Reply #4: I use a different notation.

[Igel]

Assume P = {R, D), where pi is any partisan and |p| is an
operator returning values R or D for any pn.  S = {+, -, 0}, +
standing for "positively appraised", - standing for
"negatively appraised", 0 is "undefined".

Further, assume set {E}, E being an event or proposition, real
or imaginary.

We define three-place function, E(x, y), x in {E} and y in
{P}, which maps members of {E} into S.

1.  Show that if E(xi, -y) = -E(xi, y), xi real, E must map xi
to 0.

2.  Show that OP is a special case of E(x,y), and discuss
whether the defining and discussion of E(x,y) is itself a
member of {E}.

3.  Show that E(xi, -y) = -E(xi, y) can only return values -
and + for xi in the subset of imaginary events or
propositions.

4. extra credit (required for grad students) For all real xi,
show that E(xi, y) returns values in the subset (+, -}.

5.  (required for grad students).  Let H be the set of all
American voters.  Discuss the relationship between H and P,
and whether the answers for (1) and (2) are generally valid
for H.

(not even on edit:  I think I got that right, but I have to go
so it's likely that one or two revisions I made make it even
more nonsensical than I intended.  :-)