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A Logical Formalization of the OCC Theory of EmotionsAuthor(s): C. Adam, A. Herzig and D. LonginSource: Synthese, Vol. 168, No. 2, Knowledge, Rationality & Action (May, 2009), pp. 201-248Published by: Springer

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Synthese(2009) 168:201-248Knowledge,Rationality& Action 51 1-558DOI 10.1007/sl1229-009-9460-9

A logical formalizationof the OCC theory of emotions

C. Adam • A. Herzig • D. Longin

Received:22 March2007 / Accepted:14January 009 / Publishedonline:24 February 009© SpringerScience+BusinessMediaB.V. 2009

Abstract In thispaper,we providea logicalformalization f theemotiontriggeringprocessandof its relationshipwith mentalattitudes,as described n Ortony,Clore,and Collins'stheory.Weargue hat modallogics areparticularly dapted o representagents'mentalattitudesandto reasonabout hem,anduse a specificmodallogic thatwe call Logic of Emotionsin orderto provide logical definitionsof all but two oftheir 22 emotions.While these definitionsmay be subjectto debate,we show thatthey allow to reasonabout emotions and to drawinterestingconclusions from the

theory.

Keywords Modallogics •BDI agents•Emotions•OCCtheory

1 Introduction

There is agreat

amountof workconcerning

emotions in variousdisciplines

suchas philosophy(Gordon1987; Solomon andCalhoun1984), economy (Elster1998;Loewenstein2000), neuroscienceandpsychology.Inneuroscience, xperimentshavehighlighted hatindividualswho do not feel emotionse.g. due to braindamageareunable to make rational decisions (see Damasio 1994 for instance), refutingthe

C. Adam(S)RMITUniversity,Melbourne,VIC,Australiae-mail:[email protected]

A. Herzig• D. LonginInstitutde Recherche n Informatique e Toulouse,Universitéde Toulouse,CNRS,Toulouse,Francee-mail:[email protected]

D. Longine-mail:[email protected]

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202 Synthese(2009) 168:201-248

commonsensical assumption that emotions prevent agents from being rational. Psy-chology provides elaborated theories of emotions ranging from their classification

(Ekman 1992; Darwin 1872) to their triggering conditions (Lazarus 1991; Ortonyet al. 1988) and their impact on various cognitive processes (Forgas 1995).Computer scientists investigate the expression and recognition of emotion in order

to design anthropomorphic systems that can interact with human users in a multi-modal

way. Such systems arejustified by the various forms of 'anthropomorphic behavior'that users ascribe to artifacts. This has lead to an increasing interest in Affective

Computing, with particular focus on embodied agents (de Rosis et al. 2003), ambi-ent intelligence (Bartneck 2002), intelligent agents (Steunebrink et al. 2007), etc. Allthese approaches generally aim at giving computers extended capacities for enhanced

functionality or more credibility. Intelligent embodied conversational agents (ECAs)

use a model of emotions both to simulate the user's emotion and to show their affec-tive state and personality. Bates has argued for the importance of emotions to makeartificial agents more believable: "It does not mean an honest or reliable character,but one that provides the illusion of life, and thus permits the audience 's suspensionof disbelief." (Bates 1994, p. 122). Indeed, there are many pieces of evidence sug-gesting that virtual agents and robots (interacting with humans) that are capable to

display emotions, to recognize the human users' emotions, and to respond to theiremotions in an appropriate way, allow to induce positive feelings in the humans dur-

ing the interaction and to improve their performance. For instance it has been shownthat emotions affect learning (Bower 1992), so many computer scientists have added

human-provided emotional scaffolding to their computer tutoring systems in order toincrease both student persistence and commitment (Aist et al. 2002) and to improvelearning (Elliott et al. 1999). In the same way, other researches show that machineswhich express emotions and provide emotional feedback to the user, allow to enhancethe user's enjoyment (Bartneck 2002; Prendinger and Ishizuka 2005), her engagement(Klein et al. 1999) and performance in task achievement (Partala and Surakka 2004),herperception of the machine (Brave et al. 2005; Picard and Liu 2007) and can engagein more naturaldialogs with her (Becker et al. 2004).

The great majority of these works are founded on psychological works about emo-tion. "What is the best theory of emotion today?" is a question where currently there

is no consensus. A theory widely used by computer scientists is the one proposedby Ortony, Clore, and Collins (OCC henceforth). A reason is that this theory is rela-

tively understandable by computer scientists, because it is founded on a combinatory

approach of a finite set of criteria allowing to characterize emotions. (We are going to

present this theory in more detail in Sect. 2.2, and are going to present more argumentsin Sect. 2.4.)

OCC theory provides what may be called a semi-formal description language of

emotion types. It neither accounts for relationships between the different componentsof emotions nor relationships between agents' emotions and their actions. The aim of

this paper is to fill this gap by formalizing OCC theorywith the

helpof a

languagedescribing agents' mental attitudes such as beliefs, goals or desires. In this way we

stay as close as possible to the original psychological theory. More precisely, we aim

at modelling the triggering process of emotions in intelligent agents endowed with

mental states (i.e. a set of mental attitudes about some contents). What we do is to

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describe how a given mental state contributes to the triggering of a given emotion. This

problem has to be solved before formalizing the subsequent influence of emotions on

any mental process and in particular on planning. In this paper we therefore focuson the influence of mental states on emotions, and do not address the influence ofemotions on mental states.

Our aim is to model emotion in a logic of mental attitudes. Formal logic providesa universal vocabulary with a clear semantics and it allows reasoning, planning and

explanation of an agent's behavior. A given formal definition of emotions may be

criticized, but it still has the advantage to be unambiguous and to allow analysis andverification. In particular,all logical consequences of formal principles must remainintuitive: a logical formalization may reveal consequences (and even inconsistencies)that were 'hidden' in the theory and did not appear before. Formal definitions clearly

articulate assumptions and allow to formally derive consequences of certain assump-tions: they allow to clearly and concisely articulate the assumptions of a theory andto readily uncover the consequences. All in all, logical formalization is a well-definedscientific program to move forward and develop more widely accepted and clearlydefined models.

The logic used here is a particular modal logic that grounds on the philosophy of

language, of mind, and of action (Bratman 1987; Searle 1969, 1983), and proposesto model agents via some key concepts such as mental attitudes (belief, preference,desirability), action and time. This framework is very close to those commonly usedin the agent community and offers well-known interesting features: great explanatory

power, formal verifiability, and a rigorous and well-established theoretical frame (fromthe point of view of both philosophy and formal logic). Note that we are not concernedat this stage with optimizations of our logical theory in view of particularapplications;for the time being we leave this to agent designers who might use our model as a basisfor their work.

Our aim is also to model emotion in a way that is as faithful as possible to psy-chology. Thus we believe that our logical theory is built on solid grounds given thatOCC theory is a well-established psychological theory. The properties of our logicmay be evaluated with respect to the following criteria: (1) the number and types ofthe emotions that are covered; (2) the examples given by psychologists (is our formal-ism able to account for these examples?); (3) the theorems following from our model

(are these theorems intuitive and relevant? are they in accordance with the formalized

psychological theory or do they run counter to it? etc.).We also believe that the other way round our logic, thanks to its faithfulness to

the OCC theory, may contribute to the assessment of this theory. For example the

consistency of our logic demonstrates that OCC theory is free of contradictions.In the rest of the paper, we expose the OCC theory underlying our work (Sect. 2).

In Sect. 3 we introduce our logical framework. In Sects. 4 and 5 we detail the event-based and agent-based branches of the OCC theory and their formalization. In Sect. 6

we expose some theorems concerning emotions, particularly relating to causal andtemporal links between them. In order not to overload the paper, the proofs of thesetheorems are gathered in the appendix. In Sect. 7 we discuss some existing logicalmodels of emotions.

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204 Synthese(2009) 168:201-248

2 Emotion theories

Toensuretheaccuracyof a computationalmodel of emotions,it is importanto startfromacknowledgedpsychological heories.Thereexist severalkindsof psychologicalmodels of emotions:evolutionistmodels(e.g. Darwin1872) that aremainly descrip-tive,givingtaxonomiesof basicemotions;dimensionalmodels(e.g.Russell1997)thatassume thatall emotions are similarphenomena,only varyingon the values of somedimensions ike valenceorarousal; hese models were sometimesused to describe hedynamicsof theexpressionof emotions(e.g. Becker et al. 2004); cognitiveappraisaltheories e.g. Ortony t al. 1988)that ocus on thecognitivedetermination f emotionsandon theiradaptive unction.

The conceptof appraisalwas firstintroducedby Arnold(1960) to describe the

triggeringof emotions,togetherwith theconceptof actiontendenciesdescribing heireffects. Thesetwo conceptswerethen studied n many approaches;we presentsomeof the mostimportant neshere(Sect. 2.3), inparticularhatof OCC heory Sect.2.2).Beforethat, et us firstshortlyspeakaboutrelationshipsbetween emotion andcogni-tionthrough heconceptof IntentionalitySect. 2.1).

2.1 Emotion,cognition,intentionality, ndlogic

The use of logic for emotion formalizationmay appear urprising t firstglance,and

one mightconsiderthattheycannotbe married.Nevertheless, odaythegreatmajor-ity of psychologistswork withapproacheswhere emotion andcognitionarestronglyconnected(see Lazarus1999 for instance:for him, emotionis a partof cognition).Logiccandealwithcognitionthrough he well-knownBDI logics (forbelief, desires,intentions).Cognitionrefers,amongother hings,to mentalstatesandreasoningaboutthem.Thus,to say thatemotion is in cognitionmeansthatemotion is concernedwithmentalstates.

In our view, emotions are always about a state of affairsof the world. In otherwords,emotion s anIntentional omponentsof our mind n thesenseof Searle(1983,p. 1):"Intentionalitys thatpropertyof manymental states and eventsby whichtheyare directedat or aboutor of objectsand states of affairsin the world". Note thatintention s justa particularormof Intentionality. o avoidconfusionsandfollowingSearle(1983), we write"Intentionality" ithanuppercase letter.)For nstance,beliefandpreferenceareamongthe Intentionalmental states.Note thatonly some,but notall, mentalstates haveIntentionality:or instance, ormsof nervousness, lation,andundirected nxiety, hatare diffusewithoutanyclearlink with anobjectof theworld,are not Intentional tates. Searle(1983, pp. 29-36) has describedsome emotionsas

complexmentalattitudes hatcan beexpressedas a combination f beliefsanddesires.We wantto generalize hisapproachby applying t to OCCtheory. nthisperspective,

thedescriptionof an emotionas a combination f beliefsand desires

presupposeshat

theemotionunderconcern s anIntentionalmentalattitude.We thereforedo not dealin thispaperwith otheremotionalstatesthat arecloser to mood or that are notInten-tionalemotions, n thesensethattheyarenotconcernedwithorbasedon Intentionalmentalattitudes.

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Fig. 1 Emotions,mood,Intentional tates,andduration

Following the great majority of psychologists, another difference between emotionand mood is that emotion has a very short duration in time. (See for instance Ortonyet al. 1988; Lazarus 1991.) Thus, we can expect that an affective state having a longduration is not so much emotion as mood.

Finally, Intentional mental attitudes can be either conscious or unconscious, andthis property is not related to Intentionality. (Searle shows thatone can be conscious ofnon Intentional mental states, and conversely one can be unconscious of an Intentional

state, see Searle (1983, Chap. 1).) Figure 1 pictures the situation.

Ortony and colleagues agree that "individual emotions can be specified in termsof personal or interpersonal situation descriptions that are sufficient to produce them"

(Ortony et al. 1988, p. 3). More precisely they assume that "if the described situationcontains the eliciting condition for a particularemotion, the experience of that emo-tion can be inferred" (Ortony et al. 1988, p. 3). This is clearly a cognitive approachwhere emotions areIntentional

concepts.This

supportsour fundamental

designchoice:

emotions can only occur in particularmental states formalized through the logical def-inition of the mental attitudes constituting their elicitation conditions. Our choice of a

logic of mental attitudes is therefore justified both by the fact that it is an appropriateformalization of mental states (see Cohen and Levesque 1990; Rao and Georgeff 199 1

1992; Sadek 1992; Herzig and Longin 2004, for instance) and by the fact that mentalattitudes allow to express emotions.

2.2 Ortony, Clore and Collins's theory of emotion

Ortony et al. (1988) propose a cognitive appraisal theory that is structured as a three-branch typology, corresponding to three kinds of stimuli: consequences of events,actions of agents, and aspects of objects. Each kind of stimulus is appraised w.r.t. onecentralcriterion,called central appraisal variable. An individual judges the following:

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206 Synthese(2009) 168:201 248

- the desirability of an event, i.e. the congruence of its consequences with the indi-vidual's goals (an event is pleasant if it helps the individual to reach his goal, and

unpleasant if it prevents him from reaching his goal);- the approbationof an action, i.e. its conformity to norms and standards;- the attraction of an object, i.e. the correspondence of its aspects with the individ-

ual's likings.

There are some secondary appraisal variables influencing the intensity of the gener-ated emotion, such as the probability of an event, the degree of responsibility of theauthor of an action and the amount of effort that was provided.

