FOM: arithmetic, geometry, natural science, formal systems, ...

[Neil Tennant]

Steve Simpson writes:
_________________________
I want to suggest that the correct logic for natural
science is classical rather than intuitionistic.  Identity,
non-contradiction, excluded middle: these are bedrock for me, because
reality is real, independent of our understanding of it.
Intuitionistic logic may be useful for other purposes, e.g. as a
description of *our understanding of reality at a given point in
time*, because if we are not now in a position to confirm or deny X,
then neither X nor not X is now known.  But natural science is the
study of what's real, not the study of *our current understanding of*
what's real.  That's why I focus on classical rather than
intuitionistic logic.
__________________________

Wouldn't this require an argument to the effect that only classical (but not
intuitionistic) logic can handle certain inferences that are needed by the
natural scientist?

But it seems that there is a good argument to the contrary. If you're a
classicist, you will not mind if anyone re-writes all your laws of nature
that have the form "All F's are G's" as ones in the form "It is not the case
that there is an F that is not a G". (Indeed, Popper once recommended doing 
exactly this, in order to appreciate what a law of nature of so-called
"universal" form is "really saying". What it's really saying, according to 
Popper, is that we won't find a counterexample: an F that is not a G.) 

In other words, you (the classicist) won't mind if the language of science
does without the universal quantifier. Let's call the resulting language L.

Informal Claim: All that Logic is needed for in the course of doing natural
science is to produce such *refutations* as there may be of falsifiable
theories about the external world. (Using the logic, we extract predictions
from the hypotheses; then we compare these with the results of observation. 
If prediction and observation clash, absurdity results.)

Note: We allow for the use of mathematics within the pursuit of these 
absurdities.  So you may help yourself to all the (classical) math you like,
in the form of theorems telling you what the solutions are for your
differential equations, etc.

Metatheorem: Any classical proof of absurdity from assumptions X in L can be
matched by (indeed: effectively transformed into) a proof in *intuitionistic 
relevant logic* of absurdity from (some subset of) X.

Informal Conclusion: There's no compelling reason to use classical logic
to develop natural science, no matter how much of a `realist' you happen
to be in your philosophical outlook.  Whatever you do (refutationally)
with classical logic, you can do with intuitionistic relevant logic.

Strange realization: All those strictly classical *mathematical* theorems
that you might have used in your refutation of a bad empirical theory turn
out not to be necessary staging-posts in the passage to absurdity from the
empirical hypotheses being refuted (ultimately, modulo your mathematical
axioms). The classical mathematical theorems sort of "melt away" and vanish
from the proof, in IR, of the main refutational result.  [They had been
`cut formulae' within the overall proof of absurdity, and they just
do not survive the elimination of cuts.] Strictly classical math is therefore 
a ladder that can be kicked away at the end of the day. Or, perhaps more aptly, a crutch that we can throw away.

Steve, what's your reaction to this "Amahl" phenomenon?  As a tough-minded
fellow, shouldn't you too throw away your crutches (i.e., the law of
excluded middle)?

Neil Tennant

PS In the story `Amahl and the Night Visitors' the little boy Amahl is
able to throw away his crutches as the result of some epiphany.