Jean-François Bonnefon
email: bonnefon@univ-tlse2.fr.

Pragmatics, Mental Models, and one Paradox of the Material Conditional

Most instantiations of the inference 'y; so if x, y' seem intuitively odd, a phenomenon known as one of the paradoxes of the material conditional. A common explanation of the oddity, endorsed by Mental Model theory, is based on the intuition that the conclusion of the inference throws away semantic information.

We build on this explanation to identify two joint conditions under which the inference becomes acceptable: (a) the truth of x has bearings on the relevance of asserting y; and (b) the speaker can reasonably be expected not to be in a position to assume that x is false. We show that this dual pragmatic criterion makes accurate predictions, and contrast it with the criterion defined by the mental model theory of conditionals, which we show to be inadequate. 1 p 2 Models Pragmatic 1 0 0 0 No No ? I took an aspirin; so, if I had a headache, I took an aspirin. 2 0 0 1 No No ? Nate passed the exam; so, if the exam was easy, Nate passed the exam. 3 0 1 0 No No ? This child is crying; so, if she's ours, this child is crying. 4 0 1 1 No Yes Amy dyed her hair red; so if you've not seen her yet, she dyed her hair red.

5

Introduction

One well-known 'paradox' of the material conditional consists of accepting that 'if x, y' follows from y by a valid rule of inference. This inference is deemed paradoxical because most of its instantiations in natural language are intuitively odd. For example:

(1) ?Today is Monday; so, if it is snowing outside, today is Monday. 1 Importantly, though, other instantiations of the inference appear intuitively acceptable, for example:

(2) Viv did not play soccer; so, if she played a game, Viv did not play soccer.

This fact is given prominent importance in one major psycholinguistic theory of conditionals, mental model theory [START_REF] Johnson-Laird | Conditionals: A theory of meaning, pragmatics, and inference[END_REF]. One critical argument commonly advanced in favor of the mental model theory of conditionals is its ability to predict which instantiations of the paradoxical inference seem 1 All through this article, we will use the formulation 'if x, y' for conditionals, and the formulation 'y; so if x, y' for the paradoxical inference. Natural language offers a number of other possible formulations of the conditional. Some of these formulations strengthen or weaken the paradox (e.g., 'if x then y' and 'even if x, y', respectively) to the point where it becomes impractical to generate acceptable or unacceptable examples, respectively. We settle on the formulation that apparently offers the greatest flexibility for the purpose of generating acceptable and unacceptable examples. We content ourselves with noting that the linguistic formulation of the conditional is a factor in the acceptability of the paradoxical inference, whose interaction with our variables of interest is left for future research.

acceptable, and why. The mental model account of the paradox remains controversial, though [START_REF] Johnson-Laird | The mental model theory of conditionals: A reply to Guy Politzer[END_REF][START_REF] Politzer | Reasoning with conditionals[END_REF][START_REF] Politzer | Could it be the case that if I am right my opponents will be pleased? A rejoinder to Johnson-Laird, Byrne and Girotto[END_REF], and is still waiting for a systematic assessment. In this article, we contrast the mental model account of the paradox to our own pragmatic account. We show that the mental model criterion is neither necessary nor sufficient to predict the acceptability of the paradoxical inference, which is adequately predicted by our dual pragmatic criterion. 2 While we focus on mental model theory in the course of the article, we will also discuss the contribution of our analysis to other psychological theories of natural language conditionals, which have more drastically distanced themselves from interpreting the conditional as a material implication.

In Section 2, we introduce the mental model account and its criterion for the acceptability of the paradoxical inference, which we express as a binary condition that we call condition m. In Section 3, we introduce the fundamental hypothesis underlying our pragmatic account. In a nutshell, we argue that a conditional sen- 2 The paradox that we address in this article is usually presented alongside with a second paradox, that of inferring 'if x, y' from '¬x'. We will not address this second paradox in the present article. We believe that the acceptability of these paradoxical inferences is determined by pragmatic, conversational principles; and that the conversational principles that govern the acceptability of the first inference do not necessarily govern the acceptability of the second. Since we do not believe in a single, unified theory of the acceptability of both paradoxical inferences, we do not feel compelled to offer one. Mental model theory does offer a unified account of the two paradoxes, but this unified account is necessarily incorrect if, as we intend to show, it fails on at least one of the paradoxes. Finally, we note that although it offers a unified theory of both paradoxes, mental model theory restricts itself to the first paradox when it comes to offering examples of acceptable instantiations.

tence 'if x, y' uttered in the context of y, or asserted as following from y, seems odd when the speaker is asserting that y is unconditionally true, and then that the truth of x has bearings on the truth of y. The paradox should disappear, then, when the conditional does not express that the truth of x has bearings on the truth of y, but rather that the truth of x has bearings on the relevance of asserting y (where relevance can derive from epistemic utility or practical utility).

This hypothesis leads us to predict the dissolution of the paradox from a dual criterion that we express under the form of two binary conditions p 1 and p 2 . In Section 4, we examine the eight possible cases that correspond to the satisfaction (or non-satisfaction) of the binary conditions m, p 1 , and p 2 . We show that the paradox disappears only when p 1 and p 2 are both satisfied (including the case where m is not satisfied), and remains in all other cases (including the cases where m is satisfied).

Before we proceed, we wish to exclude one specific case from our analyses, that of when y is known to be false. It is correct that the inference from ⊥ to 'if

x then ⊥' (where ⊥ denotes the contradiction) is both logically valid and intuitively odd, and could thus be considered as a regular case of the paradox. The logical validity of this inference, though, is unrelated to the interpretation of the conditional, and is rather due to the ex falso quodlibet principle of classical logic.

Anything follows validly from ⊥, including indeed 'if x then ⊥.' In this article, we are interested in a paradox that arises from the interpretation of the conditional, and not in the paradoxes that arise from the ex falso quodlibet principle; and we will accordingly discard that case from our subsequent analyses.

The Mental Model Account

The mental model theory of conditionals [START_REF] Johnson-Laird | Deduction[END_REF][START_REF] Johnson-Laird | Conditionals: A theory of meaning, pragmatics, and inference[END_REF] assumes that the paradoxical inference is counterintuitive because it throws away semantic information: That is, the conditional 'If x, y' introduces a possibility that was ruled out by the original premise y. Consider for example the premise 'Today is Monday.' From this premise, it would seem anomalous to infer that 'If we have pasta for lunch today is Monday':

(3) ?Today is Monday; so, if we have pasta for lunch, today is Monday.

According to the mental model theory, the conditional 'If we have pasta for lunch then today is Monday' refers to three possibilities: Let us now illustrate the workings of condition m with Conditional (4):

pasta
(4) If Viv played a game, Viv did not play soccer.

The model < ¬game, soccer > cannot be part of the representation of (4), because it is semantically inconsistent: one cannot play soccer without playing a game. Therefore, the models of (4) are:

game ¬soccer ¬game ¬soccer

Because none of these models contains soccer, the inference (5) should not seem anomalous:

(5) Viv did not play soccer; so, if Viv played a game, Viv did not play soccer.

And indeed, (5) does not sound as odd as (3). The fact that the mental model account could generate an acceptable instance of the paradox is an impressive feat, which speaks powerfully for the account. In the rest of this article, though, we are going to claim that the mental model account of the paradox is incorrect.

We will show that condition m is neither necessary nor sufficient to make 'y; so if x, y' an intuitively acceptable inference; and we will offer as a replacement a pragmatic account which will predict when and why the paradox disappears.

The Pragmatic Account

The fundamental idea behind the mental model account is that it is a waste of information to derive the less informative conclusion 'If x, y' from the more informative premise y. This idea concurs with our basic hypothesis presented in the introduction. Pragmatically speaking, the inclusion of the conditional in ( 6) is bound to be a waste of the listener's time and attention:

(6) ?I had bacon this morning; so, if I had juice, I had bacon this morning.

In the language of relevance theory [START_REF] Sperber | Relevance, communication and cognition[END_REF][START_REF] Edgington | On conditionals[END_REF], the second part of the statement requires additional processing effort that does not deliver additional cognitive effects. 5 Generally speaking, when y is already asserted as unconditionally true, no additional information is communicated by asserting that the truth of x has bearings on the truth of y.

