Pascal Engel
email: pascal.engel@unige.ch

Epistemic norms and the limits of epistemology

Keywords: hinges, certainty, epistemology, norms, rules, epistemic norms, Wittgenstein, Crispin Wright

I raise a dilemma for an epistemology based on the idea that there are hinge propositions or primitive certainties: either such propositions are norms or rules in the 'grammatical' sense, but they cannot regulate our inquiries since they are not genuine propositions obeying truth or evidential standards, or they are epistemic norms, but compete with the classical norms of belief and knowledge. Either there are hinges, but they have nothing to do with epistemology, or hinges are part of our knowledge, and their epistemology is part of ordinary epistemology.

Hinge epistemology: where the wild things are

A number of philosophers, among whom Husserl, Scheler, Ortega y Gasset and Wittgenstein, have argued that there are primitive certainties that stand at the limits of the epistemic domain of our ordinary beliefs and knowledge [START_REF] Mulligan | Certainty, Soil and Sediment[END_REF]. Various metaphors are used to designate their status with respect to our knowledge: soil, sediment, strata, ground, basis, framework, pillars, etc. Wittgenstein (OC) talks of hinge propositions (Angeln). Some phenomenologists take these as kinds of belief, and talk of creencias (Ortega), Urglauben (Husserl), but most writers on this theme insist on the disanalogy between these primitive certainties and ordinary doxastic attitudes such as beliefs, judgments, convictions, acceptances, suppositions or hypotheses, and knowledge attitudes such as perceptions, seeings or noticings. Although these are described generically as certainties, they are also said to be associated with such attitudes as presupposing, taking for granted, or holding fast. If, however, we compare the hinge propositions and the attitude of primitive certainty with our ordinary doxastic attitudes, their domain seems to extend beyond the reach of the theory of knowledge. In epistemology, as it is ordinarily conceived, one deals with how beliefs can be justified, revised or cancelled, what are their sources, whether they have degrees and whether these are probabilities, or one asks about their relations to evidence and their inferential connections and under what conditions they can become knowledge or lose that status. In contrast primitive certainties and hinge propositions do not lend themselves to such questions. They seem to be insulated from them. If so, how can they play, as Wittgenstein often suggests, a grammatical role of general presuppositions for our knowledge, rather than the role of foundations for knowledge? They occupy a kind of a priori status, although they are very often empirical. They are norms or rules for the language game of believing and knowing, rather than objects of belief or of knowledge. This normative status of hinge propositions raises, however, a problem. For it has been argued that there are some general epistemic norms for belief and knowledge, such as the norm that one ought to believe propositions which are true or based on appropriate evidence, and more specific norms for inductive and deductive reasoning. What is the status of the 'grammatical' norms expressed by hinge propositions with respect to the general epistemic norms of belief and knowledge? As Wittgenstein and others insist, they are not 'known', either in the internalist sense of our having access to them, or in the externalist sense of their being based on causal and reliable process to which we do not have access. On the one hand, they seem to conflict, and if they do, one system of norms must be prior or more fundamental than the other. So the question arises: which of these two systems of norms is more basic? On the other hand, if they do not conflict, it seems hard to suppose that the epistemic domain is governed by two very distinct systems of norms. The only way to resolve this issue is to try and examine in what sense Wittgenstein's 'grammatical' hinge propositions can be norms for knowledge and belief, in comparison with the more usual knowledge norms. Another way of asking the same question consists in asking how there can be a hinge epistemology and in what sense it can be normative.

I shall first compare the grammatical norms which are supposed to be in place with hinge propositions to the standard norms of belief and knowledge. I shall argue that there cannot be two kinds of epistemic norms. Grammatical norms are not epistemic norms, since they are, in the end, practical, based on our forms of life and of action. Hinge propositions are not normative at all, and this means, in so far as epistemology is supposed to be normative discipline, that hinge epistemology is wild. This is not to deny the existence of hinge propositions, but the role they play. In this respect I shall suggest that there is some ground in saying that hinge propositions can be the object of knowledge.

Two kinds of epistemic norms?

A number of different things are called epistemic norms 1 : (i) epistemic values such as truth, knowledge or rationality, (ii) very high profile norms for belief and knowledge such as the norm that one ought to believe what is true, (iii) general principles of rationality, such as those of deductive logic, of inductive logic, of probability theory, of the logic of belief revision, (iv) particular norms or conceptual roles attached to concepts (for instance logical connectives or kind concepts) (v) methodological rules of inquiry and of scientific discovery (such as standards like simplicity, or rules of inductive support, of theory acceptance, or heuristic rules), (vi) paradigms or framework principles in the sense of Kuhn. The list is not exhaustive, and depends on whether one actually takes a number of principles as normative for the epistemic domain, and to what extent. For instance if one is a non-classical logician, one will not accept (i)-(v) as genuine norms and one will adopt some non-classical principles instead. If one distinguishes norms from values, the latter will not feature in the list, and if one takes norms to be rules or conventions, which can change depending on the context, very high profile norms will not be included in the list. Nevertheless, what all these different kinds of norms have in common (except, for reasons to which I shall come back immediately, the paradigm sense (vi)) is that they all concern the epistemic appraisal of our beliefs in terms of their relations to truth, knowledge or justifiedness. To say that they are epistemic is to say that they are not, at least prima facie, norms of conduct, or rules for actions or from practical reasoning, but that they are meant to govern our cognitive states, which have a mind to world direction of fit, even if they can regulate, in a broad sense our cognitive practices. To say that they are norms entails that they are not mere empirical generalizations about what we think or know, but that they stand out as principles, which tell us what we ought to believe. A norm, whether in the sense of a specific rule or in the sense of a general principle, is supposed to guide our conduct or our thought, or to regulate a certain practice. Some epistemic norms, such as the 'rule of priority' in science, which says that the first to publish a discovery has priority for credits and rewards2 , or certain heuristic rules, are supposed to have direct effects on epistemic conduct. But type (ii) or (iii) norms are not necessarily imperatives or prescriptions that can offer direct guidance. They tell us, at least ideally, what one is supposed to believe or how one is supposed to inquire, if one is minimally rational3 . To say that they are normative is to say that they all share the feature of telling us what it is correct to believe and the conditions of this correctness. Correctness has an external sense which is its condition of satisfactiona belief is correct if and only if it is trueand an internal sensea belief is correct if it is formed in a rational way4 . Alternatively one can say that norms in the (i)-(v) sense are normative in the sense that they circumscribe the space in which it is possible to give reasons for our beliefs. This is not to say that they are themselves reasons for believing something. Thus it could be absurd to say that one's reason for believing such and such is that it is correct to believe it if and only if it is true, or justified. A reason is a piece of evidence, a fact or a consideration which favours our believing such and such. The norms in question are not reasons but, so to say, principles for reason-giving, hence for the 'space of reasons'5 . A further feature of epistemic norms in senses (i)-(v) is that they are not only principles of appraisal for the truth and rationality of our beliefs, but also that they can themselves be appraised as true or false, as correct or incorrect. Some of the norms listed above can be criticized and amended, and it makes sense to say that we can evaluate them with metanorms, through which we evaluated our normative appraisals. But some other norms, the most basic ones, seem to be unrevisable. Thus the external norm of correctness for a beliefthat it be trueseems to be primitive. Thus high profile norms of type (ii) are the norms of truth, of knowledge and of evidence: Type (ii) norms: (Norm of Truth) One ought to believe P iff P (Norm of Knowledge) One ought to believe P iff one is in position to know P (Norm of Evidence) One ought to believe P iff one has sufficient evidence that P

