First-Order Modal Logic

Origin. Carnap, Kripke (1960s). Quantifiers meet modalities. Varying domains. Rigid vs non-rigid designators. Foundation for quantified intensional logic.

Models. Possible worlds with domains. Constant vs varying domains. Rigid designators: same object across worlds. Actualist vs possibilist quantification.

Formalism.

Syntax: φ ::= P(t₁,...,tₙ) | ¬φ | φ ∧ ψ | □φ | ∀x.φ

Kripke semantics: M = (W, R, D, d, I) W: worlds, R: accessibility D: total domain or D: W → domains d: world-relative domains I: interpretation at worlds

Constant domains: Same objects at all worlds. Barcan formula: ∀x□φ → □∀x.φ Converse Barcan: □∀x.φ → ∀x□φ

Varying domains: d(w): objects existing at w. Barcan fails. Actualist: quantify over d(w). Possibilist: quantify over D.

Rigid designators: I(c, w) = I(c, v) for all w, v. Names refer to same object. Kripke's argument.

De re vs de dicto: □∃x.Fx: de dicto (necessarily, something) ∃x.□Fx: de re (something necessarily) Scope matters.

Problems: Transworld identity. Contingent existence. Substitutivity failures.

Actualist quantification: ∀x.φ true at w iff φ[a/x] true for all a ∈ d(w). Existential import.

Symbols.

Symbol	Unicode	Name	Meaning
□	U+25A1	Necessity	Modal box
∀x	U+2200	Universal	Quantifier
D(w)	—	Domain	World-relative

Metatheory. Various systems: constant/varying, rigid/non-rigid. Barcan formula distinguishes. Completeness for many systems. Undecidable generally.

Applies to. Metaphysics. Semantics. Ontology. Necessary existence. Intensional contexts.

Limitations. Complexity. Many competing systems. Transworld identity. Philosophical disputes.