Would it be desirable to carry out a deflationary research programme in modal logic? In other words, would it be desirable to re-think modal logic without the possible worlds semantics? The original hierarchy of modal logics was originally based on Lewis's axioms (1914) interpreted by Kripke in terms of possible worlds in 1950s. But that seems rather technical and removed from intuition. Is it avoidable?
Yes, there is, and for a number of reasons it is also desirable to rethink modal logic without the possible world semantics altogether, including Kripke's models. Forster in Modal Aether gives a detailed analysis of presuppositions behind it and shows that most of them are not satisfied in the "possible world talk" by philosophers:
"I can well imagine that many philosophers reading this will irritably exclaim that these assumptions are distorting oversimplifications which they do not make when applying possible world semantics to their concerns... the point is being conceded that possible world semantics are not applicable to their concerns".
Girard in Transcendental Syntax writes that "what people do with Kripke models and similar constructions [is] they take the language as it is, call it reality, and state a completeness theorem". Kahle in Modalities Without Worlds lists other philosophical and technical criticisms:
"The ontology explodes. Next to the actual world, one needs additional possible worlds to interpret modalities... If we do not consider nested operators, modal logic does not provide more than a box in front of the derivable formulas. Thus, the power of modal logic is located only in the nesting of operators... In fact, also outside of logic, we are not aware of any practical examples where modal logic or possible worlds semantics helps us to determine a necessary truth, which was not already (explicitly or implicitly) built in by certain axioms or constraints on the variety of worlds".
Kahle traces the problem to Lewis's axioms, especially S4, and his modal realism that embraces "exploding ontology" in semantics. He then discusses proof theoretic alternatives to possible world semantics. Possibility is defined as independence from a specified set of assumptions, and the usual definition of necessity as negation of possibility is discarded. Unconditional necessity in natural languages hardly occurs at all, and most uses are of the form "p is necessary for q", e.g. "team must win today to win the league", this leads to an alternative treatment of necessity. He also suggests that, as in natural languages, applicability of axioms should be context dependent. For another approach see Divers's Possible-Worlds Semantics Without Possible Worlds and Fine's Counterfactuals without Possible Worlds.
One can always topologise:
As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.
AWODEY, S., & KISHIDA, K. (2008). TOPOLOGY AND MODALITY: THE TOPOLOGICAL INTERPRETATION OF FIRST-ORDER MODAL LOGIC. The Review of Symbolic Logic, 1(2), 146-166. doi:10.1017/S1755020308080143
We present the main ideas behind a number of logical systems for reasoning about points and sets that incorporate knowledge-theoretic ideas, and also the main results about them. Some of our discussions will be about applications of modal ideas to topology, and some will be on applications of topological ideas in modal logic, especially in epistemic logic. [...]
In the area of applications of topological ideas in epistemic logic, we include a section on the following topics: a topological semantics and completeness proof for the logic of belief KD45.
Parikh R., Moss L., Steinsvold C. (2007) Topology and Epistemic Logic. In: Aiello M., Pratt-Hartmann I., Van Benthem J. (eds) Handbook of Spatial Logics. Springer, Dordrecht
This paper presents a survey of topological spatial logics, taking as its point of departure the interpretation of the modal logic S4 due to McKinsey and Tarski. We consider the effect of extending this logic with the means to represent topological connectedness, focusing principally on the issue of computational complexity. In particular, we draw attention to the special problems which arise when the logics are interpreted not over arbitrary topological spaces, but over (low-dimensional) Euclidean spaces.
Kontchakov, R., Pratt-Hartmann, I., Wolter, F., & Zakharyaschev, M. (2008). Topology, connectedness, and modal logic. Advances in Modal Logic, 7, 151-176.