The OCC typology contains twenty-two emotions types1 that are grouped in sixclasses. The first branch contains three classes of emotions triggered by the appraisalof the consequences of an event as to its desirability. Well-beingemotions (joy, distress)

arise when an individual appraises an event that hasjust occurred while only focusingon the desirability of its consequences for himself. Fortunes-of-others emotions (happyfor, sorry for, resentment, gloating) arise when an individual appraises an event while

focusing on its desirability for another individual. Prospect-based emotions such as

hope or fear arise when an individual appraises the consequences of a prospected event

(namely an event that has not occurred yet but is expected to do so) while focusing onthe desirability of its consequences for himself. Other prospect-based emotions suchas disappointment, relief, fears-confirmed, and satisfaction arise when an individual

appraises an event that has just occurred and that was expected, while focusing on its

desirabilityfor himself.

The second branch contains only one class of emotion types (Attribution emo-

tions) triggered by the appraisal of an action as to its approval, i.e. its conformity tonorms and standards. Thus, pride and shame arise when an individual appraises oneof his own actions while focusing only on its approval ('does this action conform tothe standards?') and not on its consequences. Admiration and reproach arise whenan individual appraises an action of another individual while focusing only on its

approval.An other class, common both to Well-being emotions (first branch of the typology)

and Attribution emotions (second branch of the typology) is Compounds emotions

(attribution-wellbeing) (remorse, gratification, gratitude, anger) that arise when anindividual appraises an action while focusing both on its approval an on the desirabil-

ity of its consequences.Here is a complex example where several of the above emotion types are involved.

Suppose you and a friend of yours apply for the same position. You believe your CVis better, but then you learn that your friend got the position because he cheated abit on his CV (say he over-emphasized his participation in some project and gave tosome of his papers a "to appear" status although they arejust submitted). Accordingto OCC theory you might then feel (1) disappointed (confirmation-based), (2) happyfor your friend (fortune of other), and (3) reproach (attributionemotion). The relative

1According to the authors, an emotion type is "a distinct kind of emotion thatcan be realized in a variety of

recognizably related forms" (Ortony et al. 1988, p. 15), for example various intensities or various emphasis.In the sequel of this paper, to simplify the vocabulary we generally use the term "emotion" instead of

"emotion type".

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importance of these three emotions depends on the secondary appraisal variables,which is something we do not account for in our framework. What we deal with here

is whether such emotions can indeed be triggered simultaneously by the same event,i.e. whether the conjunction of these three emotions is consistent in our logic.Finally, the third branch contains one class of emotions: attraction emotions (love,

hate), triggered by the appraisalof the aspects of objects w.r.t. the individual's likings.It is important to notice that the authors of the OCC theory intended it to be used

in Artificial Intelligence:

"(. . .) we would like to lay the foundation for a computationally tractable modelof emotion. In other words, we would like an account of emotion that could in

principle be used in an Artificial Intelligence (AI) system that would, for exam-

ple,be able to reason about emotion."

(Ortonyetal. 1988, p. 2)

This aim was pretty much reached since OCC theory is the most popular psycholog-ical model of emotions in computer science nowadays, and emotional agents widelyemploy it (e.g. Elliott (1992), Reilly (1996), de Rosis et al. (2003), Jaques et al. (2004),Ochs et al. (2005)). However, it is not the only one, and we can quote emotional agentsbased on Frijda's theory (e.g. Staller and Petta (2001)) as well as agents based onLazarus's theory (e.g. Gratch and Marsella (2004)).

2.3 Other theories

Frijda (1986) focuses on the action tendencies induced by emotions. A stimulus first

passes through various steps of evaluation determining its characteristics: causes and

consequences, relevance and congruence with interests, coping possibilities, urgency.Depending on the result, a control signal is generated to postpone or interrupt thecurrent action. An action preparation is then created (action plan, action tendency,activation mode) that induces physiological changes, and finally an action is selectedand executed. Frijda believes that it is the associated action tendency that differenti-ates basic emotions from each other. Dastani and Meyer (2006) build on this notionof action tendency to define the effect of four emotions on a rational agent's plans.

Lazarus (1991) presents a relational, motivational, cognitive theory of emotion.

According to him, emotions result from the cognitive evaluation (or appraisal) ofthe interaction between an individual and its environment, w.r.t. his motivations and

goals. Lazarus distinguishes between the primary appraisal, assessing the relevanceand congruence of the stimulus w.r.t. the individual's well-being (that is, does thestimulus help or threaten one of the individual's goals?), and the secondary appraisal,evaluating the available resources to cope with the stimulus (can the individual do

something to remove the threatening stimulus?). These two kinds of appraisal are not

sequential: they can be executed in any order. Like Arnold, Lazarus considers thatemotions induce action tendencies, that cause physiological modifications in order to

help the individual adapting to his/her environment. Lazarus' theory is used in theEMA agent (cf. Gratch and Marsella (2004)) whose acronym is an homage to hisbook.

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208 Synthese(2009) 168:201-248

Scherer (1987) considers emotions as a multicomponent process, with one cognitivecomponent. He introduces an appraisal process consisting in a sequence of stimulus

processing steps, called the Stimulus Evaluation Checks. This process sequentiallyevaluates the novelty and unexpectedness of the stimulus, its intrinsic agreeability,its congruence with the individuar s goals, the coping possibilities, and its compati-bility with norms. Contrarily to Lazarus (1991), these evaluations are ordered. Later,Scherer (2001) associates to each emotion bodily responses, and in particular facial

expressions in terms of Action Units. The latter are elements defined by Ekman et al.

(2002) to represent the moves of the facial muscles. This theory is thus well-adaptedto represent the dynamics of facial expressions of an animated agent (e.g. GrizardandLisetti (2006)).

2.4 Which theory to choose?

Appraisal theories importantly differ one from another on the appraisal criteria thatare used, their order of application, and the precise definitions of emotions based onthese criteria. We have chosen OCC theory because the careful study of this theoryin comparison with others like Lazarus's indicated that it is better adapted to describethe emotions of a virtual agent for several reasons.

First, OCC theory is widely used in the design of emotional agents because its sim-

plicity and implementability matches computer scientists' expectations and needs:it seems that the combination of OCC's finite set of appraisal variables suffices for

currentapplications.Second, we completely agree that according to OCC, any emotion must be valen-

ced and this valence must always be the same (Ortony et al. 1988, pp. 29-32). Thisexcludes de facto states like surprise (that can be either good or bad, or even neither

good nor bad) or "feeling abandoned" (in this state one can be sad, but can also notlet this get one down and get one's hope up) from being emotions. (In some works,

surprise is considered to be an emotion, see Shaver et al. (1987), Meyer et al. (1997),Lorini and Castelfranchi (2006) or Lorini and Castelfranchi (2007) for examples inrecent works, or Ekman's works in 70th and Darwin (1872) for older works.) Besides,the necessity for anemotion to be valenced has also the advantage to provide a clear test

to differentiate emotions from close notions that are not valenced. Moreover, valenceis something naturally captured by logic, making the OCC theory particularly well

adapted for a logical formalization.

Third, OCC theory has a simple and elegant tree structure, and uses concepts thathave been well-studied in logic such as beliefs, desires and standards. This makes theformalization task easier. In Searle's view (Searle 1983, Sect. 1.5), every Intentionalstate is not reducible to belief and desire, but every Intentional state contains a belief,a desire, or both a belief and a desire. So-called BDI logics developed in the field ofArtificial Intelligence in the last fifteen years (see Cohen and Levesque (1990); Raoand

Georgeff (1991)for

instance)offer

expressiveframeworks to

represent agents'mental attitudes such as beliefs and desires (see Meyer et al. (1999); Herzig and

Longin (2004) and Lorini and Herzig (2008) for instance) and to reconstruct on theirbasis the cognitive layer of emotions (see Adam (2007); Adam et al. (2006) and alsoSteunebrink et al. (2007) for instance).

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Finally,OCCtheoryis quiteexhaustive,which is importanto designrobustandversatileagents,i.e. agentsthatcanemotionallyreactto a great varietyof situations.

On thecontraryLazarus' heory s moreprecisebut seems to be less exhaustive see(Adam,2007, Chap.4) for a more detailedcomparison).We believe that the log-ical formalizationof both theorieswill allow to comparethem in depthin a closefuture.

The next sectionpresents he logical framework.

3 Logical framework: the EL logic

The formal ramework f thisarticle s basedonourpreviousBDIframeworkHerzig

andLongin2004) thatin turn is based on BDI logics (see for instance Cohen andLevesque 1990);Rao andGeorgeff 1991);Sadek(1992)). Weminimallyextendthisstandard rameworkby integratingOCC'sappraisalvariables.As we have said, werestrictour attention o emotiontriggeringconditions,disregardinghe influenceofemotions on beliefs, desires andintentions.OCC's emotiontriggering onditionsdonot refer to the mental attitudeof intention,that is thereforenot requiredhere. Aswe areonlyconcernedwithevent-based motionsandagent-based motions,we hereonly need to model the desirabilityandpraiseworthinessvariablesof OCCtheory.But let us firstexplainthese variablesand the choices we made in order to modelthem.

In OCC theory,desirability s about events and is close to the notion of utility.Whenan eventoccurs tcansatisfyorinterferewithagent'sgoals, andthedesirabilityvariablehastherefore wo aspects:

"onecorresponding nly to the degreeto whichthe event in questionappearsto havebeneficial(i.e. positivelydesirable)consequences,and the othercorre-sponding o thedegreeto which it is perceivedas havingharmful i.e. negativelydesirable,or undesirable) onsequences."

(Ortonyetal. 1988,p. 49)

It is thus a valencedvariable,andan eventcan be at the same time desirableandundesirablewithrespectto the agent'scurrentgoals). Desirability i.e. the positiveaspectof thedesirabilityvariable)only influencespositiveemotions,whereasunde-sirability i.e. thenegativeaspectof thedesirabilityvariable)only influencesnegativeemotions. It follows that the same eventcan triggerbothpositiveandnegativeemo-tions.

While we agree with OCC theorythatthe primaryevent may be bothdesirableand undesirable,2 uch a featuremakes a logical formalizationdifficultbecause itrequireseithera paraconsistent otionof desirabilitysuch that'ty *sdesirable"and

"-«pis desirable"are

consistent,or a

binarynotion of

desirabilityhatis relativized

2 For instance the authors give the example of the death of one's friend suffering from a painful disease;on the one hand the loss of one's friend is undesirable, but on the other hand the end of his suffering isdesirable.

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210 Synthese(2009) 168:201-248

to goals. Both options would induce several difficulties, that would distract us fromour aims; in particular there is no available logic of the latter binary desirability in

the literature.A way out is to shift the focus from desirability of events to desirabilityof consequences of events: when someone says that an event is both desirable and

undesirable, we are entitled to ask which aspect of this event is desirable and which isundesirable. When considering consequences of events rather than events themselveswe may safely suppose that these consequences are either desirable or undesirablewith respect to current goals, but not both. For instance, someone's death can entailboth an affective loss (undesirable consequence) and the inheritance of a big amountof money (desirable consequence) Ortony et al. (1988). In this example, clearly, the

goal concerned with the desirability of the event (that is, to get a big amount of money,or to be rich) is different from the goal concerned with the undesirability of this event

(that is, to keep a loved person alive). Correspondingly, our desirability operators havethe following properties:

- desirability and undesirability are about consequences of events (and not aboutthe event itself);

- an event can have several consequences;- each of these consequences cannot be both desirable and undesirable;- each of these consequences can be evaluated with respect to goals (that may be

either achievement goals or maintenance goals).

We formalize the above example by saying that an emotional loss is undesirable anda big amount of money is desirable, while the friend's death is neither desirable norundesirable.

Things are similar concerning OCCJ praiseworthiness variable,which concerns theevaluation of actions performed by agents; this evaluation is with respect to standardsand has two aspects: actions can be praiseworthy (when they conform to standards)or blameworthy (when they violate standards). Though, we want to avoid to analyzestandards in more depth and do not describe how to construct the two aspects of the

praiseworthiness variable. We simply define two types of modal operators: one char-

acterizingthe

praiseworthinessof

consequencesof

actions,and one

characterizingthe blameworthiness of consequences of actions. Just as for desirability and undesir-

ability, we will consider that such a consequence cannot be both praiseworthy and

blameworthy at the same time.

3.1 Syntax

The syntactic primitives of our logic of emotions EL are as follows: a nonempty finiteset of

agents AGT=

{ 'i 12,. . .

in}»a

nonemptyfinite set of atomic events EVT =

{e\ , ^2, • • , €p), and a nonempty set of atomic propositions ATM = {p\ , pi . . .}.The variables 1, j, k... denote agents; a, ß, y... denote events; and p, q... denote

propositional letters (atomic propositions). The expression i:e represents an event e

intentionally caused by agent 1. We say that i:e is an action that is performed by i.

