[FOM] Platonism and undecidability

[Neil Tennant]

Some members of the list have written to ask me privately to explain, on
the list, why it is that it is inconsistent for an intuitionist to maintain,
concerning any sentence S, that it is known that the truth-value of S
cannot be found.

The basic idea is simply that if an intuitionist is in a position to say
that S *cannot* be shown to be true, then the intuitionist must
possess some reductio ad absurdum of the assumption that S can be
shown to be true, i.e. of the assumption that there is a proof of S.
(This is the characteristic feature of intuitionism: the truth of a
sentence consists in the existence of a proof.) Thus S has ipso facto been
shown to be false.  So the intuitionist cannot then also say that S
cannot be shown to be false. 

Formally, one can point to the following intuitionistic proof (call it
D) of absurdity (#) from the assumption ~(Sv~S):

	__(1)

	S
       ____

       Sv~S  ~(Sv~S)
       _____________

	    #
	   ____(1)

	    ~S
	   _____

	   Sv~S   ~(Sv~S)
	   ______________

		#

which uses only the introduction and elimination rules for negation and
the introduction rule for disjunction. The classicist, of course, can take
a final step of classical reductio, and conclude to (Sv~S), a.k.a. the Law
of Excluded Middle. But all that the intuitionist will conclude is
~~(Sv~S).

Using the proof D just given as the subordinate proof for existential
elimination allows us to form the proof

			_______(2)

			~(Sv~S)			
		   	   D
	(Ep)~(pv~p)	   #
	_____________________(2)

		   #

This proof reveals the intuitionistic absurdity of claiming that there
exists a counterexample to the law of excluded middle. But of course
the intuitionist cannot conclude from this that (forall p)(pv~p), since
the moves needed in order to obtain that result from ~(Ep)~(pv~p) are
strictly classical.

The matter is discussed more fully in Chapter 6 of my book "The Taming of
The True":

http://www.oup.com/us/catalog/general/subject/?view=usa&ci=0198237170

Neil Tennant