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 $M_{r} = (F_{0} ⊔ F_{1}, Q_{r})$ 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. $Q_{i} (w_{i}) > 0$ ), 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 $0 \leq α \leq 1$ .

Then there exists a generalised model $M_{r}^{α} = (F_{0} ⊔ F_{1}, Q_{r}^{α})$ , with $Q_{r}^{α} = 0$ off of $r \subset W_{0} \times W_{1}$ (this $Q_{r}^{α}$ is not necessarily uniquely defined). This has natural functional morphisms $r_{0} : M_{r}^{α} \to M_{0}$ and $r_{1} : M_{r}^{α} \to M_{1}$ .

Those $r_{i}$ push forward $Q_{r}^{α}$ to $M_{i}$ , such that for the distance metric $L$ defined on morphisms,

$| r_{0} (Q_{r}^{α}) - Q_{0} |_{1} = α L (r)$ ,
$| r_{1} (Q_{r}^{α}) - Q_{1} |_{1} = (1 - α) L (r)$ .

By the definition of $L$ , this is the minimum $| r_{0} (Q_{r}^{α}) - Q_{0} |_{1} + | r_{1} (Q_{r}^{α}) - Q_{1} |_{1}$ we can get. The proof is in this footnote^[1].

Accuracy of models

If $α = 0$ , 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 $α = 1$ , then it's reversed: $M_{0}$ is an approximation, and $M_{1}$ a correct model. For $α$ between those two value, we see both $M_{0}$ and $M_{1}$ as approximations of the underlying reality $M_{r}$ .

Measuring ontology change

This...

Generalised models