## Archive for August 19th, 2010

### Determination of Truth

August 19, 2010

§2.1 is about determination.  Hellman begins by introducing the determination of truth about one set of facts by another.  If a collection of facts, A, determines a separate collection of facts, B, then the truths about B cannot change without a change in the truths about A. The task is to spell this out model-theoretically to be able to evaluate the connection between determination and reduction.

We have at our disposal a family of languages where the interpretation of terms appearing in more than one language remains fixed across languages.  We have two sets of non-logical symbols, $\phi$ and $\psi$ and a set $\alpha$ of structures that represent what is scientifically possible.

In elementary model theory two models m and m’ are elementary equivalent (m $\equiv$ m’) if they make the same sentences true.  Also, the reduct of a model m to a given vocabulary, L (m $\vert$ L) is the structure obtained form m by excluding the interpretation of all the terms not appearing in L.

Set out the following: In structures $\alpha$, $\phi$ truth determines $\psi$ truth iff $(\forall m)(\forall m') ((m, m' \in \alpha \ \wedge \ m \vert \phi \equiv m' \vert \phi) \rightarrow m \vert \psi \equiv m' \vert \psi)$.  This says that once you have fixed the $\phi$ facts, the $\psi$ facts are also fixed -or that once a complete description of things has been given in $\phi$ terms, then there is only one correct way to describe things in $\psi$ terms.

This notion of determination of truth becomes more engaging when $\alpha$ is a subset of the models of a theory T, composed of lawlike truths and both $\phi$ and $\psi$ are each subsets of the language used to articulate T and $\psi \not\subset \phi$.  If T contains sentences with indispensable occurrences of both $\phi$ and $\psi$, then T serves to connect the $\phi$ and $\psi$ terms.  Such sentences link the determining phenomena with the determined phenomena.  If all the models of T are elementarily equivalent or if all of the models of T are isomorphic then the determination is trivial.

Having set out the determination of truth, Hellman will then introduce the determination of reference before moving onto evaluating reduction.