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Mathematical logic and foundations Definition

Modal logic

Builds on Propositional logicTautologyModus ponens

Definition

The modal formulas are built from propositional variables p,q,…p, q, \dots and the constant ⊥\bot with the connectives ¬\neg, ∧\land, ∨\lor, →\to and the unary operator □\square. The formula ◊φ\lozenge\varphi abbreviates ¬□¬φ\neg\square\neg\varphi. A normal modal logic is a set Λ\Lambda of modal formulas with these properties.

  1. Λ\Lambda contains every substitution instance of a tautology of propositional logic.
  2. Λ\Lambda contains every formula □(φ→ψ)→(□φ→□ψ)\square(\varphi \to \psi) \to (\square\varphi \to \square\psi), the axiom K\mathrm{K}.
  3. If φ∈Λ\varphi \in \Lambda and φ→ψ∈Λ\varphi \to \psi \in \Lambda, then ψ∈Λ\psi \in \Lambda, the rule of modus ponens.
  4. If φ∈Λ\varphi \in \Lambda, then □φ∈Λ\square\varphi \in \Lambda, the rule of necessitation.
  5. If φ∈Λ\varphi \in \Lambda, then each formula obtained from φ\varphi by substituting formulas for its variables is in Λ\Lambda.

The smallest normal modal logic is called K\mathrm{K}.

Remarks

Example: the logic S4\mathrm{S4} is the smallest normal modal logic containing □p→p\square p \to p and □p→□□p\square p \to \square\square p. It satisfies the five clauses by construction, as the intersection of all normal modal logics that contain these two formulas.

Non-example: the set of all tautologies of propositional logic, read as modal formulas, is not a normal modal logic. It fails clause 2, since □(p→q)→(□p→□q)\square(p \to q) \to (\square p \to \square q) is not an instance of a tautology.

Other standard normal modal logics are T\mathrm{T}, given by □p→p\square p \to p, and S5\mathrm{S5}, given by □p→p\square p \to p and ◊p→□◊p\lozenge p \to \square\lozenge p. The word "modal logic" also names the whole field that studies such systems, including non-normal ones.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Proposition, concept, no evidence levelPropositionconceptTautology, concept, no evidence levelTautologyconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionModus ponens, concept, no evidence levelModus ponensconceptPropositional logic, concept, no evidence levelPropositional logicconceptModal logic, definition, no evidence levelModal logicdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proposition concept
  2. Tautology concept
  3. Conjunction definition
  4. Disjunction definition
  5. Implication definition
  6. Negation definition
  7. Biconditional definition
  8. Modus ponens concept
  9. Propositional logic concept
  10. Modal logic definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Modal_logic
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Changed2 November 2026

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