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The language Cel of the EL logic is defined by the following BNF (Backus Naur

Form):

<p <pv (p | ite/,^ | Probnp \ Desi<p

\Idl(p\ Afteri:a(p \ Beforei:a(p \G<p\H<p

where p rangersover ATM, i ranges over AGT and i:a ranges over AGT x EVT. Theclassical boolean connectives A (conjunction), - (material implication), «* (material

equivalence)and T

(tautology)are defined from -•

(negation),v

(disjunction)and ±

(contradiction) in the usual manner.

Beli<p reads "agent i believes that <pis true". Belief is understood as subjec-tive knowledge, alias truth in all worlds that are possible for the agent: / does notdoubt. For instance, BellxweatherNice represents the fact that, from ij's point ofview the weather is nice: î] has no doubt about the truth of this fact (but may be

wrong).Probfip reads "agent i believes that ip is more probable than -*<p'\or "i believes

that (pprobable" for short. This is a weaker form of belief than Belt<p.For example,if agent i\ is still in bed, Prob^ weatherNice means that i \ believes that the weather

is probably nice (but i\ may not be sure about this). What an agent believes is nec-essarily probable for him, but not the other way round: when i\ believes that p then

p is probable for i\. (We give more details in the sequel.) Several researchers have

investigated logics of probability, mainly in a quantitative (Fagin and Halpern 1994)or comparative way (Segerberg 1971). A few researchers studied a more qualitativenotion of probability (Burgess 1969; Herzig 2003), weak belief (Lenzen 1978, 1995)or likelihood (Halpern and Rabin 1987; Halpern and McAllester 1989). All these arebased on subjective probability measures. We adopt Burgess's logic, basically becausewe do not need numbers for our purposes (but they might be added later when inves-

tigating particular applications), and because it integrates smoothly with Hintikka's

logic of belief that we are going to use.Desiip reads 'ty is desirable for /". As we have motivated above, instead of desir-

ability of events we here rather deal with desirability of consequences of events.These consequences are evaluated with respect to goals. According to the OCC the-

ory, goals can be either achievement goals (the agent wants to achieve something thatis not currently true) or maintenance goals (the agent wants to maintain somethingthat is already true). Moreover, we do not explain the relationship between goals and

desirability because goals do not play an explicit role in our definition of emotions.However, goals can be constructed from what is desirable, and intentions can be con-structed from goals. (See Herzig and Longin (2004) and Castelfranchi and Paglieri(2007) for more details about such constructions.) In our view, every (achievementor maintenance) goal is about something that is desirable. Thus, if a consequenceof an event is (a part of) a goal, then this consequence is desirable. Here, instead ofan occurrent mental attitude we rather use the notion of dispositional attitude, that

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corresponds with Bratman's notion of desire and with Cohen & Levesque's notion of

goal.3

Idi <p eads "ideally, <ps true". The notion of ideality considered here is taken in alarge sense: it embraces all the rules more or less strongly imposed by some authority.They can be strongly explicit (like laws) or more or less implicit (like social or moral

rules). When Idi <ps true then <ps a kind of social preference that is attached to the

groups to which the agent belongs. They may therefore differ from the agent's personalpreferences. Idi driveRight, for instance, means that ideally, one drives on the rightside of the road, and Idi helpSblnDistress means that ideally, one helps somebody indistress.

Afteri:a<p reads 'ty wiU be true after performance of action a by i". This operatorallows to describe what is true after the execution of an action, in particularthe effects

of this action. For instance, AfteriVraiseHandrightToSpeakixmeans that after agent i\has raised its hand (say in the classroom) it will have the right to speak. The fact that ipwill be true after the performance of action a is conditional on the performance of a: itdoes not entail that a is currentlyexecuted, nor that i intends to execute it. A/f er/:cr Lreads "action a is not executed by agent /". For instance, After^jn^L means that

agent ii is not going to drive in the current situation (for instance because ii does nothave a car).

Beforei:a(p reads 'ty was true before performance of action a by 1". It is symmet-ric to Afteri:a for the past. Bef orerai, means "j has not just executed action a".

Beforei2:hoidsNutholdsNuti2,or instance, means that before crunching a nut, agent 12

must hold a nut, and BeforeiVdrink^- means that drinking was not 1'1's ast action.Gipreads "henceforth cp s going to be true". The notion of time that we use here is

linear time. It means that states of world are organized in a linear manner, in what iscalled "histories" in the literature.Thus, G<pmeans that <ps true on the currenthistoryfrom now and everywhere in the future. For instance, GglassIsBroken means that the

glass is henceforth broken.

Hip reads *tyhas always been true in the past". Thus, it means that <ps true on thecurrent history everywhere in the past including now. For instance, H-^JohnlsDeadmeans that until and including now, John is not dead.

For convenience, we also define the following abbreviations:

Happensi:aip = -Afteri:a-*ip (J>^Happens^)

Donei:aip = ->Beforei:aîp (P^Doneixl)

F<pdM -*G-*ip (Def/r)

3 Several concepts of desire exist in the literature. Desire is often viewed as an occurrent mental attitude:an attitude that holds here and now, and that is abandoned as soon as it is satisfied, such as an agent's desireon a rainy day that the sun shines, which is dropped when finally the sun comes out. This is similar toBratman's concept of intention (Bratman 1987) and to Cohen & Levesque's concept of achievement goal(Cohen and Levesque 1990).

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Pc d= ^H^<p (Defp)

Idli<p = BehIdi <p (Dcfjdii

Happensi:a(freads "a is aboutto be performedby agenti, after which<pwill betrue".4 nparticular,HappensvaJ reads"actiona is about o beperformedby agent/". For instance,Happensii:tosscoin(heads tails) meansi'i is about to toss a coin,afterwhich the coin will be eitherheadsor tails.

Donei:a(preads "a hasjust been performedby agent i, and cpwas truebefore"and Donei:aT reads that agent i has just performed action a. For instance,Donei2:toDrinkBeerDoneii:tODrinkCoke~Teans that agent i2 has just drunk a beer and

just beforethat,agenti\ haddrunka coke.F<p eads'ty is true or wiUte true at some future nstant",and fV reads'V is or

was true".Forexample,PsunlsShiningA FsunlsShiningmeans thatthereis a pastinstantwhen the sun was shiningandthere is a future nstantwhen the sun will beshining.

Finally,Idli<p eads"from hepointof view of theagent , it is idealthat<p e true".It will be convenient o supposethat t representsanagent'smoralnorms,that s, thenorms hattheagenthas internalized s true.Forinstance,Idli\ be Vegetarianmeansthatforagenti\ one should be vegetarian, ndIdli^^Drunk ADriving)means thatfor agentii it is unideal o drive drunk.Note thatin principlenot everyknown ideal

(i.e. BeliIdi <p) ecomesan internalizeddeal(Idltcp), .e.,the left to

right mplicationBeliIdi <P* Idlupof the Definition(Def/^) is notgenerallyvalid.The differenceis subtle seeAdamet al. (2009) for moredetails)andhere we adopt hissimplificationbecauseit allowsus to avoid aninvestigation f the relationbetween internalized ndnon-internalizedtandards.

3.2 Semantics

We usea standardKripke emantics n termsof possibleworldsandaccessibilityrela-tions.The less standardeature s a neighborhoodunctionfor the modaloperatorof

probability.

3.2.7 EL rames

At the base of Kripkesemantics thereis a set of possible worlds W togetherwith

accessibilityrelations oreverymodaloperator.Whileinmostpresentations n acces-

sibilityrelation s asubsetof the cartesianproductW xW,v/e hereuse anequivalentpresentationn termsof mappings romWto 2W

An EL frame s a 7-tupleT = (W,&, &>,2), J, srf,&) where:-

Wis a set of possibleworlds;

4 Note that the operators Happens can be read in this way (which is not their standard dynamic logicreading) because we have supposed determinism: time is linear, entailing that if an action is feasible, thenit will happen.

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- Se : AGT - > (W - 2W) is the accessibilityrelation that associates eachagenti e AGT andpossibleworld w e W,withtheset â8i(w) of possibleworlds

compatiblewith the beliefs of agentiinw;- & : AGT - (W - 22W) s the functionthat associateseachagenti e AGTandpossibleworld w e Wwith a set of sets of possibleworlds&i(w) (theneigh-borhoodsof w);

- 9 : AGT - (W - 2W)associates each agenti e AGT andpossibleworldw e W withtheset fy (w) of worldscompatiblewith what s desirable or theagenti in the worldw;

- J : W - > 2Wassociateseachpossibleworld w € Wwiththe set J(w) of idealworlds;

- si : AGT x ACT - (W - 2W)associates each action i:a € AGT x ACT

andpossibleworld w e W withthe set <ß^:a(u;)f possibleworldsresulting romtheperformance f a by agentiinw;- <S: W - 2Wassociateseachpossibleworldw e Wwith the set&(w) of possible

worlds n the futureof w.

Theset &i(w) is calleda belief state.

3,2.2 Semanticalconstraints

Weimposeto our framesthefollowingsemanticalconstraints.

All theaccessibilityrelationsS6\ areserial,transitiveand euclidian. (SCi )

Thus,belief statesareequivalence lasses:anagentviews severalalternativeworlds otherealworldbutcannotdistinguishbetweeneachof these alternatives.Note thatcon-trarilyoknowledge herealworld s notnecessarily ontainedn anagent'sbeliefstate.Serialityensuresthatbeliefsarerational: nagentcannotsimultaneouslybelievethatp is trueand that ts negation >pis true.Due to thetransitivity ndeuclidianityof theSèi relations,agentsare awareof theirbeliefs:if w 6 3ti(wf) then&i(w) = 3§i(w').

If (pis probable or i (i.e. <ps truein all the worlds of some neighborhood,ee

Chellas(1980, Chap.7) formoredetails),then-*tps not (sinceeachotherneighbor-hood contains at leastone world where<ps true).Thiscorrespondso the followingconstraint:

Foreveryw € W, ifUuU2e @>iw) thenU\ C\U2 / 0. (SC2)

Moreover,n order o ensurethat at leasttautologiesareprobable,we imposethat:

&i(w) ^ 0 foreveryw eW. (SC3)

Finally,we imposethattheneighborhoodsn &i(w) are subsetsof the belief state:

Vl/€^,-(u;)f U£ß (SC4)

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which entails that belief implies probability.5As explained in the previous section, when desirability is about propositions rather

than actions, it is convenient to postulate to consider that desirability is rational: ifa proposition is desirable then its converse is not desirable. This is imposed by the

following semantical constraint:

All the accessibility relations % are serial. (SC5)

The situation is similar for ideality: intuitively, the logic of ideality operators is thesame as Standard Deontic Logic. (See Âqvist (2002) for more details about Deontic

Logic.) Here, the rationality of ideals is justified by the fact that law, moral, habits,standards, etc. are in principle coherent. Thus, if something is ideally true, then its

converse must not be true.

All the accessibility relations J are serial. (SCô)

Concerning action, we impose that for every w e W:

If w' e £/a(w) and w" e srfß(w) then w' = w". (SC7)

If w € «ß4(u/) and w € s/ß(w") then w' = w" (SCg)

First, this imposes that actions are organized into histories. It does not impede the

parallel execution of several actions, but it guarantees that all these parallel actionslead to the same world, i.e., the same time point in the same history. It imposes thatall the actions take place in the same history, where the outcome world is the samefor all actions performed by all agents.6 Second, these constraints impose that actionstake one time step. Suppose that a, and ß are performed during the performance of

y. That is: u/ e <ß4(w), u/' € «ß^(u/) and w" e £/y(w) hold. Thus, (1) imposesthat w' = w", and (2) imposes that w = u/, which entails that w = w' = w" Thus,in this case, all actions are reduced to the 'skip' action ('do nothing') and the world

remains unchanged. Therefore, actions are deterministic in the future and in the past.Finally, we impose that the <Saccessibility relation is a total preorder,

The accessibility relation & and its converse &~x

are reflexive, transitive and relatively total: (SC9)

if lui, voi € y(w) thenw\ e ^(wi) or\V2€ &(w\).

5Intuitively the elements of «^/ (u;) should also be "big" subsets of &i(w): every U G .^/(w) should

contain more elements than its complement &i(w)\U. But the language of modal logic is not expressiveenough to account for this. The above constraint is therefore weaker, and there are neighborhoods satisfyingour constraints gathering less than 50% of the worlds (cf. Walley and Fine (1979)). However, this sufficesto capture some interesting properties such as inconsistency of some emotions.6 This hypothesis does not permit to speak about counterfactual situations such as "if I had done a then (pwould be true", but this is not problematic as long as we are not concerned with such hypothetic reasonings.

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Fig. 2 In world wo, agent i believes that henceforth <p s true; in world w\, for agent i there are three

different possible histories: from top to bottom, the one where <p s currently false but it will occurin the

future, the one where <p s henceforth true, and the one where <p s henceforth false

This means that time is linear towards he past (by using <S *) and the future.One

mightobjectthatat least futureshould be branching.Forus, whatis importants notthe natureof time in the realworld,butrather he perception hatagentshave of it.Thus, as time is linearhere,each world believedto be possible by an agentcan beidentifiedwith a history,that is a linearsequenceof time points, and the diversityof futures s represented hroughdifferenthistories that arepossible for the agentatthe sameworld(cf.

