Modal Semantics

Entry conditions

Use modal semantics only when you have a Kripke frame and a valuation for atomic propositions.

Definitions

In a Kripke model $(W,R,V)$, the modal operators are defined by:

• $\Box \varphi$ is true at $w$ if $\varphi$ is true at every world $v$ with $wRv$.
• $\Diamond \varphi$ is true at $w$ if $\varphi$ is true at some world $v$ with $wRv$.

Vocabulary (plain language)

• Necessity: true in all accessible worlds.
• Possibility: true in at least one accessible world.

Symbols used

• $\Box$: necessity
• $\Diamond$: possibility

Intuition

Modal operators talk about what is true across related worlds, not just within one world.

Worked example

If $w_0$ accesses only $w_1$, and $p$ is true at $w_1$, then $\Box p$ is true at $w_0$.

How to recognize the structure

• You have a valuation $V$ for atomic propositions.
• You can evaluate formulas at each world using $R$.

Common mistakes

• Using $\Box$ or $\Diamond$ without specifying $R$.