It is not always the case, though, that a statement of the form 'If x, y' expresses that the truth of x has bearings on the truth of y. Some conditionals express instead that the truth of x has bearings on the relevance of the assertion that y. This is typically the case in 'biscuit' conditionals [START_REF] Austin | Ifs and cans[END_REF][START_REF] Derose | Conditional assertions and "biscuit" conditionals[END_REF][START_REF] Siegel | Biscuit conditionals: Quantification over potential literal acts[END_REF]):

(7) a. There are biscuits on the sideboard if you want them;

b. If you are thirsty now, there are beers in the fridge;

c. If you need help, my name is Bob.

Clearly, Bob does not mean by (7-c) that the listener's need for help has bearings on whether his own name is Bob. What Bob means is rather that whether the listener needs help has bearings on the relevance of asserting that his own name is Bob. Similarly, the speaker who is uttering (7-b) means that whether the listener is thirsty has bearings on whether it would be relevant to assert that there are beers in the fridge. Note now that the felicity of these interpretations requires that the speaker is not in a position to assume that the antecedent of the conditional is false. This condition accounts for the difference between (8-a) and (8-b):

(8) a. If you are thirsty now, there are beers in the fridge;

b. ?If I am thirsty now, there are beers in the fridge.

We have claimed so far that a conditional 'If x, then y' does not always express that the truth of x has bearings on the truth of y, as some conditionals express instead that the truth of x has bearings on the relevance of asserting y. We now note that the relevance of asserting y can derive either from the basic epistemic utility of learning that y, or additionally from the practical utility of knowing that y.

In example (7-a-c), the relevance of asserting y derives from the practical utility this information has for the listener. Indicating the whereabouts of the beers, for example, would have practical utility for a listener whose goal is to quench her thirst. Accordingly, (7-b) expresses that the listener's thirst has bearings on the practical relevance of asserting that there are beers in the fridge. In other cases, the relevance of asserting y derives from the mere epistemic utility of learning that y. This happens when there is no immediately manifest goal of the listener that would be served by knowing that y, bar the generic goal of augmenting one's knowledge about the world. Consider for example:

(9) If you've not seen Amy yet, she's dyed her hair red.

Conditional (9) does not, of course, express that whether the listener has seen Amy has bearings on the fact that Amy dyed her hair red. It rather expresses that whether the listener has seen Amy already, has bearings on the relevance of asserting that Amy dyed her hair red. Indicating that Amy has dyed her hair red would have epistemic utility for a listener who would not already be in a possession of this piece of knowledge, typically because this listener would not have seen Amy already. 6Once again, the felicity of this interpretation would be compromised if the speaker was in a position to assume that the antecedent of the conditional is false, which accounts for the difference between (10-a) and (10-b):

(10) a. If you've not seen Amy yet, she's dyed her hair red;

b. ?If we've not seen Amy yet, she's dyed her hair red.

We are now in a position to get back to the paradox. We have remarked that the statement 'y; so if x, y' is pragmatically incongruous when it expresses that the truth of x has bearings on the truth of y. Exploiting the remarkable case of biscuit conditionals, we have pointed out, however, that the conditional 'If x, y' can have pragmatic import over and above the assertion that y, when it expresses that the truth of x has bearings on the relevance of asserting y. In line with Gricean and neo-Gricean pragmatics, we have shown further that this pragmatic import can be obtained when the relevance of asserting y derives from its epistemic utility for the listener, or from its practical utility for the listener; and that it requires that the speaker can plausibly ignore whether x is false. We suggest that the statement 'y; so if x, y' is pragmatically acceptable when these conditions are met, and we thus suggest replacing condition m with conditions p 1 and p 2 :

Definition 2 (Conditions p 1 and p 2 ). The inference 'y; so if x, y' is intuitively acceptable if and only if (p 1 ) the truth of x has bearings on the relevance of asserting y ; and (p 2 ) the speaker can reasonably be expected not to be in a position antecedents appropriate. For example, 'If you placed a bet on Amy's new hair colour, she's dyed her hair red.'

to assume that x is false.

Teasing Out the Two Accounts

In order to tease out the mental model account and the pragmatic account, we must review the eight cases defined by the satisfaction or violation of conditions m, p 1 , and p 2 . As displayed in Table 1, there are four cases wherein the two accounts make similar predictions, two critical cases wherein they make different predictions, and two special cases that cannot be constructed due to the logical relation between condition m and condition p 1 , which we will address later on.

Cases of Agreement

When all three conditions m, p 1 , and p 2 are met, both accounts naturally predict that the statement 'y; so if x, y' sounds acceptable. This is Case 8 of tentatively add 'at all' to the antecedent of the conditional, as in ( 12):

(12) Viv did not play soccer; so, if she played a game at all, Viv did not play soccer.

Finally, (12) gives no indication that the speaker might be in a position to assume that Viv did not play a game, thus satisfying condition p 2 .

So far, we have only addressed one case of agreement between the two accounts, wherein they both predict the paradox to vanish. Table 1 also displays three cases wherein both accounts predict that 'y; so if x, y' is intuitively incongruous. Case 1 denotes situations that violate all three conditions m, p 1 , and p 2 , as in ( 13):

(13) ?I took an aspirin; so, if I had a headache, I took an aspirin.

Condition m is violated because the possibility < ¬headache, ¬aspirin > is entirely consistent with common knowledge. Condition p 1 is violated because the informational or practical value of asserting 'I took an aspirin' does not depend in any immediately salient way on the fact that I had a headache. Condition p 2 is violated because I, the speaker, am in a position to know for a fact whether I did not have a headache prior to taking the aspirin.

Case 2 denotes situations wherein condition m and p 1 are violated, although condition p 2 is satisfied, as in ( 14), where nothing indicates that the speaker might assume that the exam was in fact hard:

(14) ?Nate passed the exam; so, if the exam was easy, Nate passed the exam.

Finally, Case 3 denotes situations where condition m and p 2 are violated, although condition p 1 is satisfied, as in ( 15):

(15) ?This child is crying; so, if she's ours, this child is crying.

Condition m is violated because it is clearly possible that a child who is not that of the speaker and the listener is not crying. Condition p 1 is met because whether the child is that of the listener clearly has bearings on the practical relevance of asserting that the child is crying: The listener plausibly has the general goal of monitoring the well-being of her child, but not as plausibly that of monitoring the well-being of other parents' children. Finally, condition p 2 is violated because the speaker should know whether the crying child is his. Note now the difference between ( 15) and ( 16):

(16) This child is crying; so, if she's yours, this child is crying.

Example ( 16) does not sound as odd as example ( 15), for reasons that the pragmatic account can explain, but that elude the mental model acccount. It reflects one of the critical cases that we will scrutinize in the next section.

Critical Cases

There are two critical cases wherein the mental model account and the pragmatic account make opposite predictions. In case 4, both conditions p 1 and p 2 are met, but condition m is violated. Accordingly, the model account predicts that'y; so if x, y' sounds unacceptable, whereas the pragmatic account predicts that 'y; so if x, y' sounds acceptable. Conversely, in case 7, condition m is met, and so is condition p 1 , but condition p 2 is violated. Accordingly, the pragmatic account predicts that 'y, so if x then y' is incongruous, whereas the model account predicts that it is acceptable.

We first consider case 4, of which (17-a-c) are three examples:

(17) a. Amy dyed her hair red; so, if you've not seen her yet, Amy dyed her hair red;

b. This child is crying; so, if she's yours, this child is crying.

c. There is beer in the fridge; so, if you want some, there is beer in the fridge.

In all these examples, the model < ¬x, ¬y > is perfectly consistent with world knowledge, and is thus part of the representation of the conditional 'if x, y.' Indeed, there is nothing intrinsically implausible, let alone impossible, with the situ-ations captured by the model < ¬x, ¬y >: Amy, whose hair is not dyed red, being seen by someone; a child who is not crying, and who is not that of a listener; a fridge that is devoid of beers, at a moment someone has no desire for beers.

Thus, the possibility < ¬x, ¬y > has to be part of the representation of the conditionals featured in (17-a-c), and the conditional 'If x, y' is bound to throw away semantic information as compared to the premise 'y.' As a consequence, the mental model account predicts that all three examples (17-a-c) should sound incongruous and unacceptable. Clearly, this prediction is not supported by our intuitive reaction to these examples.