We can ask whether one of these norms is more fundamental than the others, and what their relations are with other norms such as those of rationality in sense (iii) or with conceptual norms in sense (iv) or with methodological rules of type (v). But type-(ii) epistemic norms seem to be in some sense basic. In spite of their apparent triviality, none of these norms is unproblematic, and their proper interpretation is much debated6 . We can also appraise their truth in some way. But even if we can question the proper formulation of these norms, they are the best candidates for being the most primitive constitutive principles governing what it is to believe or to know. It is not clear that someone who would reject them would actually be talking about belief or knowledge. This is why they can be said to be a priori, at least in the sense that they register the basic commitments of our epistemic concepts. This constitutive character, however, does not entail that these norms dictate a particular account of what it takes for a belief to be knowledge or a certain account of justification. The fact that they are foundational or constitutive does not imply that one has to subscribe to a foundationalist conception of knowledge or of justification. In spite of the fact that there is much disagreement in contemporary epistemology about the nature of norms of the kind (ii), I shall consider that these are paradigmatic and that they capture a rather well understood and legitimate conception of epistemic normativity, which one can call, in a somewhat Pickwickian manner, the normativist account of epistemic norms7 . Now contrast these epistemic norms with type (vi) norms, which Kuhn calls 'paradigms', and which are often referred to as 'framework principles', 'schemes' or 'structures' governing either large periods of the development of science or specific regions of knowledge. As many epistemologists who accept such notions have emphasized, although these principles are, like norms (i)-(iii), those through which we appraise our epistemic claims, they differ from these in fundamental ways. First they are not themselves appraisable as true or false, justified or not. They are rather rules for our use of epistemic norms. They cannot be proved wrong, although they can be rejected by the advent of other framework principles which replace them. As such they are largely tacit, and hard to make explicit under the form of propositions which one could follow on the basis of reflection. Perhaps they cannot be been spelled out. Second, although they govern our epistemic claims or assertions, they are not per se epistemic. They are often described as a background of practices, of technical or institutional settings which make the pursuit of knowledge possible, rather than as propositions that could themselves be believed or known.

The contrast is even stronger between type (ii) epistemic norms and what Wittgenstein calls 'hinge propositions', the system of which has often been compared to Kuhnian paradigms and for which Wittgenstein sometimes has similar metaphors ('scaffoldings', 'yardsticks', 'river bed'). Most of Wittgenstein's commentators attribute to hinge propositions the following characteristics. First, in so far as they are propositionssuch as 'I have hands', 'I have a body', 'I am on earth', 'There are no men on the moon', 'The world was not created five minutes ago', etc.they cannot be doubted, and are beyond doubt. There is no reason to inquire about their evidence or their truth. They cannot be assessed as true. They cannot even be objects of belief or of knowledge. It does not even make sense to say that I believe, or for that matter, know that I have hands, for it does not make sense to deny this. Second there are not in any particular way epistemic, in the sense that they would have contents which could be evaluated as justified or not. As Wittgenstein repeatedly insists, they pertain to the way we act, and are based on our practices. As Wittgenstein says, these are propositions that we 'hold fast', perhaps presuppositions, but not hypotheses or truth-evaluable commitments. Third, they do not belong to the space of reasons: we cannot give reasons for them, or against them, and we cannot even consider them as grounding or regulating the space of reasons.

Can we nonetheless say that hinge propositions play an epistemological role? The vocabulary used by Wittgenstein and other writers who speak of 'primitive certainties', of propositions 'immune to doubt', when they are proposed as a solution to the problems of scepticism either in general or regarding specific propositions (about the external world, about other minds or about the past), or who speak of 'groundless beliefs', suggests that they are beliefs, or at least a certain kind of doxastic state. But many Wittgenstein commentators insist that it would be a mistake to take these propositions as beliefs or as candidates for knowledge. To even call them 'epistemic' would be wrong, since the question of their being known or believed does not arise, and to call them groundless suggests wrongly that there is something epistemological at stake8 . Although hinge propositions are often called 'rules', they contrast strongly with norms of the form (i)-(iii) in two important respects. First they are not a priori in the epistemological sense of propositions which are known independently of experience or which are in some sense true in virtue of the concepts or meanings involved. On the contrary hinge propositions are empirical propositions which, to use Wittgenstein's metaphor, were flowing in the river of belief but have come to belong to the riverbed. They have a certain persistence, but they are nevertheless susceptible of being overcome or withdrawn. They can hardly be called necessities. They are rules, although they are conditioned by 'very general facts of human nature'. Secondly, they are very contextual, and most of the time completely lack the kind of universality which is aimed at, at least with principles like (i)-(iii). Thus, in most contexts, 'I have two hands' is a hinge proposition. But in the context of my awakening on a bed hospital after an accident it may be a bona fide empirical proposition. Some hinge propositions may be such for a long period of time, and cease to be so afterwards. Many writers in epistemology who are influenced by Wittgenstein claim that epistemic evaluations in general make sense only within a context, hence that universal doubt does not make sense, since it refers to propositions which are supposed to be true or false outside any context, hence which turn upon no 'hinge' at all.

There are basically three main readings of Wittgensteinian hinges. On a first reading [START_REF] Mcginn | Sense and certainty: A dissolution of scepticism[END_REF][START_REF] Moyal-Sharrock | Understanding Wittgenstein's on Certainty[END_REF][START_REF] Coliva | Extended Rationality: A Study in Hinge Epistemology[END_REF], hinge propositions are rules, and thus are normative, in the sense of what Wittgenstein calls 'grammatical'. They do not describe any state of affairs or any system of beliefs, but they give us instructions to deal with these. On this view the sceptic's challenge is meaningless, because it stands within the space of reasons whereas hinges do not. This reading licenses a comparison with epistemic norms of type (i)-(v), but we have seen that hinge propositions are normative in a very different sense. On a second reading, which one may call naturalist [START_REF] Strawson | Scepticism and Naturalism, some Varieties[END_REF][START_REF] Williams | Unnatural Doubts: Epistemological Realism and the Basis of Scepticism[END_REF], hinge propositions are the expression of our natural upbringing within a community, and this why to doubt them would be 'unnatural'. On a third reading, which we may call epistemic [START_REF] Wright | Warrant for Nothing? and Foundations for free?[END_REF]): hinges are propositions that cannot be evidentially justified, but to which we can confer a weaker epistemic status, that of being 'entitled' to them. Although entitlement is not justification, it is, at least in part, an epistemic standing. On this view the sceptic's challenge makes sense, but it cannot be answered in evidential terms. This is not the place to assess these various interpretations. It is not clear that they are incompatible. Actually all of them accept the idea that hinges are in some sense 'normative', rule-like or 'grammatical'. For this reason we can call them 'grammatical norms'. But if this is so, what can be their relation to epistemic norms in the normativist sense? One can envisage three possibilities:

(a) Grammatical norms are identical, or strongly overlap, with epistemic norms ; (b) Grammatical norms are distinct from epistemic norms, and the former are actually more basic than, or govern, the latter; (c) Grammatical norms are utterly different, and have nothing to do with epistemic norms.

