I think that focusing on systems of truth (like formal logics) might be missing the forest for the trees a bit. There are lots of other things we use reasoning for, like decision making and navigating uncertainty, that are arguably just as valuable as establishing truthiness. Mathematicians are very careful to use words like "implication" and "satisfaction" (as opposed to words like "reasoning") to describe their logics, because the philosophers may otherwise lay siege to their department.

A model that is mathematically incorrect (i.e. has some shaky assumptions and inference issues) but nevertheless makes good decisions (like "which part of this codebase do I need to change?") would still be very valuable, no? I think this is part of the value proposition of tools like Claude Code or Codex. Of course, current agentic tools seem to struggle with both unless you provide a lot of guidance, but a man can dream =P