The concept of preference spans numerous research fields, resulting indiverse perspectives on the topic. Preference logic specificallyfocuses on reasoning about preferences when comparing objects,situations, actions, and more, by examining their formal properties.This entry surveys major developments in preference logic to date.Section 2 provides a historical overview, beginning with foundational work byHalldén and von Wright, who emphasized the syntactic aspects ofpreference. InSection 3, early semantic contributions by Rescher and Van Dalen are introduced.The consideration of preference relations over possible worldsnaturally gives rise to modal preference logic where preferencelifting enables comparisons across sets of possible worlds.Section 4 introduces von Wright’s distinction between intrinsic andextrinsic preferences and presents a two-layer structure for extrinsic(or reason-based) preferences, a major focus of development over thelast two decades. This entry reviews the primary frameworks in thisarea.Section 5 is about preference change, which can happen for various reasons, andit explores different logical approaches to these dynamics.Section 6 discusses various interpretations ofceterisparibus preferences, provides a review of its underlying logic,and examines conditional preference networks (CP-nets), which areclosely related to this concept.Section 7 briefly examines the connections between preference logic and otherresearch areas.Section 8 concludes this entry.

• 1. Introduction
• 2. Syntactic approaches to preference logic
  • 2.1 Halldén: two theories, A and B
  • 2.2 Von Wright’s logic of preference
• 3. Semantic approaches to preference logic
  • 3.1 Rescher-Van Dalen semantics
  • 3.2 Preference over possible worlds
  • 3.3 Preference lifting
• 4. Reason-based preference
  • 4.1 Intrinsic vs. extrinsic preferences
  • 4.2 Two-layer structures
    • 4.2.1 Priorities and preference
    • 4.2.2 Salient properties and preference
    • 4.2.3 Reasons and preference
• 5. Dynamics of preference change
  • 5.1 Intrinsic preference revision
  • 5.2 Preference change driven by belief update
  • 5.3 Two-layer models of preference change
    • 5.3.1 Graph updates: basic operations
    • 5.3.2 Tracking dynamics
    • 5.3.3 Belief-driven preference change in two-layer structures
• 6.Ceteris paribus preference and CP-nets
  • 6.1 Formalizingceteris paribus preferences in logic
  • 6.2 Computational models of ceteris paribus preferences
• 7. Preference across research fields
  • 7.1 Preference and conditional logics
  • 7.2 Preference and deontic logics
  • 7.3 Preference in decision theory and game theory
  • 7.4 Preference in social choice theory
• 8. Conclusions
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

Preference logic is inherently interdisciplinary, with representationsand analyses of preference varying across different fields. But thereis a distinct common feature—preference is concerned withcomparing situations, objects, or actions. The binary relationsinvolved in such comparisons are: “A is better thanB” (\(>\)), “A is equally as good asB” (\(\equiv\)), and “A is at least asgood asB” (\(\geq\)).

Preference logic began with syntactic approaches, which focused on theformal principles governing preference relations. In contrast,semantic approaches aim to provide interpretations of these relations,which enables the identification of conditions under which suchprinciples hold. In particular, possible world semantics offers asystematic framework for exploring preference logic from a semanticperspective. The transition from syntactic to semantic approaches isreflected in this entry.

When agents make comparisons, the resulting preferences may eitherreflect their own tastes, without any external justification, or theymay derive from specific reasons that are externally justified.Preferences formed purely on the basis of the agent’s own tastesare calledintrinsic preferences. In contrast, preferencesthat can be explained or justified by external reasons are referred toasextrinsic preferences. In this entry, this distinction,famously made by von Wright (1963), serves as a central theme tointroduce both classic and contemporary works in preference logic.

Extrinsic preferences require additional elements in a model in orderto represent both preferences and the reasons that justify them. Suchreasons can include priorities, object properties, or values. Severalmajor frameworks have been proposed in logic and rational choicetheory, where preferences were traditionally treated as fixed in bothdisciplines. The term “two-layer structure” will be usedto describe these frameworks. This richer modeling approach enablesreasoning about not only the preferences themselves but also theunderlying reasons. Interestingly, a parallel development can beobserved in deontic logic, where norm-based semantics introduces anadditional structure to deontic notions, thus aligning closely withthe ideas underpinning extrinsic preference models.

Another key topic in preference logic is the study of preferencechange, aligning with broader trends in incorporating new information.Just as attitudes such as knowledge, belief and intention may need tobe updated, preferences may evolve with new information, changes inthe agent’s state, or other factors. This entry assumesfamiliarity with belief revision theory (see SEP entry onpreferences) and dynamic epistemic logic (DEL; see entry ondynamic epistemic logic) and focuses specifically on the unique aspects of preferencechange.

In AI systems where decisions must be made, preferences often need tobe represented concisely. One central notion in this context isceterisparibus preference, which enables reasoningabout preferences such as “A is preferable toB, all else being equal”. This concept has been studiedextensively in logic and also forms the basis for frameworks likeCP-nets (conditional preference networks) in the computer scienceliterature. CP-nets is a well-known visual language that balancesexpressive power with computational efficiency. Accordingly, thisentry devotes significant attention toceterisparibus preferences.

2. Syntactic approaches to preference logic

Pioneering logicians such as Halldén (1957) and von Wright(1963) proposed logics for preferences that focused on comparisonsbetween situations. These logics extended propositional language witha binary connective, \(>\), denoting the concept of preference or“betterness”.

2.1 Halldén: two theories, A and B

Halldén’s (1957) bookOn the Logic of“Better” is often taken as the starting point for alogical analysis of preference. It is a point of reference for allother preference logics. Halldén extended the propositionallogic with two new connectives—what he called “ethicalconnectives”: “better” (\(>\)) and “equallyas good as” (\(\equiv\)). He distinguished between the positiveand negative sense of \(\equiv\), which then led to Theory A andTheory B. Theory A is based on thepositive sense and isformally given by \(p\equiv q\) iff for allr,\(r>p\) iff \(r>q\), and \(p>r\) iff \(q>r\). Theory B isbased on thenegative sense, where \(p\equiv q\) simply means\(\neg(p>q)\land \neg(q>p)\).

The sentences, built up from propositional variables and ordinarytruth-functional connectives as well as the two new connectives, arecalled “A-formulas”. They follow the usual convention ofleaving out brackets, with the reading then based on decreasingbinding order for \(\neg\), \(>\), \(\equiv\), \(\land\), \(\lor\),\(\rightarrow\), \(\leftrightarrow\).

Halldén proves that Theory A is consistent. He presents severaltheorems of theory A such as contraposition: \(p>q \rightarrow\negq>\neg p\). He also considers fourteen optional postulates that canbe added to Theory A. For example, there is the distributionpostulate, \(p>(q\land r)\rightarrow p>q \land p>r\), and theconnectedness postulate, \(p>q \lor p\equiv q \lor q>p\).

