#### Supplement to Epistemic Utility Arguments for Probabilism

## Proof of Theorem 12

We wish to prove the following theorem (Greaves and Wallace 2006):

Theorem 12.Strong Propriety for Pentails that, for allb,b′ inPandEinF, ifb(E) > 0 andb′ ≠b(• |E) thenGExp_{U, E}(b(• | E) |b) < GExp_{U, E}(b′ |b)

That is, if our epistemic disutility function satisfies
**Strong Propriety for P**, conditionalizing on a piece of
evidence *E* minimizes expected disutility by the lights of the
agent's original credence function *b* and in the presence of
*E*.

We use the notation Σ_{A} to denote the sum
over *v* in **V** that make proposition *A*
true. And we use the notation Σ to denote the sum over all
*v* in **V**.

By **Strong Propriety for P**, we have

GExp_{U, ⊤}(b(• | E) |b(• | E)) < GExp_{U, ⊤}(b′ |b(• | E))

for *b*′ ≠ *b*(• |
*E*). That is,
prior to any evidence, the conditionalized credence function
*b*(• |
*E*) expects itself to be better than it
expects any other credence function to be. Thus, we have

Σb(v|E)U(b(• |E),v) < Σb(v|E)U(b′,v)

by the definition of GExp_{U, ⊤}. Thus,
since *b*(*v* |
*E*) = *b*(*v* &
*E*)/*b*(*E*), we have: *b*(*v* |
*E*) = 0, if *v* does not make *E* true; and
*b*(*v* |
*E*) =
*b*(*v*)/*b*(*E*) if *v* does make
*E* true. Substituting this into the previous inequality, we
get

(1/b(E))Σ_{E}b(v)U(b(• |E),v) < (1/b(E))Σ_{E}b(v)U(b′,v)

Multiplying both sides by *b*(*E*), we get

Σ_{E}b(v)U(b(• |E),v) < Σ_{E}b(v)U(b′,v)

And this gives

GExp_{U, E}(b(• | E) |b) < GExp_{U, E}(b′ |b)

as required.