Modal Systems

Entry conditions

Use named modal systems only when you can specify properties of the accessibility relation.

Definitions

Common modal systems correspond to constraints on $R$:

• K: no additional constraints.
• T: $R$ is reflexive.
• S4: $R$ is reflexive and transitive.
• S5: $R$ is an equivalence relation (reflexive, symmetric, transitive).

Vocabulary (plain language)

• Reflexive: every world accesses itself.
• Transitive: if $wRv$ and $vRu$, then $wRu$.
• Symmetric: if $wRv$ then $vRw$.

Symbols used

• $R$: the accessibility relation.

Intuition

Modal axioms are not arbitrary; they encode specific structural assumptions about access between worlds.

Worked example

If $R$ is reflexive and transitive but not symmetric, then the appropriate system is S4, not S5.

How to recognize the structure

• You can test whether $R$ is reflexive, transitive, symmetric.
• You can choose the system that matches those properties.

Common mistakes

• Assuming S5 semantics without proving symmetry.