Reasonably Rational: Reasoning about Reasons Behind Preferences Using Modal Logic

There has been a significant interest in modal logics with intersection, prominent examples including epistemic and doxastic logics with distributed knowledge, propositional dynamic logic with intersection, and description logics with concept intersection. Completeness proofs for such logics tend to be complicated, in particular on model classes such as S5 used, e.g., in standard epistemic logic, mainly due to the undefinability of intersection of modalities in standard modal logic. A standard proof method for the S5 case uses an “unraveling-folding” technique to achieve a treelike model to deal with the problem of undefinability. This method, however, is not easily adapted to other logics, due to its reliance on S5 in a number of steps. In this paper we demonstrate a simpler and more general proof technique by building a treelike canonical model directly, which avoids the complications in the processes of unraveling and folding. We illustrate the technique by showing completeness of the normal modal logics K, D, T, B, S4 and S5 extended with intersection modalities. Furthermore, these treelike canonical models are compatible with Fischer-Ladner-style closures, and we combine the methods to show the completeness of the mentioned logics further extended with transitive closure of union modalities known from PDL or epistemic logic. Some of these completeness results are new.

Inspired by an interesting quotation from the literature, we propose four modalities, called ‘sane belief’, ‘insane belief’, ‘reliable belief’ and ‘unreliable belief’, and introduce logics with each operator as the modal primitive. We show that the four modalities constitute a square of opposition, which indicates some interesting relationships among them. We compare the relative expressivity of these logics and other related logics, including a logic of false beliefs from the literature. The four main logics are all less expressive than the standard modal logic over various model classes, and the logics of sane and insane beliefs are, respectively, equally expressive as the logics of unreliable and reliable beliefs on any class of models. The logics of reliable and unreliable beliefs are then combined into a bimodal logic, which turns out to be equally expressive as the standard modal logic. Despite this, we cannot obtain a complete axiomatization of the minimal bimodal logic, by simply translating the axioms and rules of the minimal modal logic $\textbf{K}$ into the bimodal language. We then introduce a schematic modality which unifies reliable and unreliable beliefs and axiomatize it over the class of all frames and also the class of serial frames. This line of research is finally extended to unify sane and insane beliefs and some axiomatizations are given.

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article, we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as 𝒜ℒ𝒞𝒪, 𝒜ℒ𝒞ℋ, and 𝒜ℒ𝒞ℋ𝒪ℐ and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime -complete in the presence of an ontology or the universal modality, and coNExpTime -complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case, the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions.

There are two typical ways to formalize reasoning over time. The first is in a modal logic where time is characterized implicitly through tense operators. The second addresses time explicitly in a first-order theory, where the reasoning is itself essentially first order. In this paper we shall introduce an amalgamated temporal logic which combines the tense operators of the modal approach with the referential aspects of the first-order approach.The modal operators of the new logic, called IQ-C, include the familiar tense operators plus a few new ones; together they allow complex temporal propositions to be represented in a compact form. Unlike standard modal temporal logics, IQ-C contains terms which refer to times; this allows the logic to characterize certain temporal properties and operators which cannot be axiomatized or defined by conventional modal operators. IQ-C is expressively complete over any connected flow of time, and has several other interesting theoretical properties, some of which will be developed in this paper.Most interval logics take one of two approaches to temporal assertions; either they analyze assertions for intervals in terms of assertions at points, or they treat intervals as primitive, allowing assertions for intervals but not at points. IQ-C allows assertions both for intervals and at points, and treats them as independent; this yields a special degree of semantic discrimination and at the same time simplifies the representation of several temporal properties. In this paper we shall outline the IQ-C framework, specifying its special features and defining its semantics. We shall also extend a method of skolemization, which is designed to support a resolution based approach to the proof theory. This will be sketched in some detail, with a view to providing a preliminary account of the basic proof theory. A complete specification of the deductive theory, together with relevant proofs, will be developed elsewhere.KeywordsIntervalsPointsTemporal Representation and ExpressivenessHomogeneitySkolemizationand Modal Theorem Proving

In this paper I give conditions under which a matrix characterisation of validity is correct for first order logics where quantifications are restricted by statements from a theory. Unfortunately the usual definition of path closure in a matrix is unsuitable and a less pleasant definition must be used. I derive the matrix theorem from syntactic analysis of a suitable tableau system, but by choosing a tableau system for restricted quantification I generalise Wallen's earlier work on modal logics. The tableau system is only correct if a new condition I call “alphabetical monotonicity” holds. I sketch how the result can be applied to a wide range of logics such as first order variants of many standard modal logics, including non-serial modal logics.