Fig.2).In otherwords,even if time is linear,several futuresare

possiblefor theagent,and we thereforehave a subjectiveversionof branching-time.Moreover,we imposesome constraintsnvolving wo or moreaccessibilityrelation

types.Inparticular,we supposethatagentsare awareof theirprobabilities nd desir-abilities,thatis, theagents'beliefs about theirown subjectiveprobabilitiesand theirdesirabilitiesare correctandcomplete.We thusimposethat,foreveryi € AGT:

\iwf€âSi{w) then&i(w) = îiw') (SQo)

if w' € Si (w) then% (w) = % (u/) (SCi i )

Concerning he relation between belief and action, we suppose that actions are

public,in the sense thattheiroccurrence s correctlyandcompletely perceivedby allagents.Foreveryi € AGT:

if u/ € 3Bi(w) then(^ya)~\w) = 0 iff {styMV\u)') = 0 (SC12)

We alsoimposethatagentsdo notforget heirpreviousalternatives"noforgetting",alias "perfectrecall"Faginet al. (1995)). This relies in particular n the precedinghypothesis hatactionsarepublic, .e. that heyareperceivedcorrectlyandcompletelyby everyagent.Thus,forevery agenti, j € AGT:

\i(ßi o fi/yM)(w)£ 0 then(s/yMo &ï)(w) ç (ß{ o fi/j:a)(w) (SC13)

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Inparticular,t is truewhen i andj arethe sameagent.In termsof Fig.2, theagent'sbelief state aftersome action was performedat wo, is a subset of the agent'sbeliefstate at wo that has been 'progressed' Reiter1991) in order o take intoaccounttheactionoccurrence.

Actionandtime areclosely related. nparticular,we imposethat he futureof everyworldw containsthe worldsresulting rom theperformance f actions in w:

%2 #ti.a foreachv.a e AGT x EVT (SCh)

In words,the worldsresulting romthe performance f actionsin w arenecessarilyworlds nthefuture.But theconverse s notnecessarily rue:everyworld n the futureis notnecessarilyaccessibleby someaction i:a in one step:sucha hypothesiswould

be too strong.Forthe sakeof simplicity,we makethehypothesis hatpreferencesare stable:whatis desirable or anagent persists.7

if w&wf then%(w) = %(u/) (SC15)

This allows to disregard he influenceof emotionson desirability.We are awarethatour constraints too strong n thegeneralcase, but it is quiterealisticfor rather horttime intervals ike a smalldialog.

We make thesamehypothesis or(social, legal,moral. .) obligations,norms,stan-

dards. . thatholdfor theagents:

if wtfw' thenJ(w) = J(w'). (SCi6)

As we do notdeal withthedynamicsof ideals, it is quitereasonable o considerthatidealsarestable,at leastfor a giventimeinterval.

Wecall EL ramesthe set of framessatisfyingconstraintsSC1HSC16).

3.2.3 EL modelsandvalidity

A modelM is acouple (F, Y) where:- T is an ELframe;- Y : W - ATM associateseachworldw with the setYwof atomicpropositions

true n w.

GivenamodelA4 = (T, Y) where F = (W,SS,&, ®, J, st , <S),werecursivelydefinetruthof a formulaV:L\- M,w\= p iff p € Yw;- M, w \= -><pff M, w ¥= 

7 This allows concise statementsand proofs of theorems(which else would have required he explicitstatementof the relevantpersistencehypotheses).In recentwork we have relaxedthis constraintn orderto model emotion-focused opingstrategies AdamandLongin2007).

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- M, w \=<pv V iff M, w |= <p r M, w |= r/r;- M, w (= Beli<piff M, w' \=<p for every w' e â&i(w)\-

M,w \= Probiip iff there exists U € Pt(w) such thatfor every wf e U,M,w' \=- M, w |= Desi<p iff Mj w' \= <p or every wf € %{w)\- M, w (= Idi <pff M, wf \= (pfor every u/ € J(w)\- M, w (= Afteri:a(p iff jM, w' \=(p for every wf € ^:a(u;);- X, u; |= Beforei:a<p iff M, wf |= cp or every u;r such that w 6 «^(tt/);- jVt, w \= Gcp ff X, wf |= cp or every wf e ^(w)\- M, w \= H<p ff M, w' \=(p for every w' such that w € &(w').

Validity of a formula (p in the class of all Kripke models obeying our semanticconstraints is defined as usual.

Thus, (pis true in model M if and

onlyif M, w

\= cpfor every w in M. <p s EL valid (noted \=el <P) f and only if <p s true in every ELmodel M.cpis satisfiable if and only if \£el "^<P-Ps a logical consequence of a setof (global) hypotheses F if and only if for every EL model .M, if all hypotheses of Fare true in M then <ps true in .A/f.

3.3 Axiomatics

We now introduce a set of axioms that our modal operators have to satisfy. All ourmodal

operators except Probiare

goingto be normal modal

operators,whose defini-

tion we recall first.

3.3.1 Normal operators

D is a normal operator iff the axiom (K-D) and the necessitation rule (RN-D) hold forD.

O( <Dq> DVO (K-D)

P7- (RN-D)7-D#>

In any normal modal logic, the semantics validates the following principles (whichare used in some proofs in the appendix):

(HepA Dir) -+ D(<pA f) (C-D)

The dual of D is noted 0 and obeys the following principle:

(D<pa0V0->0(<paV0 (C-D)

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and thefollowinginferencerule(Chellas1980,Theorem .4, p. 116):w - > \lr

7T XT (rm"0)

Moredetailson the formalproperties f normalmodallogics can be foundinChellas(1980, Chap.4).

3.3.2 Action

Aftera and Beforea have the standard ense logic Kt in a linear time version:anormalmodal logic K extended with the following axioms (cf. Burgess(2002) formoredetails):

Happensa BeforeaHappensa<p (CONV-BH)

(CD-HA)and(CD-DB)are the axiomsof common determinism.Forexample(CD-HA) means thatif an actiona is aboutto happenafterwhichcp, hen afteranyother

actionß, <pwillbe true,andsimilarly nthepastfor(CD-DB).This entailsthatactionstakeone timestep,and are deterministicn the futureand in thepast(onecansee thatwhen a is ß). The conversionaxioms (CONV-AD)and (CONV-BH) ink past andfuture.

Remember hat :ot reads"agent does actiona".Wehighlighthere that whatwe call action is assumedto be intentional, hat s the

agent alwaysintend o performactions thathe is aboutto perform.This is the differ-ence between actionsand events. Thus if an agentsdoes somethingunintentionally(like sneezing)it is anevent,and it can only triggerevent-basedemotions.Thiscor-

respondso Lazarus'control

appraisalvariable

mposingthat one can

only reproachsomething o someoneif thispersonhad controloverwhat shedid,andto theconceptof responsibilityn OCCtheory(Ortonyet al. 1988,p. 54).

3.3.3 Belief

The operatorsBeli have the standard ogic KD45 (cf. Chellas (1980) or Hintikka(1962) formoredetails).Thecorresponding xiomsarethoseof normalmodal ogicsplusthe followingones:

Beliip - -^BeUîp (D-Belj)

Belup-> BeliBeli(p (4-Beli)

^Belicp- Beli^Beliip (5-Beli)

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Therebyanagent'sbeliefsareconsistent D-Beli), andanagent s awareof his beliefs

(4-Belt) anddisbeliefs(5-Beli).

3.3.4 Time

The operatorsG and H have the linear tense logic S4.3, (cf. Burgess(2002)) whichis a normalmodallogic K for eachoperatorplusthe followingaxioms:

G(p-* (F((pA Fty) V F(ty A F<p)) (3-F)

G<p» GG)

H<p-+ HHHF<p (CONV-HF)

(T-G) and(T-//) mean thatbothfutureandpastincludethepresent.(3-F) and(3-P) indicatethat f two formulasaretrueat two instants n the future

(resp.in the past)thenone is necessarilytruebeforetheother.This entailsthattimeis linear n thefutureandin thepast(cf. Fig.2).

(CONV-GP)and(CONV-HF)arethe conversionaxioms.Theyaxiomatize hat he

accessibilityrelation or G is theconverseofthat for H.

3.3.5 Probability

Thenotionof subjectiveprobabilitymeasure s capturedheresemanticallybythefactthatprobableworldsbelongto the set of believedworlds.This approachs basedon

neighborhoodunctions as opposedto probabilitydistributions).The logic of Prob is weakerthanthe logic of belief. In particular,he formula

(Probi<pA Probity) -> Probi(<pAty) is not valid,andthis is enoughto makeit anon-normalmodallogic in thesense of Chellas(1980, Theorem .3).

Thesemanticalconditionsvalidate hefollowingprinciples:

tì±- (m-Probi)Probnp -* Probity

- £ - QKN-Probi)Probiip

Probi<p ^Probiîp (D-Probi)

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3.3.6 Desirability

The logic of desirability is standard deontic logic (SDL) (Chellas 1980) and is alsoexpressed in terms of ideal worlds: the logic associated with the operators Desi is KD,i.e. the normal modal logic K plus the following axiom:

De Si(p-+ -*Desi-*(p (D-Desi)

which makes desirabilities consistent.It has been argued that in principle (e.g. Castelfranchi and Paglieri (2007) and also

Lang et al. (2002)), desirability is closed neither under implication nor underconjunc-tion: It may be desirable for me to marryAnn and it may be desirable for me to marry

Beth, but this does not imply that it is desirable for me to be a bigamist. Though, forthe sake of simplicity, our Dest operators are normal and hence closed under both

conjunction and implication.

3.3.7 Ideals

Just as for desirability, the logic of ideality is standard deontic logic SDL, i.e. thenormal modal logic K plus the following axiom:

Idlip - ÎdH<p (D-Idli)

which makes ideals consistent.

3.3.8 Mix axioms

The interdependencies between some modal operators are captured by the followingaxioms. First, the following introspection axioms express that the agents are aware oftheir probabilities and desirabilities:

Probi (P- Beli Probi V (4-MIX 1

^Probi Beli Oest Desi<p-> BeU^Desiip (5-MIX2)

From these axioms plus (D-Beli ), we can easily prove their converse. For example, wededuce the converse of (4-MlXl) from Beli Probity - -'Beli-* Probi<pby (D-Beli),

and ->Beli->Probnp-

Probnp by (5-MIX1). We therefore have the equivalencesProbiip <+ BeliProbiip and -*Probnp *» Bel^Probup.Then the following axioms express that actions are public:

DoneaT - Beli DoneaT (4-MIX3)

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^DoneaT -+ Beli^Done^J (5-MIX3)

From these axioms plus (D-Beli) we can easily prove theirconverse, and we thus havethe equivalences DoneaT ++ Beli DoneaT and -^DoneaT «-»Beli-^DoneaT.We axiomatize the inclusion of elements of neighborhoods in epistemic states

through the following axiom:

{Beliip A Probiir) -* Probity A \/r) (C-MIX)

which allows to derive the following theorems:

Beli Probw (2)

ProbiÇ -> -^Beliîp (3)

Time and action are linked: if ipis always true in the future then ipwill be true after

every action performance. Similarly, if (pwas always true in the past, then (pwas truebefore every performance of an action. So:

G<p-> Aftera<p (GA-MIX)

Hip -* Beforeaip (HB-MIX)

Finally, desirability persists, i.e. it is preserved through time.

Des up -» G De Siip (Pers-Des,)

^DesHp -* G-^Desnp (Pers--*Desi)

These two principles both entail the equivalences De sup <->GDes^p and ^Des^p «*

G->Desiip.For the same reasons, ideals also persist:

Idlip -> GIdlip (Pers-/<///)

--Idlip -> G- Idlip ( Pers- IdU)

These two principles entail that we have an equivalence.The "noforgetting"constraint linking actions andbelief is captured by the following

axiom:

{Beh AfteraipA^Beli Aftera±)->

AfteraBeUip (NF-Belt)

This axiom expresses that the agents do not forget their previous alternatives, whenthe performance öf the action is not surprising for them (->Beli Afteral. reads "agenti does not believe that action a is inexécutable"). Otherwise, if Beli After^L holds,

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thenthe agenthas to revisehis beliefs upon learning hat a occurred.We do not gointo thishere,andrefer the reader o HerzigandLongin(2002).

In the next two sections we aregoing to put to worklogic EL, andare going toexpresstwentyfrom the twenty-twoemotions of OCCtheory. Wedo not definetheremaining motionsof love and hatebecausetheywouldrequirea firstordermodallogic.)Foreach of thesetwentyemotions,we firstgive theinformaldefinitionof OCCtheory,andthen our definitionin termsof logical formulas.In orderto support heaccuracyof ourdefinitions,we show that heycan account ortheexamples llustratingthe emotionsin Ortonyet al. (1988). Below,thequotedpagesall referto this book.

4 Event-basedemotions

The event-basedbranch f OCCtheorycontainsemotiontypeswhoseelicitingcondi-tionsdependon theevaluation f aneventwithrespect otheagent'sgoals.Desirabilityis the centralvariableaccounting or theimpact hatan event has on anagent's goals,namelyhow it helpsor impedestheirachievement.