Theory B consists of the rules and postulates inFigure 2. Postulates \(B_1–B_5\) are identical to \(A_1–A_5\), \(B6\) can bederived from Theory A, and \(B_7\) represents connectedness.

Theory B has been proven to be consistent. Interestingly, a procedureis provided to decide whether a given formula is a theorem of B. Itconsists of two steps calledunivocal expansion andmultivocal expansion.

In the first phase, logical principles are applied to expand theformula. For example, \(p\equiv q\) is replaced by \(p\equiv q \land q\equiv p\), and \(p> q\land q>r\) is replaced by \(p> q\landq>r\land p>r\). Consider the following sequence of replacementsfrom Halldén (1957) to determine whether

is a theorem:

The last formula is contradictory, thus we can conclude that \(p\equiv\neg p\land q\equiv \neg q \rightarrow \neg (p>q)\) is atheorem.

If the first phase is not decisive, each formula is expanded into aset of more complex formulas in a second phase. For example,\(p>q\land \neg q> \neg p\) cannot be further expanded usingunivocal expansion, but it may be expanded into this set offormulas:

reflecting all possible relations betweenp and\(\neg p\). Interested readers can find more details on how it worksin Halldén’s (1957) book.

2.2 Von Wright’s logic of preference

The second point of reference isThe Logic of Preference byvon Wright (1963). In this book, von Wright distinguishes betweenthree groups of concepts related to preferences while also noting thatthere are no sharp divisions between them. The relevant conceptgrouping is: deontological concepts (right, duty, command,prohibition), axiological concepts (good, evil, betterness), andanthropological concepts (need, want, decision, choice). These notionsare explored in deontic logic and decision theory, and they intersectwith preference logic. The connection will be discussed inSection 7. Von Wright also makes key distinctions that are relevant to the laterdevelopment of preference logic:

• preferences between states of affairs vs. preferences over meansor methods,
• intrinsic preference (intrinsic betterness) vs. extrinsicpreference (preference with reasons),
• qualitative preference vs. measurable value (e.g., utility),and
• risk-free preference vs. preference under uncertainty.

More concretely, the formal language adopted by von Wright is again afragment of Halldén’s A-formulas, and it consists of onlythose sentences that do not contain nested occurrences of \(>\) and\(\equiv\). However, in contrast to the language of Theory B, if\(\varphi>\psi\) or \(\varphi\equiv\psi\) occurs in a formula, then\(\varphi\) and \(\psi\) may contain conjunction or disjunctionsymbols.

Unlike Halldén’s two theories, the logic of preference isnot presented as an axiomatic calculus. Instead, it serves as atechnique for determining whether a given preference expression(P-expression) represents a logically true proposition.

The logic of preference is based on five key principles, shown inFigure 3. The first two—asymmetry and transitivity—describerelations among preferences and are derived fromHalldén’s systems. The other three—expansion,disjunctive distribution, and amplification—allow anyP-expression to be transformed into a standard form, which enablesapplication of the decision technique. This technique is illustratedwith an example. Consider the P-expression

Using expansion we obtain

As there is no disjunction in the atomic preferences, distribution isvacuous. The third step, amplification, gives the normal form

This formula is a tautology, which can be explained as follows. Theformula would be false if and only if the first four constituents

were all true and \((p\land \neg q>\neg p \land q)\) were false.But this conflicts with transitivity.

Looking at the five principles concerning preference, one has to becareful because they are not closed under substitution. The reasoningbehind this limitation is explained below.

Another contribution by von Wright is his attempt to understand themeaning of preference in terms of “changes”. His analysisreveals a unique way of viewing preference. When can one say that onestate of affairs is preferable to another? To answer this question,consider two generic preference states,p andq.There are four possible cases: \(p \land q\), \(p\land \neg q\), \(\neg p \land q\), and \(\neg p \land \neg q\). Anyworld must necessarily fall into one of these four possibilities. Thisexplains not only the first three principles, which had already beenadopted by Halldén, but also the principle of disjunctivedistribution, illustrated by \(V_4\) in the table above.

Principle \(V_5\) is the so-calledholistic property ofpreference. Let \(\varphi\) be a formula, and let \(PL(\varphi)\) bethe set of propositional letters that occur in \(\varphi\), which vonWright calls the universe of discourse. Suppose \(r \not\inPL(\varphi>\psi)\), then replace every formula \(\varphi> \psi\)with the conjunction

This principle, also calledamplification, is applied toeveryr in the complement of \(PL(\varphi >\psi)\) with respect to the set of propositional letters.Amplification ensures that everyr in the universeof discourse that is not directly relevant to the evaluation of apreference subformula will remain constant.

While \(V_1\), \(V_2\), \(V_3\) and \(V_4\) appear in many axiomaticpreference logics, this is not the case with \(V_5\). The reason isthat \(V_5\) cannot be combined with substitution.

Since Halldén, numerous principles have been explored in theliterature. For a comprehensive review see S. O. Hansson’s(2001) chapter on the subject, which includes the followingexamples:

Both Halldén and von Wright emphasize the generality of thepreference principles by adopting a syntactic or algorithmic approach.Later researchers have often used examples to support or challengethese principles, and many preference properties remain unsettled.

3. Semantic approaches to preference logic

This section provides a semantic perspective on preference byintroducing an earlier proposal by Rescher (1967) and Van Dalen (1974)as well as possible world semantics, which has been central to laterresearch in this area.

3.1 Rescher-Van Dalen semantics

Rescher (1966, 1967) introduced the first semantics for preferencelogic where models are real-valued functions on formulas. Van Dalen(1974) generalized this to partial orders, and he showed that thepreference logic of Rescher is a special case. The semantic structuresdirectly reflect the preferences expressed in the language. We presenthere the semantics and axiomatizations of Van Dalen.

The language coincides with the language of von Wright, i.e., it is anextension of propositional logic with the binary connective \(>\),except in this case, it lacks nested preference statements.Indifference is defined in terms of preference as \(\varphi\equiv \psi= \neg (\varphi>\psi)\land \neg (\psi>\varphi)\).

In order to assign truth values to formulas of the form\(\varphi>\psi\), Van Dalen prescribes an order relation betweenpropositions. He simultaneously considers a truth valuationvand a measure of goodnessm. Inthis framework,m is a mapping of the set ofpropositions into a partially ordered setA, andv is a mapping of propositional symbols (atoms)into \(\{0,1\}\), which is extended to all formulas by the followinginductive definition:

1. \(v(\neg \varphi) = 1-v(\varphi)\)
2. \(v(\varphi\lor\psi) = \textrm{max}\{v(\varphi),v(\psi)\}\)
3. \(v(\varphi P\psi) = 1\) if \(m(\varphi) \lt m(\psi)\) onA,or 0 otherwise

For a set \(\Gamma\) of formulas, the satisfaction relation is definedby:

Van Dalen calls the semantics described above irrational semantics,mainly because logically equivalent propositions can lead to differentpreferences. He therefore considers three extra conditions to themeasurem or the partially setAin order to get semantics of a more restricted kind:

1. If \(\varphi\equiv \psi\) is a tautology, then \(m(\varphi) = m(\psi)\).
2. A is linearly ordered.
3. A is the set of reals and\(m(\neg \varphi)=-m(\varphi)\).

