modal logic


Modal logic is the logic of “necessity” and “possibility”. It is an extensionPlanetmathPlanetmath of the classical logic, with two additional unary logical connectives and . The well-formed formulas of modal logic is then the smallest set satisfying the following:

  1. 1.

    the propositional variables are well-formed formulas,

  2. 2.

    if A,B are (well-formed) formulasMathworldPlanetmathPlanetmath, then so are AB, AB, ¬A, AB,

  3. 3.

    if A is a formula, so are A and A,

  4. 4.

    and if the logic is first-order, then for any formula A, so are xA and xA.

Let L be the set of all well-formed formulas described above, and L0 the set of all formulas formed with condition 3 removed. Then L0L is just the set of all formulas of the classical logic (the underlying classical logic), and is part of the reason why we call modal logic an extension of the classical logic. The other reasons have to do with semantics and deductionsMathworldPlanetmathPlanetmath, to be discussed below.

A is usually interpreted as “it is necessarily true that A”, and A as “it is possibly true that A”.

In order to study the logical truth and valid deductions in modal logic, we need to know how to interpret and . So just what are precisely the meanings of “necessity” and “possibility”? Because of the lack of exact mathematical interpretationsMathworldPlanetmathPlanetmath of these vague terms, and and more generally, there are a great many kinds of modal logics, than the other well-known logical systems, such as classical logic and intuitionistic logicMathworldPlanetmath.

In fact, one can take and to be qualifiers attachable to any propositionsPlanetmathPlanetmathPlanetmath (or sentencesMathworldPlanetmath). So could mean “in South Africa” and “in the U.S.”. Then, if A is the sentence “July is a summer month”, then A and A are read “July is a summer month in South Africa” and “July is a summer month in the U.S.” respectively. Some of the common interpretations of A are:

  • (alethic logic) A is necessarily true

  • (provability logic) A is provable in PA

  • (deontic logic) it ought to be that A

  • (epistemic logic) it is known that A

  • (doxastic logic) it is believed that A

  • (temporal logic) A will always be true

Additionally, for most of the modal logics, is ‘defined” in terms of :

A:=¬¬A. (1)

In the next two subsections, we restrict our attention to only propositional modal logic.


As mentioned previously, a modal logic is an extension of the classical logic. As such, tautologiesMathworldPlanetmath and tautological consequences of valid formulas should be valid in any semantical model of modal logic. More precisely, call a formula in a modal logic L prime or quasi-atomic if it is

  • either a propositional variable, or

  • of the form A, where A is in L.

It is easy to see that any formula in L can be built up from quasi-atoms using just (or other connectives, depending on the choice of axiom system for classical propositional logic). In this sense, L can be thought of as a classical logic where the propositional variables range over quasi-atoms. Now, let M be any semantic model of the L, we write


to mean A is true in the model M. Then,

  • if A is an instance of a tautology (viewing L as the classical propositional logic here), then MA,

  • if A1,,An tautologically implies A, and if MAi for each i, then MA.

We call a formula in a modal logic L a tautology if it is an instance of a tautology of L viewed as the classical propositional logic over its quasi-atoms. Therefore, AA is a tautology, since it is an instance of the tautology XX. Also, if A and AB are both valid, so is B, since B may be deduced by modus ponensMathworldPlanetmath.

For the majority of the modal logics (known as normal modal logics), the principal semantic model is the Kripke semantics, or possible-world semantics. It is a generalizationPlanetmathPlanetmath of the truth-value semantics for the classical propositional logic. A Kripke model M consists of a triple (W,R,T) where W is a non-empty set of elements called possible worlds, R is a binary relationMathworldPlanetmath on W (called accessibility relation), and T is a function from the set of atomic propositions (propositional variables) to P(W), the powerset of W. Semantic truth condition of a proposition A is written


to mean “A is true in possible world α in model M”, or αA if M is clear from the context.

The relation is defined recursively as follows:

  1. 1.

    if A is atomic, then αA iff αT(A)

  2. 2.

    α for no αW

  3. 3.

    αAB iff if αA then αB

  4. 4.

    αA iff for all β where αRβ, βA

The truth conditions of ¬A,AB,AB, and A can be derived. For example, αA iff there is some β where αRβ, such that βA.

If A is true in every possible world in M, we say that A is true in M, and write MA. When A is true in every model, then we say that A is valid, and write A. The formula schema K, for example, is a set of valid formulas. Also, if A is valid, it is easy to see that so is A valid.

In additionPlanetmathPlanetmath to Kripke semantics, there are also a number of semantic tools, some are lattices with operators, some are systems of open neighborhoods in topological spacesMathworldPlanetmath, while others are just variations or generalizations of the Kripke models, such as including more than one binary relation on the possible worlds, or having a subset of the possible worlds called “impossible worlds”.

Axiom Systems

As explained in the previous subsection, tautologies and tautological consequences of valid formulas are valid, in any semantic interpretation of a modal logic. Similarly, in any axiom system for a modal logic:

  • tautologies are to be considered axioms (or deducible from axioms), and

  • modus ponens as an inference rule.

By including additional schemas of formulas as axioms, and imposing additional inference rules, one arrives at axiom systems for the different kinds of modal logics. Some of the common axioms schemasMathworldPlanetmath are listed above: K, 4, 5, D, T, B, and W. Some common inference rules (and their abbreviations) are

  • RK: (A1An)A(A1An)A, n0

  • RR: (AB)C(AB)C

  • RM: ABAB

  • RN (necessitation rule): AA

For example, a widely studied class of modal logics is the class of normal modal logics. A modal logic is normal if its axiom system consists of the axiom schema K and the rule RN.


  • Like other logical systems, issues of soundness and completeness of a model with respect to an axiom system, as well as various decidability questions are also studied in modal logic. In addition, questions regarding on the equivalence of axiom systems are also very interesting.

  • Extending modal logic, one has the notion of multi-modal logic, where the languagePlanetmathPlanetmath of the logical system contains several, sometimes infinitely many, modal connectives. From the discussion above, a modal logic where connectives and are not inter-definable (and therefore must be non-normal) is really an example of a bimodal logic.


  • 1 B. F. Chellas, Modal Logic, An Introduction, Cambridge University Press (1980)
  • 2 R. Goldblatt, Logic of Time and Computation, 2nd Edition, CSLI (1992)
  • 3 G. Priest, An Introduction to Non-Classical Logic, Cambridge University Press (2001)
Title modal logic
Canonical name ModalLogic
Date of creation 2013-03-22 19:13:54
Last modified on 2013-03-22 19:13:54
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 34
Author CWoo (3771)
Entry type Topic
Classification msc 03B45
Related topic KripkeSemanticsForModalPropositionalLogic
Defines quasi-atomic
Defines quasi-atom