FOM: Interpreting double negation
twilson at grianne.engr.csufresno.edu
Sun Mar 8 17:41:09 EST 1998
> Second, we need to remember that the double negation transform is
> purely formal does not respect philosophical motivation.
> I beg to contradict. Double negation as a transformation is much more
> than formalism. Look at the meanings of intuitionistic formulas, which
> form a Heyting algebra. Double negation translates those meanings upwards
> to the nearest element of the Boolean algebra canonically embedded in
> that Heyting algebra.
> A philosophically well motivated interpretation of this motion is that it
> carries the truth value of the proposition "A" to that of the proposition
> "surely A", where "surely" satisfies
> surely surely = surely
> A -> surely A (but not conversely)
> surely not = not surely (so this is a strong form of surely)
> To see the philosophical motivation here, think what "not not A" says:
> it says that the absurdity of A is absurd. While this is not quite
> as strong as flatly asserting A, it is nevertheless a strong vote of
> confidence in A that can be succinctly put as "surely A".
An alternate way to read "not not A" is: "A is globally consistent".
As you say, what is being asserted is the impossibility of refuting A,
which, classically, is the same as asserting that A is consistent.
However, compared with mere consistency, double negation asserts more,
since, because of the identity
not not (A /\ B) = (not not A) /\ (not not B) ,
it follows (using Compactness) that any set of globally consistent
statements is, taken together, consistent in the usual sense that
adding them as axioms doesn't force true = false. This is of course
much stronger than the usual notion of consistency, where two
statements can each be consistent with a common theory (like CH and
not-CH are with ZF) but are inconsistent when taken together.
Computer Science Department
California State University, Fresno
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