Determination of Truth

§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.

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This entry was posted on August 19, 2010 at 8:54 AM and is filed under Reductionism, Supervenience and Determination. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Logicians do it with Models

Determination of Truth

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