Temporal, Deontic, and Epistemic Logics

The relational machinery of modal logic generalizes to time, obligation, and knowledge, yielding logics of what was and will be, what ought to be, and what is known.

Definition

These are non-alethic modal logics in which the accessibility relation is reinterpreted: as temporal ordering for tense logic, as access to deontically ideal worlds for deontic logic, and as compatibility with what an agent knows for epistemic logic.

Scope

This topic covers the principal intensional logics that share modal logic's possible-worlds apparatus: temporal (tense) logic with past and future operators evaluated over an ordering of times; deontic logic with operators for obligation, permission, and prohibition; and epistemic logic with operators for knowledge and belief evaluated over epistemic alternatives. It includes their characteristic puzzles, such as the paradoxes of deontic logic and the problem of logical omniscience.

Core questions

• How should past- and future-tense operators be axiomatized over different structures of time?
• Can obligation and permission be treated as modal operators, and how are the deontic paradoxes resolved?
• Does treating knowledge modally force the implausible thesis of logical omniscience?
• What unifies these logics, and where do their analogies break down?

Key concepts

• tense operators (past/future)
• obligation, permission, prohibition
• deontically ideal worlds
• epistemic alternatives
• logical omniscience
• deontic paradoxes

Key theories

Standard deontic logic

von Wright models obligation as truth in all deontically ideal accessible worlds and permission as truth in some, giving an axiomatic system that captures basic inferences but generates well-known paradoxes.

Epistemic logic and possible worlds

Hintikka analyzes 'a knows that p' as p holding in all worlds compatible with a's information, founding the modal logic of knowledge and belief and exposing the logical-omniscience problem.

History

von Wright launched deontic logic in 1951 and Prior developed tense logic through the 1950s and 1960s, culminating in Past, Present and Future (1967). Hintikka's Knowledge and Belief (1962) extended the modal framework to the epistemic operators, and all three logics later found extensive application in computer science and game theory.

Debates

The problem of logical omniscience

Whether the possible-worlds analysis of knowledge wrongly entails that agents know all logical consequences of what they know, and how to model resource-bounded or fragmented knowledge without abandoning the modal framework.

Key figures

• Arthur Prior
• Georg Henrik von Wright
• Jaakko Hintikka
• Robert Stalnaker

Seminal works

• vonwright1951
• prior1967
• hintikka1962

Frequently asked questions

Are these really kinds of modal logic?

Yes. Temporal, deontic, and epistemic logics all use the same relational-frame semantics as alethic modal logic, differing only in how the accessibility relation is interpreted — as time order, deontic ideality, or epistemic compatibility. This shared structure is why results and techniques transfer readily between them.