The Semantics of Entailment

Quantified modal inferences interest logicians, linguists, and computer scientists, but no previous psychological study of them appears to be in the literature. Here is an example of one: All those artists are businessmen. Paulo is possibly one of the artists. What follows? People tend to conclude: Paulo is possibly a businessman (Experiment 1). It seems plausible, and it follows from an intuitive mental model in which Paulo is one of a set of artists who are businessmen. Further deliberation can yield a model of an alternative possibility in which Paulo is not one of the artists, which confirms that the conclusion is only a possibility. The snag is that standard modal logics, which deal with possibilities, cannot yield a particular conclusion to any premises: Infinitely many follow validly (from any premises) but they do not include the present conclusion. Yet, further experiments corroborated a new mental model theory's predictions for various inferences (Experiment 2), for the occurrence of factual conclusions drawn from premises about possibilities (Experiment 3) and for inferences from premises of modal syllogisms (Experiment 4). The theory is therefore plausible, but we explore the feasibility of a cognitive theory based on modifications to modal logic.

In standard modal logic, □ ≡ ∼◊ ∼ and ◊ ≡ ∼□∼. I will, first, examine why in tense-logic, Arthur Prior thinks that ∼ ◊ ∼ is weaker than □ and ∼ □ ∼ is weaker than ◊. I will, then, examine whether there are similar motivations in modal logic to take ∼ ◊ ∼ to be weaker than □ and ∼ □ ∼ to be weaker than ◊. The upshot will be that, just as certain metaphysical views within the philosophy of time (e.g., Presentism) motivate one to deny the standard tense equivalences, certain metaphysical views within the metaphysics of modality (e.g., Contingentism, nonmodalism) motivate one to deny the standard modal equivalences.

We propose a notion of focusing for nested sequent calculi for modal logics which brings down the complexity of proof search to that of the corresponding sequent calculi. The resulting systems are amenable to specifications in linear logic. Examples include modal logic \(\mathsf {K}\), a simply dependent bimodal logic and the standard non-normal modal logics. As byproduct we obtain the first nested sequent calculi for the considered non-normal modal logics.KeywordsModal LogicInference RuleModal RuleLinear LogicSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Computational social choice is a new interdisciplinary research area at the intersection of social choice theory and computer science that studies computational aspects of social choice mechanisms as well as applications of social choice ideas in the design and analysis of computing systems. Research on computational social choice now encompasses a variety of topics, including the algorithmic aspects of voting rules, computational barriers to strategic behaviour, collective decision-making in multi-agent systems, preference elicitation and communication issues in voting, fair division, computational aspects of weighted voting games, collective decision-making in combinatorial domains, logic-based formalisms for social choice problems, belief and judgement aggregation, and social software. The aim of this special issue is to build on the spectacular success of two workshops on computational social choice (COMSOC’06, Amsterdam and COMSOC’08, Liverpool), and provide a forum for reseachers in the area to present their ideas on a variety of topics. This special issue is composed of nine research articles and a short note that span a fairly wide range of issues, including voting, mechanism design, judgment aggregation, and fairness, and a similarly wide range of techniques, such as logic, experiments, and computational complexity. The paper by Agotnes, van der Hoek and Wooldridge defines a new modal logic for representing and reasoning about preference and judgment aggregation. The authors show that their logic can be used to express a variety of concepts and results from the social choice domain, including majority voting, the discursive paradox, Arrow’s theorem and Condorcet’s paradox. Baharad, Goldberger, Koppel and Nitzan propose a maximum likelihood estimation-based approach for aggregating voters’ opinions over several issues. They argue that their method is vastly superior to simple majority voting, and use both analytical and empirical evaluation to support their argument. Balan, Richards and Luke deal with the problem of selecting a sequence of actions when the actions have multiple beneficiaries. Their goal is to select the actions according to a lexicographic fairness criterion. The explore the complexity of this problem for both finite and infinite sequences of actions and present both algorithms and intractability results for this problem. The paper by Caminada and Pigozzi applies argumentation theory to judgment aggregation, and proposes aggregation operators satisfying not only the usual property of collective rationality, but another highly desirable property,

