Modal logic adds operators for "necessarily" (□) and "possibly" (◇) to regular logic, letting you reason about what must be true versus what could be true. The standard way to give these operators meaning is Kripke semantics: imagine a collection of "possible worlds" with connections between them, and "necessarily p" means p is true in every connected world. Different assumptions about those connections (whether every world connects to itself, whether connections chain transitively, etc.) give you different modal systems with names like K, T, S4, and S5. The same mathematical machinery extends beyond necessity and possibility—it's used to model knowledge ("agent A knows that p"), obligations ("it ought to be that p"), time ("it will always be that p"), and even how programs change state.
Main takeaways:
- Modal logic adds □ ("box," necessarily) and ◇ ("diamond," possibly) operators to propositional logic
- Kripke semantics interprets modal statements using "possible worlds" connected by an accessibility relation—□p holds at world w when p holds at every world accessible from w
- Properties of the accessibility relation (reflexive, transitive, symmetric, euclidean) correspond to different axiom systems (K, T, S4, S5) with different logical theorems
- The framework generalizes to model knowledge, belief, obligation, time, and computation—different interpretations of what "worlds" and "accessibility" mean
- Modal logic provides a mathematical foundation for reasoning about counterfactuals, conditionality, and state-dependent truth
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