nLab S4 modal logic

Idea

The flavor of (classical) modal logic called $S4$ is (classical) propositional logic equipped with a single modality usually written “$\Box$” subject to the rules that for all propositions $p, q \colon Prop$ we have

1. $\Box K \colon \Box(p \to q) \to (\Box p \to \Box q)$ (K modal logic)

2. $\Box T \colon \Box p \to p$ (T modal logic)

3. $\Box 4 \colon \Box p \to \Box \Box p$. (S4 modal logic)

4. (in addition in S5 modal logic one adds: $\lozenge p \to \Box \lozenge p$).

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 $\Box 4$ 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 denoted $S4(n)$. Here $\Box_i p$ is sometimes interpreted as “the $i$th agent knows $p$.”

In intuitionistic (or constructive) Modal Logic, one does not expect the possibility modality to be dual to the necessity modality, necessarily. Alex Simpson has written in his thesis the most used constructive modal logic system, but other systems exist.

Properties

Relation to Kripke frames

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.

Theorem (Soundness of $S4_{(m)}$)

$S4_{(m)}\vdash \phi \Rightarrow \mathcal{S}4(m)\models \phi.$

Proof

(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 $\mathfrak{M},u\models p$.

Related concepts

• modal logic, S5 modal logic

• epistemic modal logic, necessity and possibility

• K modal logic

• T modal logic

• Kripke frame

• Löb's axiom

References

The terminology “S4” in modal logic originates with:

• Clarence I. Lewis, Cooper H. Langford, App. II, p 501 of: Symbolic Logic (1932), Dover (2000) [App. II pdf]

The “possible worlds” Kripke semantics of S4 originates with:

• 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]

See also:

• Saul Kripke, Naming and Necessity (1980)

A natural deduction-formulation and making explicit the modal operator as a comonad:

• Gavin M. Bierman, Valeria de Paiva, Intuitionistic necessity revisited, School of Computer Science research reports-University of Birmingham CSR (1996) [researchgate, pdf]

• Gavin M. Bierman, Valeria de Paiva, On an Intuitionistic Modal Logic Studia Logica 65 (2000) 383–416 [doi:10.1023/A:1005291931660, pdf]

• Satoshi Kobayashi, Monad as modality, Theoretical Computer Science 175 1 (1997) 29-74 [doi:10.1016/S0304-3975(96)00169-7]

• Natasha Alechina, Michael Mendler, Valeria de Paiva, Eike Ritter: Categorical and Kripke Semantics for Constructive S4 Modal Logic, in Computer Science Logic. CSL 2001 Lecture Notes in Computer Science 2142, Springer (2001) 292 [doi:10.1007/3-540-44802-0_21]

See also:

• Alex Simpson, The proof theory and semantics of intuitionistic modal logic (1994) [pdf, handle:1842/407]

More on $S4$, $S5_{(m)}$ and their applications in Artificial Intelligence can be found in

• J.- J. Ch. Meyer and W. Van der Hoek, Epistemic logic for AI and Computer Science, Cambridge Tracts in Theoretical Computer Science, vol. 41, 1995.

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.