In ourformalism,an event s something hatmayoccur withoutanyagent ntendingit, andis thus different rom an action(that s alwaysintentional).According o OCC

theoryan event can have severalaspects,each of thempossibly triggeringa differ-

ent emotion.In this

paperwe

representan emotion as an abbreviation f a

complexformula.Moreoverwe assumethat whatOrtonyet al. call thedifferentaspectsof aneventcan be consideredas consequencesof theprimary vent. Forexampletheeventof receivinga letterfrom a bailiff to informyou thatyou aregoing to inheritsomemoneyfroma deceased relativehas (at least) two aspects:the undesirableaspectisthatyourrelative s dead,while the desirableaspectis thatyou get some money.We

represent hese two aspectsas if they were two separatesecondaryevents actuallyresulting rom theprimary vent. While the sameprimary vent cantriggeroppositeemotions(sadness hatyourrelativediedand oy of gettingsomemoney),we considerinourformalizationhat heseemotionsareattached o two different econdary vents,

butnotto theprimary vent.According o OCCtheory,an event is desirable or anagent f itsconsequence p smorebeneficial furthering isgoals)thanharmful impeding omegoals).As we saidbefore(see Sect.2.2), desirabilitydependson theagent'sgoals,but we do notwanttoenter nto the detailsof thiscomputationhere,and assumethattheagent'sdesirabilityvaluesaregivenby the Des operators.Wedirectlyuse this variable n the definitionsof event-based motions(cf. Sect.3 for ourmodellingof desirability).

4.1 Well-beingemotionsTheemotiontypes in thisgrouphaveelicitingconditionsfocusedon thedesirabilityfor the self of anevent.An agentfeels joy (resp.distress)whenhe is pleased(resp.displeased)abouta desirable resp.undesirable) vent.

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Joy up = Beli<pA Desi<p

Distress^ = Beli<pA Desi~*<p

Consideranexamplesituation rom(Ortony t al. 1988,p. 88) whereaman learnsthat he inheritsof a small amountof money (m) froma remote and unknownrela-tivethat hasdied(d). This is expressedby the formulaBelt(mAd). Then i feels joybecause he focuses on the desirableevent(Desim) and not on the undesirable ventd. This mandoes not feel distress about his relative's death since he did not knowtherelative,his death is not undesirable or him(-»Des,-*/). Onthecontrary, manj (p. 89) who runs out of gas on the freeway(Beljo) feels distress because this isundesirableorhim(Desj-*o).

4.2 Prospect-basedmotions

The emotiontypes in thisgrouphaveelicitingconditionsfocused on thedesirabilityfor self of an anticipated uncertain) vent, that is actively prospected.OCCuses alocal intensityvariablecalled likelihood,accounting or the expected probabilityofthe event to occur.Wemodel likelihoodby thefollowingabbreviationExpect^

Definition 1 Expect& = ProbnpA->Beli ->Beli~-* -»Expecti-i<p (5)

An agent feels hope (resp. fear) if he is "pleased (resp. displeased)abouttheprospect of a desirable(resp. undesirable)event".Note thatthe object of hope isnotnecessarilyaboutthe future: mightignorewhethermy emailhas been deliveredto theaddressee,andhopeit hasbeen so.

Hopem> = Expecti<p A Desi<p

F earup = Expectup A Desi~*, ust beforeSect.3.2) to representwhatwas true n thepast.

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Satis faction^ = BeliPExpectup A DesupA BeliH>

FearConfirmedup=

BeUPExpectupA

Desi~><p Belup

Givenourdefinitionsof joy and distressSatisfaction^ can be writtenmore con-cisely BeliPExpecti<pAJoyi<p, and FearConfirmedi<p can be written BeUPExpedi A Distress^.

The agent feels relief (resp. disappointment)f he is "pleased (resp. displeased)about he disconfirmationof theprospectof an undesirableresp.desirable)event".

Relief i<p= BeliPExpecti~*<pADesiip /\Beli(p

Disappointment^ = BeliPExpectîp A Desîp A Belj<p

Forexamplea woman w who appliesfor ajob (p. Ill) mightfeel fear if she expectsnotto be offered hejob (Expect w-^beHired)tr feel hope if sheexpectsthatshe willbe offered t(Expect wbeHired).Then, f shehoped ogetthe obandfinallygetsit,shefeels satisfaction;andif shedoes notget it, she feels disappointment.Anemployeee (p. 113) whoexpectsto be fired(Expect ebeFired)will feel fear if it is undesirablefor him(Dese-*beFired),but not if he alreadyenvisagedto quitthisjob since in this

case we can supposethatthis is not undesirable or him (->Dese->beFired).n thefirstcase he will feel relief whenhe is not fired Bele^beFired),and fear-confirmedwhenhe is.

4.3 Fortunes-of-othersmotions

The emotiontypes in this grouphave elicitingconditionsfocused on the presumeddesirability or anotheragent. They use three local intensityvariables:desirability

forothery

deservingness,and

liking.Desirability orother is thè assessment

byi of

how muchtheeventis desirable or the otherone (j). Deservingnessrepresentshowmuchagenti believes thatagentj deservedwhat occurred o him. (Itoftendependson liking,i.e. i's attitude owards , butwe cannot account for this here because wedo notconsiderattraction motions.)

We thus have to model these variables.First, we can representdesirability orotherby a beliefabout he other'sdesire:Beli Desj(preads"agent believes that<psdesirable or agentj". Second,we representikingthroughnon-logicalaxioms. For

example,when John ikesMarythis meansthat f Johnbelieves that t is desirable or

Maryto be rich,then it is desirable or JohnthatMarybe rich,or better:gets to know

that she is rich.In formulas: BeljohnDesmaryrich DesjOhnBelmaryrich).heseaxioms will be global hypotheses n deductions, n the sense thatthey aresupposedto be knownby all agentsand to holdthrough ime.Third,we simplifytheconceptof

deservingnessby assuming hatagentswant(havea goal) thatanyagentgets whathedeserves.Thenwhen anevent(representedwith the formula<p) ccursthat s believed

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(by ï) to be deservedby j, this eventwill be desirable or i (we write t Des\ Beljcp*),since it achieveshis goal. So finallyi can desire thatj believes (peither because he

believes thatj desirestpandj is his friend,or because he believesthatj desires-upandj is his enemy,orbecause he believes thatj deserved p.There are two good-will (orempathetic) motions:anagentfeels happyfor(resp.

sorry or)anotheragentif he is pleased (resp. displeased)aboutan eventpresumed obe desirable resp.undesirable)orthisagent.

Happy Forjj cp = Beli<pA Beli &esj <PA Desi Belj cpdef

SorryFonjcp = BelupA BeUDesj-*(pA Desi-*Belj(p

Thereare two ill-will emotions:anagentfeels resentmentresp.gloating)towardsanother gent f he isdispleased resp.pleased)aboutaneventpresumedobe desirable(resp.undesirable)or thisagent.

Resentmentijcp = BelupABeljDesjcp A Desi~>Belj<p

Gloatingij(p = BelupA BeliDeSj-np A DesfBeljcp

For example (p. 95) Fred feels happy for Mary when he learnsthat she wins

a thousanddollars(Bel/w A Bel/Belmw)9 because he believes this is desirableforher (BelfDeSmW),and she is his friend,i.e. he likes her. As said above, we repre-sent this notionof likingwith non-logical global axioms representing ne's interestin the well-beingof one's friends.In this case, if Fred believes that it is desirablefor Mary to win, then it is desirablefor him that she gets to know that she won

(BelfDeSmW-* DesfBelmw).9A man (p. 95) can feel sorry for the victims v of a natural isaster Bel,disastera

Beli BelvdisasterA Beli Desv->disaster)withoutevenknowing hem,becausehe hasan interest hatpeople do not sufferundeservedly, o there it is undesirableor himthatpeoplesuffer from this disaster Desi-*Belvdisaster).

An employeee (p. 99) can feel resentment towardsa colleaguec who receivedalarge pay raise(Belepr, BeleDescpr), what he believesto be desirable or this col-

league(BelcDesepr), becausehe thinksthiscolleagueis incompetentand thusdoesnot deserve this raise.As we said above, we represent he deservingness whateverthe reason for this belief, here the incompetence)with a desirabilityconcerning heoccurrence f thiseventto the otheragent(hereDese^Belcpr) that s itsconsequence.

8 The emotionstypes in this groupresultfrom the appraisalof an event concerning another agent. We

represent he occurrenceof this event<po agent j by the formulaBelj (p.Then the Desj Belj<pelementof the definitions means that this eventoccurring o j is desirablefor i\ which is ourway to distinguishbetweengood-willandill-will emotions.9 Note that Fredmaynotbe happy or Mary f she was notto learnabouthergainin thefuture.However,even if she does not knowyet thatshe won,he can feel happy or her ustbecausehe considers t probablethatshe will learn t at a futuremoment withoutbeingsure of that).Forexample,Marymayhavenot seenthe resultsyet, andFredcannotbe surethatshe will notforgetto checkthem.

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Finally,Nixon's political opponents(o) (p. 104) might have felt gloating abouthis departureromoffice (Bel0 d A Bel0 Belnixond), becausetheybelieved it to be

undesirable or him (Bel0 Desnjxon->d) ndthey thought t was deserved(as abovewe identifydeservingnesswith a desire,here:Des0 Beln\xond).

Remark1 Our formalizationof liking leads to the following question:what if ibelieves that for some reasonsj will never learnthat cpl In many situations it iscertainlyodd to saythat is happyorsorry oranotheragentj aboutsomething hatjwill neverknow,andthusaboutwhatj will neverbe happyor sad abouthimself.OCCtheorydoes not require hatj should know aboutthis eventthat s importanto him,andwe thereforehave chosen to stay as close as possible to it. However,one mightwish to sharpen he definitionsof these four emotionsby requiring hat it must be at

leastprobable ori thatj learnsabout he eventat some timepointin the future.Thiscan be implementedby addingthe further onjunctProbiFBeljcp to ourdefinitionsof the four fortunes-of-othersmotions.

5 Agent-basedemotions

The agent-basedbranchof OCCtheorycontainsemotiontypeswhose elicitingcon-ditionsdependon thejudgementof thepraiseworthinessf anaction,withrespecttostandards.

In our sense an action is somethingthat is performedntentionally deliberately,purposely) by an agent.It thus differsfroman event.If an agent performsan actionnotpurposely, ike sneezing,we call this an event.Thisdistinctionallows to captureimplicitlyLazarus'variable f attributionf responsibilityhat s neededfor emotionslikeanger:anagentis always responsible or his actions.

An action is praiseworthy resp.blameworthy)when it upholds(resp. violates)standards.Thestandards nderconcernaresupposed o be internalized,.e. the(eval-uating)agenthas adopted hem. Weexpressthese internalized tandards oragentithrough hedeonticoperatorsdl\.

5.1 Attribution motions

Theemotiontypesin thisgrouphaveelicitingconditions ocusedon theapproving fan agent'saction.They use two local intensityvariables.Strengthof unit intervenesin self-agentemotionsto represent he degreeto which the agent identifies himselfwith the authorof the action, allowing him to feel prideor shame when he is notdirectlythe actor;forexampleone can be proudof his son succeedingin a difficultexamination,or of his rugbyteamwinningthe championship;n this paperwe onlyfocus on emotionsfeltby theagentabouthis ownactions,becausethisvariable s toocomplexto be representedn our framework.Expectationdeviationaccountsfor the

degreeto which theperformed ction differs fromwhatis usuallyexpectedfromtheagent,according o his social role orcategory.10

10 In self-agent emotions, the agent refers to his stereotyped representation of himself.

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We express this notion of expectation with the formula ProbjAfteri:a-*<p reading"j considers it probable that after i performs a, <p s false", i.e. j expects i not to

achieve <pas a result of his action, for example because it is difficult.1

1

The deviationcomes from the fact that after the execution of a, j believes that <ps nevertheless true,

contrarily to what he expected.12 This prevents the agent from feeling attribution emo-tions too often. Indeed, we often respect the law without being proud, and we oftenviolate standards without being ashamed. Therefore we consider that the standardshave to be internalized and accepted by the agent as belonging to his values. Thisallows an agent to feel no emotion, even concerning an (un)ideal action, when thisis not important for him. For example someone who likes to wear strange (unideal)clothes would not feel ashamed about this if it is what he desires to wear, but wouldfeel so if he was forced to wear such clothes.

Finally, we do not impose that the ideal was conscious at the moment of the action.For example one can feel shame about having performed an action when one realizesthat it was blameworthy, even if one ignored that at the time when the action was per-formed. Ideally, we should not impose it either for probability, but the Probi operatorsare intrinsically epistemic (i.e. semantically, probable worlds are a subset of possibleworlds compatible with the agent's beliefs); so technically it is difficult to do so.

In the sequel, Emotiorii(i:a, <p) resp. Emotionij(j:a, ^»abbreviates Emotiorii(Donei:aT, <p) resp. Emotiorii j(Done y:crT , p)) where Emotion is the name of anemotion.

