Decidability of quantified propositional intuitionistic logic and S4 on trees

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \forall p, \exists p over propositions. In the context of Kripke semantics, a proposition is a subset of the worlds in a model structure which is upward closed. Kremer (1997) has shown that the quantified propositional intuitionistic logic H\pi+ based on the class of all partial orders is recursively isomorphic to full second-order logic. He raised the question of whether the logic resulting from restriction to trees is axiomatizable. It is shown that it is, in fact, decidable. The methods used can also be used to establish the decidability of modal S4 with propositional quantification on similar types of Kripke structures.Comment: v2, 9 pages, corrections and additions; v1 8 page

A Remark on Lattice Models of Second-Order Intuitionistic Propositional Logic (Theory and Applications of Proof and Computation)

We summarize a topological representation of the completely distributive algebraic lattices. This together with the discussion in [KF19] gives a definition of lattice models from an abstract viewpoint, with respect to which the completeness of the basic system of second-order intuitionistic propositional logic is ensured

Undecidability of first-order modal and intuitionistic logics with two variables and one monadic predicate letter

We prove that the positive fragment of first-order intuitionistic logic in the language with two variables and a single monadic predicate letter, without constants and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without constants and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among them, QK, QT, QKB, QD, QK4, and QS4.Comment: Pre-final version of the paper published in Studia Logica,doi:10.1007/s11225-018-9815-

Quantified intuitionistic propositional logic and Cantor space

We consider propositional quantification in intuitionistic logic. We prove that, under topological interpretation over the Cantor space, it enjoys surprising and interesting properties such as the maximum property and a kind of distribution of existential quantifier over conjunction. Moreover, by pointing to appropriate examples, we show that the set of quantified formulas valid in the Cantor space strictly contains the set of formulas provable in the minimal system of intuitionistic logic with propositional quantification

АКСИОМАТИЗИРУЕМОСТЬ НЕНОРМАЛЬНЫХ И КВАЗИНОРМАЛЬНЫХ МОДАЛЬНЫХ ПРЕДИКАТНЫХ ЛОГИК ПЕРВОПОРЯДКОВО ОПРЕДЕЛИМЫХ КЛАССОВ ШКАЛ КРИПКЕ

Рассматривается вопрос о возможности эффективного описания ненормальных и квазинормальных предикатных модальных логик, определяемых семантически посредством классов шкал Крипке с выделенными мирами. Доказывается, что любая ненормальная или квазинормальная (в т. ч. нормальная) модальная предикатная логика, полная относительно некоторого первопорядково определимого класса шкал Крипке с выделенными мирами, погружается в классическую логику предикатов. Показано, как построить соответствующее погружение, используя т. н. стандартный перевод модальных предикатных формул в формулы языка классической логики предикатов. В конце работы приводятся следствия указанного результата, а также демонстрируются возможности обобщения описанной конструкции на классы других систем, в частности, на классы полимодальных логик — темпоральных логик с парой модальностей «всегда было» и «всегда будет» и логик знания с оператором распределенного знания. Показаны некоторые границы применимости описанного метода, приведены соответствующие примеры. Указаны контрпримеры, когда условия применимости метода для полной по Крипке модальной предикатной логики не выполнены, а построение эффективного описания этой логики, тем не менее, возможн

A secondary semantics for second order intuitionistic propositional logic

In this paper we propose a Kripke-style semantics for second order intuitionistic propositional logic and we provide a semantical proof of the disjunction and the explicit definability property. Moreover, we provide a tableau calculus which is sound and complete with respect to such a semantics

Axiomatizability of propositionally quantified modal logics on relational frames

Propositional modal logic over relational frames is naturally extended with propositional quantifiers by letting them range over arbitrary sets of worlds of the relevant frame. This is also known as second-order propositional modal logic. The propositionally quantified modal logic of a class of relational frames is often not axiomatizable, although there are known exceptions, most notably the case of frames validating the strong modal logic S5 . Here, we develop new general methods with which many of the open questions in this area can be answered. We illustrate the usefulness of these methods by applying them to a range of examples, which provide a detailed picture of which normal modal logics define classes of relational frames whose propositionally quantified modal logic is axiomatizable. We also apply these methods to establish new results in the multimodal case

On the Logic of Belief and Propositional Quantification

We consider extending the modal logic KD45, commonly taken as the baseline system for belief, with propositional quantifiers that can be used to formalize natural language sentences such as “everything I believe is true” or “there is some-thing that I neither believe nor disbelieve.” Our main results are axiomatizations of the logics with propositional quantifiers of natural classes of complete Boolean algebras with an operator (BAOs) validating KD45. Among them is the class of complete, atomic, and completely multiplicative BAOs validating KD45. Hence, by duality, we also cover the usual method of adding propositional quantifiers to normal modal logics by considering their classes of Kripke frames. In addition, we obtain decidability for all the concrete logics we discuss