New Logical Perspectives on Ceteris Paribus Preference

Fenrong Liu

New Logical Perspectives on Ceteris Paribus Preference

2013, Proceedings of Socreal 2013 3rd International Workshop on Philosophy and Ethics of Social Reality 2013

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This paper explores new logical frameworks for understanding ceteris paribus preferences in the context of modal logic. It delves into semantics related to preference, defines key concepts such as priority graphs, and presents a novel axiomatization approach. By analyzing the dynamics and implications of preferences under ceteris paribus conditions, the research contributes to a more nuanced understanding of decision-making processes in varying contexts, particularly in economics.

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.

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].

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