Eric Caumes
email: eric.caumes@psl.aphp.fr

Loïc Epelboin

France Leturcq

Phyllis Kozarsky

Peter Clarke

Tintin's travel traumas: Health issues affecting the intrepid globetrotter Les problèmes de santé de Tintin: plus de traumatismes que de pathologies du voyageur

Background. Despite the highly hazardous life-style led by comic book characters such as Tintin, we are unaware of any previous systematic description of the challenges and health impairments faced by Tintin in the course of his adventures.

Methods. We evaluated the spectrum of health impairments (HIs) that Tintin sustained in his 23 adventures as well as their causes, consequences, and relation to travel. We diagnosed Tintin"s HIs according to descriptive terms in the text. We then classified HIs as traumatic and non-traumatic, and distinguished between intentional (those perpetrated by others) and unintentional events.

Resultats. Nous avons trouvé 236 evénements conduisant à 244 PS, 13 kidnappings, six hospitalisations et deux actes chirurgicaux. Le nombre médian de PS est de 8/aventure (1-30/aventure). Le nombre moyen d"événements par aventure était significativement plus élevé avant 1945 (14.9 v 6.1; p 0.002), de même pour les kidnappings (11 v 2; p 0.001). Parmi les 244 PS, 191 étaient traumatiques (78.3%) et 53 non traumatiques (21.7%). Les plus frequents des traumatismes étaient les traumatismes craniens (62%), et les plus frequents des PS non traumatiques étaient les problèmes de sommeil (15.1%), le smanifestations anxieuses (13%), et les intoxications au gaz ou au chlorophorme (13%). Nous avons trouvé 46 pertes de connaissance (PC) : 29 d"origine traumatique et 17 nont raumatiques. Parmi les 236 événements, 69 (29%) sont intentionnels, perpétués par des tiers contre Tintin (dont 55 tentatives d"homicides), et 167 (71%) non intentionnels (dont 69 sont la conséquence des actions de Tintin). Conclusion. Les qualités presque surhumaines de Tintin font de lui un personnage de fiction, très résitant aux traumatismes. Tintin ne présente pas plus de PS liés aux voyages. Mais il semble être très vulnérable aux problèmes présentés par ses compagnons de voyage et notamment Milou, son fidèle compagnon de voyage.

Results. We found 236 events leading to 244 HIs, 13 kidnappings, six hospitalisations and two surgical procedures. There was a median of eight HIs/adventure (range 1-30/adventure).

The mean number of HIs per adventure was much greater before 1945 than subsequently (14.9 v 6.1; p 0.002), which was also true of the number of kidnappings (11 v 2; p 0.001). Of the 244 HIs, there were 191 cases of trauma (78.3%) and 53 non-traumatic problems (21.7%).

The most common form of trauma was concussion (62%), whereas the most common forms of non-traumatic problems were sleep problems (15.1%), depression/anxiety (13%), and gas or chloroform poisoning (13%). Overall, we found 46 losses of consciousness (LoC), including 29 traumatic and 17 non-traumatic LoCs. Of the 236 events, there were 69 (29%) perpetrated by others against Tintin (including 55 homicide attempts), and 167 (71%) events that were not (including 69 events related to Tintin"s actions).

Conclusion. Tintin"s almost superhuman qualities, a luxury afforded him by his fictional status, make him highly resistant to trauma. He is also not susceptible to the usual travelrelated illnesses but is easily influenced by his friends and Snowy, his faithful hound.

Résumé.

Interet. Malgré la vie hasardeuse menée par les héros de bandes dessinées comme Tintin, nous n"avons pas connaissance d"études ayant évalué les événements et les problèmes de santé présentés par Tintin pendant ses aventures.

Methodes. Nous avons évalué le spectre des problèmes de santé (PS) que Tintin a présenté au cours de ses 23 Aventures, ainsi que leurs causes, conséquences et lien avec le voyage. Les PS ont été diagnostiqués en fonction de leur description dans le texte puis classés en distinguant les problèmes traumatiques des problèmes non traumatiques. Les causes ont été classées en distinguant les événements intentionnels et non intentionnels.

Introduction

Tintin is the eponymous hero of the Adventures of Tintin, a series created in 1929 by Georges Prosper Rémi (better known as "Hergé"), a Belgian cartoonist (figure 1). Twenty-three Tintin adventures appeared between 1930 (Tintin in the Land of the Soviets) and 1976 (Tintin and the Picaros). Tintin is a boy reporter, explorer, investigator, and traveller [START_REF]The Adventures of Tintin[END_REF]2]. These swashbuckling roles understandably exposed him to a wide range of health impairments (HIs). Surprisingly, HIs have only rarely been described even for popular comic book characters such as Tintin. A PubMed search found only three articles focusing on "Tintin", including two research articles [START_REF] Cyr | Acquired growth hormone deficiency and hypogonadotropic hypogonadism in a subject with repeated head trauma, or Tintin goes to the neurologist[END_REF][START_REF] Medrano | Stigma and prejudice in Tintin[END_REF], and one editorial [START_REF] Castillo | Tintin and colleagues go to the doctor[END_REF]. The Cyr family from Canada (aged five, seven, and 33) identified 50 losses of consciousness (LoC) in 16 of the 23 Tintin adventures [START_REF] Cyr | Acquired growth hormone deficiency and hypogonadotropic hypogonadism in a subject with repeated head trauma, or Tintin goes to the neurologist[END_REF]. We should note that pictures of Tintin apparently suffering LoCs were identified by the two Cyr children (whose average age was six), which suggests the LoCs in Tintin should be evaluated by a physician. Medrano et al also analysed appearances of, or allusions to, mental illness in the Tintin series but only to see how psychologically-impaired characters were portrayed [START_REF] Medrano | Stigma and prejudice in Tintin[END_REF]. Therefore the entire spectrum of HIs in Tintin"s adventures remained to be examined.

Only two of Tintin"s adventures, The Castafiore Emeralds and The Secret of the Unicorn, occurred entirely in his native Belgium [START_REF]The Adventures of Tintin[END_REF]2]. In all other adventures, Tintin"s home, initially Brussels and later Marlinspike Hall ("Chateau de Moulinsart", in French), was merely his launch pad or landing strip for each overseas odyssey. Tintin"s voyages ultimately took him to a total of 23 real and fictional countries, not forgetting the moon and a meteorite. Tintin"s travels regularly took him to hostile environments, whether the high seas (e.g. the Red Sea, Caribbean, and Arctic Ocean), deserts (the Sahara and Arabian Deserts), forests (the Amazon and Indonesian rain forests), or mountains (Himalayas and Andes). He took every conceivable means of transport: spaceships, planes, ocean-liners, all manner of boats, cars, motorbikes (figure 1), bicycles, trains (figure 2), horses, elephants, yaks and donkeys. Amazingly, although Tintin was exposed to a plethora of travel-related health hazards, they have never previously been characterized.

We assessed the full spectrum of HIs revealed in Tintin"s adventures and their causes, consequences, and relation to travel.

Methods

We scrutinised all 23 completed volumes of Tintin"s adventures. Focussing on Tintin, the protagonist, we assessed all the HIs that arose throughout his stays in Belgium and during his travels to tropical and non-tropical countries.

Tintin travelled to four of the world"s five inhabited continents, visiting 23 countries plus the moon and a meteorite. We distinguished tropical from non-tropical countries using geographic and climatic criteria. Among the 18 real countries Tintin visited, eight are tropical and ten non-tropical (Table I). Tintin also visited five fictitious countries. We classified Borduria and Syldavia as non-tropical, and Khemed, Nuevo Rico and San Theodoros as tropical countries, totalling 11 tropical and 12 non-tropical countries. The only destinations most readers could not expect to visit, space, the moon (Explorers on the Moon), and a meteorite (The Shooting Star), were classified as "undetermined".

We diagnosed the types of HI according to the clinical description of Tintin, or his response, in the frames of the strip where each new HI occurred, and in the subsequent frames that often amplified the medical nature of the event. We primarily distinguished traumatic and nontraumatic HIs. We classified HIs as BTwCs (Body trauma without consequences other than pain), visible injuries (bumps, scratches and burns), concussion, loss of consciousness (LoC), and any other major HIs acknowledged by Tintin or the authors. BTwCs other than pain were defined by symbols revolving around Tintin"s body (except his head) following a significant trauma without further visible consequences. "Concussion" was defined as any head traumatic lesion and was termed "grade I" (stars spinning round Tintin"s head), "grade II" (squiggles, candles, bolts of lightning surrounding his head), "grade III" (LoCs lasting no more than one frame), and "grade IV" (LoCs lasting at least two frames). LoCs included traumatic LoCs (TLoCs) (i.e. grade III and IV concussion, and asphyxia), and non-traumatic LoCs (NTLoCs) (LoCs from non-traumatic causes like gas poisoning).

We excluded events that might have resulted in an HI with any normal mortal, but not Tintin.

