We've already seen that if $M_0=(F_0,Q_0)$ and $M_1=(F_1,Q_1)$ are generalised models, with the relation $r\subset W_0\times W_1$ a $Q$-preserving morphism between them, then there is an underlying model $M_r=(F_0\bigsqcup F_1,Q_r)$ between them.

Since $r\subset W_0\times W_1$, $Q_r$ is defined on $r$; indeed, it is non-zero on $r$ only. The underlying model has functions $r_0$ and $r_1$ to $M_0$ and $M_1$, which push forward $Q_r$ in a unique way - to $Q_0$ and $Q_1$ respectively. Essentially:

• There is an underlying reality $M_r$ of which $M_0$ and $M_1$ are different, consistent, facets.

Illustrated, for gas laws:

Underlying model of imperfect morphisms

But we've seen that relations $r$ need not be $Q$-preserving; there are weaker conditions that also form categories.

Indeed, even in the toy example above, the ideal gas laws and the "atoms bouncing around" model don't have a $Q$-preserving morphism between them. The atoms bouncing model is more accurate, and the idea gas laws are just an approximation of these (for example, they ignore molar mass).

Let's make the much weaker assumption that $r$ is $Q$-birelational - essentially that if any $w_i$ has non-zero $Q_i$-measure (i.e. $Q_i\left(w_i\right)>0$), then $r$ relates it to at least one other $w_j$ which also has non-zero $Q_j$-measure. Equivalently, if we ignore all elements with zero $Q_i$-measure, then $r$ and $r^{-1}$ are surjective relations between what's left. Then we have a more general underlying morphism result:

Statement of the theorem

Let $r$ be a $Q$-birelational morphism between $M_0=(F_0,Q_0)$ and $M_1=(F_1,Q_1)$, and pick any $0\le \alpha \le 1$.

Then there exists a generalised model $M_r^{\alpha }=(F_0\bigsqcup F_1,Q_r^{\alpha })$, with $Q_r^{\alpha }=0$ off of $r\subset W_0\times W_1$ (this $Q_r^{\alpha }$ is not necessarily uniquely defined). This has natural functional morphisms $r_0:M_r^{\alpha }\to M_0$ and $r_1:M_r^{\alpha }\to M_1$.

Those $r_i$ push forward $Q_r^{\alpha }$ to $M_i$, such that for the distance metric $L$ defined on morphisms,

1. $|r_0\left(Q_r^{\alpha }\right)-Q_0{|}_1=\alpha L\left(r\right)$,
2. $|r_1\left(Q_r^{\alpha }\right)-Q_1{|}_1=(1-\alpha )L\left(r\right)$.

By the definition of $L$, this is the minimum $|r_0\left(Q_r^{\alpha }\right)-Q_0{|}_1+|r_1\left(Q_r^{\alpha }\right)-Q_1{|}_1$ we can get. The proof is in this footnote^[1].

Accuracy of models

If $\alpha =0$, we're saying that $M_0$ is a correct model, and that $M_1$ is an approximation. Then the underlying model reflects this, with $M_0$ a true facet of the underlying model, and $M_1$ the closest-to-accurate facet that's possible given the connection with $M_0$. If $\alpha =1$, then it's reversed: $M_0$ is an approximation, and $M_1$ a correct model. For $\alpha$ between those two value, we see both $M_0$ and $M_1$ as approximations of the underlying reality $M_r$.

Measuring ontology change

This approach means that $L\left(r\right)$ can be used to measure the extent of an ontology crisis.

Assume $M_0$ is a the initial ontology, and $M_1$ is the new ontology. Then $M_1$ might include entirely new situations, or at least unusual ones that were not normally thought about. The $r$ connects the old ontology with the new one: it details the crisis.

In an ontology crisis, there are several elements:

1. A completely different way of seeing the world.
2. The new and old ways result in similar predictions in standard situations.
3. The new way results in very different predictions in unusual situations.
4. The two ontologies give different probabilities to unusual situations.

The measure $L$ amalgamates points 2., 3., and 4. above, giving an idea of the severity of the ontology crisis in practice. A low $L\left(r\right)$ might be because because the new and old ways have very similar predictions, or because the situations where they differ might be unlikely.

For point 1, the "completely different way of seeing the world", this is about how features change and relate. The $L\left(r\right)$ is indifferent to that, but we might measure this indirectly. We can already use a generalisation of mutual information to measure the relation between the distribution $Q$ and the features $F$. We could use that to measure the relation between $F_0\bigsqcup F_1$, the features of $M_r^1$, and $Q_r^1$, its probability distribution. Since $Q_r^1$ is more strongly determined by $Q_1$, this could^[2] measure how hard it is to express $Q_0$ in terms of $F_1$.