The pragmatic account, in contrast, predicts examples (17-a-c) to be acceptable, because they all meet conditions p 1 and p 2 . In (17-a), whether the listener has seen Amy already, has bearings on the epistemic relevance of asserting that Amy dyed her hair red. In (17-b), whether the child is that of the listener has bearings on the practical relevance of asserting that the child is crying. Likewise, in (17-c), whether the listener is thirsty has bearings on the practical relevance of asserting that there is beer in the fridge. Thus, condition p 1 is satisfied in all three examples.

In parallel, condition p 2 is also satisfied in the three examples. Nothing in example (17-a) indicates that the speaker is in a position to know or assume that the listener did not see Amy already; and, mutatis mutandis, the same can be said about examples (17-b-c). Because conditions p 1 and p 2 are met for all three examples (17-a-c), the pragmatic account (correctly) predicts that they should be intuitively acceptable.

We now turn to case 7 in Table 1, wherein condition m is met, as well as condition p 1 , but not condition p 2 . In such a situation, the model account predicts that we should not flinch at the statement 'y; so if x, y,' whereas the pragmatic account predicts the opposite. Now consider ( 18):

(18) ?I am not pregnant; so, if I am a woman I am not pregnant.

Condition m is unambiguously met by example ( 18). Indeed, the model < ¬woman, pregnant > is ruled out by common knowledge, and is not part of the representation of 'If I am a woman I am not pregnant.' Thus, the conditional does not throw away information as compared to the premise 'I am not pregnant,' and ( 18) should not sound incongruous according to the model account. The oddness of ( 18), though, is beyond reasonable doubt. The pragmatic account predicts this oddness, based on the violation of condition p 2 , i.e., based on the common expectation that speakers are in a position to know for a fact about their gender. In contrast, p 2 is satisfied if one changes (18) minimally into ( 19):

(19) This person is not pregnant; so, if this person is a woman, she is not pregnant.

Accordingly, and as predicted, ( 19) is not incongruous. Note that condition p 1 of the pragmatic account is met by examples ( 18) and ( 19), because the statement 'I am not pregnant' would have no epistemic relevance if the speaker was not a woman; and mutatis mutandis for (19). As we have hinted on several occasions, the satisfaction of p 1 is in fact entailed by the satisfaction of m. We turn to that issue in the next section, where we consider the last two, special cases in Table 1, which cannot be constructed due to the logical relation between m and p 1 .

Special Cases

Table 1 does not report any prediction of either account for cases 5 and 6. The specificity of these two cases is that they denote situations wherein condition m is satisfied whilst condition p 1 is not. These situations, however, cannot occur, and cases 5 and 6 cannot be constructed, because the satisfaction of condition m entails that condition p 1 is satisfied, too.

Let us assume that condition m is satisfied, i.e., that < ¬x, ¬y > is ruled out by common knowledge as a model of 'If x, y.' That is, common knowledge dictates that x and y cannot be simultaneously false: one at least must be true. In such a case, the truth of x is bound to have bearings on the epistemic relevance of asserting y, for this epistemic relevance is null when x is false. This entails, by definition, the satisfaction of p 1 .

Note that, in contrast, the satisfaction of p 1 does not entail the satisfaction of m. This was already illustrated by agreement case 3 and critical case 4, in which p 1 was satisfied because the truth of x had bearings on the practical (rather than epistemic) relevance of asserting y. It is not entirely clear, though, whether mental model theory has the same position on that issue, as one paragraph in Johnson-Laird and Byrne (2002) may suggest otherwise. In the course of discussing how the semantic contents of a conditional, as well as the context of its utterance, can block the construction of some possibilities, [START_REF] Johnson-Laird | Conditionals: A theory of meaning, pragmatics, and inference[END_REF] define what they call the 'relevance' interpretation of a conditional:

The relevance interpretation. Content or context precludes the possibility in the conditional interpretation in which the consequent does not occur.

Typical examples occur when the antecedent asserts merely a condition to which the consequent may be relevant, for example, "If you are interested in Vertigo, then it is on TV tonight." Individuals know that the antecedent possibility has no bearing on the occurrence of the consequent, and so the consequent holds in any case. (page 662)

This definition may be read as suggesting that the possibility < ¬x, ¬y > is discarded when x is a condition to which y may be relevant (or more precisely, we believe, the assertion of y). Thus, this definition may be interpreted as suggesting that p 1 entails m. What is not clear, under closer examination of the paragraph above, is why this should be the case. The crux of the definition is that individuals come to the assumption that y holds in any case, based on some feature of the conditional: Either that 'the antecedent possibility has no bearing on the occurrence of the consequent' or that 'the antecedent asserts merely a condition to which the consequent may be relevant'. None of these two possibilities stands correct upon closer examination.

First, one cannot come to the conclusion that y holds in any case (whether or not x is the case) from the assertion that 'if x, y' and the belief that x has no bearing on the occurrence of y. That would be a recipe for disaster:

(20) a. If we have pasta on Monday, the Mafia has weapons of mass destruction.

b. If it is cold on Tuesday, it is a good idea to put all our money in the stock market.

Clearly, we are not ready to conclude that the Mafia has weapons of mass destruction whether or not we have pasta on Monday, simply because we know that having pasta on Monday has no bearing on the Mafia having weapons of mass destruction.

Second, the fact that x is a condition to which the assertion of y is relevant cannot be taken as a guarantee that y is true whether or not x. This point is treated in great detail by [START_REF] Siegel | Biscuit conditionals: Quantification over potential literal acts[END_REF], who offers in particular the following example: Note that a serious problem arises from the very idea that the context in which the conditional is uttered might block the possibility < ¬x, ¬y >. Quite uncontroversially, the context of a conditional includes what has been uttered just before the conditional itself was asserted. Let us suppose that y is asserted just before 'if

x, y' is asserted. Because y is part of the context of the assertion that 'if x, y', context precludes that < ¬x, ¬y > is included in the interpretation of the conditional. As a consequence, 'if x then y' does not throw away semantic information as compared to y. Therefore, on the own terms of mental model theory, allowing for the contextual blocking of < ¬x, ¬y > implies that every instantiation of the paradoxical inference should become acceptable.

This being pointed out, whether or not the mental model characterisation of 'relevance' conditionals should be fixed or abandoned is not a question we can solve in this article. Our aim was rather to note that, under a superficial reading, mental model theory might be taken as suggesting that condition p 1 entailed condition m; but that there was no solid ground for that reading of the theory.

Biscuits and Paradoxes

In developing our dual pragmatic criterion, we have capitalised on the characteristics of 'biscuit' conditionals. In this section, we consider the possibility that all disappearances of the paradox may be imputable to the use of a biscuit conditional, and the related possibility that there does not exist any true exception to the paradox. 8We have argued that the absence of the paradox entails that p 1 is satisfied:

The truth of x has bearings on the relevance of asserting y. test, even though it does so for semantic reasons. If we make that choice, then the disappearance of the paradox (and thus the satisfaction of p 1 ) does entail the fact that the conditional is a biscuit; and the disappearance of the paradox can be added to the linguist's toolbox for diagnosing biscuit conditionals.

If really the disappearance of the paradox always involves a biscuit conditional, or a close cousin such as (23-a), then the possibility arises that there exists no true exception to the paradoxical character of 'y; so if x, y.' One agreed-upon characteristic of biscuit conditionals is that their consequents do not merely translate as the proposition they express. That is, the consequent y in a biscuit 'if x, y' is a different semantic object from an independent occurrence of the same sentence 'y' [START_REF] Siegel | Biscuit conditionals: Quantification over potential literal acts[END_REF]. Hence, the biscuit version of the argument 'y; so if x, y' might only have the superficial appearance of the paradox, but not its structure, for its structure would more appropriately be noted as 'y 1 ; so if x, y 2 '.

Thus, if all acceptable instances of the paradox involve biscuit, or biscuit-like conditionals; if specifically all these conditionals are such that their consequent is interpreted as a different semantic object than the independent occurrence of the same sentence; then the conclusion may be reached that there does not exist any true exception to the paradox, once the apparent exceptions are described at an appropriate level of semantic detail. This speculation, though, is left for future linguistic inquiry and not addressed further in this article, for it does not bear on our primarily psycholinguistic purposes.