Van Dalen presents several theories—PREFIR,PREF,PREFL, andPREFR—outlined inFigure 5. He demonstrates thatPREFIR is complete for irrational semantics,PREF is complete for semantics withcondition (i),PREFL is complete for linear semantics, andPREFR is complete forRescher semantics.

Moreover, Van Dalen shows that all systems have the finite modelproperty and are decidable. He also extends the language with monadicoperators for good, bad and neutral, and he considers nestedpreference statements. Other authors like Chisholm and Sosa (1966)consider first-order extensions of such preference logics.

3.2 Preference over possible worlds

In an era of rapid advancements in modal logics (see entry onmodal logic), binary preference relations have been incorporated naturally intopossible world semantics to enable the comparison of differentsituations or worlds in various contexts. This subsection and the nextreview modal preference logic and discuss various methods for liftingpreferences in order to compare sets of possible worlds.

Dyadic modal logic has been developed since the late 1960s to modelconditional obligations (B. Hansson 1969) along with relatedformalisms for conditionals (Lewis 1973; Makinson 1993). In theseframeworks, a preference structure underlying possible world semanticswas employed, particularly in studies of deontic logic andconditionals.

Starting in the late 1980s, a widely adopted monadic modal approachemerged that explored preferences within non-monotonic logic andbelief revision (Boutilier 1990; Lamarre 1991) as well as deonticlogic (Tan & Van der Torre 1996). While the motivations behindthese works vary, the monadic modal approach in some cases offers amore straightforward path to axiomatizing preference logic.

Research on preference logic has frequently been linked tounderstanding agent choices in dynamic environments. A central focusis how agents rank possible worlds in order to express theirpreferences. This allows for more nuanced models of decision-makingand normative reasoning. The integration of preference orders intomodal logic, inspired by earlier developments in conditional logic andbelief revision, extends the expressive power of logical systems byincorporating preference structures over possible worlds.

A modal preference model \(\mathcal{M}\) is a triple \((W, \succeq,V)\) whereW is a set of possible worlds,\(\succeq\) is a binary preference relation over these worlds, andVis a valuation assigning truth values topropositional letters in these worlds. Depending on the context, weassume the preference relation \(\succeq\) has certain properties. Forinstance, it can be a (total) pre-order, or a partial order, etc. Ifwe choose (total) pre-order, a strict preference relation“\(w\succ v\)” can then be defined as “\(w\succeqv\land v \nsucceq w\)”, and an indifference relation“\(w\approx v\)” can be defined as “\(w\succeqv\land v\succeq w\)”.

On the syntactic side, a modal preference language is given to expressthe basic preference relations and their properties. The language isan extension of propositional logic, with two new operators\(\lozenge^{\geq}\varphi\) and \(\lozenge^{>}\varphi\) (VanBenthem, Girard, & Roy 2009). Their truth conditions are definedbelow:

The theory shown inFigure 6 below is an axiomatization of the modal preference logic given by VanBenthem, Girard, and Roy (2009) where \(\succeq\) is a transitive andreflexive relation and E is an existential modality, interpreted asfollows:

3.3 Preference lifting

The modal preference language described above can express theproperties of preference relations between possible worlds. Butaddressing relationships between sets of possible worlds presents alifting problem (Thomason & Horty 1996) as it involves extending aproperty that applies to individual worlds so that it also applies tosets of worlds. There are various ways of achieving this. The threeexamples below use only two quantifiers.

On the syntactic side, to express the lifting, the basic modalpreference language is extended with the universal modality\(U\varphi\) (and its dual \(E\varphi\)) (Boutilier 1992). The truthdefinition for this new modality is:

\(\mathcal{M}\), \(w \models U \varphi\)    iff   for allv  , \(\mathcal{M}\),\(v \models \varphi\).

With this modality, the language gains greater expressive power todefine generic preference notions related to the lifting problem. Weprovide one example:

We can read \(\varphi\geq^{\forall\exists}\psi\) as “for each\(\varphi\)-world, there exists a \(\psi\)-world which is as good asthat \(\varphi\)-world”. We can define other notions of genericpreference in a similar manner. For instance,\(\forall\exists\)-preference and \(\exists\exists\)-preference can bedefined in the language as follows:

1. \(\varphi\geq^{\forall\exists}\psi \coloneqU(\varphi\rightarrow\langle\geq\rangle\psi)\)
2. \(\varphi\geq^{\exists\exists}\psi \coloneqE(\varphi\land\langle\geq\rangle\psi)\)

Van Benthem, Girard, and Roy (2009) argue that\(\geq^{\forall\forall}\) underceteris paribusrepresents the notion of “preference” intended by vonWright (1963) in his seminal work on preference logic and offers anaxiomatization for it. The logic ofceteris paribuspreference will be discussed inSection 6.

Other attempts to axiomatize logics for preference relations over setsof possible worlds have emerged across various research areas. Notablecontributions include the works of Van Fraassen (1973), Lewis (1973),Spohn (1988), Boutilier (1990) and Halpern (1997) among many others.An illustrative example is provided below.

Halpern (1997) proposed a complete axiomatization for the logic of“relative likelihood”. It was structurally similar to thelogic of preference, or simply “preference” as he termedit. Unlike earlier axiomatizations, he accounted for both total andpartial pre-orders. The language used is the same as that employed byvon Wright—it excludes nested preference formulas—exceptthat the atomic preference formulas \(\varphi>\psi\) may includeconjunctions and disjunctions.

Given a preference model \(\mathcal{M}=(W, \succeq, V)\), thepreference formula is interpreted as follows:

\(\mathcal{M} \models \varphi > \psi\) iff \(\llbracket \varphi\rrbracket _{\mathcal{M}}\) is nonempty, \(\forall v \in \llbracket\psi \rrbracket _{\mathcal{M}}\exists u \in \llbracket \varphi\rrbracket _{\mathcal{M}}\) such that \(u\succ v\) andudominates \(\llbracket \psi \rrbracket_{\mathcal{M}}\)

andu dominatesV if there is no\(v' \in V\) such that \(v'\succ u\).

The complete axiomatization of Halpern (1997) when \(\succeq\) is apartial order consists of the axioms and rules listed inFigure 7. Axiom 2 corresponds to the reflexivity of \(\succeq\), and axiom 3corresponds to the transitivity of \(\succeq\). In the case of a totalpreorder, the axiom \(\varphi_1>\varphi_2\rightarrow(\varphi_1>\varphi_3)\lor(\varphi_3>\varphi_2)\) must beadded.

