# [FOM] Intuitionists and excluded middle

Keith Brian Johnson joyfuloctopus at yahoo.com
Mon Oct 17 23:32:19 EDT 2005

```> > (Let us call such proofs proofs by *strong* reductio, in
> contradistinction from proofs which proceed by *weak* reductio. The
> latter proceed from the assumption that a proposition is true, via a
> contradiction, to the conclusion that this proposion is false. Note:
> if
> one accepts proofs by strong reductio, then one is also committed to
> accepting proofs by weak reductio. Proofs by weak reductio are
> acceptable in intuitionistic propositional logic; proofs by strong
> reductio are not)

I'm puzzled as to how one might accept one but not the other.  It seems
to me that if I accept that every sentence either has a truth-value
(possibly more than one) or does not; and if I more strongly accept
that every sentence either has exactly one truth-value or has none;
then I get the following four cases:

(1)  If all sentences have truth-values, so that by the above
assumptions every sentence has exactly one truth-value; and if the only
truth-values are "true" ("T") and false ("F"); then for any sentence p,
"not Tp"="Fp" and "not Fp"="Tp", since p has *some* truth-value and
since the only two available are T and F.  Then both reductios work:
Tp                   Fp
.                    .
.                    .
not Tp               not Fp
Fp                   Tp

(2)  If not all sentences have truth-values, so that for some sentences
we have "TVp" and for others we have "not TVp"; and if the only
truth-values are still T and F; then neither reductio works, although
both work in a modified way:
Tp                   Fp
.                    .
.                    .
not Tp               not Fp
Fp v not TVp         Tp v not TVp

(3)  If all sentences have truth-values but if the available
truth-values are not only T and F, then for any sentence p we will have
Tp or Fp or Ap ("A" for "alternative", where "Ap" might stand for the
disjunction of the possibilities other than Tp and Fp), and the
conclusions will be modified as "Fp v Ap" and "Fp v Ap", and again
neither reductio works without modification;

(4)  If not all sentences have truth-values and if the available
truth-values are not just T and F, then the conclusions become "Fp v Ap
v not TVp" and "Tp v Ap v not TVp", and again neither reductio works
without modification.

So, I'm puzzled as to how one could accept one form of reductio but not
the other.

As for the law of the excluded middle, I do see that I'm using it to
get from, for example, "Tp v Fp" and "not Tp" to "Fp" (in case (1)),
or, again, from "Tp v Fp v not TVp" to "Fp v not TVp" (in case (2));
but although I can see how "not true" might not mean the same as
"false"--there might be other truth-values than just "true" and
"false", or some sentences might not have truth-values at all--I can't
see how, given the stipulations I've made, "Tp v not Tp" (as opposed to
"Tp v Fp") could possibly not hold.

Do intuitionists deny either (a) that every sentence either has a
truth-value (possibly more than one) or does not; or the stronger
(b)that every sentence either has exactly one truth-value or has none?

Finally, someone said that Aristotle took "necessarily p" not to be
logically equivalent to "not possibly not p" in his Analytics; you
don't happen to know exactly where to find it, or what his reasoning
was, do you?  (This interests me particularly because I use the
equivalence of the two in arguing that a standard criticism of fatalism
is misguided, although I hadn't really thought of it that way before
seeing the post referred to.)

Keith Brian Johnson

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