Proving the consistency of the model theory
Phil Johnson-Laird developed a new informal proof of the consistency of the model theory of sentential reasoning (Khemlani, Byrne, & Johnson-Laird, 2018). Here’s a relevant introductory excerpt:
is the model theory of sentential reasoning consistent in the preceding sense? The theory postulates that an intuitive process in system 1 evaluates a given conclusion, but that a subsequent deliberative process in system 2 sometimes corrects this evaluation. Does this difference between the two systems make the theory inconsistent? Not as long as the two systems have separate analyses. On the one hand, intuitions are the source of erroneous inferences in some cases: they can err in treating inconsistent premises as consistent, and vice versa – thereby creating compelling illusory inferences. On the other hand, deliberation in ideal circumstances should not err: it is normative and should be consistent.
The proof is available for download here.