A new semantical approach to the logic of preference
Sven Hansson1989, Erkenntnis
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Abstract
A possible world semantics for preference is developed. The remainder operator (_1.) is used to give precision to the notion that two states of the world are as similar as possible, given a specified difference between them. A general structure is introduced for preference relations between states of affairs, and three types of such preference relations are defined. It is argued that one of them, "actual preference", corresponds closely to the concept of preference in informal discourse. Its logical properties are studied and shown to be plausible.
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For Better or for Worse: Dynamic Logics of Preference
2009
In the last few years, preference logic and in particular, the dynamic logic of preference change, has suddenly become a live topic in my Amsterdam and Stanford environments. At the request of the editors, this article explains how this interest came about, and what is happening. I mainly present a story around some recent dissertations and supporting papers, which are found in the references. There is no pretense at complete coverage of preference logic (for that, see Hanson 2001) or even of preference change (Hanson 1995). Agency, information, and preference Human agents acquire and transform information in different ways: they observe, or infer by themselves, and often also, they ask someone else. Traditional philosophical logics describe part of this behaviour, the 'static' properties produced by such actions: in particular, agents' knowledge and belief at some given moment. But rational human activity is goal-driven, and hence we also need to describe agents' evaluation of different states of the world, or of outcomes of their actions. Here is where preference logic have come to describe what agents prefer, while current dynamic logics describe effects of their physical actions. In the limit, all these things have to come together in understanding even such a simple scenario as a game, where we need to look at what players want, what they can observe and guess, and which moves and long-term strategies are available to them in order to achieve their goals. There are two dual aspects to this situation. The static description of what agents know, believe, or prefer at any given moment has long been performed by standard systems of philosophical logic since the 1950s -of course, with continued debate surrounding the merits of particular proposals. But there is also the dynamics of actions and events that produce information and generate attitudes Overview This paper is mainly based on some recent publications in the Amsterdam environment over the last three years. Indeed, 'dynamics' presupposes an account of 'statics', and hence we first give a brief survey of preference logic in a simple modal format using binary comparison relations between possible worlds -on the principle that 'small is beautiful'. We also describe a recent alternative approach, where world preferences are generated from criteria or constraints. We show how to dynamify both views by adding explicit events that trigger preference change in the models, and we sketch how the resulting systems connect. Next, we discuss some entanglements between preference, knowledge and belief, and what this means for combined dynamic logics. On top of this, we also show how more delicate aspects of preference should be incorporated, such as its striking 'ceteris paribus' character, which was already central in Von Wright 1963. Finally, we relate our considerations to social choice theory and game theory. Preference is a very multi-faceted notion: we can prefer one individual object, or one situation, over another -but preference can also be directed toward kinds of objects or generic types of situation, often defined by propositions. Both perspectives make sense, and a bona fide 'preference logic' should do justice to all of them eventually. We start with a simple scenario on the object/world side, leaving other options for later. In this paper, we start with a very simple setting. Modal models M = (W, ≤, V) consist of a set of worlds W (but they really stand for any sort of objects that are subject to evaluation and comparison), a 'betterness' relation ≤ between worlds ('at least as good as'), and a valuation V for proposition letters at worlds (or, for unary properties of objects). In principle, the comparison relation may be different for different agents, but in what follows, we will suppress agent subscripts ≤ i whenever possible for greater readability. Also, we use the artificial term 'betterness' to stress that this is an abstract comparison relation, making no claim yet concerning the natural rendering of the intuitive term 'preference', about which some people hold passionate proprietary opinions. Still, this semantics is entirely natural and concrete. Just think of decision theory, where worlds (standing for outcomes of actions) are compared as to utility, or Here move is the union of all one-step move relations available to players, and * denotes the reflexive-transitive closure of a relation. The formula then says there is no alternative move to the BI-prescription at the current node all of whose outcomes would be better than the BI-solution. Thus, modal preference logic seems to go well with games. 3 But there are more examples. Already Boutilier 1994 observed how such a simple modal language can also define conditional assertions, normally studied per se as a complex new binary modality (Lewis 1973), and how one can then analyze their logic in standard terms. 4 For instance, in modal models with finite pre-orders (see below), the standard truth definition of a conditional A ⇒ B reads as 'B is true in all maximal A-worlds' -and this clause can be written as the following modal combination: with [] some appropriate universal modality. While this formula may look complex at first, the point is that the inferential behaviour of the conditional, including its well-known non-monotonic features, can now be completely understood via the base logic for the unary modalities, say, as a sub-theory of modal S4. Moreover, the modal language easily defines variant notions whose introduction seems a big deal in conditional logic, such as existential versions saying that each A-world sees at least one maximal A-world which is B. Of course, explicit separate axiomatizations of these defined notions retain an independent interest: but we now see the whole picture. 5 Constraints on betterness orders Which properties should a betterness relation have? Many authors like to work with total orders, satisfying reflexivity, transitivity, and connectedness. This is also common practice in decision theory and game theory, since 3 This, and also the following examples are somewhat remarkable, because there has been a widespread prejudice that modal logic is not very suitable to formalizing preference reasoning. 4 This innovative move is yet to become common knowledge in the logical literature. 5 There still remains the question of axiomatizing such defined notions per se: and that may be seen as the point of the usual completeness theorems in conditional logic. Also, Halpern 1997 axiomatized a defined notion of preference of this existential sort.
