On Oct 24, 11:01 am, nukeymusic <nukeymu...@gmail.com> wrote: |Is the following a correct interpretation of what Brouwer said |concerning the principle of the excluded third?: |there exist p : (p or not(p)) is not equal to 1
Brouwer was deeply suspicious of the possibility of formalism becoming an end in itself. So we should be careful to know what you mean to say by this.
There is a sense in which for a statement to be "equal" to 1 or T (verum) just means that it is true. Being not equal could well mean being false or absurd. But Brouwer did not assert the absurdity of "p or not p" for any p.
In formal terms, ~(p or ~p) is unacceptable in intuitionistic logic; one can prove in fact for each p that ~~(p or ~p). There are formalizations of intuitionism in which "~(for all p)(p or ~p)" is a theorem in fact, even though "(for all p)~~(p or ~p)" is also a theorem.
I think it's a much better description of Brouwer's point of view to say that it's not justified to assert "p or not p" for every p. If you can find his collected works try an early paper titled, ""The untrustworthiness of the principles of logic" and get it just the way that he put it himself.