Entry conditions
Use named modal systems only when you can specify properties of the accessibility relation.
Definitions
Common modal systems correspond to constraints on :
- K: no additional constraints.
- T: is reflexive.
- S4: is reflexive and transitive.
- S5: is an equivalence relation (reflexive, symmetric, transitive).
Vocabulary (plain language)
- Reflexive: every world accesses itself.
- Transitive: if and , then .
- Symmetric: if then .
Symbols used
- : the accessibility relation.
Intuition
Modal axioms are not arbitrary; they encode specific structural assumptions about access between worlds.
Worked example
If is reflexive and transitive but not symmetric, then the appropriate system is S4, not S5.
How to recognize the structure
- You can test whether is reflexive, transitive, symmetric.
- You can choose the system that matches those properties.
Common mistakes
- Assuming S5 semantics without proving symmetry.