None of these options seems plausible. Although both grammatical norms (GN) and epistemic norms (EN) are normative in some sense, they differ, as we saw, strongly with respect to their epistemic status, to their place with respect to the space of reasons, and with respect to their being truth-evaluable. On some readings GN are not propositions at all, they are not epistemic and they stand outside the space of reasons. How could they be norms for our beliefs if they are not in the business of epistemology at all? So option (a) is implausible. For the same reason, option (c) is not plausible: how could GN govern or dominate EN if they are utterly different? Option (b) seems more plausible, but it is also hard to see how a hinge proposition, for instance that I have a body, could govern my believings in general. It may govern my beliefs about my location in space or about my pain in my foot, but it is less clear that it is presupposed when I think about mathematics or abstract poetry. So, then, there is a conflict between grammatical norms and epistemic norms, but for there to be a conflict they have to both be epistemic norms. But that's problematic, because if they are norms at all, both should regulate or guide our epistemic life. But they conflict if they ask us to go in distinct directions: belief and knowledge norms ask us to look for truth and evidence, whereas grammatical norms do nothing of the sort. They actually do not guide us in any way (although they regulate our use of words and what makes sense for us). My primitive certainty that the Earth has existed for a long time is a presupposition of my believing certain things about the Earth (for instance when I encounter fossils or do geology), but it does not guide or rule my beliefs about past matters in the sense of telling me how to go with getting more beliefs or justifying them. How could norms which cannot be assessed as true or false, which cannot be doubted or evaluated epistemically, and which are not rational play the role of rational requirements governing our epistemic endeavours? The only option left is that they do not guide it in the same way. We would then have two distinct set of norms. But that is problematic for any kind of norm, local or global. Suppose that there is a norm prescribing to eat sausages on Monday only, and another one forbidding to eat sausages on the first day of the month. How to go when the first day of the month is a Monday? Norms cannot conflict; unless one kind of norm takes precedence, the normative system cannot function. The same is true of general presuppositions such as hinges. If my system of beliefs presupposes that the Earth did not come in existence five minutes ago, it cannot presuppose at the same time that it came in existence four minutes ago.

Can hinges be epistemic?

In order to bring grammatical norms and epistemic norms together, we would need to understand hinge certainties as real propositions and beliefs, or at least as sufficiently close to being propositional and belief-like to be capable of serving as norms for beliefs. But this is what a number of interpretations of hinges do not allow. The naturalist or quasi-naturalist interpretation proposed by Strawson and Williams says that hinges are the expression of our natural endowment, most of it acquired through socialization. Moyal-Sharrock interprets hinges as expressions of our involvement in actions, along the line of an enactivist analysis of action (2004,2013). On this view, primitive certainties are neither believed nor known as propositions: they are constituted by our animal reactions and bodily involvement. So they are completely insulated from the normative space of reasons: hinges are not reasons and we cannot give reasons for them. On Coliva's view (2015), hinges are rules, but not in the sense of norms that can be epistemically appraised. They are like rules of a gameneither true nor false, neither justified nor not justifiedbut that we have to accept as a matter of brute fact if we want to play the game at all. All views of this sort attempt to meet the sceptic's challenge, but their basic line consists in dismissing it as meaningless, since the sceptic's question is framed in terms of knowledge and evidence. On naturalistic and grammatical views of hinges, there cannot be evidence for hinges, and for this reason it does not make sense to doubt them. Sceptical doubts cannot even arise. On Wright's view, by contrast, hinges are genuine propositions that can be assessed as true or false, and on the basis of which one can attempt to answer the sceptic. Hinge propositions, however, do not enter the circle of justification or knowledge. They enter another epistemic relation, entitlement. The notion of entitlement has been invoked in recent epistemology for a number of purposes, and there are a number of distinct notions of this sort, such as default justification or prima facie justification. So let us look at it more precisely, since it is the only kind of hinge epistemology which can lend itself to an epistemic reading.

As [START_REF] Wright | Warrant for Nothing? and Foundations for free?[END_REF] describes it, and to simplify, entitlement falls into two main categories. There is, first, strategic entitlement, which is a form of pragmatic justification in the sense in which Reichenbach talks of a 'pragmatic vindication' of induction. It is a kind of action, namely an acceptance 9 , done for a practical purpose, most of the time contextual and tied to specific circumstances, involving a taking for granted of a particular cornerstone propositionin the case at hand that our inductions are reliableand obeying the criteria of a rational strategic decision, which is what is usually known as the principle of dominance: an act dominates another if it has a better outcome for the agent. This kind of attitude may have nothing to do with actual belief. Thus in getting on a plane, you assumeacceptthat it will not crash in the absence of a belief to the contrary; Robinson Crusoe accepts that the fruits he finds on his island are edible although he may have doubts about it, because it is the dominant strategy. Acceptance in this sense is committing oneself to act on a certain assumption for the sake of a certain good. The good in question can be practical or theoretical, but the attitude of acceptance which serves it is clearly of a prudential nature. Wright, however, points out that this first kind of entitlement will not do as an answer to the sceptical challenge. For that challenge bears, both in the Cartesian or in the Humean kind of scepticism, on the ontology of the material worldthat is, on our reliance on the deliverances of our senses or of our reasoning practiceswhereas strategic entitlement is purely methodological. It fails to meet the challenge, because it addresses it only as a kind of pragmatic challenge, and answers on that terrain, without seeing it, as it should, as a theoretical challenge, about our knowledge of the external world (Wright 2004: 187). In other words, the sceptic asks a question concerning our beliefs about the external world and our cognitive right to them. Answering that we are pragmatically entitled to them, because the strategy is dominant, is to change the subject, since pragmatic acceptances are, by definition, not beliefs, but actions. So strategic entitlement and the associated attitude of pragmatic acceptance cannot be the appropriate response to the sceptical challenge.

In view of this Wright introduces another kind of entitlement, entitlement to a cognitive project; which obeys three conditions: (i) excluding doubt (and open-mindedness) from the outset, (ii) being a default position; (iii) being such that if we were to seek an evidential kind of warrant for P, we would end up in an infinite regress of justifications (Wright 2004 191-192). The difference with strategic entitlement is that the project is cognitive. The kind of acceptance involved in being entitled to a cognitive project is of the hypothetical kind. The first condition (i) excludes doubt (and open-mindedness) from the outset. The second condition (ii) is negative and amounts to saying that the kind of entitlement is a default position. The third condition (iii) actually says that if we were to seek an evidential kind of warrant for P, we would end up in an infinite regress of justifications. Here the kind of acceptance involves trust, and not only, as in the strategic entitlement, action on the assumption that. It is, in this sense, a distinctively epistemic attitude. But although the attitude is epistemic in kind, the kind of warrant involved is not. Wright claims that cornerstones, or hinge propositions for a given domain, can be warranted, but not epistemically and not evidentially.

