Formal · III

Modal logic

logica modalis · ‘the logic of modes of being’

Extends classical logic with two operators: □ for necessity and ◇ for possibility. Kripke semantics interprets them over possible worlds; the axioms below select which modal facts hold by constraining the structure of those worlds.

Operators

Necessity and possibility

Modal logic adds two unary operators to the propositional (or predicate) language: □P for ‘necessarily P’, and ◇P for ‘possibly P’. The two are interdefinable: ◇P ≡ ¬□¬P. Saying that P is possible is the same as saying that it is not necessary that ¬P.

Beyond this duality, the operators are silent about which modal facts hold. To decide whether □P → P — does necessity entail truth? — we must adopt axioms. Different choices of axioms give different modal systems, each appropriate to a different reading of □.

Axioms

The six standard axioms

Each axiom below corresponds to a structural condition on the accessibility relation between possible worlds (next section). Modal systems are built by selecting which to include.

K
□(P → Q) → (□P → □Q)
Necessity distributes over implication. Present in every normal modal logic.
T
□P → P
What is necessary is the case. The principle of reflexivity — necessity implies truth.
4
□P → □□P
What is necessary is necessarily necessary. The transitivity of necessity.
5
◇P → □◇P
What is possible is necessarily possible. The Euclidean condition on accessibility.
B
P → □◇P
If something is the case, it is necessarily possible. The symmetry of accessibility.
D
□P → ◇P
What is necessary is at least possible. The seriality of accessibility — appropriate for deontic readings.

Systems

K · T · S4 · S5 · D

A modal system is fixed by selecting which axioms above to include alongside classical propositional logic and the rule of necessitation (if ⊢ P, then ⊢ □P).

KK
The minimal normal modal logic. Adds the K axiom and the rule of necessitation (⊢ P ⊢ □P) to classical propositional logic.
TK + T
Adds reflexivity. Appropriate for any reading on which necessity entails truth — alethic and epistemic logics, in particular.
S4K + T + 4
Adds transitivity. The system of provability in many constructive settings; an attractive choice for epistemic logics with positive introspection.
S5K + T + 4 + 5
Accessibility is an equivalence relation. Necessity is the same in every world — appropriate for metaphysical necessity on a possible-worlds reading.
DK + D
Deontic logic. □ is read as 'it is obligatory that' and the D axiom (□P → ◇P) forbids the simultaneous obligation of P and ¬P.

Semantics

Kripke models

A Kripke model is a triple <W, R, V> where W is a non-empty set of possible worlds; R is an accessibility relation on W; and V assigns truth-values to atoms at each world.

The modal operators are then read as follows: □P is true at world w iff P is true at every world accessible from w; ◇P is true at w iff P is true at some world accessible from w. The accessibility relation captures which other worlds arerelevant from the standpoint of w.

The axioms above correspond to structural conditions on R:

Axiom	Condition on R
T	reflexive: w R w
4	transitive: w R v and v R u ⇒ w R u
5	Euclidean: w R v and w R u ⇒ v R u
B	symmetric: w R v ⇒ v R w
D	serial: every w sees some v

Readings

What □ can mean

One formal apparatus, many interpretations. Each row below assigns a different gloss to □ and ◇, with the appropriate choice of axioms in tow.

Alethic: □PIt is necessarily the case that P.◇PIt is possible that P.
Epistemic: □PThe agent knows that P.◇PThe agent does not rule out that P.
Deontic: □PIt is obligatory that P.◇PIt is permissible that P.
Temporal: □PIt will always be the case that P.◇PIt will at some time be the case that P.
Doxastic: □PThe agent believes that P.◇PThe agent does not disbelieve that P.

Lineage

From Lewis to Kripke

C. I. Lewis revived modal logic in the early twentieth century with A Survey of Symbolic Logic (1918) and Symbolic Logic (1932, with C. H. Langford), partly in dissatisfaction with the material conditional of classical logic. Lewis introduced the S-systems (S1 through S5) on syntactic grounds; their semantics was a long-standing puzzle.

Saul Kripke supplied that semantics in ‘A Completeness Theorem in Modal Logic’ (1959) and a series of follow-ups. Possible-worlds semantics quickly became the standard tool not only for alethic modality but for epistemic, temporal, deontic, and doxastic logics — each an application of the same underlying machinery. The expressive convenience of possible-worlds talk, in turn, became a fixture of analytic philosophy from David Lewis onward.

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