In a paper entitled "Contrastive Logic" (Logic Journal of the IGPL 3 (1995), 725–744), Nissim Francez introduced something he called bilogics, which are logics intepreted over a pair of structures instead of a single structure, in order to study word such as but and already. The idea in the case of but is that one must simultaneously consider two states of affairs, namely the actual state of affairs and the "expected" state of affairs. A later paper by J.-J. Ch. Meyer and W. van der Hoek, A modal contrastive logic: The logic of ‘but’ (Ann. Math. Artif. Intell. 17 (1996), 291–313) showed how more or less the same idea could be captured using an extension of the well-known modal logic S5, which provides a framework for analyzing possible worlds.

There is a small literature on related topics that you can find by searching for "contrastive reasoning."

In a paper entitled "Contrastive Logic" (Logic Journal of the IGPL 3 (1995), 725–744), Nissim Francez introduced something he called bilogics, which are logics intepreted over a pair of structures instead of a single structure, in order to study words such as but and already. The idea in the case of but is that one must simultaneously consider two states of affairs, namely the actual state of affairs and the "expected" state of affairs. A later paper by J.-J. Ch. Meyer and W. van der Hoek, A modal contrastive logic: The logic of ‘but’ (Ann. Math. Artif. Intell. 17 (1996), 291–313) showed how more or less the same idea could be captured using an extension of the well-known modal logic S5, which provides a framework for analyzing possible worlds.

There is a small literature on related topics that you can find by searching for "contrastive reasoning."