Final Words

We have offered a pragmatic account of when and why the inference from y to 'if x, y' is paradoxical. We have shown that even though mental model theory [START_REF] Johnson-Laird | Conditionals: A theory of meaning, pragmatics, and inference[END_REF] explains the oddity of the paradox in terms of loss of semantic information, the way the explanation is operationalised in terms of mental models is inadequate, as the condition so formulated is neither necessary nor sufficient to explain the appearance and disappearance of the paradox.

In this article, we have focused on the mental model account as a contrast to our own pragmatic account. Mental model theory has a special burden of explaining the paradox, because the 'core meaning' it assigns to the natural language connective 'if' is akin to the material conditional. Theories that do not endorse that interpretation of 'if', however, do not necessarily endorse as valid the inference from y to 'if x, y', and do not share the burden of explaining the paradox. Indeed, the paradoxes of material implication have always been, and remain, a part of the motivation for other accounts of conditionals, such as the Stalnaker-Lewis possible words semantics [START_REF] Stalnaker | A theory of conditionals[END_REF], or psycholinguistic approaches (derived from philosophical work by, e.g., [START_REF] Adams | The logic of conditionals: An application of probability to deductive logic[END_REF][START_REF] Edgington | On conditionals[END_REF] such as the Bayesian account [START_REF] Oaksford | The probabilistic approach to human reasoning[END_REF][START_REF] Politzer | Reasoning with conditionals[END_REF] or the Suppositional account [START_REF] Evans | Hypothetical thinking: Dual processes in reasoning and judgment[END_REF][START_REF] Evans | If[END_REF]. These two major psycholingustic approaches allow a conditional 'If x, y' to be uncertain, assigning it the conditional probability Pr(y|x). Because the probability of the conditional 'If x, y' (the conclusion of the inference) may be smaller than that of y (the premise of the inference), the inference from y to 'If x, y' is not valid, which accounts for its intuitive oddity.

However, a case of interest is that in which the premise y is certain. In that case, 'If x, y' must also be certain, and the inference is valid. This specific case remained to be accounted for by probabilistic approaches to conditionals; they can now rely on our pragmatic account.

  to hear a big fat lie, George W. and Condi Rice are secretly married.Example (21) is a perfectly acceptable conditional, whose antecedent asserts merely a condition to which the assertion of its consequent is relevant, and whose antecedent has no bearing on the occurrence of the consequent. Nonetheless, it does not lead people to assume that George W. and Condi Rice are secretly married no matter what.

Table 1

 1 is ruled out by common knowledge about the semantics of 'game' and 'soccer.' Condition p 1 requires that the truth of x has bearings on the epistemic or practical relevance of asserting y. It is clearly the case here that if x were known to be false (i.e., if Viv was known not to have played any game), the assertion of

	Example		Not applicable because m entails p 1	Not applicable because m entails p 1	? I am not pregnant; so, if I am a woman I am not pregnant.	Viv didn't play soccer; so, if she played a game, she didn't play soccer.
	'y; so if x, y' ?		n/a n/a	n/a n/a	Yes No	Yes Yes
	Case Conditions	m p	1 0 0	6 1 0 1	7 1 1 0	8 1 1 1

. The classic example offered by the mental model account falls in this category: (11)

Viv did not play soccer; so, if she played a game, Viv did not play soccer.

Condition m is met because the possibility < ¬x, ¬y >, here < ¬game, soccer >,

Table 1 :

 1 The eight cases defined by the satisfaction or violation of conditions m Pragmatic account), together with the corresponding predictions made by the two accounts about whether 'y; so if x, y' is an intuitively acceptable inference, and together with one example of such an inference. y (she did not play soccer) would have no epistemic relevance. 7 Example (11) thus meets condition p 1 of the pragmatic account, one quick test of which is to

	1 , and
	(Mental Model account), p

  The question is, now, whether the satisfaction of p 1 entails that the conditional 'if x, y' is of the biscuit variety. Let us consider two diagnostic tests used to distinguish normal condi-

	makes any sense.
	For all the conditionals in this article that satisfy p 1 , the answer to both ques-
	tions is 'No'. For example:
	(22)	a. If you've not seen Amy yet, she's dyed her hair red.
		b. ?And what if I have seen her?
		c. ?If Amy has not dyed her hair red, you have seen her already.
	(23)	a. If Viv played a game, Viv did not play soccer.
	While (22-a) and (23-a) both fail the question test and the contraposition test, they
	seem to do so for qualitatively different reasons.
	More precisely, (23-a) fails both tests for semantic reasons. The fact that ¬y
	is known to entail x (soccer is a game) is responsible for the incongruity of asking
	whether y would be true if x was false, and also for the incongruity of asserting
	tionals from biscuit conditionals (DeRose & Grandy, 1999). The first consists of that ¬y entails ¬x. No comparable semantic reasons can account for the fact that
	checking whether the question 'And what if ¬x ?' makes sense upon hearing 'if (22-a) fails both tests. It is an open question whether we should call (23-a) a bis-
	x, y' ; the second test consists of checking whether the contraposition 'if ¬y, ¬x' cuit conditional on the grounds that it fails the question test and the contraposition

b. ?And what if Viv did not play a game? c. ?If Viv played soccer, she did not play a game.

Based on these diagnostics, it could be tempting to consider that all conditionals that satisfy p 1 qualify as biscuit conditionals. This conclusion, however, might be hasty. What is indisputable is that these conditionals do not qualify as normal conditionals; what is disputable is whether this makes them biscuit conditionals.

Mental model theory offers the same explanation of why the inference from x to the disjunction x or y can feel intuitively anomalous: One model of the disjunction is < ¬x, y >, which was ruled out by the original premise x.

Although this last quote unambiguously expresses the necessity of condition m, we ackowledge the possibility that it might have gone beyond the intention of the authors, who did not construe m as a necessary condition for the acceptability of the inference. If this is the case, then the mental model account does not make any prediction as to whether the paradox appears when condition m is violated, and thus does not make any prediction in four of the eight cases that we will review in the rest of this article. Our own account still does, of course.

Our intention is not to introduce neo-Gricean pragmatics as a general alternative to mental model theory. We are agnostic about the general level of consistency between mental model theory and neo-gricean pragmatics. Nevertheless, the account of the paradox that we offer in this article does derive from Gricean and neo-Gricean pragmatics, and is in this specific case pitted against the mental model account.

One could imagine contexts wherein knowing that Amy dyed her hair red would have practical utility for the listener in addition to epistemic utility, and such contexts would make other

The fact that y contains a negation is not alien to the appeal of example (11). Because they cannot spend their time denying everything they know to be false, people are especially sensitive to the pragmatic presuppositions required for negations to make conversational sense. For conditional reasoning experiments capitalising on these pragmatic presuppositions, see[START_REF] Bonnefon | Modus Tollens, Modus Shmollens: Contrapositive reasoning and the pragmatics of negation[END_REF].

The arguments in this section owe greatly to one anonymous referee.


Michelle Schatzman

Michel Bercovier

Numerical Approximation of a Wave Equation with Unilateral Constraints

The system Utt-Uxx 3 f, x E (O,L) X (O,T), with initial data u(x,O) = uo(x), Ut(x, O) = u1 (x) almost everywhere on (0, L) and boundary conditions u(O, t) = 0, for all t 2: 0, and the unilateral condition ux(L, t) 2: 0, u(L, t) 2: ko, (u(L, t)-ko)ux(L, t) = 0 models the longitudinal vibrations of a rod, whose motion is limited by a rigid obstacle at one end. A new variational formulation is given; existence and uniqueness are proved. Finite elements and finite difference schemes are given, and their convergence is proved. Numerical experiments are reported; the characteristic schemes perform better in terms of accuracy, and the subcharacteristic schemes look better.

1. Introduction. Consider the following problem: a linear rod vibrates longitudinally; one end of the rod is fixed, and the other one is free to move, as long as it does not hit a material obstacle. This obstacle may constrain the displacement of this extremity either to be greater than or equal to some given number, or to be smaller than or equal to some number. We can describe this situation mathematically as follows: assume that the material of the rod is elastic, homogeneous, linear, and make the approximation of small displacements. Let x be the spatial coordinate along the rod, with the origin at the fixed end; let u(x, t) be the displacement at time t of the material point of spatial coordinate x at rest. Let I denote a density of exterior forces, depending on space and time. With an adequate scaling, the velocity of waves in the rod is one and the length of the rod at rest is L. The displacement u satisfies the following equation:

(1.1)

Du = Utt-Uxx =I in Qr = (O,L) X (O,T).
The boundary conditions are detemined as follows: at the fixed end we have a Dirichlet boundary condition (1.2) u(O, t) = 0.

