T Modal Logic

Origin. Feys (1937), von Wright. Reflexive frames. Truth axiom. Veridicality. Foundation of factive modality.

Models. Reflexive accessibility. What's necessary is true. Factive readings. Knowledge base.

Formalism.

Axioms: K: □(A → B) → (□A → □B). T: □A → A. Truth axiom.

Rules: Modus ponens. Necessitation.

T axiom readings: Necessity implies truth. What must be, is. Veridicality. Factivity.

Frame condition: Reflexivity: ∀w.wRw. Every world accesses itself. Self-accessibility.

Dual: A → ◇A. What is true, is possible. Actuality implies possibility.

Epistemic reading: □A as "knows A." KA → A: knowledge is factive. Can't know falsehoods. T = minimal epistemic.

Not valid in K: □A → A fails in K. Needs reflexivity. Provability: □Prov(A) ↛ A. Consistency statements.

Extensions: KT4 = S4. KTB: + symmetric. S5 = KT4B = KT5.

Relation to B: B: A → □◇A. Symmetry. KTB = Brouwerian system.

Symbols.

Metatheory. Reflexive frames. Factivity. Completeness. Decidable.

Applies to. Knowledge. Necessity. Truth. Factive attitudes.

Limitations. Not for provability. Positive introspection absent. Knowledge debates. Minimal.