This note follows up an earlier paper in which a possibility of defining logical constants within abstract logical frameworks was discussed, by using duals as a general method of applying the idea of ...invariance under replacement as a criterion for logicality. In the present note, this approach is applied to the discussion on logicality of generalized quantifiers. It is demonstrated that generalized quantifiers are logical constants by this criterion.
Abstract Interpretability logic is a modal logic that can be used to describe relative interpretability between extensions of a given first-order arithmetical theory. Verbrugge semantics is a ...generalization of the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. The Van Benthem Correspondence Theorem establishes modal logic as the bisimulation invariant fragment of first-order logic. In this paper we show that a special type of bisimulations, the so-called w-bisimulations, enable an analogue of the Van Benthem Theorem for interpretability logic w.r.t. Verbrugge semantics.
A class of Kripke models is modally definable if there is a set of modal formulas such that the class consists exactly of models on which every formula from that set is globally true. In this paper, ...a class is also considered definable if there is a set of formulas such that it consists exactly of models in which every formula from that set is satisfiable. The notion of modal definability is then generalized by combining these two. For thus obtained types of modal definability on the level of Kripke models, we give characterization theorems in the usual form, in terms of algebraic closure conditions. As some consequences of these, various preservation results are presented. Also, some characterizations are strengthened by replacing closure under ultraproducts with closure under ultrapowers.
A possibility of defining logical constants within abstract logical frameworks is discussed, in relation to abstract definition of logical consequence. We propose using duals as a general method of ...applying the idea of invariance under replacement as a criterion for logicality.
We can formalize judgments as (consistent sets of) logical formulas. Judgment aggregation deals with judgments of several agents, which need to be aggregated to a collective judgment. There are ...several logical formalizations of judgment aggregation. This paper focuses on a modal formalization which nicely expresses classical properties of judgment aggregation rules and famous results of social choice theory, like Arrow's impossibility theorem. A natural deduction system for modal logic of judgment aggregation is presented in this paper. The system is sound and complete. As an example of derivation, a formal proof of Arrow's impossibility theorem is given.
We prove an existential analogue of the Goldblatt‐Thomason Theorem which characterizes modal definability of elementary classes of Kripke frames using closure under model theoretic constructions. The ...less known version of the Goldblatt‐Thomason Theorem gives general conditions, without the assumption of first‐order definability, but uses non‐standard constructions and algebraic semantics. We present a non‐algebraic proof of this result and we prove an analogous characterization for an alternative notion of modal definability, in which a class is defined by formulas which are satisfiable under any valuation (the so‐called existential validity). Continuing previous work in which model theoretic characterization for this type of definability of elementary classes was proved, we give an analogous general result without the assumption of the first‐order definability. Furthermore, we outline relationships between sets of existentially valid formulas corresponding to several well‐known modal logics.
We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau ...procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.
Interpretability logic is a modal formalization of relative interpretability between first‐order arithmetical theories. Verbrugge semantics is a generalization of Veltman semantics, the basic ...semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. We study various notions of bisimulation between Verbrugge models and develop a new one, which we call w‐bisimulation. We show that the new notion, while keeping the basic property that bisimilarity implies modal equivalence, is weak enough to allow the converse to hold in the finitary case. To do this, we develop and use an appropriate notion of bisimulation games between Verbrugge models.
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