In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatio-temporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u, which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with multi-dimensional formalisms and discuss possible directions for further research.

In classical quantificational logic, every individual constant is assigned a value from the domain of discourse, thus ensuring that every instance of \(\exists x(x=\alpha )\) is valid and so a theorem of a complete logic. Standard tense and modal logics validate a rule of necessitation, according to which, crudely, every theorem is always and necessarily true. Combining these two generates the result that everything always and necessarily exists. In a number of works from the late 1950s through to his death in 1969, Prior worked to develop tense and modal logics that avoided these results. Prior’s key idea was to reject the rule of necessitation. In this essay, I present and criticize Prior’s account of the contingency of existence and offer a more satisfying account that shares many of the virtues of Prior’s account.

Computational social choice is an interdisciplinary field of study at the interface of social choice theory and computer science, with knowledge flowing in either direction. On the one hand, computational social choice is concerned with importing concepts and procedures from social choice theory for solving questions that arise in computer science and AI application domains. This is typically the case for managing societies of autonomous agents, which calls for negotiation and voting procedures. On the other hand, computational social choice is concerned with importing notions and methods from computer science for solving questions originally stemming from social choice, for instance by providing new perspectives on the problem of manipulation and control in elections. This Dagstuhl Seminar has been devoted to the presentation of recent results and an exchange of ideas in this growing research field.

We study nonlinear connectives (exponentials) in the context of Type Logical Grammar(TLG). We devise four conservative extensions of theDisplacement calculus with brackets, \DbC, \DbCM, \DbCb and \DbCbMr which contain the universal and existential exponential modalities of linear logic (\LL). These modalitiesdo not exhibit the same structural properties as in \LL, which in TLG are especially adapted for linguistic purposes. The universal modality \univexpfor TLG allows only the commutative and contraction rules, but not weakening, whereas the existential modality \exstexp allows the so-called (intuitionistic) mingle rule, whichderives a restricted version of weakening called \emph{expansion}. We provide a Curry-Howard labelling for both exponential connectives. As it turns out,controlled contraction by \univexp gives a way to account for the so-called parasitic gaps, and controlled Mingle \exstexp iterability, in particular iteratedcoordination. Finally, the four calculi are proved to be Cut-Free but decidability is only proved for $\DbCb$, whereasfor the rest the question of decidability remains open.

We introduce a general problem about bribery in voting systems. In the $\mathcal{R}$-Multi-Bribery problem, the goal is to bribe a set of voters at minimum cost such that a desired candidate wins the perturbed election under the voting rule R. Voters assign prices for withdrawing their vote, for swapping the positions of two consecutive candidates in their preference order, and for perturbing their approval count to favour candidates. As our main result, we show that $\mathcal{R}$-Multi-Bribery is fixed-parameter tractable parameterized by the number of candidates for many natural voting rules $\mathcal{R}$, including Kemeny rule, all scoring protocols, maximin rule, Bucklin rule, fallback rule, SP-AV, and any C1 rule. In particular, our result resolves the parameterized complexity of $\mathcal{R}$-Swap Bribery for all those voting rules, thereby solving a long-standing open problem and "Challenge #2" of the "Nine Research Challenges in Computational Social Choice" by Bredereck et al. Further, our algorithm runs in single-exponential time for arbitrary cost; it thus improves the earlier double-exponential time algorithm by Dorn and Schlotter that is restricted to the uniform-cost case for all scoring protocols, the maximin rule, and Bucklin rule.