In a paper entitled "<a href="https://doi.org/10.1093/jigpal/3.5.725">Contrastive Logic</a>" (<i>Logic Journal of the IGPL</i> <b>3</b> (1995), 725–744), Nissim Francez introduced something he called <i>bilogics</i>, which are logics intepreted over a pair of structures instead of a single structure, in order to study words such as <i>but</i> and <i>already</i>. The idea in the case of <i>but</i> 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 href="https://doi.org/10.1007/BF02127972">A modal contrastive logic: The logic of ‘but’</a> (<i>Ann. Math. Artif. Intell.</i> <b>17</b> (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."