The problem with this second notion of entitlement is that it does not seem to be different from the first kind in this latter respect. The kind of warrant that is provided, is, like that of strategic entitlement, of a non-evidential kind. As Wright himself notes (ibid: 197) given that the sceptic's concern is theoretical, or ontological, involving our claims to know, it is not a kind of warrant governed by epistemic reasons. If I take for granted certain propositions, and accept them as a matter of trust, I do not attend to their truth or to their justification, but to their relevance or their importance for my inquiry. But the sceptic's question bears precisely on the justification and on the truth of these propositions. The sceptic's challenge is epistemic in nature. In this sense entitlement to an epistemic project and the hypothetical acceptance which underlies it will fail to provide a satisfactory answer to the sceptic. But as everyone familiar with the dialectic involved in the answer to scepticism, are we bound to always obey the demands of the sceptic, which are, by definition, almost impossible to satisfy? Isn't it enough that we can calm down his worries, and lay him to rest? If, however, we understand the answer to scepticism to involve particular inquiries, and not a general warrant for all enquiries, Wright suggests that entitlement to a cognitive project will do 'for the purposes of a particular enquiry'. The basic idea would then be that:

If a cognitive project is indispensable, or anyway sufficiently valuable to us in particular, if its failure would at least be no worse than the costs of not executing it, and its success would be betterand if the attempt to vindicate (some of) its presuppositions would raise presuppositions of its own of no more secure an antecedent status, and so on ad infinitum, then we are entitled tomay help ourselves to, take for grantedthe original presuppositions without specific evidence in their favour. (2004: 192) It would seem that under these conditions, the relevant kind of entitlement is epistemic, as well as the kind of acceptance involved. If we suppose that, in a specific domain of inquiry, truth is our aim, and that we deliberate about the best strategy to achieve that aim, the idea is that we can make a kind of cost/benefit analysis. But three questions arise in this respect. First, is it clear that the strategy of trust prescribed by the project of attaining truth in a domain is the dominant strategy? In other words, is it correct that we will be better off epistemically by adopting the cornerstone or the presupposition than by simply taking the best strategy to avoiding error, hence by simply, as the sceptic recommends, suspending judgement? A calculus of cost-benefits will not necessarily favour the strategy. Second, is it clear that we can in such situations always sort out the pragmatic from the epistemic considerations? After all, if the choice of a cornerstone is supposed to be relative to our aims, and to the kind of values that are relevant in a given circumstance, how can we be sure that truth is our only aim? Can't there be cases where we have to balance epistemic aims with practical ones? Third, if the reasons to which the kind of acceptance / entitlement are answerable are epistemic, and if the ultimate aim is truth, why insist that the proper attitude is one of acceptance rather than belief?

It is not clear that Wright's framework can give us an answer to such questions. In particular the third question is the most pressing, since belief is supposed to be the attitude which is by its very nature constitutively governed by a norm of truth: a belief is correct if and only if it is true. To adapt a well-known argument by Bernard [START_REF] Williams | Deciding to Believe' in Problems of the Self[END_REF], a person who would adopt a certain proposition for the sake of a certain aim, but who would not grant that it is correct because it is true (but because it is , say, desirable) would not take herself as entertaining a belief but another attitude, which cannot be belief. Indeed it is perfectly possible for such a person to take herself as accepting the proposition for a particular objective, though not as believing it. If one grants this point, the third question comes to this: can we be sure that our epistemic or hypothetical acceptances, those which are oriented towards a cognitive project, do not presuppose beliefs? Hardly. I cannot accept certain cornerstones for a certain epistemic benefit, or judge that a cognitive project is valuable, in the complete absence of beliefs, not only about its outcome, but also about the conditions of its realisation. This is even true in the case of the supposedly purely pragmatic concept of acceptance. Consider the example above of someone who plans to build a house and accepts, for pragmatic reasons, that the price will not exceed a certain amount. He takes a kind of decision, and indeed he does not believe that the price will be as high as the one he takes for granted. But he hardly operates in the absence of subjective probabilities or credences: for instance he may estimate that there is an 80% chance that the prices, when the house is finished, will turn out to be higher than those of the initial contract. He actually needs to have such credences in order to take his decision. Now credences are not beliefs, but degrees of belief. But whatever our conception of the relationship between partial and full belief can be, the kind of acceptance involved here is evidentially warranted. It is basically an epistemic attitude and not simply a pragmatic one. In this sense, hypothetical acceptance presupposes the epistemic attitude of belief, and it cannot be non-evidentially warranted. If we could characterise the kind of hypothetical acceptance which underlies entitlement to an epistemic project independently from belief, and if we could grant that it can be entertained in the absence of belief, we could perhaps admit that we are dealing with a distinctive kind of entitlement or warrant. Strategic entitlement does this, but we just saw that it cannot help us with the sceptical challenge. So the very idea of a warrant 'for nothing', as Wright calls it, for no epistemic reason at all or for pragmatic reasons only, is indeed a kind of conceptual solecism. The main difficulty is still the same: we are looking for an epistemic entitlement, but characterise it in terms which are, by definition, either non epistemic or only partly epistemic. The strategy can work only if we can characterise a form of attitude which would have the appropriate features of warrant, which would be non-evidential but epistemic nonetheless. Is there such an attitude? Wright further suggests that the kind of entitlement he has in mind is entitlement to trust, and the form of acceptance he intends to highlight is a form of rational trust (2004: 194). I shall not here examine this notion, but it is clear enough that it raises the same sort of difficulties as those which I have raised for the notion of a pragmatic acceptance: is trust a doxastic attitude based at least in part on evidence and epistemic reasons, or is it a kind of faith, based on no reasons at all and driven by the desirability of its object? 10If the foregoing is correct, a propositional reading of hinge proposition, which would fall under the relation of entitlement, could not grant them a status of epistemic norms which, along with solution (b) above, could dominate the classical epistemic norms of type (ii). Rather it would be those epistemic norms themselves which dominate or govern our hinge commitments. EN would be more basic that GN. But this is clearly not the kind of result which a defender of hinge epistemology inspired by Wittgenstein would accept.