We only considered significant events revealed at least by symbols (candles, stars, squiggles) revolving around any part of Tintin"s body (head, limbs or trunk). Where symbols surrounded two people, including Tintin, in a single frame, we only logged the incident if Tintin clearly appeared as a victim of the HI. We defined an LoC"s duration by the number of elapsed frames before Tintin reverted to his usual appearance or behaviour.

For each HI we assessed: geographical location (distinguishing between Belgium, nontropical and tropical sites, and undetermined ones), cause (distinguishing intentional from unintentional events), and consequences (surgery, hospitalisation). Intentional HIs perpetrated by others included attempted homicide or assault, arrest, and events provoked by Tintin"s friends (termed "friend-related"). Unintentional causes of HIs include environment-related events (animal exposure, natural disasters, extreme temperatures), transport accidents, events resulting from Tintin"s acts, or following his lapses of judgement (termed "Tintin-related"), and other unintentional events. Each category included one to nine items.

We assessed the number of HIs per adventure, those occurring in Belgium and elsewhere, distinguishing between tropical, non-tropical, and undetermined countries. We summarised categorical variables using percentages, and made comparisons by using the Fisher exact test.

We summarised continuous variables using means +/-standard deviations and compared using the Student test.

Results

We found 244 HIs [mean 8.4/adventure (standard deviation +/-5.8), median 8/adventure; range 1-30/adventure] and 13 kidnappings (Table I). Half the adventures were written between 1930 and 1945 (median = 1944). There was at least one HI per adventure. The mean number of HIs per album was significantly higher before 1945 than afterwards (14.9 ± 7.1 v 6.1 ± 4.7; p 0.002). This was also true of the number of kidnappings (11 v 2, e.g.1.0 per album ± 0.7 v 0.09 ± 0.3; p 0.001). Kidnapping was associated with HIs ten times, six resulting from traumatic attack and four from gas or chloroform poisoning. Three kidnappings were not associated with any HI (Table I).

Of the 244 HIs that occur in the 23 adventures, there were 191 traumas (78.3%) and 53 nontraumatic problems (21.7%). The most common forms of trauma were grade I and II concussion (47%), BTwC (28%) and grade III and IV concussion (16%) (Table II). Overall there were 118 cases of concussion (112 concussion alone, and six associated with polytrauma) accounting for 48.4% of all HI. The other traumatic injuries included those caused by animals, burns, and gunshot wounds. The most common forms of non-traumatic problems were sleep problems (15.1%), depression/anxiety (13%), gas or chloroform poisoning (13%), and alcohol-related problems (9.4%) (Table III). We found 46 LoCs (28 grade III/IV concussions, one asphyxia related to strangulation by Captain Haddock, and 17 non-traumatic LoCs) (Table IV). However, the consequences of LoCs (the number of frames where Tintin was affected) could be evaluated in only 36 cases of LoC, with a mean of 9.5+/-11 frames per involvement/LoC. Despite leading such a dangerous life, Tintin was hospitalised only six times, including twice in one adventure (The Black Island). His stays ranged from one day in three adventures, to a few days or even weeks in three of his exploits. Hospitalisation twice follows gunshot wounds, and a car crash, fire, explosion and extra-terrestrial hypnosis once each. A car crash also brought a visit to casualty, which Tintin left without medical examination (King Ottokar's Sceptre).

Tintin twice underwent surgery after being shot (The Black Island, Destination Moon).

Fortunately, the bullets merely grazed a rib or his skull. These two gunshots also caused an LoC. Tintin was also shot twice in the Land of the Soviets and in The Blue Lotus, but presents with nothing worse than a BTwC and a minor injury leading to shoulder immobilisation. There were 236 events since eight of the events resulted in two HIs. Of these 236 events, 69 (29%) were intentional and 167 (71%) were unintentional. The two most common events were homicide attempts, accounting for 80% of intentional causes and Tintin-related events representing 41.3% of unintentional causes. The other causal events by order of frequency were environment-related events, transport accidents, Tintin"s friends" (mis) behaviour, and arrests (Table V). The most frequently-used weapon in attempted homicide or assault was a club or other object. There were 33 transport accidents, including 27 unintentional accidents and six intentionallycaused accidents resulting from attempted homicide. The main means of transport involved in accidents were cars and aeroplanes. Of the 27 unintentional transport accidents, none occurred in Belgium; 24 happened in other countries (eight in tropical and 12 in non-tropical countries), and three en route.

The commonest environment-related events were animal exposure and natural disasters (avalanches, earthquakes, storms, Ocean waves), while extremes of temperature were less frequent (Table V). Of the seven injuries (bites, scratches) caused by animals, one scratch (condor) and two bites (lion, piranha) occurred in tropical countries, while two bites (by a rat and a shark) took place en route, one (by Snowy) on the meteorite and one (by a parrot) in Belgium. The eighth case of biting was inflicted by the boy Abdallah, so we classified the causal event as friend-related.

Discussion

This first snapshot of the 244 HIs in the 23 adventures reveals that HIs are common in Tintin"s adventures, trauma accounting for 78% of HI and concussion for 48%. But Tintin is highly resilient, being hospitalised a mere six times and undergoing surgery just twice.

We witnessed a sharp decline in HIs and a marked decrease of kidnappings in his stories after 1945. This deserves an explanation. From the outset, Tintin shares his adventures with his trusty canine companion, Snowy. However, the list of intrepid travellers progressively swells to include the buffoon detectives Thomson and Thompson in 1934, followed by the Chinese orphan Chang in 1936, the old sea dog Captain Haddock in 1941, and the scatter-brained Professor Calculus in 1944. Taken together, after 1945 this supporting cast progressively experiences numerous and diverse health impairments. We might justifiably wonder whether Hergé increasingly preferred to spare Tintin the burden of health problems, and instead allowed his new travelling companions to inherit his own place on the sick list. This decline may also be ascribable to a sense of war-weariness on Hergé"s part. We know that Hergé"s war-time stories were studiedly set in places far from Belgian shores obviating any possible controversy with the censors [START_REF]The Adventures of Tintin[END_REF]2]. Could it be that his earlier pre-war depictions of violent acts and kidnapping, conceived in a climate of innocent playfulness, were no longer palatable to a man who had grown to understand the true consequences of the darker forces of human nature?

The most frequent HI was concussion (48% of all HIs). In the medical literature concussion is defined as mild traumatic brain injury, mild brain injury or mild head injury, which are all synonymous. Many systems measure the severity, or grade, of concussion but there is little agreement over the best [START_REF] Terrell | Sports Concussion Management: part II[END_REF][START_REF] Terrell | Sports concussion management: part I[END_REF]. We therefore used an "Hergé" grading system after considering the clinically evident post-traumatic signs (symbols surrounding Tintin"s head after trauma): stars (grade I), whirls or candles (grade II), and LoCs (grades III and IV according to duration). This system is simple and overlaps with the two most widely-used systems for grading concussion [START_REF] Chan | Defining traumatic brain injury in children and youth using international classification of diseases version 10 codes: a systematic review protocol[END_REF]. Kamp and colleagues assessed the 704 traumatic head injuries in the 34 Astérix adventures, where the major cause of trauma was assault and where the traumas were severe in over 50% of head wounds. But there all the characters were scrutinised, not only the heroes [START_REF] Kamp | Traumatic brain injuries in illustrated literature: experience from a series of over 700 head injuries in the Asterix comic books[END_REF].

LoCs were the severest HIs, and represent 18.8% of all HIs. They mostly related to severe (grade III and IV) concussions (60.9% of LoCs), intoxication (13 %) or G-force injuries (8.7 %) (Table IV). We reached a different total of LoCs than the Cyr family [START_REF] Cyr | Acquired growth hormone deficiency and hypogonadotropic hypogonadism in a subject with repeated head trauma, or Tintin goes to the neurologist[END_REF]; while they found 50 LoCs in 16 adventures, we detected 46 LoCs in 16 of the 23 adventures. We have reviewed all the frames where the Cyrs diagnosed LoCs [START_REF] Cyr | Acquired growth hormone deficiency and hypogonadotropic hypogonadism in a subject with repeated head trauma, or Tintin goes to the neurologist[END_REF] and disagree over the diagnosis of LoCs in 19 of their 50 cases. For instance, in 11 situations they described as LoCs, we instead classified the events as concussion grades I and II as they did not lead to LoCs. Our analysis also suggests the Cyrs missed some LoCs, for instance 3 non-traumatic LoCs in Cigars of the Pharaoh and three of the four LoCs related to G-force events travelling to or from the moon (two take-offs and two landings).