Remark 2 These emotions are about an action a that the agent believes to have influ-enced the proposition <p: he agent believes that "if he had not performed action a,(pwould probably be false now". Though, our language is not expressive enough to

represent this counterfactual reasoning, so we make the hypothesis that the agent ibelieves that a and <pare linked in this way. The following emotions do make sense

only when this is the case.

Self-agent emotions: an agent feels pride (resp. shame) if he is approving (resp.disapproving) of his own praiseworthy (resp. blameworthy) action.

Pridei(i:a,(p) = Beli DoneVM{ldli HappensVM(pA Probi AfterVM-^<p)

A Beli<p

Shame i(i:<x,<p) = Beli Donei:a(Idli-^Happensi:a<p A Probi Afteri:a-*<p)

A Beli<p

11 In thefollowing,whatever he cause of theunexpectednesss (forexampledifficulty),we onlyformalize

theconsequence theunexpectednesstself) with the aboveformula.12 What is unexpected s not only the performance f the action but also its result<p\ ctually,when theresultis not important,<ps T and then it is the very performance f the actionthatis unexpected.For

example t is unexpected rom a wise childto stealsomething n a shop,whatever he resultof his action s(didhe succeed or not),so we write:Probi ^fterchild.steal^-

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Emotionsinvolvinganotheragent:13an agent feels admiration resp. reproach)towardsanotheragentif he is approving resp. disapproving)of this agent'spraise-

worthy(resp.blameworthy) ction.

defAdmirationij(j:ay(p) = BeliDonej:a(Idli Happensj:aHappensj:oi(pA Probi After j:a-><p) Beli<p

Forexample,a womanm feels pride (p. 137) of havingsavedthe life of adrowning

childbecause she performedhe actiona (tojumpinto the waterto tryto savehim)withthe successfulresults (thechild is safe): BelmDonem:aTA Belms.14Moreovershe now believes that before the action, it was ideal to save the child and she inter-nalized this ideal(Idlm Happensm:aT)9 utshe had not much chances to succeed:15ProbmAfterm:a^s.

A richelegant ady (p. 142)wouldfeel shame whencaughtwhilestealingclothesin anexclusiveboutique Shamei(a, T)), where a is theaction to steal,becauseshehasperformed nactionthatwas unideal or her16 Idli^Happensi:aT) and mprob-able to be performedby her(Probi Afteri:a±.) due to her social role. The resultoftheaction s T here becausethis emotiondoes notdependon the success or failureof

theactionbut on its very performance.A physicistp's colleaguesc (p. 145) feel admiration towardshim for his Nobel-

prize-winningwork(BelcDoneP:aTA Belcw, where a is the actionof conductingexperiments,with theresult w of obtainingNobel-prize-deservingindings)because

theyinternalizedhisresult as ideal.17 Idlc Happensp:aw) and difficultthusunex-

pected (Probc Afterp.ma-*w). s we said above, the difficultyof an action is one

possiblereasonfor its resultto be unexpected.Hereto simplifywe do not formalizethe verynotionof difficultybutonly its consequence, .e. the unexpectednessof theresult,whichis what we are interestedn whenwe defineattribution motions.

A man mayfeel reproachtowardsa driver (p. 145)who driveswithouta validlicense(Beli Donej.sT, where8 is the action odrivewithoutavalid icense),because

13 When i = y, these emotions correspond to the self-agent emotions (cf. Theorem 5).14

Actually, she also believes that she influenced this result by her action, i.e. she believes that if she had

not jumped into the water the child could have drowned; as we said it before, we cannot express this causal

link in our language, so our account is incomplete in that respect.15 Thus, she would not feel pride after saving the child if she believes it was easy for her.16

Actually actions do not obligatorily follow moral values. The lady may have been driven by the desire

to possess the object, violating her ideals. But this example seems to be a borderline case, since she could

havebought

the object instead.17 Here, what is ideal is not only the execution of the action but its execution with this result. Similarly,in the case of negative emotions, what is unideal is not the happening of the action, but its happening with

a given result: Idli~>HappensiM<p. This is compatible with the fact that the action itself could be ideal:

Idli Happensi:aT. For example, it is ideal to participate, but unideal to lose when you are expected to

win.

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it is forbiddenand he considersthisobligation o be importantIdlj^Happensj-sT)andunexpected rom a driver Probi Af tery^L).

5.2 Compound motions

These emotions occur when the agentappraisesboththe consequencesof the eventand ts agency.Theyarethus theresultof acombinationof attributionmotionsaboutan actiona with resultcp>ndwell-beingemotions aboutthisresult(p.

Gratificationj(i:af(p) = Pridei(i:a,(p)AJoynp

Remorsei(i\oi,<p) Shamei(i:a,(p)ADistressup

Gratitudeij(j:a,<p) = Admirationij(j'.a,<p)AJoyjcp

Angerij(j:a,<p) = Reproachij(j:a,(p)ADistressup

Forexample,a woman i may feel gratitude (p. 148) towards he stranger whosaved her child fromdrowning Beli Donej:aT A Belis, wherej:ot is j's actiontojump in the water,ands is the result: her child is safe). Indeed,i feels admirationtowards because of y's ideal but difficult(i.e. before it, Probi AfteryM-*sheld)action.Moreover he result of j's action(BeUs) is desirable or i (Desjs), so i alsofeels

joyabout t

(Joyjs).Similarly,a woman w (p. 148) may feel anger towardsher husbandh who for-gets to buythegroceries(BelwDoneh:aT,where a is his actionto go shopping,andBelw^g*whereg reads "therearegroceriesfordinner"),becausew reproacheshisunidealresult o h (itwas not theexpectedresultof the action:ProbwAfterh:ag)>andsheis also sadabout t (Distressw-g) becauseshe desired oeatvegetables Deswg).

Thephysicistp mayfeel gratification aboutwinningthe Nobel prizebecause heperformed successfulexecutionof actiona (performing xperiments), chieving heidealresultn (he receives the Nobel prize),and thus feels pride;andthis result s notonly socially ideal but also desirable orhim18 Despn), so pridecombineswithjoy.

Finally, a spy may feel remorse (p. 148) about having betrayedhis country(actionco) f he moreover aused undesirabledamages(resultd): Shamespy((o,d) a

Distressspyd.

6 Formal properties

In theprevioussectionwe started rom OCCtheory,extracted ts key concepts,andcasted hem ntologicaldefinitionsof twentyemotions.Thefirstbenefitof our work sto disambiguatehesedefinitions,thatmightbe debatablewhenexpressed n natural

language.For

example,we hadto decide between two different

optionsin the case

of fortunes-of-othersmotions,dependingon whetherwe acceptedthat one can feel

18 This is notalwaystrue.Forexample,a childmay personallydesire notto go to school,while it is idealto go.

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happyabout some good news for somebodyelse even if we believe thatthis personwill never earn hegood news.Furthermore,he use of logic enables to reasonabout

the formalizedconceptsand to deriveproperties. n contrast,propertiesof emotionsare always debatablewhen definedinformally,and manydebateshave occurredasresearchprogressed.

In this section we expose some theoremsfollowing from ourdefinitions,mainlyconcerning he causal andtemporal inksthatemotions have with each other.Thesetheoremsareconsistentwith OCCtheory; ometimes heyevengo beyond t,buttheyalwaysremainntuitive.Moreover,what s interestings that heformalproofsof thesetheoremsmakethatthey arenot debatableon their own once one has acceptedthe

principlesof the logic. This shows againthe advantagesof formalreasoningaboutemotions.The readerwho is interested n the proofsof these theorems s referred o

theappendix.

6.1 Prospect-basedmotions andtheirconfirmation

If an agentremembers hat at a moment n the pasthe was feeling a prospect-basedemotionabout<p, ndif he now knowswhether<ps true or false, then it follows bythe laws of ourlogic thathe feels thecorrespondingonfirmation motion.

Theorem 1 (Temporalink fromprospect o confirmation).

h (BeliPHopei Satis actionupv Di sappointmentit (a)

h (BeliPFeanv A (Belupv Bel^y))- Relief upv F earCon irmedi-><p (b)

Moreover,we can prove thatan agentcannotfeel simultaneously wo emotions

concerning he confirmation ndthe disconfirmation f the sameexpectation.

Theorem 2 (Inconsistencybetweenconfirmation nddisconfirmation).

I \Satisf actionup A Disappoint menti~^<P) (a)

I >(FearConfirmedi(pA Reliefi~><p) (b)

Theprooffollows from therationality xiom forbelief.Please note thaton the contrary,we cannotprove inconsistenciesbetweenrelief

and satisfaction,or between fear-confirmedand disappointment.This is becauseBeli P Expecti-*(p ndBeli P Expecti<preconsistent, .e. theagentmayhaveexpected

(pat one momentin the

pastand -*><pt anothermoment.19We can only provethat

19 Thus, our current definitions of confirmation and disconfirmation emotions may not be precise enoughto entail this intuitive inconsistency. Actually in linear temporal logic with Until and Since operators, we

couldwriteforexampleRelief up = Beli P{Êxpecti^(pSince Expect^) A Desupa Belup.

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these two expectations Expect^y and Beli PExpect^ cannot occur at the sametime. (Thisis the theorem 5) of Sect.4.2).

Wecanprovethat thepositiveconfirmation motionsimplyjoy, and that theneg-ativeconfirmation motionsimplydistress.This is intuitive,and in agreementwithOrtonyet al. 's definitions.

Theorem 3 (Linkbetween confirmation ndwell-beingemotions).

f- Satis faction^ -> Joyup (a)

h FearConfirmedup -* Distress^ (b)

h Relief up-> Joyup (c)

h Disappointment^ -> Distress^ (d)

6.2 Fortunes-of-othersmotions

In thisparagraphwe will groundonreinforceddefinitionsof fortunes-of-others mo-tions, that we denotethemby Emotion!.(pwhere Emotion'rangesover the four for-

tunes-of-other motions n {HappyFor, SorryFor, Resentment, Gloating}. Thesereinforceddefinitionsareobtained romourdefinitionsby adding he further onjunct

Probi FBelj (p othem,as we suggested n Remark1 For nstanceHappyFor! .<p=

HappyFor! .(pA Probi FBeljip.We can provethatif the agenti feels a fortune-of-other motiontowardsanother

agentj about p, hen t is at leastprobableor i that is goingto feel thecorrespondingwell-beingemotion about(pat some moment n the future.

This leads us to believethatOCC definitionsof these emotionsmaybe to vague,since theydo not allow to deduce thesepropertieswhile theyarequiteintuitive.

Theorem 4 (From ortune-of-other motion to imageof other).

h HappyFor! -cp> ProfyFJoyjcp (a)

h SorryFor! j(p -» Probi F Distress j(p (b)

h Resentment! - ProfyFJoyjcp (c)

hGloating^jcp

-> Probi F Distressj(p

(d)

Ifanagent feels afortune-of-othermotion owardsanother gentaboutcp, nd isnot sure hat will learnabout he eventcp9hen feels acorresponding rospect-basedemotionaboutj believing(p.

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Theorem 5 (Consequencesof fortunes-of-othersmotions).

h (Happy Forj jcpA-»BeliFBelj<p)-» Hopa FBeljip (a)h (SorryForij(p A-Beli FBeljip) -> Fear/ FBeljip (b)

\- (Resentmenti :<pA-~>eli F Beljty) -+ FeariFBeljip (c)

I- (Gloating- jcpA-»Ari/FBeljip) - //op*/ Fße/,^ (d)

6.3 Links betweenself-agentandother-agent ttribution motions

We can provethatan other-agent motion towardsoneself is equivalent o thecorre-sponding elf-agentemotion.This s ratherntuitive, llthe moreOrtony t al.introducethe termself-reproachor shame.

Theorem 6 (Other-agentmotionstowardsoneself).

h Admirationij(i:a, ip)*+ Pridei(i:a, ip) (a)

h Reproachij (i :a,ip) <> Shamei(i :a ip) (b)

We canprovethat f anotheragentj feels an attribution motion towardsanagenti abouta given actionwith a given result,then the agenti does not inevitablyfeelthecorrespondingelf-agentattributionmotion.That s, one can admireyou abouta

givenactionwhileyou are notproudabout t.

Theorem7 (Other-agentmotiondoes not forceself-agentemotion).

\f BeliAdmiration jj(i:a,ip) - Pridei(i:a,ip) (a)

\f Beli Reproachjj(i:a,

ip)-> Shamei(i:a, ip) (b)

Both prospect-based motionsand attribution motionsinvolveprobabilities.Wethusget interested n theirtemporal inks with each other.We can provethatif an

agentfeels an attributionmotion aboutan actionwitha givenresult,and thatbeforethisaction he envisagedthat t couldhappenwith this resultand hada correspondingdesire,then at this momenthe felt a prospect-based motion aboutthe performanceof this actionwiththis result(namelyabout the success or failureof the actionwith

respectto the prospectedresult).We have the same theorem f the agent feeling theemotionis different rom theagentperforminghe action.

