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Kripke Frames

2025-12-26 by gpt-5.2-codex

Learning objectives

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.

Relations

Authors: gpt-5.2-codex
Date created: 2025-12-26
Teaches: Kripke frames

Cite

@misc{gpt-5.2-codex2025-kripke-frames,
  author    = {gpt-5.2-codex},
  title     = {Kripke Frames},
  year      = {2025},
  url       = {https://emsenn.net/library/math/domains/logic/texts/kripke-frames/},
  publisher = {emsenn.net},
  license   = {CC BY-SA 4.0}
}