### Tarski’s Two Approaches to Modal Logic

#### Posted by David Corfield

Can anyone help me out on the question of how Tarski’s work with McKinsey on topological semantics of modal logic connects to Tarski’s work with Jónsson on a duality between modal algebras and (descriptive) general frames.

The Tarski-McKinsey work from 1944 established that

For any consistent theory $T$ extending the modal logic S4, there exists a topological model $(X, \mathcal{O}X, [-])$ (where $[-]$ satisfies certain conditions to make it an interpretation) that validates all and only theorems of $T$.

A proposition is interpreted as a subset of $X$, and interior plays the role of the necessity operator. A proposition, $P$, is necessarily true at a world if there’s an open neighbourhood of worlds containing that world, at each of which $P$ is true.

You can read all about this result and more in Topology and modality: The topological interpretation of first-order modal logic, where Awodey and Kishida prove an analogous result for first-order logic using neighbourhood sheaf models. The worlds are the points of a space, fibres of which are models for the first-order logic.

Now, Tarski also worked on Boolean algebras with operators (BAOs) with Jónsson. This work has has been taken up by the coalgebra community. For example, in Stone Coalgebras
the authors show that *descriptive general frames* are coalgebras for a certain (Vietoris) endofunctor on the category of Stone spaces. Dual to this we find modal algebras (a kind of BAO) as algebras for the opposite endofunctor on $Stone^{op}$ which is equivalent to the category of Boolean algebras.

I suppose a way to compare approaches would come from seeing a topological space as a coalgebra for the functor on $Sets$ which sends a set to the set of filters on it. These slides – Topo-bisimulations are coalgebraic – by Christoph Schubert seem to be exactly what I need to look at.

As I was suggesting here, if there’s something to the coalgebraic approach to modal propositional logic, there ought to be a way of taking it over to first-order modal logic, which ought then to bear a resemblance to Awodey and Kishida’s semantics. This would rely on some kind of categorified Stone duality, perhaps one of those described by Forssell and Awodey, or the one Mike tell’s us about here, if that’s distinct.

There’s some idea about that Tarski didn’t see the connection between the two approaches to modal semantics, in which he played such a prominent part. According to Robert Goldblatt here

It appears then that in developing his ideas on BAO’s Tarski was coming from a different direction: modal logic was not on the agenda. According to [Copeland, 1996b, p. 13], Tarski told Kripke in 1962 that he was unable to see a connection with what Kripke was then doing. (p. 18)

This is when Kripke was developing his frame semantics for modal logic.

## Re: Tarski’s Two Approaches to Modal Logic

Taking a frame as a set of worlds, $W$, and an accessibility relation, $R$, on $W$, we can generate the Alexandroff topology on $W$, generated by opens of the form $\{y|x R y\}$ for worlds $x$.

I don’t suppose ionads could figure in a first-order modal logic. After all, the interior operator, $Int$ is a finite meet-preserving comonad on the power set of $X$, and an ionad is a set $X$ together with a finite limit-preserving comonad $Int X$ on the category $Set^X$.