## Beth’s Theorem: Is Determination Equivalent To Reducibility?

Hellman sets out the problem posed by Beth’s theorem like this.  All of the $\psi$ terms are implicitly defined by the $\phi$ terms in a theory $\textup{T}$  just in case that $(\forall m)(\forall m')((m, m' \in \textup{M}od (\textup{T}) \wedge m \vert \phi = m' \vert \phi) \rightarrow m \vert \psi = m' \vert \psi)$.  This last expression is an instantiation of Det-R, and is consequently equivalent to:

In $\textup{M}od (\textup{T})$-structures, $\phi$ reference determines $\psi$ reference.

Similarly, all of the $\psi$ terms are explicitly defined by the $\phi$ terms in a theory $\textup{T}$ just in case that:

In $\textup{M}od (\textup{T})$-structures, $\phi$ reduces $\psi$.

Here $\textup{T}$ is first-order and has finitely-many non-logical terms.  By establishing the equivalence of implicit and explicit definitions, what Beth’s theorem shows is that determination of reference of $\psi$ by $\phi$ is equivalent to $\psi$ reduction to $\phi$.  So for such a theory, and in regard of structures that are are all, and only, models of that theory, determination of reference is equivalent to reducibility. But this conclusion (which is to this day a commonly held view) is precisely what Hellman wants to avoid.

There’s a way out.  In the more general case where the set $\alpha$ contains non-standard models of $\textup{T}$, determination of reference is not equivalent to reducibility.  While physical reductionism, PR entails determination of reference, Det-R,  the converse does not hold –and it does not hold in cases wehre some models of the laws of science do not represent scientific possibility (i.e., they are non-standard).  What this means is that one can uphold the principles of physical determination without needing to uphold that all scientific facts are reducible to the mathematical-physical.  Scientific possibility itself can be specified as the subset $\alpha$ of the laws of science where some predicates are standard (e.g., those of pure arithmetic) and fix the structures that model scientific possibility such that this set does not capture all and only the models of a first order theory with finitely-many non-logical symbols. In general, mathematical concepts can be given their standard interpretation giving a set of structures that are not all and only the models of the theory.  A(nother!) great insight by Hellman here is that the very question of which models of the laws of science should be excluded to fix the structures modeling scientific possibility is a scientific question.  Science is a thing that changes and so its models must also change.

We should be able to finish off this paper in the next set of notes.