One solution to the misrepresentation problem says that the representations r₁, r₂, r₃, … belong to some “type” R. Each representation in R means whatever is meant by the “standard” member of R, which we’ll denote r*. Another way to refer to r* is by saying it is the member of R that occurs under “optimal conditions”. That is, a representation r means whatever it would mean if it arose under optimal conditions. While this proposal may help with misrepresentation, it presents us with at least two more problems: how to tell what “type” some representation falls into, and how to determine the “optimal condition” for that type.

If we are allowed to make some assumptions about how a mind works, the first problem is fairly easy to solve. By definition, every representation in a type means the same thing, namely whatever r* means. It is plausible, if not probable, that every representation that means the same thing has the same corresponding mental structure. After all, the mind doesn’t make exceptions for false beliefs, because it thinks they are true beliefs. It makes sense, then, that the same belief will involve the same mental structure, regardless of whether it is true or false. If we want to know what is meant by a representation r, we merely need to know what the same mental structure would mean if it occurred under “optimal conditions”.

Unfortunately, defining these optimal conditions is much more difficult. It appears as though we have to presuppose what r is about in order to know what kinds of conditions would be optimal. The belief that I just saw a horse on the side of the road (when, in fact, it was a cow) may occur simultaneously with the belief that I am in an area smelling of horse manure (when, in fact, it is cow manure that I smell). The optimal conditions for the belief that I saw a horse include such things as proper lighting, proper distance, and sufficient time for visually determining the type of animal before me. The optimal conditions for the belief that I am in an area smelling of horse manure include such things as absence of other smells, a clear and properly-functioning olfactory system, and sufficient time for nasally determining the type of animal waste before me. Unless we presuppose that the one belief is about something seen and the other is about something smelled, there is no way to know which set of optimal conditions goes with which belief (and this is assuming we have already managed to narrow it down to two possible sets of conditions).

We can make some headway with this problem if we assume that the structure of a mental representation gives some clue as to which sense(s) it involves. For instance, if the “activation” of a particular representation in a human mind involves some activity in or near the visual cortex, it is likely that the representation is at least partially about something visual. (This claim, taken alone, is plausible even for representations that are not about anything that we have seen before). If we discover that r is related to something visual, we can “check” what r would mean under certain optimal viewing conditions (i.e. adequate lighting and so on). Unfortunately, this is not enough information to completely determine the optimal conditions. The optimal viewing distance for any earthbound belief about the sun would be approximately 93 million miles, whereas the optimal viewing distance for a belief about my head would be much closer to 12 inches. So while this proposed solution helps with misrepresentation, it is no easy task to solve the new problems it presents.

Furthermore, it deals only with representations that actually have optimal conditions. I have some beliefs about unicorns, which are defined as mythological creatures. If a horse with a single horn actually existed, then, it would be a one-horned horse, and not a unicorn. Unicorns, by definition, cannot actually exist, so there can be no optimal conditions for perceiving them. Even if the solution proposed above could somehow unambiguously determine the optimal conditions for representations of cows and horses and manure, it can say nothing about beliefs regarding things that cannot exist. This is true even if those beliefs are all true beliefs. That is, I can hold the true belief that unicorns do not exist, but I still have a belief that is about unicorns, and no such belief is covered by the counterfactual solution to the misrepresentation problem.

Another impossible thing, which is perhaps more interesting than unicorns, is the Berry number, which is defined along the lines of “the smallest integer that cannot be expressed in fewer than twenty English words”. At first glance, it appears that this sentence should refer to some specific number. Then we realize that “the smallest integer that cannot be expressed in fewer than twenty English words” has just been expressed in thirteen English words. So now we might think that the definition doesn’t actually refer to a specific number. But there are only finitely many combinations of fewer than twenty English words, and there are infinitely many integers, so there must be infinitely many integers that cannot be described in fewer than twenty English words. Thus, it seems, we have a paradox on our hands.

But is the Berry number really paradoxical? A paradox must involve a contradiction both ways, otherwise it is simply a false statement. If we assume the Berry number exists, we prove that it doesn’t, and if we assume that it doesn’t exist, we find ourselves with the apparent contradiction of finitely many sentences somehow applying to infinitely many integers. However, it is not entirely clear that “the set of integers not expressible in fewer than twenty English words” is a well-defined set. To examine the problem further, let us abstract away from this particular example. Instead of “the set of all arrangements of fewer than twenty English words that refer to specific integers”, let us use an arbitrary set A. Instead of “the set of all integers”, let us use another arbitrary set B that has at least one more member than A. Instead of whatever set of definitions we use to assign sets of English words to specific integers, let us use the arbitrary set F, which has the same cardinality as A and is a mapping of every member of A to some specific member of B.

The analogue to the Berry number must be some member of {B–F(A)}. That is, it must be one of the members of B that is not mapped from any member of A by a member of F. Let us call this element b. The problem arises when we claim that some member of A can possibly refer to b. It is clearly impossible for any member of F(A) to be a member of {B–F(A)}. Going back to the particular example, this means that it must be impossible for any set of fewer than twenty English words to refer to any integer that is inexpressible in fewer than twenty English words. Therefore, it seems like it must be the case that there is no smallest integer that cannot be expressed in fewer than twenty English words, because there are clearly some integers that cannot be expressed in fewer than twenty English words. If we are to give up the assumption that there is a smallest member of this set, we must also give up the assumption that the set is well-defined.

Many other members of this set (in addition to the Berry number itself), once discovered, can no longer be members of the set. This is because we can replace the word “smallest” in the definition of the Berry number with “first-discovered”, “second-discovered”, and so on. The actual members of the set seem to keep receding from our investigations. Any set that does this cannot be epistemologically well-defined, but what about ontologically? If the set is somehow ontologically well-defined, there is no way that fact can actually be demonstrated when it is not epistemologically well-defined.