Modal Logics and Group Polarization (original) (raw)
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Polarity Semantics for Negation as a Modal Operator
Studia Logica
The minimal weakening N0 of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of N0 with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of N0. These logics have the finite model property and they are decidable.
Extended Modal Logics for Social Software
This is a short motivational report on work that we have done in two specific areas of Social Software, i.e. Coalitional Games and Judgement Aggregation. We argue that Extended Modal Logics prove to be particularly successful in the modeling and axiomatisation of reasoning problems in those areas. Here, we restrict ourselves to a description of the domains: technical details are to be found in [1] and [2], respectively.
A New Semantics for Positive Modal Logic
Notre Dame Journal of Formal Logic, 1997
The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering ≤ on the set of possible worlds and an accessibility relation R connected to the quasi ordering by the conditions (1) that the composition of ≤ with R is included in the composition of R with ≤ and (2) the analogous for the inverse of ≤ and R. This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions of the minimal system of normal positive modal logic.
Advances in Modal Logic, 2024
Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices L equipped with an antitone operation ¬ sending 1 to 0, a completely multiplicative operation □, and a completely additive operation ◊. Such lattice expansions can be represented by means of a set X together with binary relations ◁, R, and Q, satisfying some first-order conditions, used to represent (L, ¬), □, and ◊, respectively. Indeed, any lattice L equipped with such a ¬, a multiplicative □, and an additive ◊ embeds into the lattice of propositions of a frame (X, ◁, R, Q). Building on our recent study of fundamental logic, we focus on the case where ¬ is dually self-adjoint (a ≤ ¬b implies b ≤ ¬a) and ◊¬a ≤ ¬□a. In this case, the representations can be constrained so that R = Q, i.e., we need only add a single relation to (X, ◁) to represent both □ and ◊. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures (X, ◁, R).
A Van Benthem Theorem for Modal Team Semantics
We study the expressive power of variants of modal dependence logic, MDL, formulas of which are not evaluated in worlds but in sets of worlds, so called teams. The logic can then express that in a team, the value of a certain variable is functionally dependent on values of some other variables. The formulas of MDL, and all of its variants studied in the literature so far, are invariant under bisimulation when lifted from single worlds to sets of worlds in a natural way. In this paper we study the problem whether there is a logic that captures exactly the bisimulation invariant properties of Kripke structures and teams. Our main result shows that an extension of MDL, modal team logic MTL, extending MDL (or even only ML) by classical negation, is such a logic. We also give two further alternative characterizations for the bisimulation invariant properties in terms of extended modal inclusion logic with classical disjunction, and an extension of ML by so-called first-order definable generalized dependence atoms.
Sequent Systems for Negative Modalities
Logica Universalis, 2017
Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate semantics and proof systems, whose philosophical interpretations and computational properties are found wanting. In this paper we investigate congruential non-classical negations that live inside very natural systems of normal modal logics over complete distributive lattices; these logics are further enriched by adjustment connectives that may be used for handling reasoning under uncertainty caused by inconsistency or undeterminedness. Using such straightforward semantics, we study the classes of frames characterized by seriality, reflexivity, functionality, symmetry, transitivity, and some combinations thereof, and discuss what they reveal about sub-classical properties of negation. To the logics thereby characterized we apply a general mechanis...
A Modal Logic for Network Topologies
Lecture Notes in Computer Science, 2000
In this paper, we present a logical framework that combines modality with a first-order quantification mechanism. The logic differs from standard firstorder modal logics in that quantification is not performed inside the states of a model, but the states in the model themselves constitute the domain of quantification. The locality principle of modal logic is preserved via the requirement that in each state, the domain of quantification is restricted to a subset of the entire set of states in the model. We show that the language is semantically characterised by a generalisation of classical bisimulation, called history-based bisimulation, consider its decidability and study the application of the logic to describe and reason about the topologies of multi-agent systems.
On Strictly Positive Fragments of Modal Logics with Confluence
Mathematics
We axiomatize strictly positive fragments of modal logics with the confluence axiom. We consider unimodal logics such as K.2, D.2, D4.2 and S4.2 with unimodal confluence ⋄□p→□⋄p as well as the products of modal logics in the set K,D,T,D4,S4, which contain bimodal confluence ⋄1□2p→□2⋄1p. We show that the impact of the unimodal confluence axiom on the axiomatisation of strictly positive fragments is rather weak. In the presence of ⊤→⋄⊤, it simply disappears and does not contribute to the axiomatisation. Without ⊤→⋄⊤ it gives rise to a weaker formula ⋄⊤→⋄⋄⊤. On the other hand, bimodal confluence gives rise to more complicated formulas such as ⋄1p∧⋄2n⊤→⋄1(p∧⋄2n⊤) (which are superfluous in a product if the corresponding factor contains ⊤→⋄⊤). We also show that bimodal confluence cannot be captured by any finite set of strictly positive implications.