Overview
Modal Logic extends standard logic to handle modes of truth: not just what is true, but what must be true (necessity) or what could be true (possibility). It uses the operators $\Box$ (Box/Necessity) and $\diamond$ (Diamond/Possibility).
Core Idea
The core idea is to reason about alternatives to the actual world. A statement is “necessary” if it is true in all possible worlds, and “possible” if it is true in at least one possible world.
Formal Definition
Modal logic adds unary operators $\Box$ and $\diamond$ to the syntax. Semantically, it is often defined using Kripke structures: a set of possible worlds $W$, an accessibility relation $R$ between them, and a valuation function.
- $\Box P$: True in world $w$ if $P$ is true in all worlds $w’$ accessible from $w$.
- $\diamond P$: True in world $w$ if $P$ is true in at least one world $w’$ accessible from $w$.
Intuition
Standard logic deals with facts: “It is raining.” Modal logic deals with the status of facts: “It might rain” or “It must rain.” This allows us to model knowledge, time, and obligation.
Examples
- Alethic Modality:
- $\Box P$: It is necessarily true that $P$ (e.g., $2+2=4$).
- $\diamond P$: It is possible that $P$ (e.g., It is raining).
- Epistemic Modality:
- $\Box P$: Agent A knows $P$.
- $\diamond P$: For all Agent A knows, $P$ is possible.
- Deontic Modality:
- $\Box P$: It is obligatory that $P$.
- $\diamond P$: It is permissible that $P$.
Common Misconceptions
- Misconception: “Possible” means “Probable.”
- Correction: In logic, “possible” just means “not contradictory.” A flying pig is logically possible (no contradiction), even if highly improbable.
- Misconception: There is only one modal logic.
- Correction: There are many systems (K, T, S4, S5) depending on the properties of the accessibility relation (reflexive, transitive, etc.).
Related Concepts
- Possible Worlds: A complete way things could have been.
- Deontic Logic: The logic of obligation and permission.
- Temporal Logic: The logic of time (using “always” and “sometime” as modalities).
Applications
- Computer Science: Used to verify software and hardware (Temporal Logic), and in AI for reasoning about knowledge (Epistemic Logic).
- Philosophy: Central to metaphysics (necessity/contingency) and ethics (obligation).
- Linguistics: Used to analyze the semantics of modal verbs (can, must, might).
Criticism and Limitations
- Ontological Baggage: Quine criticized modal logic for committing us to “suspicious” entities like possible worlds.
- Complexity: Modal logics are often computationally harder (PSPACE-complete or worse) than classical logic.
Further Reading
- Naming and Necessity by Saul Kripke
- Modal Logic by Patrick Blackburn et al.