A recent paper by Shi and Sun (2021) explores an alternative method ofpreference lifting known as the Egli-Milner order. This bidirectionalapproach, formally defined below, considers both the best and worstelements across sets, aligning with the maximin principle of favorablybalancing the worst outcomes:

where \(\llbracket\varphi\rrbracket_w\) denotes the set ofw-accessibleworlds that satisfy \(\varphi\).Traditionally applied in fields like denotational semantics, Shi andSun (2021) extend the Egli-Milner order to preference logic bydeveloping a complete axiomatic system for the “convexorder” derived from a total pre-order.

4. Reason-based preference

Over the last two decades, the study of the logic of reason-basedpreference has been a primary focus of several research effortssurveyed in this section.

4.1 Intrinsic vs. extrinsic preferences

The concept of reason-based preference originates from a well-knowndistinction between intrinsic and extrinsic preferences introduced byvon Wright (1963). This distinction hinges on whether a preference isgrounded in an explicit reason or justification.

Anextrinsic preference arises when one option is preferredover another because it is regarded as better in a specific respect.For example, in von Wright’s example, one person prefers claretover hock because it is beneficial for her health. Here, thepreference for claret is extrinsic since it is based on an objectivejudgment about health benefits. Extrinsic preferences are thus tied tosome underlying rationales, often involving external factors likehealth, practicalities, or expert advice.

In contrast, anintrinsic preference is a preference basedpurely on personal liking, without any need for an explicit reason orjustification. For instance, one person may prefer claret to hocksimply because she likes the taste. This preference is intrinsicbecause it is driven solely by subjective enjoyment, with no referenceto any external rationale.

In formal models of extrinsic preference, atwo-layerstructure is typically adopted to account for the reason base andthe preference.

4.2 Two-layer structures

4.2.1 Priorities and preference

In the context of deciding between options, the notion of prioritieswas introduced (Liu 2008, 2010; De Jongh & Liu 2009). Prioritiesrepresent important properties ordered according to anindividual’s considerations, which in turn determine preferenceamong choices.

A two-layer structure consists of a priority base and preferences. Apriority base is a set of properties ranked according to anagent’s considerations. Consider the example of Alice, who isdeciding which house to buy. She evaluates each house based on threecriteria—cost, quality, and neighborhood—in that order ofimportance. For Alice, the cost of a house is good if it fits herbudget; otherwise, it is bad. Her decision is thus guided by whetherthe house meets her prioritized criteria and the relative importanceof those criteria.

To formalize this, a first-order logic is used with constants \(d_0,\)\(d_1,\)…, variables \(x_0,\) \(x_1,\)…, and predicates\(P,\) \(Q,\) \(P_0,\) \(P_1,\)…. For simplicity, this setupassumes a finite domain with monadic predicates and simple, usuallyquantifier-free formulas.

Formally, a priority base is an ordered sequence of formulas, writtenas

where each \(C_m(x)\) (\(1 \leq m \leq n\)) is a formula with exactlyone free variablex. The earlier priorities areconsidered lexicographically more significant than later ones,creating a preference order. For example, \(C_1 \land \neg C_2 \land\dots \land \neg C_m\) is preferred over \(\neg C_1 \land C_2 \land\dots \land C_m\).

To illustrate this, Alice’s priority base might look like

where \(C(x)\), \(Q(x)\), and \(N(x)\) represent thatxhas a low price (cost), high quality, and a goodneighborhood, respectively. Given two houses \(d_1\) and \(d_2\) withproperties \(C(d_1),\) \(C(d_2),\) \(\neg Q(d_1),\) \(\neg Q(d_2),\)\(N(d_1)\), and \(\neg N(d_2)\), Alice would prefer \(d_1\) over\(d_2\) according to this priority base.

The linearly-ordered priorities in the two-layer structure can also begeneralized to partially ordered priority graphs (which allows certainpriorities to be non-comparable) or even to unordered sets ofproperties (Kratzer 1981; Liu 2011b).

In deontic logic, the two-layer structure (where priorities establisha preference or obligation ordering) has also been widely recognized,particularly in systems that prioritize commands (see the work ofHansen 2008 and Van Benthem, Grossi, & Liu 2010), which will bediscussed inSection 7.

Belief as reasons

The above method of defining preference from a priority basepresupposes complete information about whether an alternativepossesses the properties in the base. In cases of incompleteinformation, beliefs play a crucial role, and agents’preferences depend on them. This situation requires the introductionof formulas such as \(BC(x),\) \(\neg BC(x),\) or \(B\neg C(x),\)where \(C_m\) represents priorities. This approach enables complexaspects to be considered, e.g., determining whether certain propertiesfrom the priority base apply or, more fundamentally, forming apriority base based on beliefs. Addressing such uncertainties callsfor a combination of doxastic and preference languages; see the workof Liu (2011b) for various procedures for defining preference underuncertainty. In this setting, beliefs about whether an alternative hasor doesn’t have a certain priority serve as reasons forpreference.

4.2.2 Salient properties and preference

Rational choice theory typically assumes that agents have fixedpreferences but does not explain the origin of these preferences.Dietrich and List (2013a,b) started to address this gap. They proposeda framework for modeling preference formation and change by suggestingthat preferences are based on the “motivationally salient”properties of alternatives rather than the alternatives themselves.This approach allows preferences to be reason-based, evolving asdifferent properties become salient.

To formalize this: an agent’s preference order over alternativesdepends on the properties of these alternatives, which aremotivationally salient in a given state.

LetX be a set of alternatives. LetPbe the set of all possible properties an alternativemight have. A motivational state \(M \subseteq P\) includes all theproperties deemed salient by the agent in a given context.

Preferences are defined by a binary relation \(\succeq_M\) overX,where \(x \succeq_M y\) means alternativexis weakly preferred toy in stateM.

To account for varying preferences across motivational states,Dietrich and List (2013a,b) introduced theweighing relation\(\geq\) overproperty combinations:

• Each alternativex can be described by the setof properties it has, denoted as \(S_x\).
• For any two sets of properties, \(S_x \geq S_y\) indicates thatthe properties in \(S_x\) are preferred over those in \(S_y\).
• The weighing relation thus acts as a stable underlying structurethat supports the agent’s preference ordering across differentmotivational states.

Formally, ifx andy arealternatives, then the agent’s preference \(x \succeq_M y\) instateM can be represented as follows:

where \(S_x = \{ P \in M : x \text{ has } P \}\) and \(S_y = \{ P \inM : y \text{ has } P \}\). This relation ensures that theagent’s preferences in any motivational stateMreflect the agent’s stable evaluation ofproperty combinations across contexts.

The works of Dietrich and List (2013a,b) explore rationality bycontrastingstructural rationality, which focuses on internalconsistency in preferences (as in traditional economics), withsubstantive rationality, which evaluates the content orworthiness of preferences. Formal rationality demands logicalcoherence, such as the preference ofA overCifA is preferred overBandB overC.Substantive rationality, however, considers preferences to be“irrational” if they are self-destructive or harmful, evenif they are consistent, as it assesses the value of the preferences,not just their coherence. An earlier discussion on these philosophicalissues can be found in the work of Pettit (2002).

