Determination of Reference

We’ve set out the determination of truth:

Det-T: In structures $\alpha$ , $\phi$ truth determines $\psi$ truth iff $(\forall m)(\forall m') ((m, m' \in \alpha \ \wedge \ m \vert \phi \equiv m' \vert \phi) \rightarrow m \vert \psi \equiv m' \vert \psi)$ .

Now, the fact that two elementary equivalent structures are indiscernible in point of reference if each term has the same reference in each leads to a determination of reference principle similar to Det-T.

Det-R: In structures $\alpha$ , $\phi$ reference determines $\psi$ reference iff $(\forall m)(\forall m')((m, m' \in \alpha \ \wedge \ m \vert \phi = \ m' \vert \phi) \rightarrow m \vert \psi = m' \vert \psi)$

Det-R says that if two $\alpha$ structures agree in what they assign to the $\phi$ terms, then they agree on what they assign to the $\psi$ terms. In terms of the structure of the models we’re discussing, if the set $\alpha$ is closed under automorphism, then Det-R is the condition that any bijection between the domain of m and m’ that is a $\phi$ -isomorphism is a $\psi$ -isomorphism.

And regarding the connection between Det-T and Det-R, Hellman points out that model-theoretically they are independent principles. We have the set $\alpha$ , the $\phi$ and the $\psi$ such that in $\alpha$ structures, $\phi$ reference determines $\psi$ reference, but $\phi$ truth does not determine $\psi$ truth.

Hellman continues: if $\alpha$ is the set of structures representing scientific possibility, $\phi$ the vocabulary of mathematical physics and $\psi$ the (broader) vocabulary in which truths can be stated, then Det-T and Det-R are principles of physical determination. Since we want $\alpha$ to represent scientific possibility, every structure in $\alpha$ must (at least) model all of the laws of science. But if having each member of $\alpha$ model all of the laws of science is a sufficient condition for $\alpha$ representing scientific possibility then, if T represents all scientific theory, the term ‘ $\alpha$ ‘ could be uniformly replaced in every instance with ‘{m : m models T}’.

Hellman points out that since it seems possible to formulate T in language with only a finite number of non-logical symbols, T has models which go against the principles of physical determination. Such models must be excluded if we are to hold onto these principles. In §2.2 Reduction and Determination, Hellman will set out ways to exclude these models while articulating more precisely the notion of scientific possibility.

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Logicians do it with Models

Determination of Reference

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