The commonest non-traumatic HIs were sleep disturbances (nightmares, insomnia), intoxication (gas, chloroform), and manifestations of anxiety or depression (tears), as highlighted in another study which focused solely on mental disorders in the characters who appear in Tintin adventures [START_REF] Medrano | Stigma and prejudice in Tintin[END_REF]. Sleep disturbances might reasonably be attributed to Tintin"s highly stressful life. Tintin even bursts into tears when the lives of his dearest companions (Snowy and Chang) are in peril or when he is separated from them. Despite his super-human qualities, Tintin is also a "regular guy", which unquestionably boosts his popularity [2].

There was no clear relationship between the type of HI and travel destination when we focused on tropical destinations alone. Tintin presents none of the commonest travel-related health impairments such as diarrhoea, respiratory tract infections, sunburn, insect bites or fever [START_REF] Hill | Health problems in a large cohort of Americans traveling to developing countries[END_REF]. For instance, when travelling by boat he is not seasick although this is the commonest HI on ships [START_REF] Shaw | Illness and injury to students on a school excursion to Peru[END_REF]. When scaling the Himalayas or the Andes he never suffers altitude sickness although this is a common HI among travellers to Peru [START_REF] Schutz | Pattern of passenger injury and illness on expedition cruise ships to Antarctica[END_REF].

Although in Tintin"s adventures transport accidents are non-fatal, road accidents are the leading cause of non-natural deaths suffered by travellers abroad and are more frequent than intentional injuries [START_REF] Sherry | Rates of intentionally caused and road crash deaths of US citizens abroad[END_REF]. Southeast Asia produces the highest unintentional injury death rates for Americans abroad owing to the high death rate from motorbike accidents, especially in Thailand and Vietnam. Some of the HIs resulted from exposure to animals, (bites, scratches, shaking, shocks, bumps, strangulation) primarily Snowy (who is presumably vaccinated against rabies), but also with a vast menagerie of non-canines (horses, lions, sharks, condors, electric rays, parrots, piranhas, rats and yaks). The outcome was a bite in six cases and scratches in one. Although the animals are not typical rabies carriers, travellers who lead quite such adventurous lives should certainly consider getting vaccinated prior to their travels [START_REF] Gautret | Animal-associated injuries and related diseases among returned travellers: A review of the GeoSentinel Surveillance Network. GeoSentinel Surveillance Network[END_REF].

We found five alcohol-related HIs in Tintin, including two in his first adventure and one in his final escapade. The first two relate to heavy drinking in Russia, and the last results from Professor Calculus"s experimental anti-alcohol pill. This pill was intended to wean Captain Haddock off alcohol, but Tintin inadvertently swallows it and suffers its effects.

Tintin"s sometime profession of journalism entails risks such as kidnapping. This is truly an escalating concern as the number of journalists kidnapped more than doubled in 2013 [START_REF] Taylor | Number of Journalists kidnapped more than doubled in 2013: report[END_REF].

Our hero is kidnapped no fewer than 13 times. He is taken hostage in just one adventure (Land of Black Gold) in the Middle East. No ransom is demanded and he is rescued four times by his faithful companion, Snowy. Perhaps thereafter Tintin becomes less vulnerable since he was kidnapped only once after 1950 (Table I). Could he have gained some wisdom with age? Tintin also presents HIs that result from aberrant behaviour by his closest companions. It is hard to escape the conclusion that they behave bizarrely because of significant underlying health problems [START_REF]The Adventures of Tintin[END_REF]2]. Captain Haddock is an alcoholic, Professor Calculus is stone deaf, and Thomson and Thompson are just plain stupid. This study has some limitations. We may have missed some health events or kidnappings.

The dividing line between a modest trauma and a significant trauma leading to a HI is sometimes narrow since Hergé also uses symbols to accentuate a narrative event. In some instances it was also hard to differentiate between a simple homicide and a kidnapping when the aim of the kidnapping was homicide. Also, none of the authors is a traumatologist. Finally Hergé"s scenario are so rich in content that it was a challenge to classify some of the HIs. It was especially tough with events where there is strong interaction between Tintin, his friends, Snowy, means of transport, animal or natural disasters. We hope that our passion for Tintin has nevertheless overcome most of these limitations.

In conclusion, Tintin is clearly an extraordinary character whose adventures continue to delight his many readers 85 years after his first appearance. Children and adults alike are enchanted by the globe-trotting adventures of a boy, his dog, and his eccentric companions. It remains for the sober physician to raise a horrified eyebrow at the characters" insouciant failure to prepare for their hazardous voyages or to manage their many ailments properly.

Nevertheless, although a modicum of common sense might spare the characters most of their calamities, it"s actually far more fun to delight in this superhuman spirit who guilelessly journeys abroad effortlessly overcome the many travel-related illnesses and traumatic events we mere mortals face daily.

*

  Figure Click here to download high resolution image

Table II : Traumatic health impairments suffered by Tintin

 II

	Concussion alone
	Grade I and II	84
	Grade III	2
	Grade IV	24
	Grade III /IV	2
	Total Concussion alone	112
	Polytrauma*
	BTWC
	Back
	Bottom**
	Foot
	Upper limbs
	Neck
	Shoulder
	Trunk
	Total BTWC
	Burns
	Gunshot injury***
	Bite****
	Miscellaneous*****
	Total

Conflict of interest: none

Legends for the figures:


Didier Dubois

Lluís Godo

Henri Prade

Weighted logics for artificial intelligence -an introductory discussion

Keywords:

Before presenting the contents of the special issue, we propose a structured introductory overview of a landscape of the weighted logics (in a general sense) that can be found in the Artificial Intelligence literature, highlighting their fundamental differences and their application areas.

Introduction

In the last decades, and especially in Artificial Intelligence (AI), there has been an explosion of logical formalisms capable of dealing with a variety of reasoning tasks that require an explicit representation of quantitative or qualitative weights associated with classical or modal logical formulas (in one form or another). The semantics of the weights refer to a large variety of intended meanings: belief degrees, preference degrees, truth degrees, trust degrees, etc. Examples of such weighted formalisms include probabilistic or possibilistic uncertainty logics, preference logics, fuzzy description logics, different forms of weighted or fuzzy logic programs under various semantics, weighted argumentation systems, logics handling inconsistency with weights, logics for graded BDI agents, logics of trust and reputation, logics for handling graded emotions, etc.

The underlying logics range from fully compositional systems, like systems of many-valued or fuzzy logic, to noncompositional ones like modal-like epistemic logics for reasoning about uncertainty, probabilistic or possibilistic logics, or even some combination of them. Sometimes the weights are not explicit and the formalisms use total or partial orderings instead.

In this short paper we present an introductory discussion organizing a landscape of weighted logics (in a general sense) that can be found in the literature, highlighting their differences and application areas. In particular we overview the main approaches in AI that deal with graded notions of uncertainty, truth, preferences and similarity, we discuss what are the logical issues behind them, and finally we also point out new emerging areas for graded settings. This is done in Sections 2 to 4.

Our aim is not to provide a full and exhaustive overview of graded formalisms in AI, but rather an informed discussion of the main general issues and approaches, that may help the reader position the papers in this special issue, whose content is presented in Section 5.

Typical graded notions

One heavily entrenched tradition in AI, especially in knowledge representation and reasoning is to rely on Boolean logic. However, many epistemic notions in commonsense reasoning are perceived as gradual rather than all-or-nothing. Neglecting this aspect may lead to insufficiently expressive frameworks and may cause some confusion. Such naturally gradual (in our opinion) notions are reviewed below.

Graded truth

Truth is a key notion in the philosophy of knowledge that is often viewed as Boolean in essence. Yet in the scope of information storage and management, this absolute view becomes questionable. When representing knowledge, one may choose a language whose primitives are Boolean, but it is also possible to use one whose primitives are many-valued entities. Indeed, as claimed quite early by De Finetti [START_REF] De Finetti | La logique des probabilités[END_REF] commenting Łukasiewicz logic, deciding that a proposition is an entity that can only be true or false is a matter of convention, as it is a matter of choosing the range of a (propositional) variable. In this sense, truth is an ontic notion, as one participating to the definition of a proposition. One may take into account the idea that in some contexts the truth of a proposition (understood as its conformity with a precise description of the state of affairs) is a matter of degree. For instance, if the height of John is known one might consider that the proposition "John is tall" is not always modeled as being just true or false. This is the view held by fuzzy logic [START_REF] Bellman | Local and fuzzy logics[END_REF]. However this notion of graded truth is restricted to an information processing perspective and has nothing to contribute to the foundations of formal philosophy.

If the truth set contains intermediary truth degrees, one issue is whether or not we can keep the truth-functionality assumption that is the key feature of classical logic. Mathematically, the answer is yes as demonstrated by the large set of multiple-valued logics that are now available. However there are a lot of unresolved issues about many-valued logics and their applications to AI such as • Why are there so few papers using multiple-valued logics as a representation of gradual properties in AI?

• How to choose among the many available truth functional many-valued systems?

• Does truth-functionality always make sense from an applied knowledge representation perspective? • How does the notion of truth-functional many-valued truth articulates with studies of vagueness?

