In the present work we investigate formal systems which are related to Kripke’s theory of truth and to its subsequent extensions via four-valued logic (Woodruff, Visser). The ground language of Peano arithmetic PA is expanded by unary predicates T(rue), F(alse); the basic theory \(BT^-\) extends PA 1) by standard axioms relating T, F with standard logical operators; 2) by conditions which embody the possibility of self- application. \(BT^-\) generalizes previous work of Feferman about the so-called reflective closure \(PA^{\hookleftarrow}\) of PA; in \(PA^{\hookleftarrow}\) T is consistent, but partial, while \(BT^-\) is compatible with the dual hypothesis (T is total but inconsistent). We also consider theories of truth with restricted number-theoretic induction. In § 3 we survey some connections with standard subsystems of Analysis, based on the iteration of the jump operator; the bounds obtained are exact by the proof theory of § 9. §§ 5-6 are devoted to semantics: given a model M of PA, the collection of M- expansions which satisfy \(BT^-\) coincides with the set \(FIX_ M\) of the fixed points of a suitable monotone operator. We show that the complete lattice \(FIX_ M\) is non-modular, it has cardinality \(2^{| M|}\) \((| M| =cardinal\) of M) and there exist \(2^{| M|}\) fixed points which are neither consistent nor complete. In §§ 7-8 we turn to recursion theory; in particular, we prove an appropriate analogue of the ordinal comparison theorem for models of the form \(MIN_ M\) (= minimum of \(FIX_ M)\) and we give some proof-theoretic applications. In § 9 we prove majorization results of the form: if T(\(\ulcorner A\urcorner)\) is provable, then T(\(\ulcorner A\urcorner)\) holds at stage \(\alpha\) (\(\alpha\) an ordinal) in \(MIN_ N\) (N standard), where \(\alpha\) is effectively extracted from the given proof.