nLab
the logic K(m)

Idea

The flavor of modal logic called $K$ is propositional logic equipped with a single modality usually written ”$\square$” subject to the rules that for all propositions $p,q:\mathrm{Prop}$ we have

• $\square K:\square (p\to q)\to (\square p\to \square q)$ (K modal logic)

Often one adds to this the following further axioms

• $\square T:\square p\to p$ (T modal logic)

• $\square 4:\square p\to \square \square p$. (S4 modal logic).

$K$ is the basic epistemic logic.

Properties

Axiomatisation

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

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

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

Derivation rules

• (MP)

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

(i.e. modus ponens);

• (Generalisation)

$$\frac{\varphi }{K_i\varphi }.$$\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, $𝔉$ is just a set, $W$ of possible worlds with $m$ relations $R_i$. A model, $𝔐=(𝔉,V)$, is a frame in that sense together with a valuation, $V:\mathrm{Prop}\to 𝒫(W)$, 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

• $𝔐,w\vDash K_i\varphi$ if and only if, for all $v\in W$ such that $R_{i}wv$, $𝔐,v\vDash \varphi$.