As for the last question, see e.g. [START_REF]Understanding Vagueness -Logical, Philosophical, and Linguistic Perspectives[END_REF]. On the other hand, the most popular many-valued logics in AI seem to be those with 3 (Kleene [START_REF] Kleene | Introduction to Metamathematics[END_REF]), 4 (Belnap [START_REF] Belnap | A useful four-valued logic[END_REF]) or 5 (equilibrium logic [START_REF] Pearce | Equilibrium logic[END_REF]) truth values, with a view to handle epistemic notions such as ignorance, contradiction, negation as failure or default knowledge, following a long tradition dating back to Łukasiewicz and Kleene [START_REF] Fitting | Kleene's three-valued logics and their children[END_REF]. However, the systematic use of truth-tables for basic connectives in these approaches looks questionable as such notions have a strong epistemic flavor closer to ideas of uncertainty, while truth is an ontic notion. For instance, Kleene suggested that the third truth value could mean "unknown", and this view has been taken for granted by many. However, "unknown" can be opposed to "certainly true" and "certainly false", not to (ontologically) true and false. This has led to confusing debates that cannot be solved without letting the representation of uncertainty and modalities enter the picture [START_REF] Dubois | On ignorance and contradiction considered as truth-values[END_REF]. However, usual epistemic logics with their relational semantics look far away from Kleene logic. Yet, a simple fragment of a KD logic, where modalities only appear in front of literals, can capture Kleene logic and other three-valued logics of partial knowledge [START_REF] Ciucci | A modal theorem-preserving translation of a class of three-valued logics of incomplete information[END_REF]. This restriction on the expressive power (in Kleene logic, it is not possible to represent the fact that p ∨ q is unknown) is the price paid for the truth-functionality of logics of the unknown.

Uncertainty

Uncertainty modeling pertains to the representation of an agent's beliefs. There are several sources of uncertainty:

• The random variation of a class of repeatable events leaves an agent unable to predict the event that corresponds to the outcome of the next trial.

• The sheer lack of information may cause an agent being uncertain about the answer to a question.

• The simultaneous presence of inconsistent pieces of information due to too many sources may equally prevent an agent from asserting the truth or the falsity of a statement.

There are two traditions in AI for representing uncertainty, that still need to be reconciled to a large extent:

• The non-graded Boolean tradition of (monotonic) epistemic logics that rely on the modal formalism, and include some exception-tolerant non-monotonic logics.

• The graded tradition typically relying on degrees of probability. Measures of uncertainty aim at formalizing the strength of our beliefs in the truth (occurrence) of some propositions (events) by assigning to those propositions a degree of belief [START_REF]Degrees of Belief[END_REF].

Whatever the tradition, it must be stressed that belief is a higher order notion w.r.t. truth, that is, a statement pertaining to belief encapsulates a Boolean proposition inside a belief qualifier: the truth degree of the statement "I believe p" does not refer to the truth of p, but to the belief itself: it is the degree of belief in p, irrespectively of p being true or not. From the mathematical point of view, a measure of uncertainty is a function that assigns to each proposition or event (understood here as a formula in a specific logical language) a value from a given scale, usually the real unit interval [0, 1], under some suitable constraints. A well-known example is given by probability measures which try to capture our degree of confidence in the occurrence of events by additive [0, 1]-valued assignments. It can be shown that such an uncertainty calculus cannot be compositional with respect to all logical connectives without leading to some form of trivialization of the construction [START_REF] Dubois | Possibility theory, probability theory and multiple-valued logics: a clarification, in: Representations of Uncertainty[END_REF]. For instance, probabilities are only compositional with respect to negation. In contrast, we have seen that degrees of truth can be truth-functional and this assumption still leads to sophisticated logical systems.

It is not easy to reconcile the probabilistic and the logical view of beliefs (see [START_REF]Special issue on Combining Probability and Logic[END_REF] for a special issue on this topic). At the most elementary level a set of Boolean formulas is often interpreted as a belief base containing propositions an agent believes. In the Bayesian probabilistic tradition, information is represented by a single probability distribution on possible worlds possibly encoded as a Bayes net. The two approaches are at odds, irrespectively of the choice of Boolean vs. gradual representations of belief:

• The logical approach leaves room for incomplete information.

• The Bayesian approach [START_REF] Pearl | Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference[END_REF] seems to be very information demanding as the lack of belief in a proposition is always equated with the belief in its negation.

On the other hand, a natural graded extension of the elementary logical approach is captured by possibilistic logic [START_REF] Dubois | Possibilistic logic: a retrospective and prospective view[END_REF][START_REF] Dubois | Possibilistic logic -an overview[END_REF] where degrees of uncertainty may be captured by means of a mere total ordering of possible worlds and some propositions can be more believed than others. This plausibility ordering can be encoded as a numerical possibility distribution if needed. It preserves the property that if an agent believes two propositions (to any degree), this agent also believes their conjunction. This property (typical of epistemic logic) is violated in probability theory.

The use of a modal language enables the syntactic expression of partial ignorance, and explicit patterns of reasoning from it, but the semantics of epistemic logic in terms of accessibility relations does not fit with the numerical tradition of representing beliefs. However, specializing this semantics to mere epistemic states and restricting its language to epistemic sentences restores the link with uncertainty theories [START_REF] Banerjee | A simple logic for reasoning about incomplete knowledge[END_REF]. Putting together the probabilistic and such simplified epistemic logic approaches to belief leads to reasoning in the setting of imprecise probabilities [START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF] which explicitly attach degrees of belief and degrees of plausibility to propositions [START_REF] Lukasiewicz | Special issue on Logical Approaches to Imprecise Probabilities[END_REF]. Such degrees are graded versions of necessity and possibility modalities; besides, possibility distributions can encode special cases of convex probability families [START_REF] Dubois | When upper probabilities are possibility measures[END_REF].

A nice by-product of reconciling probability and epistemic or possibilistic logics is to offer a logical handling of conditioning: pushing probability down to the Boolean context, in de Finetti style, leads to the three-valued logic of conditional objects [START_REF] Dubois | Conditional objects as nonmonotonic consequence relationships[END_REF], that is also a semantics for non-monotonic logics; adding weights to this construction bridges the gap between logical representations of belief and both probability and possibility theories [START_REF] Benferhat | Possibilistic and standard probabilistic semantics of conditional knowledge bases[END_REF].

It remains the issue of choosing a proper scale for grading beliefs in a given application context, namely how much numerical is it useful to be? A unified framework for reasoning about uncertainty leaves us the choice between various graded representations: ordinal, qualitative (with a finite value scale), integer-based (as with Spohn kappa functions [START_REF] Spohn | Ordinal conditional functions: a dynamic theory of epistemic states[END_REF]) or real-valued.

Preferences

Preferences are clearly not Boolean most of the time. Artificial Intelligence has developed a Boolean framework for decision-making problems based on constraint propagation and satisfaction. While many problems are amenable to a constraint-based formulation, it makes little sense to ignore the gradual nature of preferences.

The tradition in preference modeling has been to use either order relations (total or partial) or numerical utility functions, albeit with little attention to the issue of preference representation in practice. On the contrary, Artificial Intelligence has focused on compact logical or graphical representations of preferences on multi-dimensional (often Boolean domains) [START_REF] Domshlak | Preferences in AI: an overview[END_REF]. In this situation, an interpretation represents an option described by Boolean attributes. For instance, CP-nets have exploited an analogy with Bayes nets to design a graphical structure encoding ordinal preferences when local decision variables are Boolean. However CP-nets are far from capturing all possible ordering relations between possible worlds and the notion of preference handled by CP-nets is all-or-nothing. More general logical languages where a preference relation between formulas appears in the language have been used, that are more expressive. However the question of the meaning of comparing two logical formulas in terms of its consequences on a preference ordering between interpretations is not obvious, and several proposals exist that are at odds with each other. For instance do we mean that all models of the preferred formula should be preferred to all models of the other formula? Or just their best models? See, e.g. [START_REF] Kaci | Reasoning with various kinds of preferences: logic, non-monotonicity and algorithms[END_REF].

One alternative is to use weights attached to Boolean formulas. Such a weight may reflect the imperativeness of the satisfaction of the associated proposition, then viewed as a goal to reach (a prioritized constraint). When this weight is used as a penalty for the interpretations that violate the formula, the weight can be viewed as a lower bound of a necessity measure [START_REF] Benferhat | Towards a possibilistic logic handling of preferences[END_REF]: this is just another use of possibilistic logic. However, other approaches exist, e.g., [START_REF] De | Penalty logic and its link with Dempster-Shafer theory[END_REF][START_REF] Chevaleyre | Expressive power of weighted propositional formulas for cardinal preference modeling[END_REF], where the weight is a reward when satisfying the formula, and [START_REF] Lang | Utilitarian desires[END_REF] where desires or preferences have a utilitarian semantics. More generally at the semantic level one may focus on the least preferred interpretations that violate the formula or on the best preferred ones that satisfy it. Or on the contrary, an interpretation may be considered all the better (resp. worse) as the sum of the rewards (resp. penalties) attached to formulas it satisfies (resp. violates) is higher (resp. smaller). An alternative to the use of weights is the introduction of a preference relation inside the representation language, as in, e.g., [START_REF] Wright | The Logic of Preference[END_REF][START_REF] Van Benthem | Dynamic logic of preference upgrade[END_REF][START_REF] Bienvenu | From preference logics to preference languages, and back[END_REF].

