The semantics of modal logics based on Kripke frames (“geometric” in contrast to algebraic models for modal logics), often called Kripke semantics [Kripke (1959), (1963)].
In the following we say frame for Kripke frame.
Again we look at the basic model language.
A model for $\mathcal{L}_\omega(1)$ is a pair $\mathfrak{M} = (\mathfrak{F},V)$, where $\mathfrak{F}$ is a frame (for $\mathcal{L}_\omega(1)$), that is, a non-empty set, $W$, equipped with a binary relation, $R$, and $V: Prop \to \mathcal{P}(W)$, is a function, called a valuation, assigning to each atomic proposition $p$ a subset of the set, $W$, of worlds.
Informally we think of $V(p)$ as the set of ‘worlds’ in our model where $p$ is true.
The binary relation of the frame allows for the modelling of the truth of a modal proposition, $\Diamond \phi$ or $\Box \phi$, at some world in terms of the truth of $\phi$ at related worlds.
Suppose that $w$ is a state in a model $\mathfrak{M} = (W,R,V)$. We inductively define the notion of a formula $\phi$ being satisfied in the model at state $w$, as follows:
for $p\in Prop$, $\mathfrak{M},w \models p$ if and only if $w\in V(p)$;
$\mathfrak{M},w \models \bot$ never;
$\mathfrak{M},w \models \neg \phi$ if and only if it is not true that $\mathfrak{M},w \models \phi$;
$\mathfrak{M},w \models \phi \vee \psi$ if and only if $\mathfrak{M},w \models\phi$ or $\mathfrak{M},w \models\psi$;
$\mathfrak{M},w \models \Diamond \phi$ if and only if, for some $v \in W$ such that $R w v$, $\mathfrak{M},v \models \phi$.
The terminology used in talking about ‘satisfaction’ tends to interpret $\mathfrak{M},v \models \phi$ as saying the formula $\phi$ is true in $\mathfrak{M}$ at state $v$.
$\mathfrak{M},w \models \square \phi$ if and only if $\forall v\in W$ $R w n$ implies $\mathfrak{M},v \models \phi$. (The proof is fairly routine manipulation of negations.)
We say a set $\Sigma$ of formulae is true at state $w\in W$ of a model $\mathfrak{M}$, written $\mathfrak{M},w \models \Sigma$, if all members of $\Sigma$ are satisfied as $w$.
It is convenient to extend the valuation $V$ to arbitrary formulae by setting $V(\phi) := \{w \mid \mathfrak{M},w \models \phi\}$. We then get useful interpretations such as: if for all worlds $v$, $\mathfrak{M},v \models (\phi\to \psi)$ and modus ponens holds, (i.e. we have a logic as such in which we are working, then $V(\phi)\subseteq V(\psi)$. (This comment is deliberately slightly vague, but indicates a common type of argument that will be behind many results, where more precision is available, so think of it as a template for a result rather than a result as such.)
In the definition of a Kripke model the valuation is all important. It is what puts meaning onto the frame. It is very useful to push this idea around through a couple of adjunctions and reinterpretations. The process is fairly intuitive, but it pays to do things reasonably formally:
Note that the power set $\mathcal{P}(W)$ can also be written as $\mathbf{2}^W$ the set of functions from $W$ to $\mathbf{2}$, where we write $\mathbf{2}$ for the ‘set’ of classical truth values, $\{\bot \to \top\}$, or $\{true,false\}$, or $\{0,1\}$ or … . We wrote ‘set’ in inverted commas for several reasons:
Firstly an important role here is played by the multiple structures that $\mathbf{2}$ has. It is a Heyting algebra; it has a natural poset structure; it is the subobject classifier in the topos of sets, and so on. In fact it is the source of most of the logic semantic structure within this context. Its structure induces similar structures on the powerset, $\mathcal{P}(W)$, given by union etc, that made the semantics work above. (See also the discussion on ambimorphic objects in the entry on the Chu construction.)