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The paper discusses the origins and structure of Preference Semantics, a procedural and computational system for extracting the meaning structure of natural language texts, based on notions of "maximal semantic density" and coherence. The basic representational structures and procedures of Preference Semantics are described, as well as the forms these notions have taken in the work of others.
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Preference relations are intensively studied in Economics, but they are also approached in AI, Knowledge Representation, and Conceptual Modelling, as they provide a key concept in a variety of domains of application. In this paper, we propose an ontological foundation of preference relations to formalise their essential aspects across domains. Firstly, we shall discuss what is the ontological status of the relata of a preference relation. Secondly, we investigate the place of preference relations within a rich taxonomy of relations (e.g. we ask whether they are internal or external, essential or contingent, descriptive or non-descriptive relations). Finally, we provide an ontological modelling of preference relation as a module of a foundational (or upper) ontology (viz. OntoUML). The aim of this paper is to provide a sharable foundational theory of preference relation that foster interoperability across the heterogeneous domains of application of preference relations.
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Representing preferences and reasoning about them are important issues for many real-life applications. Several monotonic and non-monotonic qualitative formalisms have been developed for this purpose. Most of them are based on comparative preferences, for ...
Semantics for Modelling Reason-Based Preferences
Lecture Notes in Computer Science, 2015
In [13] the authors developed a logical system based on the definition of a new non-classical connective ⊗ originally capturing the notion of reparative obligation. The operator ⊗ and the system were proved to be appropriate for rather handling well-known contrary-to-duty paradoxes. Later on, a suitable modeltheoretic possible-world semantics has been developed [4, 5]. In this paper we show how a version of this semantics can be used to develop a sound and complete logic of preference and offer a suitable possible-world semantics. The semantics is a sequence-based non-normal one extending and generalising semantics for classical modal logics.
Necessary and possible preference structures
Journal of Mathematical Economics, 2013
A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive and complete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold in many applications, preferences are often modeled by means of weaker binary relations, dropping either completeness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders). We introduce an alternative approach to preference modeling, which uses two binary relations -the necessary preference N and the possible preference P -to fulfill completeness and transitivity in a mixed form. Formally, a NaP-preference (necessary and possible preference) on A is a pair
From Good to Better: Using Contextual Shifts to Define Preference in Terms of Monadic Value
Outstanding Contributions to Logic, 2014
It has usually been assumed that monadic value notions can be defined in terms of dyadic value notions, whereas definitions in the opposite direction are not possible. In this paper, inspired by van Benthem's work, it is shown that the latter direction is feasible with a method in which shifts in context have a crucial role. But although dyadic preference orderings can be defined from context-indexed monadic notions, the monadic notions cannot be regained from the preference relation that they gave rise to. Two formal languages are proposed in which reasoning about context can be represented in a fairly general way. One of these is a modal language much inspired by van Benthem's work. Throughout the paper the focus is on relationships among the value notions "good", "bad", and "better". Other interpretations like "tall" and "taller" are equally natural. It is hoped that the results of this paper can be relevant for the analysis of natural language comparatives and of vague predicates in general.
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Annals of Mathematics and Artificial Intelligence, 1993
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2008
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Preference and Choice
Introduction to Formal Philosophy, 2018
Preferences and choices have central roles in moral philosophy, economics, and the decision sciences in general. In a formal language we can express and explore the properties of preferences, choices, and their interrelations in a precise way, and uncover connections that are inaccessible without formal tools. In this chapter, the plausibility of different such properties is discussed, and it is shown how close attention to the logical details can help dissolve some apparent paradoxes in informal and semi-formal treatments.
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Preference Logic, Conditionals, and Solution Concepts in Games
2006
Preference is a basic notion in human behaviour, underlying such varied phenomena as individual rationality in the philosophy of action and game theory, obligations in deontic logic (we should aim for the best of all possible worlds), or collective decisions in social choice theory. Also, in a more abstract sense, preference orderings are used in conditional logic or non-monotonic reasoning as a way of arranging worlds into more or less plausible ones. The field of preference logic (cf. Hansson ) studies formal systems that can express and analyze notions of preference between various sorts of entities: worlds, actions, or propositions. The art is of course to design a language that combines perspicuity and low complexity with reasonable expressive power. In this paper, we take a particularly simple approach. As preferences are binary relations between worlds, they naturally support standard unary modalities. In particular, our key modality ♦ϕ will just say that is ϕ true in some world which is at least as good as the current one. Of course, this notion can also be indexed to separate agents. The essence of this language is already in [4], but our semantics is more general, and so are our applications and later language extensions. Our modal language can express a variety of preference notions between propositions. Moreover, as already suggested in [9], it can "deconstruct" standard conditionals, providing an embedding of conditional logic into more standard modal logics. Next, we take the language to the analysis of games, where some sort of preference logic is evidently needed ([23] has a binary modal analysis different from ours). We show how a qualitative unary preference modality suffices for defining Nash Equilibrium in strategic games, and also the Backward Induction solution for finite extensive games. Finally, from a technical perspective, our treatment adds a new twist. Each application considered in this paper suggests the need for some additional access to worlds before the preference modality can unfold its true power. For this purpose, we use various extras from the modern literature: the global modality, further hybrid logic operators, action modalities from propositional dynamic logic, and modalities of individual and distributed knowledge from epistemic logic. The total package is still modal, but we can now capture a large variety of new notions. Finally, our emphasis in this paper is wholly on expressive power. Axiomatic completeness results for our languages can be found in the follow-up paper [27].
