I need a first order modal logic, where inconsistency between formulas in not binary: a pair of formulas may be more or less inconsistent. The modal operators express uncertainty. So the formulas which would be contradictory without the modal operators can have some "fuzzy" inconsistency with the modal operators. Did somebody encounter such a logic? If not, is there something wrong with such a logic?

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    $\begingroup$ At a glance, it looks to me like this paper is about that idea, but only the propositional part of it, i.e., this paper doesn't handle predicate logic: sciencedirect.com/science/article/pii/S1570868304000497 Sorry if this isn't what you're looking for, or if that paper does not turn out to be useful. $\endgroup$
    – A.S.
    Jul 23 at 21:41
  • $\begingroup$ Actually, I meant two formulas having the same modality may have non binary consistency. $\endgroup$
    – Marina
    Jul 23 at 21:53
  • $\begingroup$ Say $\asymp \big(\varphi(x) = y \big)$ and $\asymp \big(\varphi(x) = z \big)$ may be more or less inconsistent depending on the difference $|y - z|.$ $\endgroup$
    – Marina
    Jul 23 at 21:58
  • $\begingroup$ This sounds more like probability than logic? E.g. maybe you want a belief network. $\endgroup$
    – none
    Jul 23 at 23:35
  • $\begingroup$ There are no probabilities. The modality I am talking about is interpreted as "it appears that". The only thing I care about is to evaluate inconsistency between atomic formulas with the same modality. $\endgroup$
    – Marina
    Jul 24 at 0:02

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