In the above preference representation framework, neither the presence of several criteria nor the possibility of uncertainty is considered. In this case, there may be two kinds of weights:

• Weights expressing preferences of options over other options.

• Weights expressing the likelihood of events or importance of groups of criteria.

In the case of decision under uncertainty, one way out is to use two sets of formulas, one for the knowledge base and one for the preference base, and to articulate some kind of inference technique exploiting both bases so as to encode the optimization of a given criterion mixing uncertainty and utility [START_REF] Dubois | Logical representation and computation of optimal decisions in a qualitative setting[END_REF]. This approach looks more problematic for multiple criteria decision-making, where value scales may not be commensurate, and importance weights pertain to yet another dimension of the evaluation process. Still, a logical counterpart to Sugeno integrals, a family of qualitative multiple criteria aggregation operators, has been recently obtained [START_REF] Dubois | The logical encoding of Sugeno integrals[END_REF].

Similarity

The use of the idea of similarity in reasoning may refer two different points of view: either one does not want to distinguish between objects that are found to be similar, thus building a partition, or one wants to take advantage of the existence of a metric structure on a set of objects and exploit it in some way. In the first case, we perform a granulation of the universe of discourse, while in the second case we are interested in extrapolation or interpolation.

Similarity is often a graded notion, especially when it is related to the idea of distance. It may refer to a physical space as in spatial reasoning, or to an abstract space used for describing situations, as e.g., in case-based reasoning.

The representation of spatial relations between regions often relies on first order theories based on a family of partial preorders between regions, called mereologies. Although the term "mereology" usually refers to the idea of parthood as a basic notion, the idea of connection may be also used as a primitive notion. There are eight basic relations between regions known as RCC relations (RCC stands for "Region Connection Calculus") [START_REF] Cohn | Qualitative spatial representation and reasoning with the region connection calculus[END_REF]. One may even start with a fuzzy connection relation (which might be defined from a distance or a pointwise closeness relation), and then define a "part of", or an "overlap" fuzzy relation between regions for instance, and obtain a graded extension of the RCC calculus [START_REF] Schockaert | Reasoning about Fuzzy Temporal and Spatial Information from the Web[END_REF]. Modal logics are also used for representing spatial information. Spatial interpretations of modalities have been provided for capturing various spatial concepts qualitatively with a topological or geometric flavor such as nearness or distance, for example. See [START_REF] Dupin De Saint-Cyr | An exploratory survey of logic-based formalisms for spatial information[END_REF] for a review of the logic-based representations of mereotopologies in classical or modal logics, and in fuzzy and rough sets settings, as well as modal logic representations of geometries.

Rough sets [START_REF] Pawlak | Rough Sets. Theoretical Aspects of Reasoning about Data[END_REF] provide a formal setting for the "granulation" of a universe of discourse partitioned into equivalence classes of elements that are found indistinguishable (because they share exactly the same properties among the ones that are considered). They come from the idea of information systems where the available descriptions of objects via attributes are not sufficient to distinguish between them (more attributes would be needed). Only upper and lower approximations of subsets of objects can be defined. Such clustering between indistinguishable objects also leads to eliminate inessential attributes. Modal logics have been proposed for reasoning with lower and upper approximations of sets of models in such a setting [START_REF] Del Cerro | DAL -a logic for data analysis[END_REF][START_REF] Demri | Incomplete Information: Structure, Inference, Complexity[END_REF]. Clearly the equivalence relation may be turned into a fuzzy relation, giving birth to graded rough set calculi [START_REF] Dubois | Putting rough sets and fuzzy sets together[END_REF][START_REF] Radzikowska | Characterisation of main classes of fuzzy relations using fuzzy modal operators[END_REF]. In that case the partition becomes a fuzzy covering of the set of objects. There is to-date a large technical literature putting rough sets, fuzzy sets and fuzzy similarity relations together.

Extrapolation and interpolation reasoning are based on the idea of closeness between interpretations. Thus, for instance, if the set of models of a proposition p fails to be included in the set of models of a proposition q, but remains included in the set of interpretations that are close to models of q, one may say that p → q is "close to being true". This has been advocated by different authors [START_REF] Ruspini | On the semantics of fuzzy logic[END_REF][START_REF] Klawonn | Similarity in fuzzy reasoning[END_REF] (and contrasts with non-monotonic reasoning where one requires that the preferred/normal models of p be included in the models of q). The closeness (or proximity) relation between the interpretations gives birth to a graded consequence relation, which is the basis for a logic of similarity dedicated to interpolation [START_REF] Dubois | A logical approach to interpolation based on similarity relations[END_REF], and captures fuzzy logic-based approximate reasoning. In a similar spirit, a logic allowing to reason about the similarity with respect to specific sets of prototypes has been recently proposed [START_REF] Vetterlein | A logic of the similarity with prototypes and its relationship to fuzzy logic[END_REF].

In the above approaches, similarity is graded. More qualitative approaches have been proposed, using comparative relations. The logic CSL [START_REF] Sheremet | A logic for concepts and similarity[END_REF] is based on a modal binary operator that is used for denoting the set of interpretations that are closer to p than to q. The underlying distance-based semantics can be restated in terms of preferential structures using a ternary relation expressing that "all the points in a region z are at least as close to region x than to region y" [START_REF] Alenda | Comparative concept similarity over minspaces: axiomatisation and tableaux calculus[END_REF]. Another qualitative approach, without any grade, relies at the semantic level on the conceptual spaces framework [START_REF] Gärdenfors | Conceptual Spaces: The Geometry of Thought[END_REF] where the possibility of expressing spatial-like localization such as "being in between" (by means of a ternary relation), or "parallelism" (by means of a quaternary relation) provides a basis for capturing interpolative and extrapolative reasoning respectively [START_REF] Schockaert | Qualitative reasoning about incomplete categorization rules based on interpolation and extrapolation in conceptual spaces[END_REF]. This proposal comes close to logical reasoning with analogical proportions [START_REF] Prade | Multiple-valued logic interpretations of analogical, reverse analogical, and paralogical proportions[END_REF], which are quaternary statements of the form "a is to b as c is to d" (involving Boolean, or graded, properties), but remains more cautious.

Lastly, let us also mention that apart from reasoning about similarity, what may be termed reasoning with similarity has been also proposed as a semantic basis in non-monotonic reasoning [START_REF] Godo | Graded similarity-based semantics for nonmonotonic inferences[END_REF], information updating (which relies on ternary comparative closeness relation) [START_REF] Katsuno | On the difference between updating a knowledge base and revising it[END_REF], or in distance-based information fusion, where, e.g., Hamming distances are computed with respect to set of interpretations [START_REF] Konieczny | Merging information under constraints: a logical framework[END_REF]. Another view of information fusion, recently proposed in [START_REF] Schockaert | Solving conflicts in information merging by a flexible interpretation of atomic propositions[END_REF], relies on the idea that inconsistency can be often resolved by enlarging the sets of models of the information to be fused, thanks to similarity relations.

Logical issues

In this section our aim is to lay bare the main contrasting features of logical formalisms dealing with graded uncertainty and graded truth.

Uncertainty semantics

Let us assume an agent is to reason about what the world is and assume that each of the possible states of the world is described by a complete Boolean truth evaluation of a given (finite) set of atomic propositions Var. So let Ω denote the set of Boolean truth-evaluations w : L → {0, 1} of formulas from a propositional language L built from the finite set of variables Var and with the usual connectives ∧, ∨, → and ¬. It is well known that the connectives ∧, ∨ and ¬ endow L, modulo logical equivalence, with a structure of a Boolean algebra.

We start by considering the case where the agent has complete information about the world, so he knows that the actual worlds is w 0 ∈ Ω. In this case there is no uncertainty at all, in fact, knowing what the world is, the agent is able to ascertain the truth status of every possible proposition. This corresponds to consider the agent's epistemic state as being represented by the pair (Ω, E) where Ω reflects the agent's language (that govern the precision in the description of possible states of the world) and E = {w 0 } is a singleton.

A first form of uncertainty appears when the agent's information only allows him to know for certain that the actual world w 0 is in some given subset E ⊆ Ω. This is the typical case where the agent has a theory T (a set of formulas) describing what he knows about the world. The epistemic state of the agent is then represented as (Ω, E) where E is a non-empty subset of interpretations, indeed the set of models of T , and the agent is only able to determine the truth status of some propositions, but not for some others. Indeed, a proposition ϕ is known to be true if E | ϕ (or equiv. T ⊢ ϕ), it is known to be false when E | ¬ϕ and it is unknown otherwise, i.e. when both E | ϕ and E | ¬ϕ. Therefore in this setting, propositions may be in three different epistemic statuses.

