Friday, 24 August 2012

Logical Consequence

This is all a bit basic, but worth saying because it interacts with foundational questions (for example, "what are the relata of logical consequence?" or "what is more basic, semantics or inference?"). Suppose that $A = \{a_1, a_2, a_3\}$ is an alphabet. Suppose that $\sigma_1$ and $\sigma_2$ are non-empty strings from $A$. That is, they are finite sequences of the form
$\sigma_i : [0,n] \rightarrow A$.
Consider a claim of the form:
$\sigma_2$ is a logical consequence of $\sigma_1$.
I think this claim is seriously underspecified, because logical consequence can only make sense given a class of interpretations of the strings.

Suppose that $A$ above is the alphabet of an interpreted language $L$. Suppose that this also determines a special class $Sent(L)$ of strings, and a class $\Sigma(L)$ of interpretations for $L$ in such that way that, for any $I \in \Sigma(L)$, and any $\phi \in Sent(L)$,
$I \models \phi$
is defined, meaning,
the $L$-string $\phi$ is true in $I$.
Then consider the claim,
$\sigma_2$ is a logical consequence in $L$ of $\sigma_1$.
On the Bolzano-Tarski definition, this means,
$\sigma_1, \sigma_2 \in Sent(L)$ and, for any $L$-interpretation $I \in \Sigma(L)$,
if $I \models \sigma_1$ then $I \models \sigma_2$.