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Modal logic

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In mathematical logic, modal logic is a type of formal logic that extends propositional logic, first-order logic, or higher-order logic with modalities. A specific modality of the qualities of truth. For example, a proposition like “it’s raining” can be preceded by a modality ː

  • It is necessary that it rains;
  • Tomorrow it rains;
  • Christopher Columbus thinks it’s raining;
  • It is shown that it is raining;
  • It is mandatory that it rains.

There are a variety of modal logics such as temporal logics, epistemic logic (logic of knowledge). In computer science, modal logic is used for its expressiveness and algorithmic aspects. For example, time logic is used for defined programs and then verify them.

Alethic modal logic

(The modal square: relations between the modalities of Aristotelian logic.)

In alethic modal (or Aristotelian, or classical) logic, we identify four modalities:

  • necessary (which cannot be true), noted noted ◻;
  • contingent (which may be false), denoted ¬ ◻;
  • possible (which may be true), denoted by ◊;
  • impossible (which cannot not be wrong), denoted ¬ ◊.

These 4 modalities are linked, just one is enough to define the other three.

The intuitive interpretation (not shared by the whole of the philosophical-logic community) is as follows:

  • Necessary ≡ not impossible;
  • Contingent ≡ not necessary ≡ not possible;
  • Possible ≡ not impossible.
  • Impossible = not possible.

We therefore distinguish two unary dual connectors from each other:

  • The necessary ◻;
  • The possible ◊.

◻p means that p is necessarily true, while ◊p means that p is possibly true, that is to say compatible with current knowledge.


  • ¬◻ work: it is not necessary for the students to work;
  • ¬◊ work: it is not possible for students to work;
  • ◻¬ work: it is necessary that the students do not work;
  • ◊¬ work: students may not work.

In alethic modal (or Aristotelian, or classical) logic, we can express the four operators using only one (here necessity) and the negation. So :

  • Impossible is ◻¬;
  • Possible is ¬◻¬.

A necessary proposition cannot be false without implying a contradiction, a contrario of a contingent proposition which can be false without implying a contradiction.

Different modal logics

Other types of modal logic are also used, the modes of which are:

  • epistemic (relating to knowledge):
    • known by agent i, denoted by Ci
    • questionable
    • excluded
    • plausible
    • common knowledge of group G of agents, denoted CKG
    • shared knowledge of group G of agents, denoted EKG (everyone knows)
  • deontics (moral):
    • mandatory, noted O
    • prohibited, noted I
    • permit, noted P
    • optional, denoted F
  • temporal:
    • always noted ⟨ or G
    • one day, noted ◊, or sometimes F
    • never, noted ¬◊
    • tomorrow, noted X
    • until, binary operator denoted by U
    • always in the past, noted H
    • a day passed, noted P
  • doxastic (on beliefs):
    • raw, noted B
    • common belief of the group G of agents, denoted CBG
  • counterfactuals:
    • if A were true, where we know that A is not true.
  • dynamics (effect of actions, noted a, on propositions):
    • there exists an execution of a such that after a, p is true, denoted by ⟨a⟩p
    • p is true after any execution of a, noted [a]p.

Axioms of modal logic

Each modal logic is provided with a series of axioms which define the functioning of the modalities.

We can thus construct different systems according to the admitted axioms.

  • The K system designed by Kripke and called the normal or Kripke system. He admits the following two axioms:
    • (K) ◻(A → B) → (◻A → ◻B) (Kripke’s distribution axiom);
    • (RN) (or (N) or (NEC)) If A is a theorem, then ◻A too (necessity inference rule).
  • The D system, designed by adding the axiom (D) to the K system:
    • (D) ◻P → ◊P {(in Aristotelian logic, this expresses that necessity implies possibility).
  • The T system designed by Robert Feys in 1937 by adding the axiom (T) to the K system:
    • (T) (or (M)): P → ◊P (in Aristotelian logic, this expresses that the fact implies the possibility).
  • The S4 and S5 systems defined by Clarence Irving Lewis.
    • To construct S4, we add to the system T the axiom (4):
      • (4) ◻p → ◻◻p.
    • To construct S5, we add to the system T the axiom (5):
      • (5) (or (E)): ◊p → ◻◊p.
  • The B (or Brouwérien) system, designed by Oskar Becker in 1930, by adding the axiom (B) to the T system.
    • (B): p → ◻◊p.

We say that one system is weaker than another when everything that is demonstrated in the first system is demonstrated in the second, but not vice versa.

This prioritizes, from weakest to strongest, the systems K, T, S4 and S5. Likewise, K is weaker than D and T is weaker than B.

The series of systems K to S5 form a nested hierarchy which makes up the core of normal modal logic. Axiom (D), on the other hand, is mainly used in deontic, doxastic and epistemic logics.

Modal logic models

Kripke’s models, or models of possible worlds, give semantics to modal logics. A Kripke model is the data:

  • of a non-empty set of possible worlds W;
  • of a binary relation R between the possible worlds called relation of accessibility;
  • of a valuation V which gives a truth value to each propositional variable in each possible world.

The semantics of a modal operator is defined from an accessibility relation as follows: the formula ◻A is true in a world w if, and only if the formula A is true in all the worlds accessible from w by the relation R.

Classification of modal logic systems

Modal logic systems are organized according to the rules of inference and the axioms that characterize them.

Classic modal logics

Classical modal logic systems are those which accept the following inference rule:

A ↔ B
(RE) ———–
◻A ↔ ◻B

It is customary to give such a system a canonical name of the type Eξ1ξ2∙∙∙ξn, where the ξi are the names of the axioms of the system.

Monotonic modal logics

Monotonic modal logic systems are those that accept the RM inference rule:

A → B
(RM) ———–
◻A → ◻B

The set of monotonic systems is included in the set of conventional systems.

Regular modal logics

Regular modal logic systems are those that accept the RR inference rule:

(A ∧ B) → C
(RR) ——————–
(◻A ∧ ◻B) → ◻C

The set of regular systems is included in the set of monotonic systems.

Normal modal logic

Normal modal logic systems are those that accept the RK inference rule:
(A1 ∧ ⋯An) → B
(RK) ———————–
(◻A1 ∧ ⋯◻An) → ◻B

The set of normal systems is included in the set of regular systems.

An equivalent and more common definition of normal systems is as follows: a modal logic system is said to be normal if it has the axiom (K) and accepts the rule of necessity (RN) as the rule of inference:

(K)◻(A → B) → (◻A → ◻B)

(RN) —-

The normal systems are the most used, because they are those which correspond to the semantics of Kripke. It is however possible to find semantics for non-normal classical logics, but they generally have poorer properties.

Link with other logics

Intuitionist logic can be built on alethic logic as a modal logic. Modal logic is a fragment of first-order logic.

Includes texts translated and adapted from Wikipedia

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