Carnap (1943) noted that categoricity fails even for propositional logic. He observed that some “non-normal” models exist for classical logical consequence. The immediate reaction to this phenomenon was to strengthen the formal machinery or impose semantical constraints to rule out non-normal models, and as a result, their very structure was largely overlooked. In this paper, I examine these non-normal models and explore the possibility that they assign non-classical meanings to logical connectives. By relaxing some standards regarding semantics, we will see that these models, in some sense, embed a non-deterministic matrix (Nmatrix) with interesting similarity to the strong Kleene’s 3-valued matrix. Then, I briefly explore its induced logics: two non-compositional counterparts of the well-known logics LP and K3.