To be definite, assume a lower obstacle at the other end; then (1.3) u(L, t) ~ ko.

When the rod touches the obstacle, its reaction can be only upwards (see Figure 1), so that ux(L, t) ~ 0 on the set {t I u(L, t) = ko}; when the rod does not touch the obstacle, the end is free ux(L, t) = 0 on the set {t I u(L, t) > ko}. These last two conditions, which will be entirely justified in the sequel, can be summarized as (1.4) (1.5) Ux(L, t) ~ 0, ux(L, t)(u(L, t)-ko) = 0.

Conditions (1.3)-(1.5) are usually termed "unilateral conditions". We are given initial conditions (1.6) (1.7)

u(x,O) = uo(x), Ut(x,O) = u1(x).
It has been shown in [START_REF] Lebeau | A wave problem in a half-space with a unilateral condition at the boundary[END_REF]Theorem 14] that, if u 0 belongs to the Sobolev space H 3 1 2 (0,L), u 0 (0) = 0, u 0 (L) ~ k 0 , if u 1 belongs to the Sobolev space H 1 1 2 (0,L), and if I belongs to H 3 1 2 (Qr), then (1.1)-(1.7) possesses a unique solution u in the space L 00 (0, T; H 3 1 2 (0, L)) n W 1 • 00 (0, T; H 1 1 2 (0, L)).

It has been proved in [START_REF] Lebeau | A wave problem in a half-space with a unilateral condition at the boundary[END_REF] that the energy of the solution is conserved: Relation (1.8) is a consequence of the equations, and therefore, the model considered here does not include the possibility of a loss of energy during the contact with the obstacle.

1 {L ( 2 2)
The problem described here is closely related to the problem of a vibrating string with pointwise unilateral constraints which has been studied in [START_REF] Amerio | Su un problema di vincoli unilaterali per l'equazione non omogenea della corda vibrante[END_REF], [START_REF] Citrin! | The energy theorem in the impact of a string vibrating against a point shaped obstacle[END_REF], [START_REF] Citrin! | Risultati tipici sui problema della corda vibrante con ostacolo puntiforme," I.A.C. (Istituto[END_REF], [START_REF] Schatzman | Un probleme hyperbolique du 2eme ordre avec contrainte unilaterale: Ia corde vibrante avec obstacle ponctuel[END_REF] and [START_REF] Schatzman | On the Numerical Approximation of a Vibration Problem with Unilateral Constraints[END_REF]. Let u be a solution of (1.1)-(1.7), and denote by u the function defined by u(x, t) = u(x, t), 0 ::; X ::; L, u(x, t) = u(2L-x, t), L ::; X ::; 2L; the initial conditions are symmetrized similarly. Then it is easy to check that u satisfies the following relations, where Jl is a measure, (1.9)

(1.10) (1.11) (1.12) (1.13) (1.14) Du = Jl over (0, 2L) x (0, T), u(L, t) ~ ko fort in (0, T), supp(Jl) C {L} x {t E [O,T] ju(L,t) = k 0 }, u(x,O) = uo(x), ut(x,O) = u1(x), u(o, t) = u(2L, t) = o.
Only relation (1.9) is not completely obvious: Du is supported in {L} x [O,T]; Du is not zero whenever Ux jumps; but only negative jumps are allowed, thanks to condition (1.4) and to the symmetry of u. With the minus sign before the second derivative in space, this justifies intuitively (1.9). The argument can be made rigorous with standard distribution techniques.

Problem (1.9)-(1.14) has been studied in [START_REF] Schatzman | Un probleme hyperbolique du 2eme ordre avec contrainte unilaterale: Ia corde vibrante avec obstacle ponctuel[END_REF], where a solution is obtained in larger spaces of initial data than in [START_REF] Lebeau | A wave problem in a half-space with a unilateral condition at the boundary[END_REF]. Reference [START_REF] Schatzman | Un probleme hyperbolique du 2eme ordre avec contrainte unilaterale: Ia corde vibrante avec obstacle ponctuel[END_REF] gave a variational formulation of (1.9)-(1.14), which was rather awkward. Problem (1.1)-(1.7) possesses an explicit solution (see [START_REF] Lebeau | A wave problem in a half-space with a unilateral condition at the boundary[END_REF]), because the medium is uniform. A new variational formulation is given here; it has two aims: the first is to give a good framework in which to generalize the present homogeneous problem to a nonhomogeneous one, and the second to develop numerical schemes in a natural fashion.

In Section 2 of this paper, we give our variational formulation and sketch the proof of the equivalence of the variational formulation with the original formulation.

In Section 3 we state an existence theorem for a solution by Galerkin's method of approximation in finite-dimensional space and outline its proof . In the remainder of this section, we define a family of numerical schemes with the help of a variational formulation. Let Vh be a sequence of approximation spaces of (1.15)

V = {u E H 1 (0,L) / u(O) = 0};
Vh can be a space of finite elements; the convex set Kh is the set of elements of Vh which satisfy the constraint; let (-, •) denote the scalar product in L 2 (0, L), and let a denote the scalar product in V defined by (1.16) Then, our family of schemes is defined by

{ un+ 1 E Kh; ( un+1 _ 2un + un-1 ) D.t 2
,v-u +a(un,v-un+l)

;::: (r,v-un+ 1 ), 'Vv E Kh.

(1.17)

Here, r will be a suitable approximation of f(•,ni:lt), and the initial data u 0 and u 1 of the scheme are adequately chosen. The scheme (1.17) converges under a stability condition which is the same as the condition of the linear case.

In Section 4 we turn to explicit difference schemes; they can be written in variational form and are analogous to ( 1.17), with the main difference that the mass matrix is diagonal, so that their implementation is very easy. These schemes are given by (1.18)

U k+l-2Uk + uk-1 uk 2Uk + uk J J J j+1 - j j-1 k
--"-.:...::..-...,....-''::

----""-----=--Y if J. < n, D.t2 D.x2 -J uk+I - (k 2Uk uk-1 D.t2 (uk uk) D.t2 Fk) n -max o, n -n + D.x2 n-1 -n + n •
The proof of convergence is fairly easy if the scheme is noncharacteristic (

D.t < D.x);
it is much more technical in the characteristic case (D.t = D.x).

In Section 5 we report on numerical experiments. They can be summarized as follows: if the initial data are smooth, the characteristic and noncharacteristic schemes give reasonably good results; the characteristic scheme is substantially better. This suggests that for nonhomogeneous rods, the CFL number should be taken as large as possible. If the initial data are piecewise affine, the characteristic scheme gives good results, while the noncharacteristic scheme gives very bad results. Thus dispersion does not seem to make matters easier.

We surmise that the computation of phenomena with rough data in a nonhomogeneous medium will be difficult.

2. The Variational Formulation. 2.1. Notations and Definitions. We shall need a number of spaces and sets. The first of these is V, which has already been defined by ( Observe that 22 is the space of functions of locally bounded energy; its topology is the topology of H 1 ( QT). In particular, the elements of 22 have a trace on { x = L}. Therefore, the following definition makes sense:

(2.6) (2.9)

% = {u E 22 I u(L,
u(O) = uo,
(2.10) -(ut, v(O))-loT (ut, Vt-ut) dt +loT a(u, v-u)dt;::: loT(!, v-u)dt, 'Vv E% such that there exists"'> 0 such that v = u fort;::: T-"'• At a nai've level, the equivalence between (2.8)-(2.10) and the original equations (1.1 )-(1.7) is only a matter of writing enough Green formulae. The difficulty lies in the validation of the formal computations, and in particular in the trace theorems.

It is not obvious a priori that Ux has a trace on { L} x [0, T] because we require u to be only in a space of functions of bounded energy, namely 22. Microlocal techniques could be used to prove that this trace exists. An alternative method is proposed in [START_REF] Schatzman | On the Numerical Approximation of a Vibration Problem with Unilateral Constraints[END_REF], with ample details. We give here only the main steps.