4.2.3 Reasons and preference

Osherson and Weinstein (2012) propose a logic of preference wherepreferences are not arbitrary or merely consistent but are based onidentifiable reasons. This framework introduces modal connectives thatallow the reasons for desiring a certain outcome to be expressedformally, which provides a way to model how various reasons combine toshape an agent’s preferences.

When choosing whether or not to install a fire alarm, a person mayconsider safety and cost as different reasons that support or opposethat course of action. In this example, a preference for installing afire alarm puts great value on increasing safety, so this preferenceis based on a “safety reason”. If finances are tight, thecost may serve as a reason against installing the alarm.

Each reason (e.g., safety or cost) can be quantified usingutilityscales representing how much better or worse each choice is,based on a specific criterion. Formally, a “utilityfunction”u is defined as a mapping

whereW is the set of possible worlds andSindexes utility scales such as safety or cost.

To evaluate a choice such as whether or not to install a fire alarm,the model uses a “selection function”sto identify alternative worlds that are closest to the real world forcomparison purposes. For example, it compares

• world \(w_1\), where a fire alarm is installed (a safer world),and
• world \(w_2\), where no fire alarm is installed (the current, lesssafe world).

Formally, the “selection function” \(s(w, A)\) selects arepresentative world from setA based on itssimilarity to the current worldw:

Finally, the preference relation can be formally defined using modalexpressions. For instance, \(p \succ_1 \neg p\) holds if, for safetyreasons, installing an alarm (represented byp) ispreferred over not installing one, denoted by the utility scale\(u_1\) for safety. If \(u_2\) (cost) is also considered, thepreference relation may combine both \(u_1\) and \(u_2\), therebycapturing a comprehensive view that balances safety and cost.

This logic framework enables structured reasoning that incorporatesspecific reasons behind preferences, allowing for a nuancedrepresentation of choices grounded in real-life decision-makingcontexts.

Osherson and Weinstein explore decidability issues and variousconditions for subclasses of models.

Two-layer structures have become a standard approach to modelingpreferences with reasons, with potential applications in variousfields. A recent example is preferences that stem from underlyingcausal considerations—see Xie and Yan’s (2024) work onthis topic. In a similar vein, Chen, Shi, and Wang (2024) integratespreference and dependence into cooperative games, providing a unifiedperspective on concepts like Nash equilibrium, Pareto optimality, andthe Shapley core.

5. Dynamics of preference change

Preferences can change due to various triggers, making preferencechange a natural phenomenon to consider. Logical modeling has thusbeen proposed as a way to study reasoning about these dynamics. Thissection partially follows the categorization of preference changeoutlined by Lang and Van der Torre (2008), which is based on the typeof input driving the change. Additionally, this section highlights thedynamical changes that can occur in two-layer structures.Grüne-Yanoff and Hansson (2009) discussed why preference changehas been overlooked in the social sciences, particularly in economics,and the volume of essays presents various approaches to studyingpreference change (see also the section on preference change in theentry onpreferences). This section examines the primary ideas and formal frameworks withinthe logic tradition.

5.1 Intrinsic preference revision

The first type of preference change is analogous to belief change:just as belief revision incorporates newly acquired beliefs into anexisting belief state, intrinsic preference revision incorporates newpreferences into an existing preference state. Here, preferences arerevised by other preferences to form new preferences, without beliefsintervening.

Two main methodologies are used to model preference change: theAlchourrón, Gärdenfors and Makinson (AGM) postulationalframework (see the description of the AGM belief revision theory inthe entry onlogic of belief revision) and dynamic logic-based systems (see entry ondynamic epistemic logic). Both have been widely adopted by researchers to explore various formsof preference change.

Sven Ove Hansson (1995) provided a detailed analysis of preferencechange, identifying four distinct types within the AGM framework:

• revision: acquiring a preference for \(\varphi\)over \(\psi\) with the revision statement “\(\varphi\) ispreferable to\(\psi\)”.
• contraction: removing the preference that“\(\varphi\) is preferable to\(\psi\)”.
• addition: adding a new alternative to the set ofoptions considered.
• subtraction: removing an option from the set ofoptions considered.

While belief revision assigns priorities to minimize changes to priorbeliefs, Hansson argues that priorities are unsuitable for preferencerevision. In preference revision, accommodating a new statement like\(\varphi\succeq \psi\) can involve adjusting the position of either\(\varphi\) or \(\psi\). He also proposes a similarity measure to keepthe revised preference model as close to the original as possible,along with AGM-style postulates for these operators.

Van Benthem and Liu (2007) developed an extension of dynamic epistemiclogic as a way to reason about preference change. A“suggestion” operator was introduced, denoted as\(\sharp\varphi\). The formula \([\sharp\varphi]\psi\) means“After \(\varphi\) is publicly suggested, \(\psi\) holds”,and is defined as:

where \(\mathcal{M}_{\sharp \varphi}\) is defined with the samedomain, valuation, and actual world as \((\mathcal{M}, s)\) but withthe preference relations updated by subtracting the preference linksfrom \(\varphi\)-worlds to \(\neg \varphi\)-worlds:

More generally, in this framework, suggestion \(\sharp(\varphi)\) canbe understood as a relational program in propositional dynamic logic(PDL) (cf. Harel, Kozen, & Tiuryn 2000), with

indicating that all existingR links are retainedwith the exception of those links from \(\varphi\)-worlds to \(\neg\varphi\)-worlds. This setup allows for different types of preferencechange. For example, adding preference links (making every\(\varphi\)-world preferable to every \(\neg \varphi\)-world) can berepresented as

where \(\top\) denotes the universal relation.Radical preferenceupgrade is defined as

which makes all \(\varphi\)-worlds preferable to all \(\neg\varphi\)-worlds while preserving the original ordering within eachgroup.

This dynamic preference logic extends DEL by supporting variousrelation-changing operators, and Van Benthem and Liu (2007) providecompleteness theorems for these operators. Their results leverage thePDL format, which enables reduction of dynamic modalities andrecursive computation of reduction axioms.

5.2 Preference change driven by belief update

The second type of preference change arises in response to changes inbelief and reflects the dynamic interplay between what an agentbelieves and how they evaluate options. This topic has gainedreasonable attention (see the work of Bradley 2007, Lang & Van derTorre 2008, and Liu 2011b). For instance, following the tradition ofdecision theory established by Savage (1954), Bradley argues thatpreference change can arise from a change in beliefs where“preference change is induced by a redistribution of beliefacross some partition of the possibility space”. He formalizesthis principle using a Bayesian approach. Consider the followingexample by Lang and Van der Torre (2008):

Initially, I desire to eat sushi from this plate. Then I learn thatthis sushi has been made with old fish. Now I desire not to eat thissushi. (2008: 351)

Learning that the sushi was made with old fish changes my belief,which then reverses my initial preference. Here, preference change isa consequence of belief change, making belief change the key elementto model.

