Abstract

We present a temporal logic of branching time with four primitive operators: |$\exists {\mathcal {C}}$| – it may change whether; |$\forall {\mathcal {C}} $| – it must change whether; |$\exists \Box $| – it may be endlessly unchangeable that; and |$\forall \Box $| – it must be endlessly unchangeable that. Semantically, operator |$\forall {\mathcal {C}}$| expresses a change in the logical value of the given formula in every state that may be an immediate successor of the one considered, while |$\exists {\mathcal {C}}$| expresses a change in the logical value of the given formula in some state that is a possible immediate successor. |$\forall {\mathcal {C}} $| and |$\exists {\mathcal {C}}$| are not normal operators, they are not mutually definable and have unusual properties: |$\forall {\mathcal {C}} A\leftrightarrow \forall {\mathcal {C}} \neg A$| and |$\exists {\mathcal {C}} A \leftrightarrow \exists {\mathcal {C}} \neg A$| are axioms. Operators |$\exists \Box $| and |$\forall \Box $| express endless unchangeability in some and all paths, respectively. Our axiomatization contains two infinitary rules that allow one to obtain |$\exists \Box A$| from infinitely many combinations of formulas with |$A, \neg, \forall {\mathcal {C}}, \land $|⁠, and to get |$\forall \Box A$| from infinitely many combinations of formulas with |$A, \neg, \exists {\mathcal {C}}, \land $|⁠. We show that our formalism is complete by presenting a bridge between our logic and logic |$\textsf {UB}$|⁠, which is a fragment of |$\textsf {CTL}$| in which until operators does not occur.