Some hinges are known

There is one kind of view of hinges which I have not yet examined. It is the view that hinge propositions are not rules or general presuppositions, but bona fide general propositions which we actually know, although a large part of this knowledge is tacit and not normally accessible to us. The kind of knowledge that we have of them is to be understood, unlike Wright's account of it in terms of entitlement, along externalist lines 11 . The idea would be that such propositions as 'I have hands', 'I have a body', 'The Earth has existed for a long time' are actual beliefs justified through reliable processes, although we are, most of the time, not conscious of them in any reflective or internalist way. We can, however, in some contexts, become conscious of them, if our attention is attracted to them. In my daily life, I do not even think about the fact that I have two hands. But if someone makes me remark that I have two handsor if a sceptic challenges my knowledge on this matter -I shall be able to access this proposition and to confirm that I know it. Indeed the context here is crucial, as Wittgenstein remarks (OC 125, OC 250): as I lie on a hospital bed and contemplate my hands in bandages, the question whether I have two hands makes straightforwardly empirical sense. But one may object here that precisely for this reason the hinge proposition 'I have hands' cannot be the same as the hospital proposition 'I have hands'. Their use is absolutely distinct. The first is an answer to a question which never arises. The second is an answer to a question which can arise. Compare with the kind of examples that are often given about a certain kind of tacit knowledge, which involves a disposition to assent to a certain proposition when our attention is attracted to it, but which was unnoticed before. You never thought that Kant's left earlobe is smaller than the Sea of Tranquility, nor that 'In girum imus nocte et consumimur igni' is a palindrome, but when one raises the issue, one is readily disposed to assent to this12 . But, from a Wittgensteinian perspective, one cannot say that these propositions were antecedently known to us, in a tacit way, and that these very propositions became explicit. For Wittgenstein would here say that when 'I realise' that In girum imus nocte et consumimur igni is a palindrome, I actually do not put to use an old proposition located somewhere in my mind (brain). I am actually giving it a use which it did not have before, because I was not used at all.13 I come to understand the proposition and thereby put it in my language game, but it was not previously part of my language game. So if hinge propositions and beliefs were like these tacit beliefs, they could not be known, for they would not be understood and would have no use. Indeed this is the gist of Wittgenstein's criticism of Moore: to say 'I know that I have hands' makes no sense.

Duncan Pritchard argues that the knowledge account of hinge propositions cannot work, since these cannot be said to be known through a reliable process: It doesn't seem at all plausible to suppose that we know hinge propositions in virtue of forming the target beliefs via a reliable belief forming process. Indeed, our hinge commitments do not seem to be the product of any specific kind of belief-forming process, but are rather part of the backdrop against which we acquire our beliefs in non-hinge propositions. For example, my hinge commitment to the universe having not come into existence 5min ago was not acquired via a specific cognitive process, but is rather something which is presupposed in the specific cognitive processes by which I come to acquire particular historical beliefs, such as that Napoleon's victory at Austerlitz was in 1805. (2012: 263) Pritchard is right that such general hinge propositions as 'The universe was not created 5 minutes ago' are hard to ascribe as things that people know rather than as cornerstones, pillars, or frameworks for other things that we know or can inquire about. But this does not entail that other general propositions cannot be known. Process reliabilism has a problem with 'generality': for propositions which seem to come from a very general competence, how is it possible to account for them on the basis of specific cognitive processes, and what is the degree of generality of such processes? 14 If reliabilism has this problem, this is a problem for any conception of this sort, and it does not show that any knowledge account of hinge propositions is wrong. Moreover, is it certain that hinges cannot give rise to any sort of knowledge? A number of writers inspired by Wittgenstein's view (in particular Moyal-Sharrock) have claimed that hinge propositions can be understood as kinds of practical knowledge or knowledge-how, consisting in specific abilities or capacities, giving rise to certain kinds of reflexes. To have an attitude of basic certainty is like a way of acting or know-how or reflex action (like grabbing a towel from the towel rack without thinking). Indeed, on such a view, the bits of knowledge in question cannot be propositional, and a strong separation has to be maintained between knowledge how and knowledge that 15 .

So there is room for a knowledge account of hinge propositions. I can only outline here the shape that such an account could take. The first step to make is to come back to the idea that there are various kinds of hinges. Moyal-Sharrock (2004: 102) gives a useful classification: there are linguistic hinges ('Which colour is meant by the word blue'), personal hinges ('I come from such and such a city', 'For months I have lived at address A'), local hinges ('There is an island, Australia', 'No one was ever on the moon'') and universal hinges ('The earth exists', 'There are physical objects', 'Things don't systematically disappear when we're not looking', 'If someone's head is cut off, the person will be dead and not live again', 'Trees do not gradually change into men and men into trees', 'I have a brain', 'I am a human being'). In spite of the fact that they are all characterized as general presuppositions 16 immune to doubt, it seems clear that their statuses differ. Although it is correct to say that a reliabilist account of very general hinge propositions such as 'I live on Earth' or 'The universe has not come not existence 5 minutes ago' is dubious, is such an account implausible for more specific or local hinge certainties? Local and personal hinge certainties seem at least in part based on memory, and memory is a (generally) reliable process. Even very general hinges such as 'There are physical objects' can be accounted for in terms of reliable process. A whole field of cognitive psychology discusses our 'core knowledge' about physical objects and the properties of or 'naïve physics'. Developmental psychologist claim that our concepts of a physical object are known very early in life, and may be in large part innate (see e.g. [START_REF] Carey | The Origins of Concepts[END_REF]). Indeed such a proposition as 'There are physical objects' seems to be so general that its meaning exceeds what a psychological story about its acquisition could tell us. But why should we deny that we grasp the meaning of such a proposition on the basis of some form of innate or acquired knowledge of perceived objects?

A second consideration in favour of a knowledge account of hinges concerns those propositions which Wittgenstein clearly takes as rules and to which he devotes a large amount of his writings, the basic propositions of logic and mathematics. Indeed very basic logical principles, such as the 'law of contradiction' or Excluded Middle are good candidates for being hinges. But do they have the universal character which universal hinges such as 'The Earth exists' enjoy, in the sense of their being immune to doubt? Non-classical logicians, such as dialethists and intuitionists question their universal validity, and reject classical principles such as the equivalence of double negation and an affirmation. They do not question it in terms of their being less 'hingy' than the classical principles, but in terms of their truth and consistency. It would be strange indeed to describe the conflict between classical and non-classical logicians 15 One line of argument, which someone who rejects the idea that hinge propositions can be known propositionally will resist, consists, in the manner of Stanley and Williamson 2000, in arguing that knowledge how reduces to knowledge that. If this line is correct, the knowledge account of hinges would be vindicated. But these issues are moot, and I cannot address them here. 16 Though Moyal-Sharrock insists they should not even be characterized as presuppositions in that they are not propositions (2004,55,65).

as a conflict about hinge propositions. Typically intuitionists or dialethists object to classicists about their views on truth and on inference. They do not oppose classical norms for logic to non-classical norms by arguing that theirs have a better claim to be hinges, but by evaluating their truth and consistency. So the debate about the principles of logic and mathematics is not a debate between two 'forms of life' or different 'language games' for logic, but a debate which arises at the level of epistemic norms and principles. It is not a conflict between two kinds of grammatical normssay those which are classical and those which are non-classicalbut a conflict within the domain of epistemic norms for logic. If this is so, this means that logical principles are outside the scope of our problem, which concerns the potential conflict between two kinds or norms.

