§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, and and a set of structures that represent what is scientifically possible.
In elementary model theory two models m and m’ are elementary equivalent (m m’) if they make the same sentences true. Also, the reduct of a model m to a given vocabulary, L (m 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 , truth determines truth iff . This says that once you have fixed the facts, the facts are also fixed -or that once a complete description of things has been given in terms, then there is only one correct way to describe things in terms.
This notion of determination of truth becomes more engaging when is a subset of the models of a theory T, composed of lawlike truths and both and are each subsets of the language used to articulate T and . If T contains sentences with indispensable occurrences of both and , then T serves to connect the and 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.