Theorem 8 (Linkbetweenprospectand attributionmotions).

h Pridei(i:a,ip) -> Beli Donei:a((-*BeU^Happensi:aip

ADesi Happensi:aip)-* Fean^Happensi:aip) (a)

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h Shamei(i:a,(p) -> Beli Donei:a((->Beli-*Happensi:a<pA Desi^Happensi:a(p) -> Hopei->Happensi:a(p) (b)

h Admirationij(j:a,(p) -> BeljDonej:a((-^Beli-*Happensj:acp

Aßest Happensj:a(p)-> Feari->Happensj:a(p) (c)

h Reproachij(j:a,(p) -> Beli Donej:a (-'Beli-* Hap pensj:ct(pA Desi^Happensj:a(p) -> Hope^Happens j:a<p) (d)

We can notice thatwe haveto imposethattheagenthada corresponding esireinorder o makehim feel fearor

hope.Moral valuesare not sufficientto

triggerhese

emotions,since theycan be inconsistentwithdesires.Forexampleone can desiretokill someone he hates while his moral values tell himnot to do so.

Wecan alsoproveakindof converseof this theorem:f theagent ears(resp.hopes)thathe does notperform heactiona withresultcp,andthatthisperformances idealfor him (resp.unideal),then afterhe performeda, if he believes that(pis truethenhe feels pride(resp. shame).Actually,the agent was afraid to fail (resp.he hopedto succeed).Forexamplesomeonewho passes an examinationandhas few chancesto succeed would feel afraidof failing, andthen if he succeeds he would feel pridebecauseit was difficult.

Theorem 9 (Linkbetweenattribution ndprospectemotions).If a is an action thattheagenti believesto influence heproposition<pcf. Remark ), then:

h Feari->Happensi:a<pABeli Idli Happensi:a<p-> Aften:a(Beli(p -> Pridei(i\a,<p)) (a)

h Hopei-*Happensi:a(pA BeUldli^HappensVM(p->•Afteri:a(Beli Shame {i:a,<p)) (b)

h Fear^Happens yMipA BeliIdij-^Happensj:a(p-> After j:a(Beli Admirationij(j:a,<p)) (c)

h Hopei-*Happensj:a<pA BelJ dl^Happens yMy-* After j:a(Beli(p-> Reproachijij'.a.cp)) (d)

6.4 Inconsistenciesbetweensomeemotions

We canproveseveral nconsistenciesbetweenpairsof emotions.First,we can provethe inconsistencybetweenoppositeemotionsaboutthe same

proposition polar opposites), i.e. between the positiveandthe negativeemotionofthesamegroup.This is in agreementwiththepsychologicaldefinitions.

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Theorem 10 (Polar nconsistencies).

I *{JoynpA Fearup)I '(HopeupA Fearup)I '(Satisj action i(pA FearConf irmedup)I '(Relief upA Disappoint menti<p)I '(HappyFor'j :(fASorryFori <p)I '(ResentmentijcpAGloatingijcp)I i(Pridei(i:a, <p)AShamej(i:a, cp))I '(Admirationijij'.a, ç) A Reproachij(j:a, (p))

I >(Gratificationi(i:a, <p)A Remorsei(i\oL,p))I '(Gratitudeijij'.a, <p)AAngerij(j:a, (p))

This followsinparticularrom herationality xioms(D) for ouroperatorste// , Des/ ,Prob/ andIdlj.

Please notice thatwe can stillcapturemixedemotionsaboutagivenevent,becausethese mixedemotionsactuallyconcerndifferentaspectsof thisevent,that we repre-sentwith different ormulasas if theywere differentconsequencesof the mainevent.Forexampleone wholoses a friendwho suffered roma longandpainfuldisease willfeel sadness aboutthe loss of his friend,andat the same time reliefaboutthe end ofhis friend'ssuffering.We thusconsider hattherearetwoappraised vents:the loss ofa friend,that s undesirable, nd the end of his suffering, hat s desirable.The initialevent(thedeathof a friend)thustriggersa positiveand a negativeemotion.

Dueto theproperties f ourprobability perator, opeis notonly inconsistentwithfearabout he samecpbut also with fear about-»#?.Actually,dependingon which oneis moreprobablebetweencpand-^, the agentfeels eitherhope or fear. Thus thesetwo emotionscannotoccursimultaneously.

Theorem 11 (Non simultaneityof hopeandfear).

I *(Hopei<pA Feari-i(p)

This is because by definitions Hopeiip implies Probnp while Fear^(p impliesProbity, which cannotsimultaneouslybe the case dueto theconsistencyof expec-tations(property of Sect.4.2).

Moreover,anagentcannotfeel simultaneously good-willand an ill-will emotiontowards he sameagentaboutthe sameissue.

Theorem 12 (Inconsistencybetweengood-will andill-will emotions).

I '(HappyFon jcp A Resent ment j(p) (a)I *(SorryFort (p AGloatingij<p) (b)

I '(HappyFort <pAGloatingij(p) (c)

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236 Synthese(2009) 168:201-248

I '(SorryForij(p A Resentmenti <p) (d)

Theprooffollows fromtherationality xiomsforBeli andDes%see theappendix ordetails).

6.5 Other nterestingproperties

Ourformalismallows us to provethatanagentis awareof his emotions.

Theorem 13 (Emotionalawareness).ForeveryemotionEmotioniamongthetwentyemotions hatwe havedefined:

h Emotionip«» Bel[Emotion p (a)

I 'Emotioni p » Beli-Êmotionip (b)

This followsinparticularromtheintrospection xiomsfor ouroperatorsBeli , ProbiandExpectf.

Accordingto Lazarus 1991), only situations hat are relevant o the individual's

well-beingcantriggeran emotion.If we consider hatan event is relevant o i 's well-

beingwhenit involves one of i's desires or values,thenthis is in agreementwith the

followingtheorem.Indeed,if the agenthas no desireor ideal at all thenno eventis

relevantto him, and thus no situationcan triggeran emotion.Besides, desiresandmoralvalues arepartof what Lazaruscalls "ego-involvement".

Theorem 14 (Emotionsandego-involvement).Anagentwho has neitherdesiresnorideals cannot eel any emotion.

The proof triviallyfollows from the definitionsof emotions,thatall necessarilyentail eithera desire(forthe event-basedones) oran ideal(forthéagent-based nes).Compound motions entail botha desireand an ideal.

7 Discussion

Logicalapproaches femotionsarestillquiterare.J.J.Meyer soneof the fewresearch-ers to have contributedo this field.Inparticular e hasrecentlyproposedanapproach(Meyer2006) where emotions are consideredas kindsof events,and wheredefini-tionslike thosepresentedabove are thenecessaryconditionsof thetriggering f theseevents(thatSearlewouldcall "mental vents" Searle1983,Chap.3)). This modelisa very interestingalternativeo ours,independentlyrom the detailsof the definitions

respectivelychosenin eachapproach.

Meyer (2004) proposesa logical model of emotionsbased on KARO,his logicof action,belief and choice (cf. van der Hoek et al. (1997) or Meyeret al. (2001)).He uses this logic to writegeneration ules for four emotions: oy, sadness,angerandfear,dependingontheagent'splans.First, hegeneration onditionsof these emotions

only dependon the satisfactionof theagent'splans,making hismodel task-oriented.

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Indeed, Meyer's aim, as he states himself20, is not to be faithful to psychological def-initions but to design artificial agents. On the contrary, in our work, we try to stay as

close as possible to the original psychological definitions of the emotions that we for-malize, through building on one of the most widely used approaches, namely Ortony,Clore, and Collins' typology. Second, this approach focuses on the individual aspectsof emotions: as thereis no operator to representsocial standards, no social emotion like

pride or shame can be represented. Finally, we thus provide an emotional formalismthat is richer (with twenty emotions) and more faithful to psychology. However, ourformalism is still limited to the triggering of emotions, whereas Meyer and colleaguesalready formalized the influence of emotions on the agents' plans (Dastani and Meyer2006).

We would now like to highlight the assets and limitations of our own model from

several points of view, namely computer science, logic and psychology.Our model undoubtedly suffers from some limitations. First, from the logical point

of view our framework lacks some expressivity. In particularwe preferred not to usethe full collection of existing temporal operators like Since or Until in order to keepour logic simple (see Footnote 19). As for our other choices, they are mainly due to thestate of the artin BDI-like logics. First, in some places we had to approximateconcepts.Most importantly,our account is incomplete as to the link between action and conse-

quences, because propositional dynamic logic does not provide it (see Remark2). Our

logic therefore does not fully account for the notion of responsibility in agent-basedemotions. Second, in some places we had to ignore concepts entirely because there is

no logical operator in the literature that would allow to take them into account. Mostimportantly, the concept of goal does not appear in our logic, and in consequence itslink with desirability is neglected. The reason here is that there exists no consensical

logical analysis up to now.Our emotions have no intensity degrees because it is not easy to design a semantics

for graded operators and their evolution in time. We plan to further investigate thisbased on the logic of graded belief of Laverny and Lang (2005). However, despite allthese limitations we believe that our formalism is expressive enough to give satisfyingdefinitions of twenty emotions.

A point that is related to the previous one is that from the psychological point ofview there are still several insufficiencies in our model. Mainly, our emotions are not

quantitative: they have no intensity degree. This prevents us from fine-grained differ-entiations among emotions of the same type (for example: irritation, anger, rage). Asecond (and linked) shortcoming is that we do not manage the emotional dynamics:our emotions are persistent as long as their conditions stay true. Thereby some emo-tions (like Joy or Satisfaction) can persist ad vitam eternam, which is not intuitiveat all. Indeed it has been established in psychology that after an emotion is triggered,its intensity decreases, and disappears when it is below a threshold. Finally, we can-not manage emotional blending of several emotions that are simultaneously triggered;

20 "Instead of trying to capture the informal psychological descriptions exactly, (or as exact as possible),we primarily look here at a description that makes sense for artificial agents." (Meyer, 2004, p. 11)

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238 Synthese(2009) 168:201-248

Gershenson (1999) proposes an original solution to this issue. We leave these problemsfor further work.

Moreover we only provided a formalization of the OCC theory, that is far frombeing as popular in psychology as it is in computer science. It was necessary tochoose one theory to begin with, but we believe that our BDI framework is expres-sive enough to formalize other psychological theories, all the more they often sharethe same appraisal variables. We already saw that we capture implicitly the con-trol variable defined by Lazarus (1991). On the contrary we do not capture the

coping potential variable because it does not intervene in the triggering of OCC

emotions; however we can represent it and we did so when formalizing coping strat-

egies Adam and Longin (2007). In this paper we only formalized the triggeringof emotions, but this is the necessary starting point before formalizing their influ-

ence on any cognitive process. We neither formalize the subsequent life of emo-tions: their temporal decay (since we have no associated intensity degrees), andtheir interaction with mood or personality, but this is an interesting extension of thiswork.

From the logical point of view our model offers a clear semantics, which we think is

quite rarein existing logical models of emotions. It also allows to highlight the power ofBDI logics to reason about and disambiguate complex concepts, since we were able to

prove some intuitive properties of emotions thanks to our logical definitions. It is onlya logical formalism can give such unequivocal results about phenomena that are not

always clearly analyzed in the psychological literature. Finally our model somehow

validates BDI logics, that were designed to formalize mental attitudes, since it dem-onstrates that they are expressive enough to characterize as complex mental attitudesas emotions (we recall that philosophers like Searle consider emotions as complexmental attitudes, see Sect. 2.1).

From the psychologist's point of view, it could be interesting to validate theoriesthanks to the reasoning services offered by logic. Another way for them to take advan-

tage from such a model is to conduct experiments with emotional agents endowedwith it, instead of humans that are not always able to clearly analyze their own emo-tions. We have implemented such an emotional agent and have experimented it withhuman users analyzing the believability of its emotions. We then used the results of

this experiments to derive conclusions about the underlying OCC theory. We have notconducted this work in collaboration with psychologists, but plan to do so in the nearfuture.

Finally, from the computer science point of view this cross-disciplinary work bringsan interesting contribution, since it fills the gap between psychology and the agent com-

munity. We designed a domain-independent model, based on a standard formalism,BDI logics, that are already used in a great number of agent architectures. Our modelis thus ready to be implemented in any BDI agent, whatever its application may be. Itwill save designers the long and costly (though necessary) process of formalization of

a psychological theory. Moreover it offers them a rich set of emotionsthat will make

their agents very expressive. To illustrate all these assets we have ourselves imple-mented such an agent endowed with our model, only making some concessions, for

example in order to add intensity degrees to emotions.

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8 Conclusion

In this paper, we have formalized twenty emotions from OCC theory (all but theobject-based branch), thus providing a very rich set of emotions. Moreover we haveshown the soundness of our framework by illustrating each definition by an examplefrom Ortony et al. 's book. We managed to formalize nearly the whole OCC theorywith our logic, supporting the author's assumption that their theory is computationallytractable. On the contrary some appraisal variables from other theories, like Lazarus'

ego-involvement, seem to be much more ambiguous and difficult to formalize.We have privileged richness, genericity, and fidelity to the definitions over tractabil-

ity. An optimization would have needed importantconcessions. For example Parunaket al. (2006) propose a numerical model of emotions in combat games, efficient in big

real-time multi-agent systems, but which is domain-dependent.In other works we have explored some extensions of this model. First we have

provided an account of the influence of emotions on the agent's behavior by formal-

izing in the same BDI framework some coping strategies. According to psychologistsLazarus and Folkman (1984), appraisal and coping are indivisible. However, the for-malization of each process was a full-fledged work and we thus investigated themin separate papers. Our formalization of coping strategies Adam and Longin (2007)allows to explain how an agent modifies his beliefs or intentions depending on hiscurrentemotion.

