A credence assignment *c* on a space *Ω* of situations is a function from the powerset of *Ω* to [0, 1], with *c*(*E*) representing one’s degree of belief in *E* ⊆ *Ω*.

An accuracy scoring rule *s* assigns to a credence assignment *c* on a space *Ω* and situation *ω* the epistemic utility *s*(*c*)(*ω*) of having credence assignment *c* when in truth we are in *ω*. Epistemic utilities are extended real numbers.

The scoring rule is strictly truth directed provided that if credence assignment *c*_{2} is strictly truer than *c*_{1} at *ω*, then *s*(*c*_{2})(*ω*)>*s*(*c*_{1})(*ω*). We say that *c*_{2} is strictly truer than *c*_{1} if and only if for every event *E* that happens at *ω*, *c*_{2}(*E*)≥*c*_{1}(*E*) and for every event *E* that does not happen at *ω*, *c*_{2}(*E*)≤*c*_{1}(*E*), and in at least one case there is strict inequality.

A credence assignment *c* is extreme provided that *c*(*E*) is 0 or 1 for every *E*.

**Proposition.** If the probability space *Ω* is infinite, then there is no strictly truth directed scoring rule defined for all credences, or even for all extreme credences.

In fact, there is not even a scoring rule that strictly truth directed when restricted to extreme credences, where an extreme credence is one that assigns 0 or 1 to every event.

This proposition uses the following result that my colleague Daniel Herden essentially gave me a proof of:

**Lemma.** If *P**X* is the power set of *X*, then there is no function *f* : *P**X* → *X* such that *f*(*A*)≠*f*(*B*) whenever *A* ⊂ *B*.

Now, we prove the Proposition. Fix *ω* ∈ *Ω*. Let *s* be a strictly truth directed scoring rule defined for all extreme credences. For any subset *A* of *P**Ω*, define *c*_{A} to be the extreme credence function that is correct at *ω* at all and only the events in *A*, i.e., *c*_{A}(*E*)=1 if and only if *ω* ∈ *E* and *E* ∈ *A* or *ω* ∉ *E* and *E* ∉ *A*, and otherwise *c*_{A}(*E*)=0. Note that *c*_{B} is strictly truer than *c*_{A} if and only if *A* ⊂ *B*. For any subset *A* of *P**Ω*, let *f*(*A*)=*s*(*c*_{A})(*ω*).

Then *f*(*A*)<*f*(*B*) whenever *A* ⊂ *B*. Hence *f* is a strictly monotonic function from *P**P**Ω* to the reals. Now, if *Ω* is infinite, then the reals can be embedded in *P**Ω* (by the axiom of countable choice, *Ω* contains a countably infinite subset, and hence *P**Ω* has cardinality at least that of the continuum). Hence we have a function like the one the Lemma denies the existence of, a contradiction.

**Note:** This suggests that if we want strict truth directedness of a scoring rule, the scoring rule had better take values in a set whose cardinality is greater than that of the continuum, e.g., the hyperreals.

**Proof of Lemma** (essentially due to Daniel Herden): Suppose we have *f* as in the statement of the Lemma. Let *O**N* be the class of ordinals. Define a function *F* : *O**N* → *A* by transfinite induction:

I claim that this function is one-to-one.

Let *H*_{α} = {*F*(*δ*):*δ* < *α*}.

Suppose *F* is one-to-one on *β* for all *β* < *α*. If *α* is a limit ordinal, then it follows that *F* is one-to-one on *α*. Suppose instead that *α* is a successor of *β*. I claim that *F* is one-to-one on *α*, too. The only possible failure of injectivity on *α* could be if *F*(*β*)=*F*(*γ*) for some *γ* < *β*. Now, *F*(*β*)=*f*(*H*_{β}) and *F*(*γ*)=*f*(*H*_{γ}). Note that *H*_{γ} ⊂ *H*_{β} since *F* is one-to-one on *β*. Hence *f*(*H*_{β})≠*f*(*H*_{γ}) by the assumption of the Lemma. So, *F* is one-to-one on *O**N* by transfinite induction.

But of course we can’t embed *O**N* in a set (Burali-Forti).