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Several sources includes catalogs of many modal logics, often arranged into a lattice of inclusion, showing increasing power, from K to S5. Naturally, for each logic there is a corresponding characterization of an accessibility relation for the relevant semantic frame that validates the axioms for each system.

While investigating the logic S4 and some of its derivatives S4.2 and S4.3 it struck me that there is no reference anywhere I can find to a logic between S4 and S5 which takes a partial order as its accessibility relation. That is, a relation which is reflexive, transitive, and anti-symmetric. As in the lattice order of a Boolean algebra, or the subset relation in an algebra of sets.

Is this a familiar issue to anyone in the field? I've searched and searched all over the Web and the library, but I'm not seeing the topic appear in any text or survey or review.

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  • I believe I recently read something on deontic logic that did have a partially ordered accessibility relation. If I get ambitious I'll try to look it up and give you a pointer, but until then you could search for accessibility functions in deontic logic. May 29, 2023 at 12:53
  • @DavidGudeman Please do share! I have been searching energetically, but coming up empty. The closest I have found is this post, where temporal logic is mentioned in a comment: philosophy.stackexchange.com/questions/59390/… May 29, 2023 at 13:02
  • I was probably thinking of this: thespaceofreasons.blogspot.com/search/label/Modal%20Logic. It's an epistemic logic, not a deontic logic, and I misremembered: it isn't strictly symmetric but it isn't anti-symmetric either. May 29, 2023 at 14:41
  • Temporal reachability relation is a partial order. Kramer uses it in Logic of Intuitionistic Interactive Proofs, p. 4 to interpret intuitionistic connectives. To clarify, did you say in the linked answer that partial order accessibility leads to p→☐p and hence to modal collapse?
    – Conifold
    May 29, 2023 at 19:03

2 Answers 2

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S4.3 gives partial ordering on the accessibility relation, but not antisymmetry.

You might like to take a look at temporal logic, which is similar in many ways to modal logic. Temporal logic treats time in an antisymmetric fashion, since we usually take it for granted that if t1 is earlier than t2 then it is not the case that t2 is earlier than t1. We also often take it for granted that there is only one timeline, so we allow that all times are reachable from a given starting point, and we may wish to allow that time has an end.

Hughes and Cresswell (A New Introduction to Modal Logic) cover the relationship between modal logic and temporal logic (S4.3 on pages 127-131 and S4.3.1 on pages 179-180). They point out that S4.3 imposes a connectedness constraint on the accessibility relation, but it allows for collections of moments that are distinct but simultaneous. There are ways to resolve this: they mention one by Segerberg.

The Stanford Encyclopedia has a good article on temporal logic.

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This is a familiar issue to me, but I also have not found it anywhere in the literature or standard references, although I have usually dealt with the conditional, rather than an accessibility relation.

It does not seem to be widely recognized that the material conditional of classical two-valued logic is just such a partial ordering, (That is, a relation which is reflexive, transitive, and anti-symmetric) and furthermore, that it is chiefly what makes deductive reasoning with extended chains of claims possible.

This is also a difficulty in multi-valued logic. An examination and survey of the many-valued systems in the standard references on many-valued logic reveals that most of the conditionals defined are not partial orderings and do not have the desired properties.

I am most familiar with the conditional relation of Lukasiewiz 3-valued logic which also fails to have these properties, Hovever, within this system it is possible to define a "definite" or "strict" conditional that does. I have not found any references which employ this approach.

The Lewis systems and their kin typically employ the strict conditional symbolized by "fishhook" as the connective of choice, but it is not clear at first glance whether it has the desired properties.

However, in the 3-valued system, it is also possible to define a conditional which satisfies the usual definition of the Lewis strict conditional and which fails as a partial ordering.

It may be interesting to note that if the Lewis strict conditional holds, so does the Lukasiewicz strict conditional, but not conversely. This raises the possibility that a truth-functional system of modal logic can be constructed on the basis of 3-valued logic. This possibility is generally dismissed in the literature I have examined as having been found unworkable, but the reasons, if they are given at all, don't hold up to close examination.

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