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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?

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

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

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

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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.

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