Modal Logic
Modal logic adds operators for necessity and possibility to classical logic, providing the formal machinery behind reasoning about what must, might, ought, or will be the case.
Definition
Modal logic is the study of inference involving the operators of necessity and possibility (and their analogues), interpreted via models in which sentences are evaluated relative to possible worlds connected by an accessibility relation.
Scope
This area covers the formal systems of modal logic and their philosophical interpretation. It treats possible-worlds (Kripke) semantics, the standard hierarchy of modal systems (K, T, S4, S5) and their characteristic axioms, the extension of modal logic with quantifiers and its connection to essentialism and de re modality, and the family of related intensional logics — temporal, deontic, and epistemic — that share the same relational-frame apparatus.
Sub-topics
Core questions
- How should the necessity and possibility operators be interpreted, and what are possible worlds?
- Which axioms govern correct modal reasoning, and how do they correspond to properties of the accessibility relation?
- How does modality interact with quantification, identity, and the existence of objects across worlds?
- Can the various intensional logics — temporal, deontic, epistemic — be unified under one framework?
Key concepts
- necessity and possibility
- possible worlds
- accessibility relation
- modal systems (K, T, S4, S5)
- de re vs. de dicto modality
- rigid designation
Key theories
- Possible-worlds (Kripke) semantics
- Modal sentences are evaluated at worlds connected by an accessibility relation: 'necessarily A' is true at a world iff A is true at every accessible world, and varying the relation's properties yields different modal systems.
- Modal realism
- Lewis holds that possible worlds are concrete, mutually isolated universes as real as the actual world, providing a reductive truthmaker for modal claims at the cost of an extravagant ontology.
History
C. I. Lewis revived modal logic in the early twentieth century to capture strict implication, but it lacked a clear semantics until Kripke (and independently Hintikka and Kanger) supplied relational possible-worlds models around 1959-1963. This sparked the metaphysics of modality, with Lewis's modal realism and Plantinga's actualism as rival accounts of what possible worlds are.
Debates
- The ontology of possible worlds
- Whether possible worlds are concrete existing universes (Lewis's modal realism) or abstract entities such as maximal states of affairs or sets of propositions (actualism), and which account best grounds modal truth.
Key figures
- Saul Kripke
- C. I. Lewis
- Rudolf Carnap
- David Lewis
- Ruth Barcan Marcus
- Alvin Plantinga
Related topics
Seminal works
- kripke1963
- lewis1986plurality
- hughescresswell1996
Frequently asked questions
- What is the difference between de re and de dicto modality?
- A de dicto modal claim ascribes necessity or possibility to a whole proposition ('necessarily, all bachelors are unmarried'), whereas a de re claim ascribes a modal property to an object independently of how it is described ('this man is necessarily human'). The distinction is central to debates about essentialism.