The AGM framework addresses belief revision by establishing postulatesfor how a belief set should adapt to new, possibly conflicting,information. Lang and Van der Torre (2008) apply a similar approach topreference change resulting from belief updates. The formal languageproposed includes: a dyadic modal operator for belief,\(N(\alpha\mid\beta)\), meaning “\(\alpha\) is normal (orbelieved) given\(\beta\)”; apreference operator, \(P(\alpha\mid\beta)\), meaning “\(\alpha\)is preferred given\(\beta\)”; and adynamic operator,\([*\alpha]\beta\), readas “after learning \(\alpha\), \(\beta\) holds”. Tocapture the properties of preference change, they propose eightpostulates. Two examples are given below:

(P1) means that learning that something is already preferred does notalter the initial preference. For instance, if I desire wealth, thenupon becoming wealthy, this preference remains. This persistencegenerally feels natural, but is debated; in Jeffrey’s decisiontheory, for example, once you become wealthy, the desire to becomewealthy may no longer apply.

(P5) implies that learning that something that was expected isactually the case should not alter existing preferences. Thisprinciple, equivalently stated as \(P\alpha \land \neg[*\beta]P\alpha\rightarrow \neg N\beta\), expresses that unexpected information isnecessary for preference change to occur.

In summary, belief updates drive preference changes, whichdemonstrates their close relationship. Frameworks like the AGM theoryand dynamic logic systems provide tools for modeling thisevolution.

5.3 Two-layer models of preference change

In a two-layer structure, the priority base can shift due to changesin the relative importance of factors naturally leading tocorresponding changes in preferences. De Jongh and Liu (2009) examinedthe interplay between a linear priority base and preferences andshowed how priority shifts influence preference relations. In arelated context, Andréka, Ryan, and Schobbens (2002) exploredmethods for deriving preferences from various priority orderingswithin a more general, non-linear graph structure. Building on theseideas, Girard (2008) and Liu (2011a) introduced basic graph updateoperations that offer a formal framework for analyzing changes inpriority structures. Christoff, Gratzl, and Roy (2021) studied theproof theory of priority merge in labeled sequents.

5.3.1 Graph updates: basic operations

Suppose there is a priority base, denoted as \(\mathscr{B}\), which isa set of prioritized propositions. When a new propositionAis added, there are straightforward options forincorporating it:

• giveA the highest priority:A can be placed at the top of the priority base tomake it the most important proposition.
• giveA the lowest priority:A can be added to the bottom to give it the lowestpriority.
• side by side:A can be placedalongside \(\mathscr{B}\) without establishing an order betweenAand the other priorities.

The set \(\alpha(\mathscr{B}, A)\) of basic graph updates is definedin a way that allows these possible changes to be captured:

Here, \(\mathscr{B}_1 ; \mathscr{B}_2\) represents the sequentialcomposition of adding \(\mathscr{B}_1\) on top of \(\mathscr{B}_2\) sothat every element in \(\mathscr{B}_1\) has a higher priority thanevery element in \(\mathscr{B}_2\). Parallel composition,(\(\mathscr{B}_1 \parallel \mathscr{B}_2\)), combines\(\mathscr{B}_1\) and \(\mathscr{B}_2\) as disjoint sets with nopriority ordering between them.

A priority base can be modified not only by inserting new propositionsbut also with deletion operations that remove priorities. For example,top deletion removes any items from the priority base that are not“dominated” by others, i.e., those that have the highestpriority level.

These basic graph updates provide simple ways to restructure a givenpriority base by adding or removing propositions.

5.3.2 Tracking dynamics

To track changes across different levels—at the world layer andthe priority layer—we define a specific technical notion. Givena priority update \(\alpha: (\mathscr{B}, A) \rightarrow\mathscr{B'}\) and a world-level map \(\sigma: (\preceq, A)\rightarrow \preceq'\), it is said that \(\alpha\)induces\(\sigma\) if:

An operation \(\alpha\) is aPDL-definable operation if itinduces aPDL-definable relation transformer \(\sigma\) byusing tests for formulas in the language, weak and strict basicpreference relationsR, and the universal relation,while allowing arbitrary unions and sequential compositions:

These are interpreted as the standardPDL program operationsoftest \(?\varphi\),sequential composition\(``;"\) andchoice \(``\cup"\).

In the two-layer framework, it is interesting to see how prioritydynamics can mirror preference dynamics. For example, it is possibleto achieve a preference change due to a “suggestion”operator either by altering the world-layer preference relation or bymodifying the priority base. If a new propositionAis added in parallel to an existing base \(\mathscr{B}\), thepreference relation is updated, as shown in the following commutingdiagram:

Diagram A: Commutative diagram where new proposition A is added in parallel to an existing base [Anextended description of Diagram A is in the supplement.]

Similarly, placing a new propositionA at the topof the priority graph \((\mathscr{B}, <)\) aligns with a radicalupgrade at the world level:

Diagram B: Commutative diagram where new proposition A is placed at the top of the priority graph [Anextended description of Diagram B is in the supplement.]

These examples illustrate cases where priority-layer transformationscorrespond seamlessly with world-layer relation updates.

While priority dynamics and preference dynamics will often align,there are limits. For instance, the deletion operation\(\textit{del}(\mathscr{B})\) is notPDL-definable (Liu2011a). Certain natural priority-layer modifications cannot be fullyrepresented at the world layer.

5.3.3 Belief-driven preference change in two-layer structures

In two-layer structures, it also makes sense to consider preferencechanges driven by belief updates. Within the framework of De Jongh andLiu (2009), various strategies are employed to derive preferences frompriorities and beliefs, as illustrated in the following example:

Alice is looking for an apartment. She regards price as more importantthan neighborhood. She believes that apartment \(d_1\) has a low pricebut is in a bad neighborhood. She hasno information aboutthe price of apartment \(d_2\) but believes it is in a goodneighborhood.

In situations involving uncertainties, the “decisionstrategy” compares two options by prioritizing the mostimportant criterion that each option is believed to satisfy. In thiscontext, Alice would prefer \(d_1\) over \(d_2\) based on herbeliefs.

However, if her beliefs change (e.g., if she learns that \(d_2\) has alow price), her preference may shift so that she favors \(d_2\) over\(d_1\). This highlights the distinction between preference changesdue to belief updates and those resulting from priority shifts, as inthe previous examples.

In conclusion, priority dynamics and preference dynamics, thoughclosely related, retain unique features, underscoring the value ofmodeling both intrinsic and reason-based preferences for acomprehensive understanding of agent decision-making.

6.Ceteris paribus preference and CP-nets

Ceteris paribus preference refers to a way of expressingpreferences under the condition of “all else being equal”.It allows agents to state that one option is preferred over anotherwhile assuming that all other relevant factors remain constant. In AIsystems for decision-making, a CP-net is a graphical model used torepresent conditional preferences. It relies on the principle thatcomparisons between options are made under the assumption that otherfactors are unchanged, which aligns closely with the concept ofceteris paribus.