A more refined representation is when the agent may associate weights to interpretations related to the likelihood of each interpretation of describing the actual world. In this case, the epistemic state can be represented as a pair (Ω, µ), where µ : Ω → [0, 1] attaches a weight to each possible world. In the probabilistic model (see e.g. early works by Nilsson [94]), µ = p is a probability distribution on Ω, hence requiring {p(w) | w ∈ Ω} = 1, that allows to rank the likelihood of any proposition of being true according to its probability measure

P (ϕ) = {p(w) | w ∈ Ω, w | ϕ}. When p(w) ∈ {0, 1}
for all w ∈ Ω, then the epistemic state reduces to a singleton E = {w 0 } where w 0 is the unique world such that p(w 0 ) = 1.

In the possibilistic setting [START_REF] Dubois | Possibilistic logic, in: Handbook of Logic in Artificial Intelligence and Logic Programming[END_REF], µ = π is a possibility distribution, where π(w) = 1 means that it is totally possible (plausible) that w = w 0 , π(w) = 0 means that w = w 0 is totally disbelieved. Then propositions are weighted according to the corresponding conjugate necessity and possibility measures: N(ϕ) = 1 -min{π (w) | w | ¬ϕ}, and Π(ϕ) = max{π (w) | w | ϕ}. Again, when π(w) ∈ {0, 1} for all w ∈ Ω, then the epistemic state defined by π corresponds to the previously considered three-valued setting with E = {w | π(w) = 1}. Other measures can also be used, like the guaranteed possibility (ϕ) = min{π (w) | w | ϕ}, which is equal to 1 whenever all models of ϕ are considered plausible. The case when (ϕ) > Π(¬ϕ) expresses a form of strong belief in ϕ [START_REF] Battigalli | Strong belief and forward induction reasoning[END_REF].

More generally, one may consider graded epistemic states of the form (Ω, µ), where µ : 2 Ω → [0, 1] is a monotonic set-function, in such a way that every proposition ϕ can be attached a likelihood or belief degree of being true as the evaluation w.r.t. µ of the set of models of ϕ, i.e. of µ({w ∈ Ω | w(ϕ) = 1}). This representation generalizes the previous probabilistic or possibilistic models, to other more general ones like for instance those defined by belief functions, upper and lower probabilities or imprecise probabilities (see e.g. [START_REF] Halpern | Plausibility measures: a general approach for representing uncertainty[END_REF][START_REF] Halpern | Reasoning About Uncertainty[END_REF] for a general approach encompassing many uncertainty models in a modal logic setting). The information cannot then be reduced to a distribution over interpretations. In the qualitative setting, a monotonic set-function can always be viewed as the lower bound of a set of possibility measures (µ( A) = min n i=1 Π i (A)) which comes down to representing epistemic states expressed by µ as a set of possibility distributions over Ω (e.g. corresponding to several agents) [START_REF] Dubois | Qualitative capacities as imprecise possibilities[END_REF].

Syntactic formats for uncertainty

From a syntactical point of view, a number of formalisms coping with graded uncertainty have been proposed in the literature. Most of them use a modality, either explicit or implicit, referring to graded belief. For illustration purposes, we mention two kinds of languages: those based on Boolean uncertainty statements restricting degrees of uncertainty of formulas into a given subset; and those expressing gradual uncertainty statement using many-valued predicates or modalities. 1An instance of the first approach is Halpern's probability logic [START_REF] Halpern | Reasoning About Uncertainty[END_REF], where formulas express constraints among the probabilities of ϕ 1 , . . . , ϕ k as linear inequalities of the form a 1 P (ϕ 1 ) + ... +a k P (ϕ k ) ≥ b, with a 1 , . . . , a k , b being real numbers and where P (ϕ i ) stands for probability of ϕ i . The same language is used to reason about other kinds of uncertainty measures like plausibility measures, belief functions, etc., see also [START_REF] Wilson | Extending uncertainty formalisms to linear constraints and other complex formalisms[END_REF] for a related general approach to reason under uncertainty using linear constraints. In the Serbian school (Markovi ć, Ognjanovi ć and colleagues) on probability logics [START_REF] Dordevi Ć | Completeness theorem for propositional probabilistic models whose measures have only finite ranges[END_REF][START_REF] Milosevi Ć | A first-order conditional probability logic[END_REF][START_REF] Raskovi Ć | A logic with conditional probabilities[END_REF], they use graded modalities of the form P ≥a ϕ, declaring that the probability of ϕ is at least a, and then these probabilistic atoms are combined by means of classical connectives. A similar approach is Lukasiewicz's probabilistic logic [START_REF] Lukasiewicz | Weak nonmonotonic probabilistic logics[END_REF] (see also [START_REF] Cozman | Probabilistic logic with independence[END_REF] for a logic allowing to express independence), where the author uses expressions of the form (ϕ)[l, u] to denote that the probability of ϕ lies in the interval [l, u], as well as van der Hoek and Meyer's approach [START_REF] Van Der Hoek | Graded modalities in epistemic logic[END_REF] on graded probabilistic modalities. Likewise, in Dubois-Prade's possibilistic logic [START_REF] Dubois | Possibilistic logic, in: Handbook of Logic in Artificial Intelligence and Logic Programming[END_REF][START_REF] Dubois | Possibilistic logic: a retrospective and prospective view[END_REF][START_REF] Dubois | Possibilistic logic -an overview[END_REF], formulas are pairs (ϕ, α), stating that the necessity N(ϕ) of ϕ is at least α. Recently, this language has also been generalized to deal with Boolean combinations of possibilistic atoms N ≥α ϕ, in a similar way to the previous probabilistic logic language [50].

A different approach by Hájek and colleagues [START_REF] Hájek | Fuzzy logic and probability[END_REF][START_REF] Hájek | Metamathematics of Fuzzy Logic[END_REF][START_REF] Godo | A fuzzy modal logic for belief functions[END_REF] has also been proposed, where the modality B used to denote graded belief (probability, necessity, etc.) is graded itself, that is, even if ϕ is Boolean, the atomic modal expression Bϕ, read as "ϕ is believed", is graded in nature (ϕ can be more or less believed, probable, necessary, etc.). In this way, the truth-degree of Bϕ can be taken as, e.g., the probability (or necessity) degree of ϕ, and then these graded atoms are combined using the rules of a suitable fuzzy logic.

Logics of graded truth

Indeed, formalisms that cope with graded truth (fuzzy logics) radically depart from the formalisms for uncertainty reasoning [START_REF] Dubois | Possibility theory, probability theory and multiple-valued logics: a clarification, in: Representations of Uncertainty[END_REF][START_REF] Dubois | Fuzzy-set based logics -an history-oriented presentation of their main developments[END_REF]. Giving up the bivalence principle and adopting a truth scale with intermediate degrees between 0 (false) and 1 (true) (usually the unit real interval [0, 1]) lead to a number of many-valued truth-functional systems with connectives extending the classical ones. The most relevant examples of formal fuzzy logic systems are the so-called t-norm based fuzzy logics [START_REF] Hájek | Metamathematics of Fuzzy Logic[END_REF]. These correspond to logical calculi with the real interval [0, 1] as set of truth-values and defined by a conjunction & and an implication → interpreted respectively by a continuous t-norm * and its residuum ⇒, and where negation is defined as ¬ϕ = ϕ → 0, with 0 being the truth-constant for falsity. In this framework, each continuous t-norm * uniquely determines a semantical (propositional) calculus PC( * ) over formulas defined in the usual way from a countable set of propositional variables, connectives & and → and truth-constant 0 (further connectives, like ϕ ∧ ψ as ϕ&(ϕ → ψ), can also be defined). Evaluations of propositional variables are mappings e assigning to each propositional variable p a A formula ϕ is a said to be a 1-tautology of PC( * ) if e(ϕ) = 1 for each evaluation e. Let the set of all 1-tautologies of PC( * ) be denoted as TAUT( * ). The main axiomatic systems of fuzzy logic, like Łukasiewicz logic (Ł), Gödel logic (G) or Product logic (Π ), syntactically capture different sets of tautologies TAUT( * ), according to the choice of the t-norm * , see e.g. [START_REF] Hájek | A complete many-valued logic with product-conjunction[END_REF][START_REF] Hájek | Metamathematics of Fuzzy Logic[END_REF][START_REF] Gottwald | A Treatise on Many-Valued Logics[END_REF][START_REF]Handbook of Mathematical Fuzzy Logic, Studies in Logic[END_REF]. Indeed one always has results of the form:

ϕ is provable in the logic Ł iff ϕ ∈ TAUT( * Ł ) (1)
where * Ł can be chosen as x * Ł y = max(0, x + y -1)

(Łukasiewicz logic Ł), x * G y = min(x, y) (Gödel logic G) and x * Π y = x • y (Product logic Π ).
It is worth noticing that, in contrast with classical logic, the algebraic structures of the set of formulas modulo logical equivalence in these systems of fuzzy logic are no longer Boolean algebras but weaker structures like MV-algebras, prelinear Heyting algebras, etc. Strictly speaking, a formula in a many valued logic is interpreted as a function from the Cartesian product of copies of the truth-set to the truth set. Each many-valued logic captures a class of functions at the ontic level, and there is no epistemic ingredient in such logics, which may explain why they are not much used in Artificial Intelligence.

