FOM: Intuitionism

[richman@acc.fau.edu]

Jesper Carlstrom wrote:
 
>>>On Fri, 17 May 2002, wiman lucas raymond wrote:

>>>> would an intuitionist accept the statment
>>>> "either p is provable, not-p is provable, or neither is provable"?

>>>The principle
>>>
>>>   If a proposition cannot be known to be true,
>>>   then it can be known to be false
>>>
>>>is defended from an intuitionistic point of view in Martin-Löf ...

 
> On Sat, 18 May 2002, richman wrote:
>> Is the principle supposed to apply to the statement?
 
> If "neither is provable", then, in particular, p cannot be known to be
> true. Hence p can be known to be false. But then -p can be known to be
> true, so -p is provable, which contradicts the assumption.

I still don't see what the content here is. If I understand your
argument, it derives a contradiction from the assumption that "neither
is provable". This reduces Raymond's statement to

   "either p is provable or not-p is provable"

so that Raymond's modification of an instance of the law of excluded
middle would be no modification at all. Was it your purpose to
demonstrate that?

Because intuitionists identify the assertion of p with the assertion
of the provability of p, they could derive a contradiction from
"neither is provable" by using the tautology "not-(not-p and
not-not-p)".

Why is the distinction between "proof" in the sense of a mathematical
object and "demonstration" in the sense of a convincing argument
relevant to a question about what intuitionists would accept?

--Fred