Cut-free formulations for a quantified logic of here and there

A predicate extension S Q H T = of the logic of here-and-there was introduced by V. Lifschitz, D. Pearce, and A. Valverde to characterize strong equivalence of logic programs with variables and equality with respect to stable models. The semantics for this logic is determined by intuitionistic Kripke models with two worlds (here and there) with constant individual domain and decidable equality. Our sequent formulation has special rules for implication and for pushing negation inside formulas. The soundness proof allows us to establish that S Q H T = is a conservative extension of the logic of weak excluded middle with respect to sequents without positive occurrences of implication. The completeness proof uses a non-closed branch of a proof search tree. The interplay between rules for pushing negation inside and truth in the “there” (non-root) world of the resulting Kripke model can be of independent interest. We prove that existence is definable in terms of remaining connectives.