The existence of a trace for derivatives will depend on results relative to strongly continuous semigroups and their duals: let X be a reflexive Banach space with norm 1111, and dual X*. Let A be an operator from D(A) C X into X. Assume that A is the generator of a strongly continuous semigroup S(t). We know that for every uo in D(A) there exists a unique function

u in C 1 ([0, T]; D(A)) n C 0 ([0, T]; X) such that (2.11) du dt (t) + Au(t) = 0 with initial condition (2.12) u(O) = uo.
Such a function is called a strong solution. As X is reflexive, it is possible to define [START_REF] Phillips | The adjoint semi-group[END_REF] nicely a dual semigroup S*(t) of S(t); S*(t) is a strongly continuous semigroup of generator A*, with

(u,A*u*) = (Au,u*), VuE D(A), Vu* E D(A*).
The dual semigroup enables us to define a notion of weak solution of (2.11 )-(2.12); for this purpose, we need a notation: the space Lfoc ( (a, b); Z) is the space of measurable functions from (a, b) to the Banach space Z, such that their restrictions to the compact subsets [c,d] are in LP(c,d).

DEFINITION 2.2. Letu belongtoLf 0 c((O,T);X) andf toL 1 (0,T;X). Thenu is said to be a weak solution of (2.11) if and only if

-loT J>(s)(u(s),u*)ds+ loT ¢(s)(u(s),A*u*)ds =loT ¢(s)(f(s),u*)ds, V¢ E £g(O,T), Vu* E D(A*).
(2.13)

Then we have the following regularity result on weak solutions.

PROPOSITION 2. 3. Let u be a weak solution of (2.11), according to Definition

2.2.
Then there exists a function u, equivalent to u modulo null functions, such

that u E C 0 ([0, T]; X).
The proof of this result is given in [START_REF] Schatzman | On the Numerical Approximation of a Vibration Problem with Unilateral Constraints[END_REF]. It follows from Proposition 2.3 that a weak solution of (2.11) has an initial value u(O). In particular, a weak solution of (2.11) is given by

u(t) = S(t)uo.
A straightforward consequence of Proposition 2.3 is COROLLARY 2. 4. Let u belong to 22, and assume that Utt -Uxx (defined in the sense of distributions on QT) is square-integrable. Then, for every positive c,

(2.14) uEC 0 ([0,T];H 1 (0,L-c)) and Ut EC 0 ([0,T];L 2 (0,L-c)), (2.15) u E C 0 ([0, L]; H 1 (c, T-c)) and UtE C 0 ([0, L]; L 2 (c, L-c)).
In order to prove this result, we must choose a convenient semigroup; we have already a partial differential operator, -fP jfJx 2 for (2.14), and -fJ 2 jfJt 2 for (2.15).

We only need boundary conditions. We choose Dirichlet boundary conditions. They are not satisfied by u, but by u¢, where ¢ vanishes for x = L, for (2.14), and ¢ vanishes fort= 0 and t = L, for (2.15). In this argument we use the symmetry of 0 with respect to time and space differentiation. Details can be found in [START_REF] Schatzman | On the Numerical Approximation of a Vibration Problem with Unilateral Constraints[END_REF].

A last technical lemma concerns 2" 00 :

LEMMA 2. 5. The space 2" 00 is included in the space of Holder continuous functions C 0 • 1 1 2 (QT), and the injectionfrom2"oo to C 0 • 1 1 2 (QT) is continuous.

The proof of this result is elementary. Details, if needed, can be found in [START_REF] Schatzman | On the Numerical Approximation of a Vibration Problem with Unilateral Constraints[END_REF].

The proof of Theorem 2.1 goes as follows: let u satisfy (2.8)-(2.10); relation (2.8) makes sense because u and v belong to 22, the elements of which have traces on [0, L] x {0} belonging to H 1 1 2 (0, L). All the terms of (2.10) make sense.

It is immediate that u satisfies Du=f in the sense of distributions.

Fix ' Yf > 0 and let 2;_E be the space of functions belonging to 22 which vanish for

C 0 ([0, L]; H 1 (0, T-'Yf)) and Ut belongs to C 0 ([0, L]; L 2 (0, T-'Yf)).
The proof of this result exploits the local energy identity inside Qr, and Proposition 2.3. The variational inequality (2.10) holds for functions v in % such that u -v is in 2;_E. By density, and the usual limiting arguments, it holds for all functions v in %.

The proofs are presented in detail in [START_REF] Schatzman | On the Numerical Approximation of a Vibration Problem with Unilateral Constraints[END_REF], to which the reader is referred for further information. The uniqueness is proved in [START_REF] Lebeau | A wave problem in a half-space with a unilateral condition at the boundary[END_REF] and in [START_REF] Schatzman | Un probleme hyperbolique du 2eme ordre avec contrainte unilaterale: Ia corde vibrante avec obstacle ponctuel[END_REF], with different techniques. The sequence of convex sets Kh is defined by

Semidiscretization in Space and Finite Elements.

(3.2)
We shall denote by M 1 ([0, T]; Vh) the space of bounded measures on [0, T] with values in Vh• The space W Given Uho and Uht in vh, we look for a function Uh which will satisfy the following functional requirements:

(3.3)

{3.4)

These conditions imply that u(t) has bounded variation in time and the limit limt!Ouh(t) = uh(o+) exists. The initial conditions are satisfied in the following sense:

{3.5) {3.6) (uhl-uh(o+), Vh-uh(O)) ~ 0, 'Vvh E Kh, iuh(o+)i = lu1hl• Condition {3.6
) means that uh(o+) satisfies a variational inequality and an energy condition. In the language of convex analysis, the set of vectors which satisfy the first relation of (3.6) is a translate of the opposite of the orthogonal cone at uh{O) to Kh.

Finally, uh satisfies an evolution variational inequality given by uh(t

) E Kh 'Vt E [0, T], {3 .7)

loT ((uh(t), vh(t)-uh(t)) + a(uh(t), vh(t)-uh(t))

-{!, vh(t) -uh(t))) dt ~ 0 for all continuous vh with values in Kh. Relation {3.7) can be written in a slightly different but equivalent form: define an operator Ah from Vh to itself by (Ahuh, Vh) = a(uh, vh), 'Vvh E vh, and a maximal monotone operator 8¢h (see [START_REF] Brezis | Operateurs Maximaux Monotones et Semi-Groupes de Contraction dans les Espaces de Hilbert[END_REF], [START_REF] Lions | Quelques Methodes de Resolution de Problemes aux Limites Non-Lineaires[END_REF], [START_REF] Rockafellar | Integrals which are convex functionals II[END_REF]) by (3.8)

{ {0} a¢h ( uh) = 0{ X I (x, Vh -uh) ~ 0, 'Vvh E Kh} if Uh E aKh, otherwise.
Then relation {3. 7) can be written as {3.9)

uh + Ahuh + 8¢h(uh) 3 fh• Of course, fh(t)
is the projection of f(t) on Vh with respect to the scalar product ( •, . ) .

We know from [START_REF] Schatzman | A class of nonlinear differential equations of second order in time[END_REF] that problem {3.9) together with conditions {3.1)-(3.6) possesses a solution which moreover satisfies the energy relation {3.10) iuh(t±W +a(uh(t),uh(t)) = luh1l 2 +a(uho,uho) +2 fot(f(s),u(s))ds.

This relation means that when the constraint imposes a jump in velocity, the magnitude of the velocity vector is conserved. Even among the solutions which conserve the energy, there is generally no uniqueness, as was shown in [START_REF] Schatzman | A class of nonlinear differential equations of second order in time[END_REF].

We denote by uh an arbitrary solution of {3.3)-(3.7) which satisfies {3.10). We choose initial data uoh and u1h which satisfy {3.11)

lim{iiuho-uoll + iuhl-u11) = 0. h!O
The convergence theorem is the following THEOREM 3 .1. Assume that (3.11) holds. Then, the sequence uh converges for the strong topology of 22 to the unique solution of (2.8)-(2.10).

Proof. The stability of the approximations uh is an immediate consequence of the energy relation (3.10) and of the subsequent Gronwall inequality which yields sup(iuh(t)i + lluh(t)ii) :5 C ( m:-x(lu1hl + lluohll), T), Vh :5 ho.

