nLab K modal logic

Idea

The flavor of modal logic called $K$ 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)

Often one adds to this the following further axioms

• $\Box T \colon \Box p \to p$ (T modal logic)

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

$K$ is the basic epistemic logic.

Properties

Axiomatisation

• (Taut) All (instances of ) propositional tautologies.

• For each $i = 1,\ldots, m$, the axiom, ($K_i$):

$$(K_i\phi \wedge K_i(\phi \to \psi))\to K_i\psi.$$

Derivation rules

• (MP)

$$\frac{\phi \quad \phi\to \psi}{\psi} \quad$$

(i.e. modus ponens);

• (Generalisation)

$$\frac{\phi}{K_i\phi}.$$

The second deduction rule corresponds to the idea that if a statement has been proved, then it is known to all ‘agents’.

Normality

This logic is the smallest normal modal logic.

Semantics

The semantics of $K_{(m)}$ is just the Kripke semantics of this context, so a frame, $\mathfrak{F}$ is just a set, $W$ of possible worlds with $m$ relations $R_i$. A model, $\mathfrak{M} = (\mathfrak{F},V)$, is a frame in that sense together with a valuation, $V \colon Prop \to \mathcal{P}(W)$, and the satisfaction relation is as described in geometric models for modal logics with just the difference implied by the fact that that page correspond to the use of $\Diamond_i = M_i$ whilst this uses $K_i$. This means that

• $\mathfrak{M},w \models K_i \phi$ if and only if, for all $v \in W$ such that $R_i w v$, $\mathfrak{M},v \models \phi$.

Related concepts

type of modal logic	relation in its Kripke frames
K modal logic	any relation
K4 modal logic?	transitive relation
T modal logic	reflexive relation
B modal logic?	symmetric relation
S4 modal logic	reflexive & transitive relation
S5 modal logic	equivalence relation

(following BdRV (2001) Table 4.1; cf. Fagin, Halpern, Moses & Vardi (1995) Thm. 3.1.5)

• modal logic, epistemic modal logic

• necessity and possibility

• T modal logic

• S4 modal logic

• S5 modal logic

• axiom K (modal logic)

References

The notation “K” for a modal operator meant to express epistemic modality originates with:

• K. Jaakko J. Hintikka, Knowledge and belief: An introduction to the logic of the two notions, Cornell University Press (1962) [ark:/13960/t9k437s65]