I really
have no idea as to what that phrase might mean. Do you mean, we can make choices? No. Do you
mean, our choices are uncaused? No, because then they'd be random or probabilistic, neither of
which helps. Do you mean, our choices defy physical laws? But then what laws do they follow,
"mental" ones? But then we're back to where we started. Do you mean, our choices are neither
random nor do they follow any kind of laws? But then what is it to make a choice, and how are we, as
the dynamic of physical neural nets, so apart from physical causality? Do you mean, we do not
understand physical causality? Now there I'd agree that is a possibility. It may indeed be that there is
another alternative here that we do not (as yet) understand, or cannot understand, although our
remarkable success with the physical sciences argues against this. And of course the existence
(which by the way I do not hold likely at all) of some sort of ESP has no bearing on the above
arguments, as you no doubt realize. Why then does Buddhism?

If you want modern critiques (which I am very skeptical of) of the sciences and that conception of
causality, I'd also recommend someone like Feyerabend or Foucault.

Steven Ravett Brown

Yes, there are lots. There are many versions of logic around nowadays, and some deal with these
sorts of issues. Paraconsistent logics allow that there can be statements A, such that A and not-A are
true. I suggest that you start by looking up the article on paraconsistent logic in the online Stanford
Encyclopedia of Philosophy at:

http://plato.stanford.edu/entries/logic-paraconsistent/

One of the authors of the article, Graham Priest, is a leading philosopher in this field. There are lots of
suggestions for further reading if you find that interesting.

Tim Sprod