Zach, Richard. 2004. “Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤ω.” Journal of Philosophical Logic 33 (2): 155–64. https://doi.org/10.1023/B:LOGI.0000021744.10237.d0.
Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \(\forall p\), \(\exists p\), where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most \(\omega\), the resulting logics are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel-Dummett logics with quantifiers over propositions.
Zach, Richard. 2004. “Hilbert’s ‘Verunglückter Beweis’, the First Epsilon Theorem, and Consistency Proofs.” History and Philosophy of Logic 25 (2): 79–94. https://doi.org/10.1080/01445340310001606930.
In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert’s Programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert’s epsilon-substitution method. There was, however, a second approach which was not reflected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert’s first epsilon theorem and a certain ‘general consistency result’ due to Bernays. An analysis of the form of this so-called ‘failed proof’ sheds further light on an interpretation of Hilbert’s Program as an instrumentalist enterprise with the aim of showing that whenever a ‘real’ proposition can be proved by ‘ideal’ means, it can also be proved by ‘real’, finitary means.