If we now consider logical rules which are, by logical standards, less fundamental than the law of non-contradiction or the excluded middle, but which nevertheless are in some sense basic, such as the modus ponens ((P→Q

) & P)→ Q) disjunctive syllogism ((P v Q) & not P) → Q))
, or Peirce's Law ((P→Q)→P)→P)), we can also ask in what sense they are hinge propositions which escape the status of being known. Consider the modus ponens as it is stageset in Carroll's famous story 'What the Tortoise said to Achilles' [START_REF] Carroll | What the Tortoise said to Achilles[END_REF]. The Tortoise is presented a simple inference in modus ponens form, with two propositions 'P' and 'if P then Q', but refuses to draw the conclusion Q, although he accepts the proposition that ((P→Q) & P) → Q). Besides the obvious mistake that the Tortoise makes in confusing premises and rules of inference, it seems clear that it does not understand that the modus ponens is a rule rather than a law or principle which has to be true. Wittgensteinians here claim that the modus ponens is a rule which is neither true nor false, but which has to be taken for granted as a cornerstone or hinge without which a lot of ordinary inferences do not make sense. Carroll's story also seems to go against any interpretation of the rule of modus ponens as a proposition. The very regress which arises when one takes it as a proposition that one needs to consider before applying it seems to entail that it is a mistake, typical of an intellectualist conception of rules, to consider that following a rule involves the explicit and reflective consideration of a proposition expressing it. On the contrary, Wittgenstein's reader wants to argue, we follow rules, including those of logic, 'blindly', without consulting the deliverances of the space of reasons. Is this so clear?

Here is not the place to assess the lessons of Carroll's paradox 17 , but there is room for quite different readings of Carroll's story. One reading consists in arguing, in the style of the externalist reading of hinge propositions, that logical rules are based on rather specific pieces of knowledge, which may be innate or acquired, and which account for our readiness to use them. If they are innate, they cannot lend themselves to doubt, and belong to our cognitive equipment. A large amount of cognitive psychology of reasoning consists in trying to understand this kind of logical knowledge. This would account for many of the features of hinges, except that they have more local character than universal hinges. They would nevertheless be based on specific reliable causal processes, which is enough, on a reliabilist conception of knowledge, for claiming that they are, at least in a large part, known. 18 The second 17 For an overview, see Engel (Forthcoming). 18 The claim that rules of logic are based on some form of tacit knowledge is not meant to imply anything about the problem of the justification of logical rules or of logic in general. A partisan of hinge epistemology might claim that the justification of a particular logical rule, say modus ponens, presupposes the general validity of the reading consists in taking these basic logical rules as norms for our reasoning practices. On this understanding of rules, one can ask how logic can move the mind [START_REF] Blackburn | Practical Tortoise Raising[END_REF]. A Humean naturalist answers: it cannot move us, unless there is a passion, feeling or natural fact about our psychology, which intervenes. A Kantian, or a Platonist about rules answers that normative reasons can by themselves move, through some sort of rational power. Clearly the Wittgensteinian has more sympathy for the former than for the latter. The issue is a deep one about the nature of rules and reasons. But here again, the very fact that there can be genuine, and not only spurious, disagreements about which logical rules to followwhether we have to accept modus ponens or disjunctive syllogisms as rules of logicshow that the issue is not simply one of arbitrating what belongs to our nonrational nature and what belongs to our rational nature. It is a conflict not between what is outside the space of reasons and what is inside it, but a conflict which lies within the space of reasons: which logical principles are true, justified and which are not.

Such considerations certainly do not show that a knowledge account of hinge propositions is correct. But at least they indicate that it is not wholly implausible.

Conclusion: cozy epistemology

I have raised a challenge for those who claim that hinge propositions are normative or can play the role of epistemic norms: if they are such, they would have to compete with classical epistemic norms such as the norm of truth or the norm of knowledge for belief, or other normative requirements. But they cannot compete with these, for they are not norms in the same sense. Either they arein particular when they are universal hingeslocated so much below or antecedent to our belief systems that they play no epistemic role at all; or, if they are more local hingessuch as logical rulesthey are also candidates for being propositions that are known, hence that stand within the domain of knowledge rather than outside of it. Either way, they cannot be epistemic norms in the usual sense. Either grammatical rules are not normative, or they are not normative for epistemology. If this is correct, the realm of hinge epistemology is unfathomable, and its pillars, cornerstones and yardsticks are wild. So why not stay within our cozy epistemological house? There is enough work to do to tidy up our epistemological home, to have to consider the wild creatures standing outside it. laws of logic, which are thus hinges for our logical thinking. Whether or not this circularity of justification holds, this question is independent from the claim, advanced here that logical rules are known. This claim is not meant to give an answer to sceptics about the justification of logic.

For an overview see Engel

See Strevens 2003.

For this reading, seeEngel 2013Engel , 2013 a. a.

For this distinction, see[START_REF] Thomson | Normativity[END_REF]

Some of them are rather called, in Broome's sense[START_REF] Broome | Normative Requirements[END_REF], 'rational requirements'. Broome opposes these to reasons.

See e.g[START_REF] Bykvist | Does thought imply ought?[END_REF][START_REF] Wedgwood | The Aim of Belief[END_REF][START_REF] Chan | The Aim of Belief[END_REF], Gibbons 2013.

This is the account defended by, among others,[START_REF] Boghossian | The Normativity of Content[END_REF], Wedgwood 2007, Engel 2013, 2013a.

On this, see Schönbaumsfeld in this issue.

For different notions of acceptance see e.g,[START_REF] Engel | Believing and Accepting[END_REF]

On some views of the notion of hinge, such as Moyal-Sharrock's, hinge propositions are not the object if beliefs, but of trust, understood as a form of belief-in.

[START_REF] Pritchard | Wittgenstein and the Groundlessness of our believing[END_REF] gives an interesting taxonomy of alternative readings of hinge propositions along internalist and externalist lines, depending upon how they fare with respect to the apparently obvious principle that their knowledge ought to transmit to further propositions that one can deduce from them. But I cannot deal with this here.

See on this classical issue e.g.[START_REF] Lycan | Tacit Belief[END_REF], Engel 2005.

Compare with Malcolm views about dreaming: dreams are events that we recall, so they cannot be brain events.

See e.g.[START_REF] Conee | Evidentialism[END_REF] on the generality problem for reliabilism.


Anne Marie Dupuy

Grégoire Pasquier

Louis Thiebaut

François Roubille

Mustapha Sebbane

Jean-Paul Cristol

Additive value of bioclinical risk scores to high sensitivity troponins-only strategy in acute coronary syndrome

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Florence Jacquey

Kevin Loquin

Frédéric Comby

Olivier Strauss

NON-ADDITIVE APPROACH FOR GRADIENT-BASED EDGE DETECTION

Keywords: Edge detection, Choquet integral, capacity, summative and non-summative kernels, possibility, fuzzy numbers

In this paper, we propose a new method to perform the first derivative estimation of a discrete intensity distribution. This approach is based on a non-additive aggregation process and provides an estimate of the gradient as intervals instead of single values. These intervals are used to threshold a gradient based edge detection and therefore discard spurious detections due to noise.

