(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
distributive lattice, completely distributive lattice, canonical extension
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
The flavor of modal logic called $S4$ is propositional logic equipped with a single modality usually written “$\Box$” subject to the rules that for all propositions $p, q \colon Prop$ we have
$\Box K \colon \Box(p \to q) \to (\Box p \to \Box q)$ (K modal logic)
$\Box T \colon \Box p \to p$ (T modal logic)
$\Box 4 \colon \Box p \to \Box \Box p$. (S4 modal logic)
Traditionally the canonical interpretation of the Box operator is that $\Box p$ is the statement that “$p$ is necessarily true.” (The de Morgan dual to the necessity modality is the “possibility” modality.) Then the interpretation of $\Box$ is that “If $p$ is necessarily true then it is necessarily necessarily true.” S4 modal logic appears in many temporal logics.
If instead of a single Box operator one considers $n \in \mathbb{N}$ box operators $\Box_i$ of this form the resulting modal logic is denote $S4(n)$. Here $\Box_i p$ is sometimes interpreted as “the $i$th agent knows $p$.”
The models for T modal logic corresponded to Kripke frames where each relation $R_i$ is reflexive.
For $S4$ modal logic they are furthermore transitive.
$S4_{(m)}\vdash \phi \Rightarrow \mathcal{S}4(m)\models \phi.$
(We show this for $m = 1$.)
Suppose that $\mathfrak{M}= ((W,R),V)$ where $R$ is a reflexive transitive relation on $W$. We have to check that $\mathfrak{M}\models (4)$.
Suppose $\mathfrak{M},w\models K p$, then, for every $t$ with $R w t$, we have $\mathfrak{M},t\models p$. Now suppose we seek to check that $\mathfrak{M},w\models K K p$ so we have $t$ with $R w t$ and want $\mathfrak{M},t\models K p$, so we look at all $u$ with $R t u$ and have to see if $\mathfrak{M},u\models p$, but as $R w t$ and $R t u$ hold then $R w u$ holds, since $R$ is transitive, and we then know that $\mathfrak{M},u\models p$.
Saul Kripke, Naming and Necessity (1980)
Natasha Alechina, Michael Mendler, Valeria de Paiva, and Eike Ritter, Categorical and Kripke Semantics for Constructive S4 Modal Logic (pdf)
More on $S4$, $S5_{(m)}$ and their applications in Artificial Intelligence can be found in
General books on modal logics which treat these logics thoroughly in the general context include
Patrick Blackburn, M. de Rijke and Yde Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001.
Marcus Kracht, Tools and Techniques in Modal Logic, Studies in Logic and the Foundation of Mathematics, 142, Elsevier, 1999.