### Modal logic (columbicubiculomania, pt.V)

Oct. 12th, 2013 11:17 pm**winterkoninkje**

It's been a while since I've posted under this tag, but I made a map of common normal modal logics based off the diagram in SEP. I don't have much to say about it, but I'm taking a course on the topic now so hopefully I'll have a bit more to say in the future.

While I was at it, I revised the map for binary relations (first published back here). Not too much has changed overall. I added the definition of "dense" (op-transitive) and "shift-reflexive" relations, since these show up in modal logics and so could be interesting elsewhere. I also tightened up the list of entailments to more explicitly capture the difference between the weak and strong definitions of (anti)symmetry. In the new versions, the entailments all hold in minimal logic assuming the substitution principle for equality (if `x=y`

and `P(x)`

then `P(y)`

, for any elements `x`

and `y`

and any property `P`

)— except for those marked with a subscript `I`

, which require intuitionistic logic. In addition, any of the entailments with strong (anti)symmetry as a premise can be strengthened to only require weak (anti)symmetry if we have decidable equality or are working in classical logic.

*Edit 2013.10.12:* Updated these two again. For the relations, just added a note that shift-reflexivity entails density. For the modal logics, added a bunch of new goodies and cleaned things up a bit.