Also, in all these systems, the implication captures the truth-ordering, since if ⇒ is the residuum of a left-continuous t-norm * (i.e., x ⇒ y = max{z ∈ [0, 1] | x * z ≤ y}), then x ⇒ y = 1 iff x ≤ y, and hence e(ϕ → ψ) = 1 iff e(ϕ) ≤ e(ψ).

Therefore, a formula ϕ → ψ actually represents that ψ is at least as true as ϕ, this is to say, t-norm based fuzzy logics are logics of comparative truth. To explicitly deal with truth-degrees in the reasoning, one may introduce truth-constants r, e.g. for all rational values in r ∈ [0, 1]. Then a formula r → ϕ expresses that the truth-degree of ϕ is at least r (see e.g. [START_REF] Esteva | Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results[END_REF]), which suggests that such many-valued logics, cast at the epistemic level, may account for weighted logics of uncertainty of the previous kind (using graded modalities).

From an epistemic point of view, in a fuzzy logic setting, the states of the world are described by complete [0, 1]-evaluations of atomic formulas. Let Ω ′ be the set of these evaluations, i.e. Ω ′ = {w | w : Var → [0, 1]}. Note that the set Ω of Boolean interpretations is indeed a subset of Ω ′ . Because of truth-functionality, a completely informed scenario thus corresponds now to a precise many-valued truth-value assignment w 0 ∈ Ω ′ to all propositional variables. Analogously to the Boolean case, different kinds of incomplete information about the world translates here to different many-valued generalizations of epistemic states. In general they are of the form (Ω ′ , µ ′ ), where µ ′ : [0, 1] Ω ′ → [0, 1] is a generalized uncertainty measure over fuzzy sets of interpretations (see e.g. [START_REF] Flaminio | On the logical formalization of possibilistic counterparts of states over n-valued Łukasiewicz events[END_REF][START_REF] Flaminio | A logic for reasoning about the probability of fuzzy events[END_REF][START_REF] Flaminio | Logics for belief functions on MV-algebras[END_REF] for some logical formalisms that are able to deal with uncertainty over non-Boolean (fuzzy) events).

New areas for graded settings and conclusion of this overview

Beyond the handling of the ideas of truth, uncertainty, preference, and similarity that may require graded settings for a proper handling, the field of deontic reasoning is another area where one encounters basic notions whose strengths may be usefully compared, stratified into layers, or even graded: one may think of graded obligations, or permissions for instance [START_REF] Liau | A semantics for logics of preference based on possibility theory[END_REF][START_REF] Dellunde | Introducing grades in deontic logics[END_REF]. One may also distinguish between weak and strong permissions [START_REF] Kaci | Permissions and uncontrollable propositions in DSDL3: non-monotonicity and algorithms[END_REF]. Priorities, or preference relations among worlds, aiming at ordering worlds from the most ideal ones to the least ideal ones, may help solving dilemmas [START_REF] Cholvy | An attempt to adapt a logic of conditional preferences for reasoning with contrary-to-duties[END_REF].

Besides, there are several domains of active research in Artificial Intelligence nowadays where several of the basic notions mentioned above can be encountered, possibly combined with other notions that also need to be graded. Let us review them briefly.

• So-called BDI agents [START_REF] Bratman | Faces of Intention[END_REF] are supposed to have beliefs about the world, desires, from which they elicitate intentions that are feasible desires. Clearly beliefs may be pervaded with uncertainty, desires may be modeled as collections of goals with different priority levels, feasibility may be a matter of cost, leading to ordered/graded intentions, see, e.g. [START_REF] Casali | A graded BDI agent model to represent and reason about preferences[END_REF]. The modeling of desires in terms of guaranteed possibility measures, already at work in the previous reference, has been further discussed and investigated in [START_REF] Dubois | Bipolar possibility theory as a basis for a logic of desire and beliefs[END_REF][START_REF] Dubois | Nonmonotonic desires -a possibility theory viewpoint[END_REF].

• Modeling the trust that can be associated with information sources or agents, as well as related notions such as distrust or reputation, is an important issue in practice. Many proposals exist -modal or numerical -where gradedness has been introduced in various ways [START_REF] Liau | A modal logic framework for multi-agent belief fusion[END_REF][START_REF] Lorini | From binary trust to graded trust in information sources: a logical perspective[END_REF][START_REF] Demolombe | Graded trust[END_REF][START_REF] Victor | Practical aggregation operators for gradual trust and distrust[END_REF][START_REF] Ben-Naim | Evaluating trustworthiness from past performances: interval-based approaches[END_REF]. Still, there is not yet a fully clear view of how graded trust may relate to beliefs and uncertainty.

• Argumentation is another area where the idea of strength seems naturally associated with arguments [5], as recently investigated [START_REF] Dunne | Weighted argument systems: basic definitions, algorithms, and complexity results[END_REF][START_REF] Gabbay | Introducing equational semantics for argumentation networks[END_REF][START_REF] Coste-Marquis | Weighted attacks in argumentation frameworks[END_REF][START_REF] Amgoud | A formal view of abstract argumentation[END_REF][START_REF] Hunter | Probabilistic argument graphs for argumentation lotteries[END_REF]. However, it is likely that a uniform view of strength is not applicable here, since the strength of an argument may refer to the uncertainty pervading the pieces of information on which it is based [START_REF] Alsinet | Formalizing argumentative reasoning in a possibilistic logic programming setting with fuzzy unification[END_REF][START_REF] Hunter | A probabilistic approach to modelling uncertain logical arguments[END_REF][START_REF] Alsinet | RP-DeLP: a weighted defeasible argumentation framework based on a recursive semantics[END_REF] and on the reliability of the source(s) of the argument, as well as on the rhetoric form of the argument (e.g. its length); moreover the argument itself may refer to a gradual view of truth (when stating for instance that "the higher the fever the more certain the child should remain in bed"). The handling of such strengths may also depend on the kind of problem to which argumentation is applied: persuasion, negotiation, or deliberation, for instance.

• Emotions, such as surprise, fear etc. have been recently modeled by means of definitions in modal logic [1]. It seems natural here again, to have these modalities complemented with grades, leading to hybrid notions that might be a compound involving more basic notions such as uncertainty, preference, or similarity [START_REF] Steunebrink | A formal model of emotions: integrating qualitative and quantitative aspects[END_REF][START_REF] Lorini | A dynamic logic of knowledge, graded beliefs and graded goals and Its application to emotion modelling[END_REF][START_REF] Livet | Rational choice, neuroeconomy and mixed emotions[END_REF].

• In weighted social networks, the degrees of connection between actors may be described in a graded logical setting, such as many-valued modal logics [START_REF] Fan | Logical characterizations of regular equivalence in weighted social networks[END_REF].

In this introduction we have briefly discussed several issues in the main approaches at work to represent and reason with fundamental notions in AI, where the use of degrees appears to be natural, such as truth, uncertainty, preferences, or similarity (but also trust, permission, obligation, desires, etc.). Specifically, the basic difference between graded truth and other graded notions has been highlighted. While the former implies a change at the ontic level (from two truth values to multiple truth-values), without any reference to epistemic knowledge or ignorance, the latter relates to intensional notions that (usually) apply to Boolean propositions, like their epistemic (belief) status, or how they compare to other propositions in terms of preference, utility, similarity, etc. This is reflected on the kind of formal models that support these graded notions, many-valued truth-functional models in the former case, Kripke-like models and graded modalities in the latter case.

Actually, many-valued logics have been seriously criticized at the philosophical level because of the confusion between truth-values on the one hand and degrees of belief, or various forms of incomplete information, on the other hand, a confusion that even goes back to pioneers including Łukasiewicz (e.g., the idea of possible as a third truth-value). Actually, due to this issue and the numerical flavor of fuzzy logic, there is a long tradition of mutual distrust between Artificial Intelligence and fuzzy logic. There are two ways to fill this gap. One is to use fuzzy logics to encode graded epistemic modalities that can account for various uncertainty theories; these graded modalities may be applied to Boolean formulas, yielding a two-tiered logic. The other way is to show how reasoning about knowledge and uncertainty can also be defined on top of fuzzy/gradual propositions by augmenting fuzzy logic with epistemic modalities. Recent works along this line may be considered as first steps towards a reconciliation between possibility theory and other theories of belief as well with fuzzy logic (in the sense of a rigorous symbolic setting to reason about gradual notions), see e.g. [START_REF] Blondeel | Fuzzy autoepistemic logic: reflecting about knowledge of truth degrees[END_REF][START_REF] Flaminio | On the logical formalization of possibilistic counterparts of states over n-valued Łukasiewicz events[END_REF].