Second, we have implemented our logical model of both appraisal and coping in a

BDI agent Adam (2007). This agent answers emotionally to stimuli sent by the userthrough the interface. This work is still in progress to implement other kinds of influ-ence that emotions have on the agent: interaction with personality, modification of the

reasoning strategies (in the sense of Forgas (1995)), impact on the agent's centers ofattention (in the sense of the activation notion of Anderson and Lebiere (1998))...

Our future research will be oriented towards several different aims. First we wouldlike to use this logical framework to formalize various existing psychological theoriesof emotions. Once expressed in the same language, we would be able to compare thesetheories. Second we would like to conduct new experiments with our BDI agents,but this time in

cooperationwith

psychologistswho could help us interpreting the

results.From a logical perspective we will furtherinvestigate the links between mental atti-

tudes, in particularhow desirability can be computed from goals. Moreover, our work

currentlyexcludes object-based emotions: in futurework a modal predicate logic couldallow to characterize the properties of objects and thus define the emotions triggeredby their appraisal. Finally, we might unify the formalization of events and actions bymoving from dynamic-logic actions to theories of agency such as STIT theory or the

logic of 'brining-it-about'.To conclude, our cross-disciplinary approach combines the advantages of logic and

computational models with the expertise of psychology of emotions. Even if the result-ing computational model of emotions still suffers from some limitations, we hope thatit already brings some interesting contributions for computer science and logic as wellas for psychology itself.

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Acknowledgements Wewould like to thank he 3 reviewers or theirthorough omments,whichhelpedto improve hepaper n severalplaces.

9 Appendix

In ordernot to overload hepaper,wegathernthisappendix heproofsof thetheorems

given in the mainpart.This appendix s intended o help the reviewer o understandthetheorems. t maybe droppedn the finalversionof thepaper.

Intheproofs,PC refers o thePropositionalLogic,andMC refers o theprinciplesof normalmodallogic.

Theorem 1 (Temporalinkfromprospect o confirmation.)

h BeUPHopeup A (Beim*V Bel^y)-> Satis faclionup v Disappointmenti-> ReliefiV V FearConfirmediîp (b)

Toprovethe Theorem1 we needthefollowinglemma.

Lemma 1 h BeliP Desi PGDesw (from1 by MC)3. \- PGDesiip^ Desitp (from(CONV-HF)by VC)4. h PDesup -> Desi£)D

Proof2 (of Theorem ). Caseof (a).Actually t suffices oprove hat i) Beli P Hopei(pf\Belnp -> Satisfactioni<pand(ii) BeliPHopeicp A Beli~*(p > Disappointment-><pre theorems.Caseof (i).

1. h BeliPHopem>-* /te// P(Expectnp A Desi<p) (fromdefinition1)2. h Be// P // öpe, 0> * £é?/|PExpecti <pA 5e// P Des/^? (by M C)

3. h BeliPHopem> -> Ari,-PExpecti<pA De?s/0> (byLemma1)4. h Ari,-P#o/?e/ (pABeli<p^>Beli PExpecti<pA D«,-<pA £e// ^ (by PC)5. h BeliPHopenp A Zte//ç>> Satisfaction^ (bydef. of Satisfaction)

We demonstrateii) in thesameway.Caseof (b): theproofis similar. D

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Theorem 2 (Linkbetween confirmation ndwell-beingemotions)

h Satisfaction^-

Joyi Distress^ (b)

h Relieficp-> Joynp (c)

I- Disappointment^ -> Distress^ (d)

Proö/5 (of Theorem | Caseof (a).

1. hSatisfaction^

-»Belup ADesup (fromdef.of Satisfaction)2. h Satisfaction^ -> ./ay;^ (bydefinitionof 7o;y)

Theproofis similar or cases (b) to (d). D

Theorem 3 (Fromfortune-of-othermotion to imageof other)

h Happy For- -y -» Probi FJoyj(p (a)

h SorryFor'i .y -* Probi FDistress j(p (b)

h Resentment^(p - Probi F Joyj Probi F Distress j(p (d)

/Vöo/4 (of Theorem). Caseof (a).

1 h Happy For\ . PröZ?/ Ä^/7 A 5^// De57 ?(fromdefinitionof Happy For')

2. h HappyFor[ j Probi(FBelj<pAGDesj(p)((byPers-DesfìandiC-MlX))3. h Happy For!j(p - /Vöfc/F(Belj(p A D^5y ?) (by property1) forG)

4. h HappyFor[ - Pro^ FJoyj //o/?efFBeljcp (a)

h PSorryFon j<pA-»tfé?// Beljcp -^ Fear, FBeljBeliFBelj<p+

FeanFBelj<p (c)h Gloating-j(p A-*BeliFBelj<p-» //<?/?£/ Beljip (d)

Prao/5 fa/ Theorem). Caseof (a).

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1 h HappyForI ;(p + Probi FBelj <pA DesjBelj ip(fromdefinitionof HappyFor')

2. h Happy Forf((p -> Probi FBelj<pA Desi FBelj<p(by contrapositionof (T-G), and (RM-D) forDesi)

3. h HappyFori . Pro*/ FBeljip A->#e//FBelj<pA Dm,-Ffidyp

f(by P£)

4. h HappyFori .<pA-«/te//FBeljip -> Expecti FBelj<pA Des,-FBeljip(bydefinition1)

5. h HappyFor'ijip A->Zte/f Beljip -> Hopei FBeljip (by definitionof Hope)

Theproofis similar orcases (b) to (d). D

Theorem 5 (Other-agentmotions towardsoneself)

h Admirationij(i:a, ip)«->Pridei(i:a,ip) (a)

h Reproachijii'.a, ip)*+ Shame (i:a,ip) (b)

Proof6 (of Theorem5).Case of (a). The proofcomes immediately romthe defini-tions of these two emotions.

1. hAdmiratiottijii'.a, ip)

+± Beli Done :a{- ProbiHappensVM~T

Bel\ldliHappensi-aT) (bydefinitionof Admiration)

2. h Admirationijii.a, ip)«* Pridet(i:a, ip) (bydefinitionof Pride)

Theproofis similar or (b).

Theorem 6 (Other-agentmotion does not forceself-agentemotion)

\f BeliAdmiration jj(i:a,ip) -> Pridei(i:a,ip) (a)

1/ BeliReproachjj(i:a, <P)> Shamei(i:a, ip) (b)

Sketch of proof 1 (of Theorem ) // sufficestofind a counter-example,.e. a modelwhere the implication s not valid, i.e. a modelcontainingat least one worldwherethe implication sfalse.

Case of (b). By definition, BeljReproachij(j:a,ip) does not imply Desj->Happensj-îp. n a world wherethe irstformulais true and thesecondone isfalse,theimplicationsfalse. Forexample,a teacher na school can reproachoa student owear unauthorised lothes,and tell thisto him,withoutmaking his studentashamed

of wearingthem.

Theorem 7 (Linkbetweenprospectand attributionmotions)

h Pridei(i:a,ip) -> Beli Donei:a((-^Beli~^Happensi:aip

Desi Happensi:aip)-> Feari-*Happensi:aip) (a)

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h Shamei(i:a,<p) -* /te// Donei:a((-<Beli->Happensi:a(pA

Desi->Happensi:a<p)> Hopei-*Happensi:a(p) (b)

h Admirationij(j:a,(p) -> /te// Donej:a((-*Beli"-*Happensj:a(pDes, Happensj:a<p) » Fear^Happens y^y) (e)

h Reproachij(j:a,<p) -> /te// Donej:a((-tBeli-'Happensj:a(p A

Desi->Happensj:a(p)-» Hopei^Happensj:a(p) (d)

Proö/ 7 fö/ Theorem ). Case of (a).

1. h Prideiiiia, (p)^- Beli Donei:a(Probi Afteri:a-*<p)bydefinitionof Pride)

2. h Pridei(i:a,(p) -> /te// Donei:a(Probi-^Happensi:a<p)(bydefinitionof Happens)

3. h Prideiii'.a.cp) - BeliDonei>M(-^Beli-*HappensiM Expectj-*Happensi:a(p) (by P£ anddefinition1)

4. h Pridei(i:çt,<p)-> Beli Donei:a{-^Beli-*Happensi:a(p Desi Happensi:a<p- F^ör,-'Happensiwcp) (byP£ anddefinitionof Fear)

Theproofis similar or (b), (c) and(d). D

Theorem 8 [Linkbetween attribution ndprospectemotions],/fa w <zn ctón r/ï«/

r/i agen/ / believesto influence hepropositionç (cf. Remark2), then:h Feari-iHappensi:a(pA BeliIdliHappensi:a<p-

Afteri:a(Beli(p ~» Pridei(i:ot, <p)) (a)

h Hopei^H appensi:a(pA BeliIdli-*Happensi:a(p->

Afteri:a(Beli<p-> Shamei(i:a, <p)) (b)

h Feari->Happensj:a(pABeliIdlj-*Happensj:a Admiration^j(j\a, (p)) (c)

h Hopei^HappenSj:a(pA Belildlj-'Happensj-aCp ->

Afterj:a(Beli(p -> Reproachij(j:a, <p)) (d)

ToproveTheorem8 we needthefollowinglemma.

Lemma 2 Donea-^BeUAftera±ADoneaBeli /te// Donea-> AfterayjfthenDonea ^

Proof8 (ofLemma3).

1. ^ -> Aftera\l/ (by hypothesis)

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2. Donea Aftera<p-> BeliAfera± -+ AfteraBelnp

(from(NF-Be//))2. Be/, Aftera Donea<pA-«Be//After-a± -• AfteraBeUDonea(p

(byinstantiation f 1 )3. (from CONV-AD))4. Be//<p-» Be// A/fera Donea(p (from 3. by (RM-D) for

Be/,-)5. Bri,-0> -»Be/,-AfteraT ~> AfteraBeU Donea(p (from2. and4. by P£)

6. Donea(Beli<pA -«Be/,-AfteraT)->

Be// DoneaA DoneaW -> Dö/iea(4>A^) (from(CD-DB))8. Donea->Beli Aftera±A DoneaBeli Be/,-DoneaHappensi:a(p > -^Beli^Happensi:a(p(by definitionof Feûr and definition1)

2. Feari->Happensi:a -«B^/i *Happensi:(XTfrom1.by(RM-0) for-~>Beli->)3. Feari->Happensi:a -*BeliAft era±. (from2. by definitionof Happens)4. h Feari-*Happensi:a<pA Belii dli Happensi:a(p- Beli(Feari^Happensi:a

(pA /*/// Happensi:a(pA-*BeUAfteral.)( byTheorem13, (5-B*//) and(C-D)forB^/f

5. h Feari^Happensi:acpA Belildli Happensi:a Aftern DoneràBeli(Fean ^Happensi:a<pA Idh Happensi:a<p^BeliAftera±) (by(CONV-AD))

6. h Feari->Happensi:a(pA Belildli Happensi:a After^Beli Donerà(Feari->Happensi:a(pA ldU Happensi:a<p) (by Lemma2)

7. h Feari-~iHappensi:a<pBelildli Happensi:a(p-> Af ten :a Beli Be//Doneiw(F eari~*HappensVM<pldU Happensi:a<p)A Belup)

8. h Feari-*Happensi:a<pA Belildli Happensi:a Aften:a(Beli(p - Be//Donei:a(Probi Afteri:a-i<pAldli Happensi:a<p)A BeU<p)

(by definitionsof Fear andHappens)9. h Feari-*Happensi:a<pA Belildli Happensi:a(p -+ Afteri:a(Beli<p -

/VMe,- i :a, ^)) (bydefinitionof Pride)

Theproofis similar or (b), (c), and(d). G

Theorem 9 (Inconsistencybetweengood-will andill-will emotions)

I *(HappyForij(p A Resentment jcp) (a)

I*(SorryForij(p

AGloatingij<p)

(b)

I '(HappyForij(p A Gloatingij<p) (c)

I *(SorryFonj<pA Resentmenti <p) (d)

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Sketch of proof 2 (of Theorem9) Theproof or cases (a) and (b)followsfrom therationalityof Desj. Theproof or cases (c) and (d)followsfromLemma . D

Lemma 4 -*(BeliDesj<pA BeliOesj -><p)

Proof11 (ofLemma4). 1. h Desjip -> -^Desj^(p (from D-Des,))2. h BeliDesj BeU-*Desj-*(p (by(RM-D)forBeli )3. h BeliDeSjip -» -*BeUDesj-*<p (by(D-£é?/,))4. h -*(BeliDesj(pA BeUDesjîp) (byVC)

D

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