This section introduces results on the logic ofceterisparibus preferences along with the fundamentals of CP-nets.

6.1 Formalizingceteris paribus preferences in logic

The core problem ofceteris paribus preference is how tointerpret “all else being equal”. For example, one man maygenerally prefer red wine over white wine,ceteris paribus,but not if the red wine is poisonous and the white wine is not. So thechallenge of formalizing aceteris paribus proviso is how todetermine whether the conditions are the same except for the color ofthe wine. This issue was discussed in the first studies of preferencelogic by Halldén (1957) and von Wright (1963). The differencebetween the equality reading ofceteris paribus and thenormality reading “all else being normal” was alsodiscussed by Van Benthem, Girard, and Roy (2009).

When interpreting the phrase “all else being equal”, anatural approach is to use equivalence classes of worlds. Thisapproach has been taken by many researchers in the field. Forinstance, Doyle and Wellman (1994) defined context-dependentequivalence relations among individuals, which is called contextualequivalence. As a consequence, this means that the equivalence classmay vary according to the context under consideration.

Formally, let \(\Omega\) be a set of states, and let\(\mathcal{E}(\Omega)\) denote the set of all equivalence relations on\(\Omega\). The contextual equivalence is defined on a set \(\Omega\)as a function \(\eta\):\(\mathcal{P}(\mathcal{P}(\Omega))\rightarrow\mathcal{E} (\Omega)\)assigning to each set of proposition \(\{p, q, \ldots\}\) anequivalence relation \(\eta(p, q, \ldots)\). For\(w\,\eta(p,q,\ldots)\, w^{\prime}\) , we write \(w\approx_{ mod\,p,q,\ldots} w^{\prime}\), meaning that with regard to \(p, q,\ldots\), it is the case thatw and \(w^{\prime}\)are the same. Doyle and Wellman (1994) thus proposed dividing thespace of possibilities into equivalence classes and ignoringcomparison links that go across these classes.

Van Benthem, Girard, and Roy (2009) applied this idea to equivalencerelations between possible worlds: truth-value equivalence for allrelevant modal formulas in some specified set. In addition, theyshowed how this notion links up with von Wright’s work, and howit applies to game-theoretic solution concepts. They introduced newmodalities of the form \(\langle \Gamma \rangle\varphi\) into thelanguage, formally defined as

where \(\Gamma\) is an arbitrary set of formulas. This way,“being equal” is part of the formal language itself.Semantically, \(\Gamma\) defines an equivalence class of possibleworlds. For instance, \(w \approx_{\Gamma} v\) indicates thatwandv are equal w.r.t.\(\Gamma\)—they satisfy the same formulas in \(\Gamma\).

Aceteris paribus preference model is a quadruple\(\mathcal{M}\)= \((W, \succeq, \trianglerighteq_{\Gamma}, V)\)where:

• W is a set of states, \(\succeq\) is a binarypreference relation,V is an evaluation
• \(\trianglerighteq_{\Gamma}\) is the binary relation given by\(w\trianglerighteq_{\Gamma}v\) iff \(w\succeq v\) and \(w\approx_{\Gamma} v\)
• the strict subrelation \(\trianglerighteq_{\Gamma}\) is defined by\(w\succ v\) and \(w \approx_{\Gamma} v\)

The relation \(\trianglerighteq_{\Gamma}\) is essentially anintersection of two relations over worlds: the basic preferencerelation and truth-value equivalence with respect to the formulas in\(\Gamma\). The truth conditions for the new formulas that involve theequivalent classes are:

• \(\mathcal{M}, w \models \langle \Gamma\rangle^{\geq}\varphi\)  iff   \(\exists v\) s.t. \(v\trianglerighteq_{\Gamma} w\) and \(\mathcal{M}, v\models\varphi\)
• \(\mathcal{M}, w \models \langle \Gamma\rangle^{>}\varphi\)  iff   \(\exists v\) s.t. \(v\vartriangleright_{\Gamma} w\) and \(\mathcal{M}, v\models\varphi\)
• \(\mathcal{M}, w \models \langle \Gamma \rangle\varphi\) iff   \(\exists v\) s.t. \(v \approx_{\Gamma} w\) and\(\mathcal{M}, v\models \varphi\)

For those cases where \(\Gamma\) is finite, Van Benthem, Girard, andRoy (2009) provide a complete axiomatization. For the features ofceteris paribus, the new set of axioms is found inFigure 8.

In the full language, \(\Gamma\) in modalities\(\langle\Gamma\rangle\varphi\) can be of an arbitrary size. Thefollow-up work of Seligman and Girard (2011) presents anaxiomatization of part of the full logic, the set of what is called“flexible” validities, and provides an alternativesemantics in which all validities are flexible and so completelyaxiomatized. The axiomatization of infinite arbitrary sets is still anopen problem.

6.2 Computational models of ceteris paribus preferences

CP-nets, short for “conditional preference networks”,essentially embody the concept ofceteris paribus, which isthe core principle underlying how the networks represent preferences.This section introduces the basics of CP-nets, highlighting theircompact, intuitive, and structured way of representing agentpreferences under theceteris paribus assumption (all elsebeing equal).

A conditional preference network (CP-net) is a graphical model definedas follows.

Let \(V = \{X_1, X_2, \ldots, X_n\}\) be a set of variables where each\(X_i\) has a finite domain \(\text{Dom}(X_i)\) of possible values.There is a directed graphG where:

• each variable \(X_i\) corresponds to a node.
• the parents of \(X_i\), denoted as \(\text{Pa}(X_i)\), are nodeswith edges directed towards \(X_i\). These parents represent thevariables that influence the preferences over \(X_i\). An edge from\(X_i\) to \(X_j\) means that the preferences for \(X_j\) depend onthe value of \(X_i\).

Instead of listing preferences for all possible outcomes (which growsexponentially with the number of variables), CP-nets use conditionalpreference tables (CPTs) to capture only relevant dependencies (i.e.,the elements that the preference depends on). Each node \(X_i\) isannotated with a CPT that specifies a total order \(\succ_{u}\) over\(\text{Dom}(X_i)\) for every possible instantiationuof \(\text{Pa}(X_i)\). For example:

if \(\text{Pa}(X_i) = \{A, B\},\) then the CPT of \(X_i\) specifiespreferences for all combinations ofA andB.

The semantics of CP-nets describe how preferences over outcomes arederived based on the structure of each network and the conditionalpreference tables.

Anoutcome is a complete assignment of values to allvariables in the CP-net. For example, if \(V = \{X_1, X_2, X_3\}\) andthe domains are

then one outcome would be \(o = \{X_1 = a_1, X_2 = b_2, X_3 =c_1\}\).

Following theceteris paribus assumption, which meanscomparing outcomes while keeping all other variables constant, if werespect the conditional preferences specified in the CPTs of eachvariable, we can get preferences over outcomes.

