Kripke Frames

Entry conditions

Use modal logic only when you can:

• Specify a set of worlds or states.
• Specify an accessibility relation between worlds.

Definitions

A Kripke frame is a pair $(W,R)$ where $W$ is a set of worlds and $R\subseteq W\times W$ is an accessibility relation.

Vocabulary (plain language)

• World: a state or context where propositions can be evaluated.
• Accessibility: which worlds are considered reachable from another.

Symbols used

• $W$: the set of worlds.
• $R$: the accessibility relation.

Intuition

Modal logic studies truth across related worlds. The relation $R$ encodes which worlds are relevant to each other.

Worked example

Let $W=\{w_0,w_1\}$ and $R=\{(w_0,w_1),(w_1,w_1)\}$. From $w_0$ you can access $w_1$, and $w_1$ can access itself.

How to recognize the structure

• You can list the worlds.
• You can list the ordered pairs that make up $R$.

Common mistakes

• Using modal operators without defining accessibility.