INTRODUCTION

Edges in images are areas with strong intensity contrasts. Gradient based edge detection consists of detecting maxima and minima in the image first derivative. However, edges are not the only features maximizing (or minimizing) this first derivative. Acquisition (or reconstruction) noise, digitalization and spurious local textures induce undesirable discontinuities in the intensity function.

Computation of the first derivative, in digital images, relies on the concept of fitting a continuous surface locally to the data. One of the most advanced approaches makes use of summative kernels whose role is to define a weighted neighborhood of each sampled location. This neighborhood provides an interplay between continuous and discrete. It also aims at using a hypothesized ergodicity to reduce the effect of random noise by providing regularization in the interpolation process.

Noise reduction and edge detection lead to antagonist criteria. The Canny operator was designed to be an optimal edge detector according to particular criteria. However there are other detectors around that also claim to be optimal with respect to slightly different criteria.

Estimation of the derivative of the continuous signal s resulting from the interpolation of discrete signal S via the summative kernel κ can easily be obtained by convolving S with the first derivative dκ of kernel κ. Most of the usual approaches lead to a derivative kernel that can be obtained by subtracting two usual kernels:

dκ(x) = η + (x + x ) -η -(x -x ). (1)
Even when a supposed optimal filter is used, noisy images produce spurious discontinuities that have to be removed by a tracking process involving an arbitrary threshold depending on the image noise. The threshold arbitrariness reduces the detection robustness. Indeed, the edge detection highly relies on the appropriateness of the hypothesized optimal threshold, especially with tomographic reconstructed images whose noise is usually assumed to be Poisson distributed.

In this paper, we present a method producing a natural threshold estimation, depending on local noise level, at each sampled location. The underlying idea of this work is the following: when estimating a piece of information from a set of noisy data with a kernel-based approach, variations due to measurement errors can be estimated by slightly varying the shape (but not the size) of the kernel. Such a variation can be simply achieved by replacing the conventional summative kernel approach by a non-summative kernel approach. We come up to a gradient estimator that produces intervals instead of single values. The length of each interval is proportional to the local noise level. A location is declared not being an edge if the value 0 belongs to the estimated interval.

SUMMATIVE AND NON-SUMMATIVE KERNEL

Summative or non-summative kernels are functions that can be thought as weighted neighborhoods.

Summative kernels are IR-valued functions κ defined on a domain Ω, satisfying the summative normalization property Ω κ(ω)dω = 1.

(2)

In image processing they are usually positive and unimodal as in the rest of this paper. Positivity implies that they can be seen as probability density functions, with whom we can associate, for any A ⊆ Ω, a probability measure

P κ (A) = A κ(ω)dω.
Estimation of a given signal S in a given summative or probabilistic neighborhood κ is the expectation of this signal according to κ. That is:

E κ (S) = Ω SdP κ = Ω S(ω)κ(ω)dω. (3)
Non-summative kernels are [0, 1]-valued functions π defined on a domain Ω, satisfying the maximitive normalization property max ω∈Ω π(ω) = 1.

(4)

They can be seen as possibility distribution functions [START_REF] Dubois | Possibility Theory An Approach to Computerized Processing of Uncertainty[END_REF], with whom we can associate, for any A ⊆ Ω, a possibility measure Π π (A) = sup ω∈A π(ω). A possibility measure is a special case of concave Choquet capacity ν. The conjugate ν c of such a capacity, defined by ∀A ⊆ Ω, ν c (A) = 1 -ν(A c ) is a convex capacity. The conjugate N π of a possibility measure Π π is a necessity measure.

Those two kinds of neighborhood definitions (summative and non-summative) are linked by the fact that a concave capacity ν can encode a special family of probability measures (or equivalently additive measures), noted core(ν c ) and defined by

core(ν c ) = {P κ | ∀A ⊆ Ω, ν c (A) ≤ P κ (A) ≤ ν(A)}. (5)
Now when the summative neighborhood is replaced by a capacity, estimation (3) is replaced by an estimation involving the Choquet integral [2, 3]

C ν (S) = (C) Ω Sdν = +∞ 0 ν({ω : S(ω) > α})dα. (6)
The encoding property of a capacity ν (5) is transferred to the estimation as stressed by these equations:

C ν c (S) = inf κ|Pκ∈core(ν c ) E κ (S), (7)
C ν (S) = sup κ|Pκ∈core(ν c ) E κ (S). (8)
See [4] proposition 3 for a proof of (7) and [2] proposition 10.3 for a proof of (8).

As mentioned above, a possibility measure is a special case of concave capacity. A transformation from probability to possibility which, for any summative kernel κ, results in a non-summative kernel π satisfying the consistency (or domination) principle: ∀A ⊆ Ω, P κ (A) ≤ Π π (A), induced by (5), is proposed by Dubois et al. in [5].

Any summative kernel κ, which is centered, bounded and symmetric on the interval [-∆, ∆] satisfies:

∀A ⊆ Ω, P κ (A) ≤ Π T (A), (9)
with T the triangular possibility distribution on [-∆, ∆]. This justifies our choice for the use of the triangular possibility distribution for the interplay between continuous and discrete.

A last point is the extension to higher dimensions of possibility measures domination (for instance triangular) over probability measures. For the dimension of 2, on a space Ω 1 × Ω 2 , a product of marginal possibility measures Π 1 and Π 2 is defined in [START_REF] Dubois | Possibility Theory An Approach to Computerized Processing of Uncertainty[END_REF] by

∀(A, B) ⊆ Ω 1 × Ω 2 , Π 12 (A, B) = min(Π 1 (A), Π 2 (B)).
Suppose that marginally, Π 1 ≥ P 1 and Π 2 ≥ P 2 , with P 1 and P 2 probability measures on, respectively, Ω 1 and Ω 2 . Then Π 12 dominates P 12 defined by

∀(A, B) ⊆ Ω 1 × Ω 2 , P 12 (A, B) = P 1 (A)P 2 (B).
min(Π 1 (A), Π 2 (B)) ≥ min(P 1 (A), P 2 (B)) ≥ P 1 (A)P 2 (B), because P ≤ 1. Therefore, by definitions of P 12 and Π 12 , Π 12 (A, B) ≥ P 12 (A, B).

