Interpreting a Structure

This post is related somewhat to the previous one, but has nothing to do what what a theory is. Rather it gives an analysis of what it means to say that a structure $\mathcal{M}$ is correct under some interpretation $\mathcal{I}$.

One cannot define correctness of a structure without specifying an $\mathit{\text{interpretation}}$ of the structure. Similarly, one cannot define truth of a linguistic string, say, "bajfgdhsalsbs" unless one specifies an $\mathit{\text{interpretation}}$ of the language to which the string belongs. So, given a language $\mathcal{L}$, one usually defines:

1. The notion of an $\mathcal{L}$-interpretation, $\mathcal{I}$.
2. The notion of a formula $\varphi$ being true under $\mathcal{I}$.

By analogy, one wants to define:

1. The notion of an interpretation $\mathcal{I}$ of a structure $\mathcal{M}$.
2. The notion of $\mathcal{M}$ being correct under $\mathcal{I}$.

The proposed definition of "interpretation of a structure" is as follows:

(D1) Suppose a structure $\mathcal{M}$ is given. An interpretation $\mathcal{I}$ of $\mathcal{M}$ is specified by three components:
(i) A domain $D_{\mathcal{I}}$.
(ii) A function $f_{\mathcal{I}}:dom(\mathcal{M})\to D_{\mathcal{I}}$. Call this the denotation function.
(iii) For each distinguished relation $R$ in $\mathcal{M}$, a relation $R^{\mathcal{I}}$. Call this the referent of $R$ under $\mathcal{I}$.

This analogous to the standard idea of interpreting a language $\mathcal{L}$. We specify a domain $D_{\mathcal{I}}$ for the quantifiers to range over, and we specify, for each primitive predicate symbol $P$, a relation $P^{\mathcal{I}}$ on the domain $D$.

Next, we can define what it means for a structure $\mathcal{M}$ to be correct, in some sense, under any given interpretation $\mathcal{I}$.

(D2) Let $\mathcal{M}$ be a structure and let $\mathcal{I}$ be an interpretation. Then $\mathcal{M}$ is correct under $\mathcal{I}$ just if $f_{\mathcal{I}}$ is a bijection and, for each distinguished relation $R$ of $\mathcal{M}$, $f_{\mathcal{I}}(R)=R^{\mathcal{I}}$.

An example is the following. Suppose we have a very simple structure $\mathcal{M}$ such that $dom(\mathcal{M})=\{0,1\}$, with a single distinguished relation $R=\{(0,1)\}$. This is more or less, a simple directed graph with two nodes, with one connected to the other. Let us specify an interpretation $\mathcal{I}$ as follows:

$D_{\mathcal{I}}=\{a,b\}$.
$f_{\mathcal{I}}(0)=a$.
$f_{\mathcal{I}}(1)=b$.
$R^{\mathcal{I}}=\{(a,a)\}$.

This interpretation treats $0$ as denoting $a$, treats $1$ as denoting $b$, and interprets the relation $R$ as $\{(a,a)\}$. Clearly, $\mathcal{M}$ is not correct under $\mathcal{I}$.

(This post is based on a talk I gave at Leeds University in 2003. Here is a photo, taken by Joseph Melia, of the talk:

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