# By convention, what are the tautologies and the consequence relation of modal logics?

When looking at a modal logic as a propositional logic, it seems like there are a few choices for which sentences form the tautologies (A, B, C below) and what the consequence relation should be (D, E below).

I'm not saying that these are all reasonable choices, just that they capture some of the intuition behind being a set of tautologies or being a consequence relation. If we treat the possible world semantics of modal logic as conceptually prior, what are the set of tautologies and the consequence relation normally considered to be?

$$\text{\varphi is a tautology if and only if it holds at the designated world 0} \tag{A}$$ $$\text{\varphi is a tautology if and only if it holds at every world that's accessible from 0 } \tag{B}$$ $$\text{\varphi is a tautology if and only if it holds at every world.} \tag{C}$$

$$\text{\Gamma \vdash \varphi holds if and only if \varphi holds at 0 whenever \Gamma holds at the topic world} \tag{D}$$

$$\text{\Gamma \vdash \varphi holds if and only if, for every possible world w, at least one \Gamma is false or \varphi is true} \tag{E}$$

We can take validity to be the limiting case of logical consequence from the empty set. So, the question reduces to what logical consequence for modal languages consists in.

That's a non-trivial issue, since we actually have two natural candidates for being consequence relations: local consequence and global consequence. Both notions have different properties. The deeper reason for this difference is that local consequence is basically a first-order notion, while global consequence is a second-order notion. Let me explain.

Assuming the standard definition of frames and models for similarity types, I will restrict myself to consequence for a basic modal language $$\mathcal{L}$$, where we have exactly one 1-place modal operator $$\Diamond$$. For $$\Phi \subseteq \mathcal{L}, \varphi \in \mathcal{L}$$ we can define:

• If $$\mathcal{M} =(W,R ,V)$$ is an $$\mathcal{L}$$-model and $$w \in W$$, let $$\mathcal{M}, w \models \Phi$$ iff $$\mathcal{M}, w \models \psi$$, for all $$\psi \in \Phi$$. If $$\mathcal{F} =(W,R)$$ is an $$\mathcal{L}$$-frame, let $$\mathcal{F} \models \Phi$$ iff $$\mathcal{F} \models \psi$$, for all $$\psi \in \Phi$$
• $$\varphi$$ is a local consequence of $$\Phi$$ ($$\Phi \models \varphi$$) iff for any $$\mathcal{L}$$-model $$\mathcal{M} = (W, R, V)$$ and any $$w \in W$$ we have: $$M, w \models \Phi$$ only if $$M, w \models \varphi$$
• $$\varphi$$ is a global consequence of $$\Phi$$ ($$\Phi \models^g \varphi$$) iff for every $$\mathcal{L}$$-frame we have: $$\mathcal{F} \models \Phi$$ only if $$\mathcal{F} \models \phi$$.

These notions of consequence have different properties. For instance for local consequence we have a deduction theorem, while this is not the case with the global notion. More importantly, choosing the global notion means to consider modal logic as a means to describe frames, while choosing the local version implies looking at modal logic as a way to talk about models. Modal logic as a frame description tool allows via its associated notion of global validity (global consequence from the empty set) to capture many essentially second-order properties like well-foundedness. This is no accident. As Thomason showed global consequence basically coincides with the consequence relation of monadic second order logic. So, global consequence and global validity are second-order concepts.

In contrast taking modal logic as talking about models means taking a first-order perspective on modal logic. Indeed the van Benthem characterization result tells us that modal logic as a means to describe models is the bisimulation invariant fragment of first-order logic. So, local consequence and local validity are first-order notions.

Of course matters a more complicated than outlined above. For instance I have said nothing on the consequence relation defined via so-called general frames. General frames yield a sort of connection between the local and the global notion. This is in parallel to the case of general models for second-order logic, which provide a first-order perspective on second-order logic. But the rough picture above should indicate the complexities involved in the notion of modal consequence, which have no parallel in classical logic.