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In this paper I shall present some of the results I have obtained on modal theories which contain quantifiers for propositions. The paper is in two parts: in the first part I consider theories whose non-quantificational part is S5; in the second part I consider theories whose non-quantificational part is weaker than or not contained in S5. Unless otherwise stated, each theory has the same language L. This consists of a countable set V of propositional variables pl, pa, ... , (...) |
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I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A. |
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Hybrid languages have both modal and first-order characteristics: a Kripke semantics, and explicit variable binding apparatus. This paper motivates the development of hybrid languages, sketches their history, and examines the expressive power of three hybrid binders. We show that all three binders give rise to languages strictly weaker than the corresponding first-order language, that full first-order expressivity can be gained by adding the universal modality, and that all three binders can force the existence of infinite models and have undecidable satisfiability (...) |
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 BASIC MODAL LOGIC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. |
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Hybrid languages are expansions of propositional modal languages which can refer to worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work has focussed on a more constrained system called $\mathscr{H}$. We show in detail that $\mathscr{H}$ is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations and a syntactic characterization. The key result to emerge is that $\mathscr{H}$ corresponds to the fragment of first-order logic which is invariant for generated (...) |
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We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and SOPML. |
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An interpolation theorem holds for many standard modal logics, but first order $S5$ is a prominent example of a logic for which it fails. In this paper it is shown that a first order $S5$ interpolation theorem can be proved provided the logic is extended to contain propositional quantifiers. A proper statement of the result involves some subtleties, but this is the essence of it. |
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We investigate the expressive power of second-order logic over finite structures, when two limitations are imposed. Let SAA ) be the set of second-order formulas such that the arity of the relation variables is bounded by k and the number of alternations of second-order quantification is bounded by n . We show that this imposes a proper hierarchy on second-order logic, i.e. for every k , n there are problems not definable in AA but definable in AA for some c (...) |
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We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible. |
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We define a propositionally quantified intuitionistic logic Hπ + by a natural extension of Kripke's semantics for propositional intutionistic logic. We then show that Hπ+ is recursively isomorphic to full second order classical logic. Hπ+ is the intuitionistic analogue of the modal systems S5π +, S4π +, S4.2π +, K4π +, Tπ +, Kπ + and Bπ +, studied by Fine. |
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We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and SOPML. |
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