|On Aug 6, 2011, at 9:14 AM, John F. Sowa wrote:|
In his modal logic, CSP also distinguished five different kinds
of modalities. In his existential graphs, he used different
colors to represent them (he had a box of colored pencils):
1. Logical possibility. A proposition p is possible iff it is not
provably false. Impossible means inconsistent or provably false.
This won't do. Provably false in what theory? Consider the continuum hypothesis (CH) that all uncountable sets of real numbers are the same size. By well known results of Gödel and Cohen, CH is independent of ZF set theory. Hence, neither CH nor not-CH is provably false in ZF. So both CH and not-CH are possible according to Peirce?
Clearly there is a sense of possibility that is independent of any theory. CH, in particular, is either true or false — either all uncountable sets of reals are the same size or some uncountable sets of reals are smaller than others. Hence, either CH or its negation is impossible, full stop, regardless of their consistency with our best theory of sets.
If you believe, as I suspect you might, that the uncountably infinite is the villain here, note that basically the same argument can be run vis-á-vis Peano Arithmetic and its Gödel sentence (relative to some coding scheme), which we in fact know to be true in the natural numbers (hence necessarily true) but independent of PA.