Moreover, it is obvious that uh belongs to.%. We can extract a subsequence, again denoted by uh, such that uh ~ u in 2" 00 weak * .

The injection C 0 • 1 1 2 ( QT) t.....+ C 0 • 01 ( QT) is compact for every a strictly smaller than 1/2. Therefore, u belongs to .% .

In order to prove that the limit u satisfies (2.10), it is necessary to take convenient test functions. The elements of .% are not smooth enough in time, and they have to be approximated before being projected onto Vh• This projection does not conserve the constraint at x = L, and therefore, the elements of .% need another approximation in order to satisfy the constraint strictly. More precisely, let v be an element of.% which is equal to u fort~ Tc. For 'fJ:::

:; c/4, define { u(x, t) + l ftt+TI (v-u)(x, s) ds + k('fJ)x¢(t) if t::::; T-'fJ, (3.12) v 11 (x t) = 11 ' u(x, t) if t ~ T-'f/•
The function¢ is nonnegative and smooth; it is equal to one on (0,

:; T-2• It is not difficult to check that, fort in [T-c/2, T-' fJ], v11(x, t) = u(x, t) + k('fJ)x¢(t),
so that vTI belongs to.%. On the other hand, vTI belongs to L 00 (0, T; V) because the time integration has a smoothing effect.

We denote by Qh the projection onto Vh, orthogonal with respect to the scalar product of H. The sequence Qh converges in the strong operator topology of H to the identity, and therefore, thanks to the Sobolev injections, there exists a sequence "th converging to zero as h tends to zero such that IIQhz-zllco ::5 "fhllzll, Vz E V.

Moreover, there exists a positive constant C such that (3.15) This property is proved by a classical computation.

The test function which will be used in ( With Vh as in (3.1) and Kh as in (3.2), we define a fully discretized scheme {

u~+ 1 E Kh; ( uk+1 _ 2 uk + uk-1 ) (3.16) h h h v _ uk+1 + a(uk v _ uk+1) > (Jk v _ uk+1) flt2 ' h h' h -h ' h ' Vv E Kh,
with Jk a suitable discretization of f and initial conditions u~ and u~ such that (3.17) ( l u1-uo I)

~rJ llu~ -uoll + h flt h -u1 = 0.
Using the notation in (3.8), relation (3.16) can be written equivalently as

+ hUh+ U'f'h Uh 3 h•

The scheme (3.16) is implicit in the constraint. It is equivalent to minimizing a coercive and twice differentiable functional on a convex set. Therefore, it defines a unique u~ at each step.

The stability condition will use the constant "'h defined by

(3.19) (Ahu, u) "'h = sup l uJ2 uEVn \{0} THEOREM 3. 2.
Assume that there exists a strictly positive number a such that (3.20) flt::::;; 2J 1 -Q' .

"'h

Then, under condition (3.17), the numerical scheme (3.16)

(Ahu, v) = 4 (Ah(u + v), u + v)-4 (Ah(u-v), u-v)
and thus

1 ,u-v' 2 ( "-htl.t 2 )
R(u,v)~4"(Ah(u+v),u+v)+ ---;5;t 1 --4 -.

Thanks to (3.20),

(3.22) 1 ,u-v' 2 R(u, v) ~ 4 (Ah(u + v), u + v) + o: ---;5;t
If we perform a discrete integration over (3.21), we obtain

k ( 1+1 1-1) R(uk uk+1) < R(uo u1) +' "" rl uh -uh tl.t h' h - h' h L....J J h' tl.t 0 1=1
Therefore,

1 luk+1 ukl2 411u~+l + u~ll + o: h tl.~ h k k 1 1+ 1

t)-u(ktl.t)).

If we substitute this value for v in (3.16) and perform a discrete time integration, we obtain The passage to the limit in this expression is obvious. In order to have strong convergence in 22, it is enough to show that the total energy of uh converges to the total energy of u. This is done by a discrete integration of (3.21). 0 Let us now compute the value of "-h for uniform P 1 finite elements, corresponding to a step 6.x = L/n. It is a classical result that 11uu 2 :5 1 ~;r, so that "-h = 12/ 6.x 2 . Therefore, the stability condition (3.20) becomes 6.t < 6.xy'f"=a.

( 1 0 ) M-1 k k-1 k k+1 k-1 k -uh-

-J3

4. Explicit Difference Schemes.

Notations and Description of the Family of Schemes.

Let n be an integer, let h = 6.x = L/n be the space step, and let 6.t be the time step. We denote by uj the solution of the finite difference scheme defined as follows:

uk+l = 2Uk-uk-1 + 6.t2 (uk -2Uk + uk ) + 6.t2 pk J J J 6.x2 J+l J J-l J (4.1) for1:5j:5n-1, uk+l - [k 2Uk uk-1 6.t2 (uk uk) 6.t2 Fk] n -max o, n -n + 6.x2 n-1 -n + n '
where Fj is a suitable discretization of f. By convention, U~ = 0.

If we define an interpolation in the space of U 3 k, and a scalar product which is an approximation of the £ 2 scalar product, this family of schemes admits a variational formulation. Namely, let The properties of sh are summarized in the following result:

(X-j~x) n (X-J•~x) u = L g r Uj' v = L g ~x Vj' J=1 J=1
the scalar product sh satisfies the following identity:

( ) ( ) ( ) ~x2
( ) U n Vn 4.5 sh u,v = u,v + - 6 -a u,v + ~x- 2 -.
Moreover, we have the inequality

(4.6) ( ) < 4 sh(u,u) a u,u _ ~x 2 , Proof. We observe that n-1 sh(u, u)-(u, u) = L ~(U} + U}+ 1 )~x + ~U~ ~x This shows that J=O n-1 -L ~(U} + UjUj+1 + U}+1) ~X j=O 1 n-1 2 1 2 =-'""'(U•+1-U) ~x+-U ~x 6L...t J J 2 n • j=O ~x2 u2 sh(u, u) = (u, u) + -6 -liull 2 + ~xf.
Relation ( 4.5) follows immediately by differentiation. The other relation is a consequence of the elementary inequality jui+1-Ujj

2 :5 2(U}+ 1 + U}).
Adding these relations with respect to j, we obtain

1 n-1 2 2 n-1 2 2 4 "ALIUi+1-Uji :5 A 2 L(Ui+ 1 +Uj)~x:5~ 2 sh(u,u). uX uX . X j=1 J=1
This proves the result. D LEMMA 4. 2. The finite difference scheme ( 4.1) is equivalent to the variational inequality Proof. According to (3.2), the convex set Kh is the set

Kh = {u E VhiUn = u(L) ~ ko}.
If we substitute v in ( 4. 7) by u + </J, with <P(L) = 0, then ( uk+1 _ zuk + uk-1 ) sh h t::.t~ h ,</J +a(u~,</J) ~sh(fk,</1), which is clearly equivalent to the first relation in (4.1). Take now v such that Vj = Uj, V j :5 n -1 and Vn ~ ko ;

then we find, using the explicit form of a( u, v) in terms of the Uj and Vj,

U k+1 zuk + uk-1 uk uk-1 n - n n (V. -Uk+1) n-n (V. -Uk+1)
f:::.t2 n n

+ f:::.x2 n n ~ F!(vn-U~+l), Wn ~ ko.
It is straightforward to check that this relation is equivalent to the second relation of (4.1). 0

To make things more precise, we shall settle for the following approximations of the exterior forces and of the initial data:

1 !j.O.x 1(k+1).0.t FJ = A A f(x, t) dxdt. ~t~x (j-1).0.x k.O.t
In the remainder of this section, we shall consider that >. = t::.t I t::.x is a fixed number belonging to [0, 1]; we shall first prove the convergence of the scheme in the noncharacteristic case ( >. < 1), and then in the characteristic case ( >. = 1).

(v-u v-u) R'(u, v) = 4 a(u + v, u + v)-4 a(u-v, u-v) + sh -z;:t' -z;:t 1 (u-v u-v) ( 2 )
2:: 4a(u+v,u+v)+sh ~,~ 1-.A .