Representing preferences as ceteris paribus comparatives
1994
Decision-theoretic preferences specify the relative desirability of all possible outcomes of alternative plans. In order to express general patterns of preference holding in a domain, we require a language that can refer directly to preferences over classes of outcomes as well as individuals. We present the basic concepts of a theory of meaning for such generic comparatives to facilitate their incremental capture and exploitation in automated reasoning systems. Our semantics lifts comparisons of individuals to comparisons of classes "other things being equal" by means of contextual equivalences, equivalence relations among individuals that vary with the context of application. We discuss implications of the theory for representing preference information.
Preference Based on Reasons
The Review of Symbolic Logic, 2012
We describe a logic of preference in which modal connectives reflect reasons to desire that a sentence be true. Various conditions on models are introduced and analyzed.
Two alternatives for handling preferences in qualitative choice logic
Reasoning about preferences is a major issue in many decision making problems. Recently, a new logic for handling preferences, called qualitative choice logic (QCL), was presented. This logic adds to classical propositional logic a new connective, called ordered disjunction symbolized by ×. That new connective is used to express preferences between alternatives. Intuitively, if A and B are propositional formulas then A ×B means: "if possible A, but if A is impossible then at least B". One of the important limitations of QCL is that it does not correctly deal with negated and conditional preferences. Conditional rules that involve preferences are expressed using propositional implication. However, using QCL semantics, there is no difference between such material implication "(KLM ×Air France) ⇒ Hotel Package" and the purely propositional formula "(Air France∨KLM) ⇒ Hotel Package". Moreover, the negation in QCL misses some desirable properties from propositional calculus. This paper first proposes an extension of QCL language to universally quantified first-order logic framework. Then, we propose two new logics that correctly address QCL's limitations. Both of them are based on the same QCL language, but define new non-monotonic consequence relations. The first logic, called PQCL (prioritized qualitative choice logic), is particularly adapted for handling prioritized preferences, while the second one, called QCL+ (positive qualitative choice logic), is appropriate for handling positive preferences. In both cases, we show that any set of preferences, can equivalently be transformed into a set of basic preferences from which efficient inferences can be applied. Lastly, we show how our logics can be applied to alert correlation.
Preference Logics: Towards a Unified Approach to Nonmonotonicity in Deductive Reasoning
Annals of Mathematics and Artificial Intelligence, 1993
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Preference, Priorities and Belief
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A model for preference
Proceedings of the third conference on European chapter of the Association for Computational Linguistics -, 1987
In this paper we address the problem of choosing the best solution(s) from a set of interpretations of the same object (in our case a segment of text). A notion of preference is stated, based on pairwise comparisons of complete interpretations in order to obtain a partial order among the competing interpretations. An experimental implementation is described, which uses Prolog-like preference statements.
Representing and Reasoning with Qualitative Preferences: Tools and Applications
Synthesis Lectures on Artificial Intelligence and Machine Learning, 2016
is book provides a tutorial introduction to modern techniques for representing and reasoning about qualitative preferences with respect to a set of alternatives. e syntax and semantics of several languages for representing preference languages, including CP-nets, TCP-nets, CI-nets, and CP-theories, are reviewed. Some key problems in reasoning about preferences are introduced, including determining whether one alternative is preferred to another, or whether they are equivalent, with respect to a given set of preferences. ese tasks can be reduced to model checking in temporal logic. Specifically, an induced preference graph that represents a given set of preferences can be efficiently encoded using a Kripke Structure for Computational Tree Logic (CTL). One can translate preference queries with respect to a set of preferences into an equivalent set of formulae in CTL, such that the CTL formula is satisfied whenever the preference query holds. is allows us to use a model checker to reason about preferences, i.e., answer preference queries, and to obtain a justification as to why a preference query is satisfied (or not) with respect to a set of preferences. is book defines the notions of the equivalence of two sets of preferences, including what it means for one set of preferences to subsume another, and shows how to answer preferential equivalence and subsumption queries using model checking. Furthermore, this book demontrates how to generate alternatives ordered by preference, along with providing ways to deal with inconsistent preference specifications. A description of CRISNER-an open source software implementation of the model checking approach to qualitative preference reasoning in CP-nets, TCP-nets, and CP-theories is included, as well as examples illustrating its use.
New Logical Perspectives on Ceteris Paribus Preference
Proceedings of Socreal 2013 3rd International Workshop on Philosophy and Ethics of Social Reality 2013, 2013
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