## Updated notes on counterpart semantics for modal predicate logic

In around 2009, I got interested in counterpart-theoretic interpretations of modal predicate logic. Lewis's original semantics, from Lewis (1968), has some undesirable features, due to his choice of giving the box a "strong" reading (in the sense of Kripke (1971)), but it's not hard to define a better-behaved form of counterpart semantics that gives the box its more familiar "weak" reading.

Wondering if anyone had figured out the logic determined by this semantics, I found an answer in Kutz (2000) and Kracht and Kutz (2002). I also learned that counterpart semantics seems to overcome some formal limitation of the more standard "Kripke semantics". For example, while all logics between quantified S4.3 and S5 are incomplete in Kripke semantics (as shown in Ghilardi (1991)), many are apparently complete in the "functor semantics" of Ghilardi (1992), which I do not understand but which is said to have a counterpart-theoretic flavour. Skvortsov and Shehtman (1993) present a somewhat more accessible "metaframe semantics", inspired by Ghilardi's approach, and claim that the quantified version of all canonical extensions of S4 remain canonical (and hence complete) in metaframe semantics. Kracht and Kutz argue that their – much simpler – counterpart semantics inherits these properties of functor and metaframe semantics.

When I thought about how all this might work, I realized that the completeness proof in Kutz (2000) and Kracht and Kutz (2002) is incorrect, and that the logic presented by Kracht and Kutz is not in fact complete with respect to their semantics. I figured out what the right logic should look like, and how to fix the completeness proof.

At this point, I abandoned the project, leaving some important questions unresolved. In particular, I would have liked to check that counterpart semantics can indeed serve as a model theory for a wide range of systems (such as those between S4.3 and S.5) that are incomplete in Kripke semantics.

I returned to this question late last year, when a (very!) talented undergraduate student here at Edinburgh asked if I had any open questions in modal logic. He worked through the canonicity transfer proof in Skvortsov and Shehtman (1993) and tried to adapt it to my counterpart semantics.

While we worked on this, I realized that (a) my Kutz-inspired construction of canonical models does not work for many important systems, and that (b) Skvortsov and Shehtman (1993) don't actually manage to show that the quantified extension of every canonical propositional logic above S4 remains canonical in metaframe semantics, at least not on the usual understanding of "canonical". (They call a logic "canonical" if it is sound wrt a Kripke frame F whenever it is sound wrt some tight and compact general frame based on F, and as Emil Jeřábek confirmed to me, this leaves out a lot of logics one would normally call canonical.)

I then spent a few days trying to come up with a completeness proof specifically for quantified S4M – a useful test case, given its nice discussion in Hughes and Cresswell (1996) –, but without success.

I now plan to abandon this project again. It still seems to me that some of the open questions would be worth resolving, but I don't have enough knowledge, skill, and time.

At least I've updated my notes on what I have managed to show and what remains to be shown. Here are the new notes.

*Journal of Symbolic Logic*56 (2): 517–38. doi.org/10.2307/2274697.

*Studia Logica*51 (2): 195–214. doi.org/10.1007/BF00370113.

*A New Introduction to Modal Logic*. London and New York: Routledge.

*Advances in Modal Logic*, edited by Frank Wolter, Heinrich Wansing, Maarten de Rijke, and Michael Zakharayaschev. Vol. 3. World Scientific Publishing Company.

*Identity and Individuation*, edited by Milton K. Munitz, 135–64. New York: New York University Press.

*Journal of Philosophy*65 (5): 113–26. doi.org/10.2307/2024555.

*Annals of Pure and Applied Logic*63 (1): 69–101. doi.org/10.1016/0168-0072(93)90210-5.