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In justification logic, given a formula A, one can have modal formulas of the form 't : A', read as 't justifies A' or 't is evidence for A'. Has there been any investigation of what the dual of such modal operators would be? I.e. has there been any investigation of a modal operator that would be read as 'it is not the case that t justifies not-A' or 't is consistent with A'?

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  • Some parts of Lakatos theory of science trend that way. You may want to see if any of that has been formalized in logic.
    – user9166
    Dec 28 '14 at 14:53
  • @jobermark what in Lakatos are you thinking of? Dec 28 '14 at 15:40
  • @ChristopherE He is the first person I read to talk about both how theories challenge one another and how they protect what is consistent with them, without treating consistency and conflict as exact opposites.
    – user9166
    Dec 28 '14 at 17:43
  • To the main point. Why choose to see this as a modal operator? It does not seem to function like the ordinary ones. Markers like "necessity" or "possibility" operate on statements, rather than relations between them. So the notion of complementary mode (which I think is the old grammar term for what you mean by dual) may not fit in here very well. It seems to me to be more of a partial deduction mechanism than a modality.
    – user9166
    Dec 28 '14 at 17:55
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There is a long tradition of understanding probability to mean a degree of partial entailment, or a degree of justification. This began with John Maynard Keynes in his "Treatise on Probability", and was taken further by Rudolf Carnap in his "Logical Foundations of Probability" and by Richard Cox in "The Algebra of Probable Inference". It is now referred to as the logical concept of probability and it is a close cousin to the bayesian interpretation, because both conceive probability as something epistemic, i.e. concerned with justification and evidence, rather than as being a property of the world.

Probability can in some ways function like a modal operator, but it is different from necessity because it admits of degrees. Whereas a proposition might be considered necessary or possible or neither, probability uses a numerical range to allow comparisons. The dual of probable is simply improbable: to the extent that A renders B probable, A renders not-B improbable, and if A is independent of B, then A is independent of not-B. So your use of "justifies" translates to renders probable, and "consistent with" to independent of.

As to how we cash out exactly what "renders probable" and "independent of" mean, one of the most fruitful approaches is to employ the concepts of relative entropy and mutual information. These allow us to quantify the shared information between probability distributions, which in turn allows us to interpret "A makes B probable" as if I have the information A then I have this much information about B. This does not by any means solve all the problems of statistical inference, but it allows us to explain what epistemic probability is without recourse to imprecise concepts such as "rational belief".

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