# [FOM] An intuitionistic query

rgheck rgheck at brown.edu
Mon Sep 7 18:24:05 EDT 2009

```On 09/03/2009 05:49 AM, Arnold Neumaier wrote:
> Alice tells Bob that she has a set A={a,b,c} of cardinality 2, but she
> is silent about any further detail. Intuition tells Bob that a, b, c
> are names for two distinct elements, so that a=b or a=c or b=c.
> Can intuitionistic logic prove this argument correct?
>
> In other words, is
>      A={a,b,c}, |A|=2    ==>    a=b v a=c v b=c          (*)
> intuitionistically provable with generic interpretations of
> the symbols on the left hand side of (*)?
>
>
I would have thought so. The claim that |A| = 2 will mean something
like: There is a 1-1 function from A onto {0,1}, and this will yield a
long disjunction of the form:
[f(a) = 0 &f(b) = 0 & f(c) = 1] v [f(a) = 0 & f(b) = 1 & f(c) = 1] v ...
v [f(a) = 1 & f(b) = 1 & f(c) = 0]
which, together with what we know about 1-1 onto functions will yield
the disjunction you want.

Richard

--
-----------------------
Richard G Heck Jr
Professor of Philosophy
Brown University

```