DERIVATION WITH NON-SUMMATIVE KERNEL

As presented in section 1, most of the usual derivative kernels can be obtained by subtracting two summative kernels η + and η -. Convolving signal S with the derivative kernel given by expression ( 1) is equivalent to summing up the convolution of S with the kernel η + and the convolution of the opposite of S with kernel η -:

DS(x) = (η + (x + x ) -η -(x -x )) * S(x) = η + (x + x ) * S(x) + η -(x -x ) * (-S(x)) Therefore let κ + (x) = η + (x + x ) and κ -(x) = η -(x -x ).
The estimate of DS(x), derivative of S at location x, is given by: While the signal to be processed being discrete, a kernel has to ensure the interplay between continuous and discrete. In our approach, this interplay makes use of triangular fuzzy numbers. The continuous signal is supposed to be known on imprecise locations defined by the fuzzy partition (H i ) i∈{1,...,N } depicted on Figure [START_REF] Dubois | Possibility Theory An Approach to Computerized Processing of Uncertainty[END_REF]. We aim at transferring the discrete knowledge of the signal to any weighted neighborhood W out of the partition. This transfer is ensured by using a Choquet integral. The signal being discrete, expression (6) can be simply computed by: S). We propose to use the pignistic transfer to define the capacity used in Equation ( 11). This capacity is concave and its expression is:

DS(x) = E κ + (S) + E κ -(-S) (10)
C ν W (S) = N n=1 S (n) [ν W (A (n) ) -ν W (A (n+1) )], (11
ν W (A (n) ) = |W A (n) | |W A (1) | . ( 12
)
Expression (10) leads to an imprecise estimation of the first derivative of S:

[DS, DS] = [-C ν + W (-S), C ν + W (S)] + [-C ν - W (S), C ν - W (-S)] = [-C ν + W (-S) -C ν - W (S), C ν + W (S) + C ν - W (-S)].
Since the capacity ν W is concave, this interval contains all estimates of the derivative provided by a probability density function dominated by ν W (i.e. all probability density functions defined by a kernel dominated by the possibility distribution defined by W ). While the length of this interval increases with the noise level, this derivative can be supposed to be null if 0 ∈ [DS, DS].

EDGE DETECTION WITH NON-SUMMATIVE KERNEL BASED GRADIENT

The discrete image representation introduces an imprecision on the grey level localization. Let P i,j be the pixel located at (i, j) on the image. Its grey level localization is unknown within the 2D interval

[i -∆ h , i + ∆ h ] × [j -∆ v , j + ∆ v ],
where ∆ h (resp. ∆ v ) is half the width of the horizontal (resp. vertical) sampling. This imprecision is due to the spatial sampling and can be modeled by using a fuzzy partition [6]. Each pixel of the image is considered as a bi-dimensional imprecise quantity. The 1-D triangular fuzzy numbers presented in section 2 are extended in 2D pyramidal fuzzy numbers (Figure 2(a)) considering the t-norm min (cartesian product of two 1-D triangular fuzzy numbers), providing a strong fuzzy partition of the image. As shown in section 2, the domination property still holds. Therefore, the properties presented in section 3 can be easily extended to provide an imprecise estimation of the gradient. We present this approach with the x-derivative. Let κ(x, y) = κ x (x)κ y (y) be a separable kernel used to perform the derivation, the 2D extension of expression (1) is given by: Therefore, an estimate of the derivative of the 2D signal S along the x-axis (GS x ) can be given by simply adding the convolution of S with the summative kernel κ +

δ δx (κ(x, y)) = δ δx (κ x (x)κ y (y)) = δ δx (κ x (x))κ y (y) = (η + (x + x ) -η -(x -x ))κ y (y).
x to the convolution of (-S) with the summative kernel κ -

x :

GS x = E κ + x (S) + E κ - x (-S). (13)
If ν + x (resp. ν - x ) is a concave capacity that dominates the kernel κ +

x (resp. κ - x ) then the domination can be easily transferred to the estimation process by using the Choquet integral (11). Then, the imprecise estimation of GS x is a simple 2D extension of the approach presented in section 3. It involves two pyramidal non-summative kernels depicted on Figure 2(b) dominating a family of kernels κ +

x and κ - x . Imprecise estimation of GS y is achieved in the same way.

Within this approach, the edge detection is performed on the median estimate of the gradient [GS x , GS x ]×[GS y , GS y ]. An extremum of this function is discarded if

(0, 0) ∈ [GS x , GS x ] × [GS y , GS y ].

EXPERIMENTAL RESULTS

In this section, we compare our approach with the Canny-Deriche [7], Shen-Castan [8] and Prewitt approaches. This comparison uses the artificial image depicted in Figure 3(a). This image has been corrupted with two kinds of noise. We have applied first a Gaussian noise with an increasing variance σ (Figure 3(b)) to test the robustness of the edge detection to the noise level and then a Poisson noise to highlight the robustness of the method when the noise level varies.

The operators performances are quantified by the parameter P 1 defined by Fram and Deutsch in [9]. P 1 measures the sensitivity of the detector in presence of noise. P 1 = 1 if the detecting of edges is optimal.

In a first experiment, edges are detected without any threshold. Figure 4(a) shows that all approaches except ours fail when σ > 3. In a second approach, we have applied an arbitrary threshold chosen, for each method, to provide the best result within the P 1 -criterion. When edge detection is applied on tomographic images, the threshold based approach usually fails because the noise is not uniformly distributed in the image. The noise in reconstructed tomographic images is usually assumed to be Poisson distributed . In the last experiment, we have corrupted the test image with a Poisson noise. Results of this experiment are reported in Table 1. It shows that the fuzzy approach is 8 times less sensitive than the Prewitt approach and 5 times less than the Canny-Deriche and Shen-Castan approaches.

CONCLUSIONS

In this paper, we have presented a new approach for gradientbased edge detection. This approach involves non-summative kernels and Choquet integrals and produces, as output, intervalvalued estimation of the gradient. When used to detect edges on a noisy image, our approach, contrarily to classical methods, does not need an arbitrary threshold to discard spurious edges, while the threshold is naturally provided by the gradient estimation. Tests on synthetic images have illustrated the good performances of this approach with respect to state of art methods. Tests on natural images have been performed but not reported here, while the applications of the edge detection cannot be measured by objective criteria.

  If κ + (resp. κ -) is dominated by a concave capacity ν + (resp. ν -) then, the valueE κ + (S) (resp. E κ -(-S)) belongs to [C ν +c (S), C ν + (S)] (resp. [C ν -c (-S), C ν -(-S)]). If κ + (resp. κ -)is a bounded symmetric positive kernel, then the triangular possibility distribution can be used to ensure this domination (see section 2).

Fig. 1 .

 1 Fig. 1. Strong triangular partition of the real line and fuzzy neighborhood W .

  ) where S n is the real positive value associated with the n th cell H n and (.) indicates a permutation such that S (1) ≤ S (2) ≤ ... ≤ S (N ) . The A (n) = {H (n) , ..., H (N) } are binary coalitions of cells whose associated values are superior or equal to S (n) . Computation of C ν c W (S) can be simply achieved while noticing that C ν c W (S) = -C ν W (-

Fig. 2 .

 2 Fig. 2. (a) Fuzzy pixel, (b) 2D non-summative kernels

  Figure 4(b) shows that

Fig. 3 .Fig. 4 .

 34 Fig. 3. (a) Original image (b) with a Gaussian noise σ = 25 and (c) with a Poisson noise

Table 1 .

 1 P 1 values on the Poisson noised image

	Approch Prewitt	Deriche	Shen	Fuzzy
	P1	0.089676 0.14975	0.141	0.69881