A standard scoring rule argument for probabilism—the doctrine that
credence assignments should satisfy the axioms of probability—goes as
follows. If *s* is a scoring
rule on a finite probability space *Ω*, so that *s*(*c*)(*ω*) is the
epistemic utility of credence assignment *c* at *ω* in *Ω*, and (a) *s* is strictly proper and (b) *s* is continuous, then for any
credence *c* that does not
satisfy the axioms of probability, there is a credence *p* that does satisfy them such that
*s*(*p*)(*ω*) is
better than *s*(*c*)(*ω*) for all
*ω*. This means that it’s stupid
to have a non-probabilistic credence *c*, since you could instead replace
it with *p*, and do better, no
matter what.

Here is a problem with the dialectics behind this argument. Let *P* be the set of all credence
assignments that satisfy the axioms of probability. But suppose that I
think that there is some nonempty set *M* of credence assignments that do
not satisfy the axioms of probability but are rationally just as good as
those in *P*. Then I will think
there is some way of making decisions using credences in *M*, just as good as the way of making
decisions using credences in *P*. The best candidate in the
literature for this is to use a
level set integral, which allows one to assign an expected value
*E*_{c}*U* to
any utility assignment *U* even
if *c* is not a probability.
Note that *E*_{p}*U* is
the standard mathematical expectation with respect to *p* if *p* is a probability.

The argument for probabilism assumed two things about the scoring rule: strict propriety and continuity. Strict propriety is the claim that:

*E*_{p}*s*(*p*) >*E*_{p}*s*(*c*) whenever*c*is a credence other than*p*

for any probability *p*. In
words, by the lights of a probability *p*, then we get the best expected
epistemic utility if we make *p*
be our credence.

Now, if I am not convinced by the argument that (1) should hold for
any probability *p* and any
credence *c* other than *p*, then I will be unmoved by the
scoring rule argument for probabilism. So suppose that I am convinced.
But recall that I think that credences in *M* are just as rationally good as the
probabilities in *P*. Because of
this, if I find (1) convincing for all probabilities *p*, I will also find it convincing
for all credences *p* in *M*, where *E*_{p} is my
preferred way of calculating expected utilities—say, a level set
integral.

Thus, if I am convinced by the argument for strict propriety, I will
just as much accept (1) for *p*
in *M* as for *p* in *P*. But now we have:

**Theorem 1.** If *E*_{p} is strongly
monotonic for all *p* ∈ *P* ∪ *M* and
coincides with mathematical expectation for *p* ∈ *P*, and (1) holds for
all *p* in *P* ∪ *M*, where *M* is non-empty, then *s* is not continuous on *P*.

(Strong monotonicity means that if *U* < *V* everywhere then
*E*_{p}*U* < *E*_{p}*V*.
The Theorem follows immediately from the Pettigrew-Nielsen-Pruss
domination theorem.)

Suppose then that I am convinced that a scoring rule *s* should be continuous (either on
*P* or on all of *P* ∪ *M*). Then the conclusion
I am apt to draw is that there just is no scoring rule that satisfies
all the desiderata I want: continuity as well as (1) holding for all
*p* ∈ *P* ∪ *M*.

In other words, the only way the argument for probabilism will be
convincing to me is if my reason to think (1) is true for all *p* in *P* is significantly stronger than my
reason to think (1) is true for all *p* in *M*, *and* I have a
sufficiently strong reason to think that there is a scoring rule that
satisfies all the true rational desiderata on a scoring rule to conclude
that (1) holding for all *p* in
*M* is not among the true
rational desiderata even though its holding for all *p* in *P* is.

And once I additionally learn about the difficulties in defining sensible scoring rules on infinite spaces, I will be less confident in thinking there is a scoring rule that satisfies all the true rational desiderata on a scoring rule.