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One thing that confuses me about modal logic is the exact meaning of the names Kripke frame, Kripke model, (un)pointed [[thing]], etc.

I'd like to know the standard names of different subtuples of $(P, W, R, V, w)$, defined below, so I can have a better understanding of textbooks and articles that talk about modal logic and so that I can explicitly distinguish the different possible consequence relations one might reasonably use in modal logic without making mistakes.

My question is threefold:

  1. What are the names for the different subtuples of $(P, W, R, V, w)$ in modal logic?
  2. Are there any variations in definitions that one might reasonably encounter in the wild and should pay special attention to?
  3. Is there a conventional way to refer to things that have a distinguished world?

As far as I know, relational models for modal logic have the following notion of truth. I choose as my primitive connectives $\neg, \to, \;\text{and}\; \lozenge$.

$$(P, W, R, V, w) \models A \;\; \textit{if and only if} \;\; \text{$V(w, A)$ is $1$ (i.e. $A$ holds at world $w$)}$$ $$ (P, W, R, V, w) \models \lnot \alpha \;\; \textit{if and only if} \;\; \text{it does not hold that $(P, W, R, V, w) \models \alpha$} $$ $$ (P, W, R, V, w) \models \alpha \to \beta \;\;\text{if and only if}\;\; \text{if $(P, W, R, V, w) \models \alpha$ then $(P, W, R, V, w) \models \beta$} $$ $$ (P, W, R, V, w) \models \lozenge \alpha \;\;\text{if and only if}\;\; \text{there exists a $u$ such that $wRu$ and $(P, W, R, V, u) \models \alpha$ } $$

I think the presentation above is standard. $P$ is, I think, seldom explicitly mentioned, but it does exist implicitly in standard presentations of the relational semantics of modal logic.

This is pretty straightforward, and looks a lot like the definition of truth in a first-order structure. I'm not using $\Vdash$ because I don't really know when to use it and I find systematically making an analogy with first-order logic helpful personally.

As far as I'm aware, the following terminology is standard:

  • $P$ is my set of primitive propositions.
  • $W$ is the set of possible worlds.
  • $R \subset W \times W$ is the accessibility relation.
  • $V$ is the valuation map. There's some (inconsequential) choice in exactly what type to give it. I like making it send pairs of worlds and primitive propositions to $\{0, 1\}$.
  • $w$ is the distinguished world. I sometimes call it the start world but I think this is nonstandard.

After this things get fuzzy for me. I've picked up a few names for the different subtuples of $(P, W, R, V, w)$.

I call $(P, W, R, V, w)$ a pointed model because it has a distinguished world.

I call $(P, W, R, V)$ an unpointed model because it does not have a distingushed world.

I call $(W, R, w)$ a pointed Kripke frame.

I call $(W, R)$ an unpointed Kripke frame.

I don't think that the pointed/unpointed distinction is standard though.

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Concerning your first question I will assume that basic modal logic (one one-place modal operator) is your main concern:

  1. Modal logicians distinguish between languages to be interpreted and the objects doing the interpretation (which I simply call 'structures'). So structures never contain the set of propositional variables as a component.

  2. The pair $(W, R)$ is typically called a frame (for basic modal logic). Triples consisting of frames and worlds are, as far as I know, not in common use and have no standard name.

  3. Maps interpreting propositional variables over frames in the way you indicate are typically called 'valuations' or 'valuation functions'. Pairs consisting of frames and valuations on them are called 'models'.

  4. Pairs consisting of models and worlds are mostly called 'pointed models' (they primarily appear in epistemic logic as far as I know).

Concerning your second question:

Terminology may vary to considerable degree on two dimensions: Firstly the application domain of modal logics and secondly the similarity type of the logic (the number of modal operators and their respective arities).

With regard to the first dimension, in computer science applications frames of basic modal logic may be addressed as 'transition systems' or simply '(directed) graphs' or, when more than on accessibility relation is involved, as 'labelled transition systems' or 'multi graphs'. Furthermore, especially when temporal logics for concurrent systems are at issue, valuations are taken to be functions from worlds to sets of propositional variables and are then called 'labelling functions'.

With regard to the second dimension, dynamic logic texts tend to call models 'frames'! The reason behind this is that in even the simplest non-trivial dynamic logic, regular propositional dynamic logic, accessibility relations cannot be defined independently from truth of formulas at points. There, these notions must be defined via mutual recursion.

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