There is a somewhat updated
(1946) interpretation of Rewritten in response to David Corfield's comment below.)
A somewhat more modern take on the Erlanger Programm from the standpoint of logic, due to F.I. Mautner, which you can find here.
Actually, and rather more famously, there is was given in Tarski's 1966 talk What Are Logical Notions? (published 1986), which is as described a bit at in the Wikipedia article on Tarski, which proposes a distinction between what is logical and what is non-logical, based on the . The idea that as one loosens the theory (say from Euclidean geometry to affine geometry to topology to...), the relevant automorphism group becomes larger and larger, so that maximal automorphism groups (symmetric groups) correspond to theories of maximal looseness, where one is left with purely logical notions.
However, it should be said that Tarski's idea was clearly anticipated by F.I. Mautner, writing in 1946; see here. For some commentary on this, see this post by David Corfield at the $n$-Category Café.
As shameless self-promotion, I'll mention that James Dolan and I dabbled a little in this as well; some results were described at the $n$-Category Café, here and here. There we give describe a Galois correspondence between subgroups of symmetric groups and complete theories, in categorical terms.