For example, if the CPT for \(X_2\) specifies that \(b_1 \succ b_2\)given \(X_1 = a_1\), then:

An algorithm can be designed to efficiently find the optimal outcomein a CP-net—see the work of Boutilier, Brafman, and others(2004); Wilson (2004); and Goldsmith, Lang, and others (2008).

By capturing only relevant conditional preferences, CP-nets avoidcombinatorial explosion from enumerating preferences over all possibleoutcomes. CP-nets have demonstrated significant value in AI-drivendecision-making with applications in areas such as recommendationsystems and product configuration. They have attracted considerableattention in the fields of computer science and artificialintelligence, as highlighted in the more recent work of Allen (2015)and Cai, Zhan, and Jiang (2023), among others.

7. Preference across research fields

7.1 Preference and conditional logics

It is well known that there has been a strong relationship betweenpreference logics \(p\geq^{\forall\exists}q\) and conditional logicssince the end of the 1960s and the early 1970s (Lewis 1973), as someof them are interdefinable.

For example, if I prefer fish over meat, and I prefer meat overvegetarian food, and I am in a steakhouse where they offer only steakor salad, then in that context, I prefer steak. Such reasoning aboutpreferences and facts relates preference reasoning to conditionalreasoning.

In possible world semantics, this can be explained from the fact thatmodels of conditional logic are also preference structures. However,these preference logics developed as counterparts of conditional anddeontic logics in the late 1960s and early 1970s, and they were ratherdifferent from the preference logics developed in the decade before byHalldén and von Wright.

If we write \(\varphi\Rightarrow \psi\) for the conditional “if\(\varphi\) then\(\psi\)”, then undersuitable conditions, we can define that \(\varphi\) is preferred to\(\psi\) if \(\varphi\lor \psi\) implies \(\varphi\) and not\(\psi\):

Then we also have that the conditional “if \(\varphi\) then\(\psi\)” is defined as a preferenceof \(\varphi\land \psi\) over \(\varphi\land \neg \psi\):

Conditional logics were more popular due to the frequent use ofconditionals in natural language. We may say that some preferencelogics became known in the literature as conditional logics and othersas deontic logics. However, it does not follow from this result thatpreference and conditional logics are the same thing. In general, therelationship between conditionals and comparatives is only partiallyunderstood (Makinson 1993).

7.2 Preference and deontic logics

Traditional deontic logic often uses possible world semantics tointerpret notions like obligation and permission, with the worldsordered by a preference relation that ranks them by ideality. Deonticformulas are evaluated in the context of the current world based ontheir accessibility to these ideal worlds: an action is obligatory ifit occurs in all the most preferred worlds accessible from the currentworld. The intricate relationship between better worlds and the bestworlds has been extensively examined by Parent (2014) and Grossi, Vander Hoek, and Kuijer (2022). This ordering reflects intuitive deonticprinciples, but it struggles with conflicts and dilemmas because asingle ordering may not fully capture multiple, potentiallyincompatible imperatives.

Makinson (1999) emphasized the distinction between norms (imperativeswithout truth values) and deontic propositions (statements aboutobligations or permissions that can be true or false). He argued thatthis distinction has been overlooked in contemporary deontic logic andcalled for a “reconstruction” of deontic logic as a systemgrounded in explicit imperatives rather than abstract truthconditions. In his 2012 book, Horty made a notable contribution whenhe proposed a framework for understanding the reasons behind actionswhich employed default logic as its foundational structure. Heelucidated how multiple reasons interact—supporting, overriding,or conflicting with one another—to determine what individualsought to do.

These concerns with reasons led to the development of imperative-basedsemantics in deontic logic, which defines deontic operators inrelation to explicit sets of imperatives. Standard deontic systems(e.g., monadic and dyadic systems) have been adapted to fit thismodel, with extensions for handling priority conflicts and dilemmasbetween imperatives. A two-layer structure has also been adopted fordeontic logic. Notable works include those of Hansen (2008, 2014); VanBenthem, Grossi, and Liu (2014); and Benzmüller, Parent, and Vander Torre (2020).

7.3 Preference in decision theory and game theory

In decision theory and game theory, the notions of“preference” and “preference relation” areused to model choices, often with a behavioral interpretation. Theformal properties of preference relations and their representationthrough utility functions have been studied extensively in theliterature, some of which even predates preference logic. This entryprimarily emphasizes the qualitative approach to preferences, whichfocuses on the formal properties of preference relations. By contrast,the quantitative approach typically relies on utility functions torepresent preferences numerically. Decision-making under uncertaintyis commonly modeled using an expected value framework, which combinesthe agent’s beliefs (expressed as probabilities) with utilitiesto capture preferences. Critical research has reexamined thefoundations of preference theory. For example, Hausman (2012)distinguishes between notions of preference as choice, welfare, andvalue, raising questions about whether utility functions canadequately capture well-being—especially when qualitativefactors shape individual preferences. Fu (2020) proposes a novelapproach that integrates both qualitative and quantitative methods tomore accurately reflect the complexities of real-worlddecision-making. To connect this entry with decision theory, it iscrucial to consider the numerical, quantitative representations ofreason-based preferences derived from the richer two-layer structurespresented inSection 4.

7.4 Preference in social choice theory

In social choice theory, the core issue is how to aggregate individualpreferences into a coherent group preference. One well-known result inthis area is Arrow’s (1951) impossibility theorem, whichdemonstrates the limitations of creating a fair aggregation rule thatmeets certain desirable conditions. Further research has exploredalternative assumptions, leading to new results. For instance,Andréka, Ryan, and Schobbens (2002) use graphs to representalternative orderings of options as a method for aggregatingindividual preferences. List (2022) (in the entry onsocial choice theory) provides a useful survey on this topic. In social settings,individual preferences are often not fully independent and areinfluenced by the views of peers. As a community, it is intriguing toconsider how social preferences evolve over time and whether stablepreferences can emerge. Seligman, Liu, and Girard (2011) and Liang andSeligman (2011) discuss logic research on this topic. This area ofstudy requires an exploration of community structures, mechanisms forupdating individual preferences, and methods for defining socialpreferences. Related insights can be drawn from works on socialbelief, such as that of Shi (2021). A similar issue is investigated inevolutionary game theory, particularly in the so-called“indirect evolutionary approach” to social preference. Forexample, Bowles and Gintis (2011) explore how prosocial preferencessuch as altruism, fairness and cooperation can evolve and achievestability.

8. Conclusions

Preference logic provides a versatile framework for understanding howagents compare situations and make decisions. This entry has reviewedkey developments in the field, tracing its evolution from syntacticand algorithmic approaches to semantic frameworks. It has alsohighlighted recent research on two-layer structures, which stem fromthe distinction between intrinsic and extrinsic preferences,emphasizing the role of reasons in shaping preferences.

The study of preference change and the concept ofceterisparibus preference highlight the dynamic and practical aspects ofpreference logic, particularly in decision-making contexts. Theseframeworks will remain a foundation for addressing future challengesand advancing research in the field.

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