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A Complete FirstOrder Logic of Knowledge and Time
, 2008
"... We introduce and investigate quantified interpreted systems, a semantics to reason about knowledge and time in a firstorder setting. We provide an axiomatisation, which we show to be sound and complete. We utilise the formalism to study message passing systems (Lamport 1978; Fagin et al 1995) in a ..."
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We introduce and investigate quantified interpreted systems, a semantics to reason about knowledge and time in a firstorder setting. We provide an axiomatisation, which we show to be sound and complete. We utilise the formalism to study message passing systems (Lamport 1978; Fagin et al 1995) in a firstorder setting, and compare the results obtained to those available for the propositional case.
THE MODAL LOGIC OF FORCING
, 2007
"... Abstract. A set theoretical assertion ψ is forceable or possible, written ♦ ψ, if ψ holds in some forcing extension, and necessary, written � ψ, ifψ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFCprovable principle ..."
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Abstract. A set theoretical assertion ψ is forceable or possible, written ♦ ψ, if ψ holds in some forcing extension, and necessary, written � ψ, ifψ holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if ZFC is consistent, then the ZFCprovable principles of forcing are exactly those in the modal theory S4.2. 1.
Modal systems based on manyvalued logics
"... We propose a general semantic notion of modal manyvalued logic. Then, we explore the difficulties to characterize this notation in a syntactic way and analyze the existing literature with respect to this framework. ..."
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We propose a general semantic notion of modal manyvalued logic. Then, we explore the difficulties to characterize this notation in a syntactic way and analyze the existing literature with respect to this framework.
Representable Cylindric Algebras and ManyDimensional Modal Logics
, 2010
"... The equationally expressible properties of the cylindrifications and the diagonals in finitedimensional representable cylindric algebras can be divided into two groups: (i) ‘Onedimensional ’ properties describing individual cylindrifications. These can be fully characterised by finitely many equat ..."
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The equationally expressible properties of the cylindrifications and the diagonals in finitedimensional representable cylindric algebras can be divided into two groups: (i) ‘Onedimensional ’ properties describing individual cylindrifications. These can be fully characterised by finitely many equations saying that each ci, for i < n, is a normal (ci0 = 0), additive (ci(x+y) = cix+ciy) and complemented closure operator: x ≤ cix cicix ≤ cix ci(−cix) ≤ −cix. (1) (ii) ‘Dimensionconnecting ’ properties, that is, equations describing the diagonals and interaction between different cylindrifications and/or diagonals. These properties are much harder to describe completely, and there are many results in the literature on their complexity. The main aim of this chapter is to study generalisations of (i) while keeping (ii) as unchanged as possible. In other words, we would like to analyse how much of the complexity of RCAn is due to its ‘manydimensional ’ character and how much of it
Nonfinitely axiomatisable twodimensional modal logics
, 2011
"... We show the first examples of recursively enumerable (even decidable) twodimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linea ..."
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We show the first examples of recursively enumerable (even decidable) twodimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nestingdepth. 1
Common Knowledge and Quantification
"... The paper consists of two parts. The first one is a concise introduction to epistemic (both propositional and predicate) logic with common knowledge operator. As the full predicate logics of common knowledge are not even recursively enumerable, in the second part we introduce and investigate the mon ..."
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The paper consists of two parts. The first one is a concise introduction to epistemic (both propositional and predicate) logic with common knowledge operator. As the full predicate logics of common knowledge are not even recursively enumerable, in the second part we introduce and investigate the monodic fragment of these logics which allows applications of the epistemic operators to formulas with at most one free variable. We provide the monodic fragments of the most important common knowledge predicate logics with finite Hilbertstyle axiomatizations, prove their completeness, and single out a number of decidable subfragments. On the other hand, we show that the addition of equality to the monodic fragment makes it not recursively enumerable. 1 Introduction Ever since it became common knowledge that the intelligent behavior of an agent is based not only on her knowledge about the world but also on the knowledge about both her own and other agents' knowledge, logical formalisms desig...
Products Of `transitive' Modal Logics Without The (abstract) Finite Model Property
"... It is well known that many twodimensional products of modal logics with at least one `transitive' (but not `symmetric') component lack the product finite model property. Here we show that products of two `transitive' logics (such as, e.g., K4 K4, S4 S4, GrzGrz and GLGL) do not have the (abstr ..."
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It is well known that many twodimensional products of modal logics with at least one `transitive' (but not `symmetric') component lack the product finite model property. Here we show that products of two `transitive' logics (such as, e.g., K4 K4, S4 S4, GrzGrz and GLGL) do not have the (abstract) finite model property either. These are the first known examples of 2D modal product logics without the finite model property where both components are natural unimodal logics having the finite model property.
STRUCTURAL CONNECTIONS BETWEEN A FORCING CLASS AND ITS MODAL LOGIC
"... Every definable forcing class Γ gives rise to a corresponding forcing modality, for which □Γ ϕ means that ϕ is true in all Γ extensions, and the valid principles of Γ forcing are the modal assertions that are valid for this forcing interpretation. For example, [9] shows that if ZFC is consistent, t ..."
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Every definable forcing class Γ gives rise to a corresponding forcing modality, for which □Γ ϕ means that ϕ is true in all Γ extensions, and the valid principles of Γ forcing are the modal assertions that are valid for this forcing interpretation. For example, [9] shows that if ZFC is consistent, then the ZFCprovably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2. In this article, we prove similarly that the provably valid principles of collapse forcing, Cohen forcing and other classes are in each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CHpreserving forcing and others are each exactly S4.2; and the provably valid principles of ω1preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise.
On the Finite Model Property of Intuitionistic Modal Logics over MIPC
, 1998
"... It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property if there exists a universal firstorder sentence 8 such that (1) L is characterized by the class of Kripke frames satisfying 8 and (2) every Kripke frame that validates L satisfies 8. Here, MIPC is ..."
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It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property if there exists a universal firstorder sentence 8 such that (1) L is characterized by the class of Kripke frames satisfying 8 and (2) every Kripke frame that validates L satisfies 8. Here, MIPC is a wellknown intuitionistic modal logic introduced by Prior (1957).
Proofs and Expressiveness in Alethic Modal Logic
, 2001
"... Introduction Alethic modalities are the necessity, contingency, possibility or impossibility of something being true. Alethic means `concerned with truth'. [28, p. 132] The above dictionary characterization of alethic modalities states the central notions of alethic modal logic: necessity, and othe ..."
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Introduction Alethic modalities are the necessity, contingency, possibility or impossibility of something being true. Alethic means `concerned with truth'. [28, p. 132] The above dictionary characterization of alethic modalities states the central notions of alethic modal logic: necessity, and other notions that are usually thought of as being definable in terms of necessity and Boolean negation: impossibility, contingency, and possibility. The syntax of modal propositional logic is inductively defined over a denumerable set of sentence letters p 0 , p 1 , p 2 , . . . as follows: A ::= p  A  (A # B)  #A The other Boolean operations (#, #, #, # and #) are defined as usual. A formula<F10.9