## Tuesday, 19 July 2011

### Adding an Interpretation to a Collection of Structures

This is the third in a series of posts discussing the notion of structural representation, and related to the question of what a scientific theory is. Normally, scientists put forward conjectures, hypothesis, assumptions, etc., and then examine them in the light of evidence and other theories. One might conjecture that there is a black hole at the centre of every galaxy; or conjecture that global warming is caused by human diet; or that there was a species of dinosaur that invented algebraic geometry, etc.

The model-theoretic conception of scientific theories states that a theory is collection $\Sigma$ of mathematical structures. As I've argued below, this implies that theories are not truth bearers. And if one tries to resolve this by defining what "$\mathcal{M}$ represents the world" means, one is lead either to Newman's Objection ($\mathcal{M}$ represents the world iff the world has large enough cardinality) or back to the standard view, along with the claim that the world $\textit{is}$ a structure $\mathbb{W}$.

The underlying problem here is that mathematical structures in $\Sigma$ are $\textit{uninterpreted}$. So, we need to introduce some notion of $\textit{interpreting}$ a structure. And the previous post has proposed an analysis of what an interpretation $\mathcal{I}$ of a structure is, and a definition of what it is for a structure $\mathcal{M}$ to be correct under an interpretation.

Suppose then that one accepts the two objections described earlier - the Truth-Bearer Objection and the Newman Objection. The resolution is to introduce an interpretation, so that we can now talk of the structures being correct or incorrect (under the interpretation). So, we revise the Model-Theoretic Conception to:
The Interpreted-Model-Theoretic Conception:
A theory $T$ is a pair $(\Sigma, \mathcal{I})$ consisting of a collection $\Sigma$ of structures, all of the same signature, and an interpretation $\mathcal{I}$, which assigns an interpretation for each $\mathcal{M} \in \Sigma$. This $\mathcal{I}$ is the intended interpretation of $T$.
(Here there is a slight unclarity: the interpretation $\mathcal{I}$ must be specified for every $\mathcal{M} \in \Sigma$. Because all have the same signature, the interpretation just assigns the same referent $R^{\mathcal{I}}$, when $R$ is the distinguished relation for the structure in question. But for each structure $\mathcal{M}$, a specific denotation function, from $dom(\mathcal{M})$ to $D_{\mathcal{I}}$, must be specified too.)
Then we can define:
(D) $T$ is true iff for some $\mathcal{M} \in \Sigma$, $\mathcal{M}$ is correct under $\mathcal{I}$.
This at least provides some sort of an answer the Truth-Bearer Objection and the Newman Objection.

1. Hi Jeff,

Just a quick question: what role do the structures in \Sigma play for someone who adopts a model-theoretic approach? (For instance, if they're used simply to derive predictions, how do they do this?)

Best,
Sam

2. HI Sam, I'd be really interested in this but there are no precise explanations given. van Fraassen suggests that, for each M \in \Sigma, there is a special substructure, say M°, which "represents" the observable things and their relations.

However, I've never seen it worked out for any specific case, or a genuine scientific theory.

Example 1. Let (M, g_{ab}, T_{ab}) be a spacetime structure. What is the empirical substructure?

Example 2. Let (R^4, V, psi) be a solution of the one-particle Schroedinger equation, with potential field V. What is the empirical substructure?

In my 2004 BJPS article "Empirical Adequacy and Ramsification", I set up the structures M to be 2-sorted, with two domains D_O and D_T, and the "empirical substructure" M° is the result of forgetting D_T and all the relations on D_T (and mixed relations between D_O and D_T). Technically, M° is a reduct of M.

Suppose T is formulated in an interpreted 2-sorted language (L, I). I.e., I is a 2-sorted model. Let the empirical reduct of I be I°.

Then define:

(Df1) T is weakly empirically adequate in (L, I) iff every O-theorem of T is true in I°.
(Df2) M is empirically correct in (L, I) iff M° is ismorphic to I°.
(Df3) A theory T is empirically adequate in (L, I) just when T has an empirically correct model M.

One can show that if T is empirically adequate, then T is weakly empirically adequate. One can give counterexamples for the converse: T might be weakly empirical adequate, but not empirically adequate.

If Ram(T) is the Ramsey sentence of T, then one can prove

(DF) Ram(T) is true iff T has an empirical correct model M whose theoretical domain has the same cardinality as I has.

(I use the label "(DF)" because the first authors to identify such results were William Demopoulos and Michael Friedman in 1985.)