Finally, we would like to mention that, although we have mainly focused on logical issues, many of the concerns discussed here have also echoes in closely related areas like description logics, logic programming and answer set programming when they come to handle uncertainty, preferences or fuzziness, see for instance [START_REF] Lukasiewicz | Description logic programs under probabilistic uncertainty and fuzzy vagueness[END_REF][START_REF] Stoilos | Fuzzy extensions of OWL: logical properties and reduction to fuzzy description logics[END_REF][START_REF] García-Cerdañaya | Fuzzy description logics and t-norm based fuzzy logics[END_REF][START_REF] Janssen | General fuzzy answer set programs[END_REF][START_REF] Madrid | On coherence and consistence in fuzzy answer set semantics for residuated logic programs[END_REF][START_REF] Bauters | Possibilistic answer set programming revisited[END_REF][START_REF] Blondeel | Complexity of fuzzy answer set programming under Łukasiewicz semantics[END_REF] for a variety of logic programming approaches coping with graded uncertainty and/or truth.

Contents of the special issue

This special issue organized by the last two authors of this introduction, originates in an international workshop on Weighted Logics for Artificial Intelligence (WL4AI'12), co-located with the 20th European Conference on Artificial Intelligence, and held in Montpellier (France) on August 28, 2012. It includes expanded versions of papers presented at the workshop as well as papers submitted in response to a call for papers on the workshop topic published in this journal. All accepted contributions have largely benefited of useful reviewers' comments which helped the authors improving their papers substantially through several rounds of revision.

The topics of these papers illustrate several of the many concerns just surveyed. Indeed the special issue gathers 13 contributions which cover a broad variety of topics: probabilistic logics, three-valued logics, description logics, answer set programming, argumentation, deontic norms, causality, and information fusion.

The two first papers deals with probabilistic logics. The first one "Hierarchies of probabilistic logics" by N. Ikodinovi ć, Z. Ognjanovi ć, A. Perovi ć, and M. Raskovi ć, studies logics with probability operators expressing that the probability of a formula is at least s (where s is a rational number), or that the probability is in a set F ranging over different recursive families of recursive subsets of the set of rationals in the unit interval, leading to distinct logics in terms of expressivity. The second paper "Conditional p-adic probability logic" by A. Ili ć Stepi ć, Z. Ognjanovi ć, N. Ikodinovi ć, presents a proof-theoretical approach to p-adic valued conditional probabilistic logics that extend classical propositional logic with conditional probability operators and where formulas are interpreted in Kripke-like models that are based on p-adic probability spaces.

The third paper, "Borderline vs. unknown -Comparing three-valued representations of imperfect information", deals with other basic concerns. Their authors, D. Ciucci, D. Dubois and J. Lawry compare the expressive power of elementary representation formats for vague, incomplete or conflicting information, namely Boolean valuation pairs, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, and supervaluations, making explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The paper aims at clarifying the similarities and differences between truth-functional and non-truth-functional approaches.

The three next papers deal with description logics, a family of knowledge representation formalisms for handling taxonomies. M. Cerami, A. García-Cerdaña and F. Esteva define fuzzy description logics on the basis of first-order many-valued fuzzy logics. Their paper "On finitely valued fuzzy description logics" is based on the fuzzy logic of a finite BL-chain in the sense of Hájek. The paper addresses the hierarchy of fuzzy attributive languages and discusses reasoning tasks and computational complexity. The second paper "Consistency reasoning in lattice-based fuzzy description logics" by S. Borgwardt and R. Peñaloza is also about fuzzy description logics with a semantics based on complete De Morgan lattices and one basic reasoning task, namely deciding whether an ontology representing a knowledge domain is consistent. The paper focuses on the fuzzy description logic L-SHI, providing a tableaux-based algorithm for deciding consistency when the underlying lattice is finite, and identifies decidable and undecidable classes of fuzzy description logics over infinite lattices. The last description logic paper "Completion-based generalization inferences for the description logic ELOR with subjective probabilities" by A. Ecke, R. Peñaloza and A.-Y. Turhan studies the logic Prob-ELOR that extends ELOR with probabilities, and presents a completion-based algorithm for polynomial time reasoning in a restricted version of Prob-ELOR. This algorithm is extended in order to provide tools for building ontologies automatically from examples by computing the most specific concept that generalizes a given individual into a concept description, and the least common subsumer that generalizes several concept descriptions into one. The feasibility of the approach is demonstrated empirically on a prototype.

The paper "Complexity of fuzzy answer set programming under Łukasiewicz semantics", by M. Blondeel, S. Schockaert, D. Vermeir and M. De Cock, illustrates another active area of research, namely answer set programming (ASP), here generalized to fuzzy ASP in which propositions may be graded. The paper analyzes the computational complexity of fuzzy ASP under Łukasiewicz semantics, and shows that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level, and a reduction from reasoning with such fuzzy ASP programs to bilevel linear programming is established.

The two next papers deal with argumentation systems, another hot topic in artificial intelligence nowadays. In their paper "Valued preference-based instantiation of argumentation frameworks with varied strength defeats", S. Kaci and Ch. Labreuche consider an extension of Dung's abstract argumentation framework where the degree to which an argument defeats another argument can be graded, and instantiate it by a preference-based argumentation framework with a valued preference relation (which may be derived from a weight function). They show under some conditions that there are less situations in which a defense between arguments holds with a valued preference relation compared to a Boolean preference relation. The paper "Postulates for logic-based argumentation systems", by L. Amgoud, focuses on argumentation systems that are based on deductive logics and that use Dung's semantics. It proposes and discusses five rationality postulates that such systems should satisfy: consistency and closure under the consequence operator of the underlying logic of the set of conclusions of arguments of each extension, closure under sub-arguments and exhaustiveness of the extensions, and a free precedence postulate ensuring that the free formulas of the knowledge base (i.e., the ones that are not involved in inconsistency) are conclusions of arguments in every extension. Postulates for weighted argumentation systems are finally addressed.

The paper "Reasoning about norms under uncertainty in dynamic environments" by N. Criado, E. Argente, P. Noriega, and V. Botti proposes a multi-context graded BDI architecture that models norm-autonomous agents able to deal with uncertainty in dynamic environments. A degree of saliency is attached to norms, and beliefs, desires and intentions are graded. This architecture has been experimentally evaluated on a fire-rescue scenario and results are reported in the paper.

In "A weighted causal theory for acquiring and utilizing open knowledge", J. Ji and X. Chen introduce a weighted counterpart to McCain and Turner's non-monotonic causal theories, the action model of a robot/agent being specified as a causal theory. The authors solve an abduction problem for finding the most suitable set of causal rules from open-source knowledge resources, so that a plan for accomplishing the robot task can be computed using the action model and the acquired knowledge. The weights associated to hypothetical causal rules are used for comparing competing explanations which induce causal models satisfying the task. This is illustrated with a service robot example.

The last two papers discuss tools for information fusion. In "Sum-based weighted belief base merging: from commensurable to incommensurable framework", S. Benferhat, S. Lagrue, and J. Rossit provide a postulate-based analysis of the sum-based merging operator when sources to merge are incommensurable (i.e., they do not share the same meaning of uncertainty scales). They show that this operator can be characterized either in terms of an infinite set of compatible scales, or by a Pareto ordering on a set of propositional logic interpretations. Finally, T. Nakama and E. Ruspini, in "Combining dependent evidential bodies that share common knowledge", establish a formula for combining dependent bodies of evidence that are conditionally independent given their shared knowledge. They show that the Dempster-Shafer formula and the conjunctive rule of Smets' Transferable Belief Model can be recovered as special cases of their combination formula.

Hopefully this special issue, by providing an overview of weighted logics in AI and gathering a rich sampling of works illustrating the interest of introducing grades in many knowledge representation and reasoning problems, will contribute to a better understanding of these issues, and to the proper use of the different possible approaches.

  truth-value e(p) ∈ [0, 1], which extend univocally to compound formulas as follows: e(ϕ ∧ ψ) = min e(ϕ), e(ψ) e(ϕ&ψ ) = e(ϕ) * e(ψ) e(ϕ → ψ) = e(ϕ) ⇒ e(ψ)

There is a third approach using statements like "ϕ is more likely than ψ ", that relies on conditional logics in the style of Lewis[START_REF] Lewis | Counterfactuals and comparative possibility[END_REF], that takes another point of view on logics of uncertainty.