Fredrik Engström: What is logic? On logicality, invariance and definability

The traditional account of what a logical consequence is says that A follows logical from T if for every reinterpretation of all the non-logical expressions in T and A; if all the sentences in T are true then so is A. This definition rests on the fact that we know how to distinguish between logical and non-logical expressions, this is the problem of identifying the logical constants.

The talk will start with a general outline on logicality and logical constans and alternative ways of defining when an operator should be considered a logical constant. I will then focus on the model theoretic answer: An operator is a logical constant if it is invariant under the most general transformations. I hope to be able to both give background material and some recent technical results (jointly with Denis Bonnay) on Galois correspondences between invariance and definability: The dual character of invariance under transformations and definability by some operations has been used in classical work by for example Galois and Klein. In this talk I will study this duality from a logical viewpoint and generalize results from Krasner and McGee into a full Galois correspondence of invariance under permutations and definability in $L_{\infty\infty}$. I will also present a similar correspondence for invariance under similarity relations and definability in $L_{\infty\infty}^-$, the equality free version of $L_{\infty\infty}$.