Next note that although we said ‘set’ we could do a lot of this in other settings. For instance we could work within a more general topos with an object of possible worlds and an object of propositions. We would need the extra Heyting algebra structure on what would there probably be written as $\Omega$, and our negation interpretation would be more subtle.
Finally we could categorify things. For the moment we will leave this aside, but note the discussion at truth value.
For convenience we will write $P$ for $Prop$, then a valuation is $V: P\to \mathbf{2}^W$, and using cocurrying this corresponds to $V: P\times W\to \mathbf{2}$. That, of course, corresponds to a subset of $P\times W$. That subset consists of all pairs, $(p,w)$, for which $w\in V(p)$ so interprets as the set of pairs in which the proposition $p$ is ‘true’ in world $w$.
We could recurry after transposing $P$ and $W$, to get $V$ to correspond to $\tilde{V}: W \to \mathcal{P}(P)$, so that $\tilde{V}(w)$ is the set of propositions true about the world $w$.
Another useful direction is to see this as giving a binary Chu space. (To be investigated later.)
Again we look at the basic model language this time with $n$ unary modal operators.
A model for $\mathcal{L}_\omega(n)$ is a pair $\mathfrak{M} = (\mathfrak{F},V)$, where $\mathfrak{F}$ is a frame (for $\mathcal{L}_\omega(n)$) and $V: Prop \to \mathcal{P}(W)$, is a function, called a valuation, assigning to each atomic proposition $p$ a subset of the set, $W$, of worlds.
This is virtually the same as in the monomodal case, except for the last case which involves the $n$ modal operators. (We give it in full repeating the earlier cases for convenience.)
Suppose that $w$ is a state in a model $\mathfrak{M} = (W,R,V)$. We inductively define the notion of a formula $\phi$ being satisfied in the model at state $w$, as follows:
for $p\in Prop$, $\mathfrak{M},w \models p$ if and only if $w\in V(p)$;
$\mathfrak{M},w \models \bot$ never;
$\mathfrak{M},w \models \neg \phi$ if and only if it is not true that $\mathfrak{M},w \models \phi$;
$\mathfrak{M},w \models \phi \vee \psi$ if and only if $\mathfrak{M},w \models\phi$ or $\mathfrak{M},w \models\psi$;
for each $i = 1, \ldots, n$, $\mathfrak{M},w \models \Diamond_i \phi$ if and only if, for some $v \in W$ such that $R_i w v$, $\mathfrak{M},v \models \phi$.
Satisfaction on a class of models is related to validity of modal formulae. In this way a class of frames can determine a logic and then the logic determines a class of frames. The axiomatisation of that class/logic is then an interesting challenge.
The concept originates with:
Saul A. Kripke, A Completeness Theorem in Modal Logic, The Journal of Symbolic Logic 24 1 (1959) 1-14 [doi:10.2307/2964568, jstor:2964568, pdf]
Saul A. Kripke, Semantical Analysis of Modal Logic I. Normal Modal Propositional Calculi, Mathematical Logic Quaterly 9 5-6 (1963) 67-96 [doi:10.1002/malq.19630090502]
Saul A. Kripke, Semantical Considerations on Modal Logic, Acta Philosophical Fennica 16 (1963) 83-94 [pdf]
Saul A. Kripke, Semantical Analysis of Modal Logic II. Non-Normal Modal Propositional Calculi, in The Theory of Models (Proceedings of the 1963 International Symposium at Berkeley) Studies in Logic and the Foundations of Mathematics (1965) 206-220 [doi:10.1016/B978-0-7204-2233-7.50026-5]
Textbook accounts:
Patrick Blackburn, Maarten de Rijke, Yde Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science 53, Cambridge University Press (2001) [doi:10.1017/CBO9781107050884]
Valentin Goranko, Martin Otto, §1 in: Model Theory of Modal Logic, in Section 5 in: Handbook of Modal Logic, Studies in Logic and Practical Reasoning 3 (2007) 249-329 [pdf, book webpage, publisher page]
Generalization beyond Kripke frames:
Last revised on July 27, 2023 at 19:31:17. See the history of this page for a list of all contributions to it.