Reduction of provability logics to Σ₁-provability logics

The Σ1-provability logic of HA

“…The authors of [1] found a reduction of the Solovay-Löb Theorem to the Visser Theorem only by propositional substitutions [1]. This result is tempting to think of applying similar method for the intuitionistic case.…”

On Provability Logic of HA

“…The method used in [Visser, 1981] essentially uses Solovay's technique in [Visser, 1981]. Then [Ardeshir and Mojtahedi, 2015] shows that in a sense, the Σ 1 -provability logic of PA is harder than the standard provability logic of PA. Later [Mojtahedi, 2021] studies reductions between provability logics and characterizes several other provability logics. Most notably, it is shown in [Mojtahedi, 2021] that the Σ 1 -provability logic of HA relative in the standard model, is the hardest known provability logic.…”

The -Provability Logic Of

“…The authors of [1] found a reduction of the Solovay-Löb Theorem to the Visser Theorem only by propositional substitutions. It is tempting to think that the method used in [1] can be carried out in the intuitionistic case.…”

“…The authors of [1] found a reduction of the Solovay-Löb Theorem to the Visser Theorem only by propositional substitutions. It is tempting to think that the method used in [1] can be carried out in the intuitionistic case. However it seems to us that there is no obvious way of doing such a reduction for the intuitionistic case, and it should be more complicated than the classical case.…”