Therefore, under the assumptions of the proposition, R' is a positive definite quadratic form over Vh X Vh which satisfies the coercivity inequality

R'(u,v) 2:: C(.A)(a(u,u) + a(v,v)),
where C(.A) is some positive constant. Then we perform a discrete time integration; we use a discrete Gronwall inequality, and we obtain the result, exactly as in the proof of stability in Theorem 3. Clearly, u belongs to K. We define v'~ as in (3.12), and v~ as in (3.24) by

v~ = u~+l + Qh(v'~(k~t)-u(k~t)).
We substitute v by v~ in (4.5); a disc1ete integration in time yields

M-1 M-1
+ ""' a(uk vk-uk+l)~t > ""' s (fk vk-uk+1)~t

L..J

hl h h -L..J h h, h h . k=1 k=l
The only difference between (4.11) and (3.25) is that the scalar product (•, •) is replaced by the scalar product sh(•, •). Therefore, if we substitute in (4.11) the scalar product sh by the regular scalar product, we commit an error given by

A 2 1 0 A U1 uo ~ (uh -Uh o _ 1) ~ ( n -n) (V.O _ U1) 6 a ~t ' V u + 2 ~t n n ~x2 M-1 (uk _ uk-1 vk _ uk+1 _ vk-1 + uk) + __ L a h h h h h h ~t 6 k=1 ~t , ~t ~x M-1 (uk _ uk-1) (v.k _ uk+1 _ v.k-1 + uk) + _ """ n n n n n n ~t 2 Lt ~t ~t k=1 (4.12) ~ 2 M-1 ~ M-1 + ~ """a(fk vk-uk+l)~t + ____: : _ """Fk(V.k-uk+1)~t 6 Lt h• h h 2 Lt n n n k=1 k=1
= T1 + T2 + T3 + T4 + T5 + T6. We have to estimate all the terms Ti, i = 1, ... ,6. For the term T 1 , we extend u1 to [L, 2L] by letting

u1(L + x) = u1(L-x)
and we observe that (4.9) can be written for 1 For the third term, we observe that

~ j ~ n as Ui -UO 1 !(j+l/2)Ax 1 1 = - u1(s)ds, ~t ~X (j-1/2)Ax so that I u1 -uo I 1 [!(j+1/2)Ax 2 ]1/2
llv~-u~+l-vz-1 + u~ll = IIQh ( v'1 (k~t) -u(k~t) -v'1 ((k-1)~t) -u((k + 1)~t)) II ~ llv'1(k~t)-u(k~t)-v'1((k-1)~t)-u((k + 1)~t)ll.
Proof. The difficulty in the proof of convergence comes from the fact that the approximate solution does not satisfy simple estimates as in the proof of Theorem 4.4. Because of the presence of staggered grids, we estimate and n iuk+2-Ukl2 2:: J J j=O ~X Let m be an integer between 0 and 3, and let m = 2b 1 + bo be its binary expansion.

We define four subgrids G~m) by

Q(m) = ([(2N+bo)n{O, ... ,n}]U{O}U{n}) x ((2N+bl)U{O}).
It is immediate that for any m = 0, 1, 2, 3, and any pair (j, k) in G~m) such that 2 ~ j ~ n -2 and 2 ~ k the following holds:

U k+2 _ uk + uk _ uk-2 j - j-2 j+2 j .
The grid G(m) defines elementary rectangles with vertices at neighboring points of the grid. We define an interpolation u~m) with respect to each grid G~m) by requiring u~m) to be continuous, to be equal to the discrete solution at the nodes:

u~m) (j~x, k~x) = UJ, V(j, k) E G~m) ,
and to coincide with a polynomial of the form a + bx + ct + dxt in each of the elementary rectangles of the grid. Then, from (4.20), u~m) is bounded in 2" 00 and by extraction, there is a subsequence, again denoted by u~m), such that

u(m) _... u(m) in L 00 (0 T• V) h ' ' ' du(m) du(m) + _. . . -;u-in L 00 (0,T;H), u~m) ~ u in C 0 •f3(Qr ), V[J < ~•
It is very easy to see that, in the limit,

Ou(m) = f in Qr.
Moreover, the initial conditions are the same for each of the u(m), and they all satisfy the boundary condition u(m) (0, t) = 0. It remains to see that they coincide

for x = L. First, because of the relation uk+l = uk + uk _ uk-1 J J-1 J+l J '
a passage to the limit shows that u(O) = u( 3 ) and u(ll = u( 2 ). Moreover, the relation

U k+1 = max(k uk + uk -uk-1) n o, n n-1 n
yields, by a passage to the limit, if n + k is even,

u(l) (L, t) = max(ko, u(O) (L, t)),
and if n + k is odd, Using the same methods as in the noncharacteristic case, and the fact that all the u(m) coincide, we can prove that the limit u satisfies the variational inequality (2.10). By an energy argument, the convergence of the u~m) is strong, and therefore, the usual interpolation uh converges to u strongly in 22. D 5. Numerical Experiments. We have taken L = 1, a contact at x = 1 and initial data such that the solution of the linear problem would be a wave propagating towards the positive x direction, and then being reflected. We use the finite difference schemes, either characteristic or subcharacteristic, with initial data which are differentiable or not differentiable.

In order to have a wave travelling to the right, the initial data must satisfy In order to have a good approximation of a wave travelling to the right, the discretization of the initial data will be UJ = uo(J"f).x), UJ = uo(J"f).x-f).t). The first experiment is with a characteristic scheme, and a time step of 1/50. The results look quite satisfactory, though a careful inspection of the returning wave, after reflection on the side with a unilateral constraint, reveals a small hollow which does not exist in the exact solution (see Figure 2). This small hollow is due to the effect of the unilateral constraint which implies a change in boundary condition, together with the change of the sense of variation of the wave.

du 0 U1 = --.
In the second experiment (see Figure 3), with the same space step, and a CFL number of 0.8, the small hollow is not apparent, but the dispersive effect becomes large at later times. If the spatial step is refined to 1/200, without changing the CFL number, the results are substantially better, as in the linear case (see Figure 4). The piecewise affine data are uo(x) = -max((1-5lx-0.61),0). With a space step of 1/50, and a characteristic scheme, we have a notch in the middle of the returning wave; it is too large to make the simulation acceptable; with a refined space step of 1/200 (see Figure 5), the notch is still there, but much smaller. Experimentally, the notch is first-order.

These same initial data, and a subcharacteristic scheme with a CFL number of 0.8, lead to noticeable oscillations where the notch was in the characteristic scheme. This is not a surprising result in view of the linear analysis of these schemes. But the dispersive effect of this scheme improves the aspect of the solution with time; see Figure 6.

FIGURE 1 (

 1 FIGURE 1 (a) lower constraint: u(L, t) 2 k 0 ; (b) upper constraint: u(L, t) ~ ko.

  2Jo iuti +iuxi (x,t)dx (1.8) = ~ foL (lu1l 2 + ~~:OI 2 )(x)dx + fotfoL Ut(x,s)l(x,s)dxds, VtE[O,T].

3 . 1 .

 31 Existence by Semidiscretization in Space. Let Vh be a sequence of finitedimensional subspaces of V such that --v(3.1) uvh = v. hThe scalar product (•, •) on Vh is the restriction of the scalar product of H to Vh.

uh

  -uh + uh A k !:>,/,. ( k+l) fk flt2

  dt ~ u in L 00 (0,T;H) weak *• 1 uh-+u inC 0 •.B(Qr), V{J<2: Now, we choose v"' as in (3.12), and we let (3.24) v~ = u~+ 1 + Qh(v"'(ktl.

(4. 2 )

 2 g(x) = max(1 -lxl, 0), and (4.3) Then u~ is an element of Vh, the space of uniform P 1 finite elements with nodes at the points jh, for 0 :5 j :5 n. The function fk is defined from Fk in a similar fashion. Let u and v be of the form ~ (x-j6.x) u = L...J Uj g 6.x , J=l ~ (x :._ j6.x) v = L...J Vj g 6.x ; J=l we define an alternative scalar product over Vh by ( 4.4) n sh(u,v) = LUjVj~X. j=1

LEMMA 4 . 1 .

 41 For all u and v in Vh of the form n

(4. 8 )

 8 UJ = uo(jflx), (4.9) { f:::.t !(j+l/2).0.x U}=UJ+~ u1(x)dx, if1:5j:5n-1, ~x (j-1/2).0.x 2flt !n.O.x u; = u~ +""A'"

4. 2 .

 2 Convergence in the Noncharacteristic Case.

PROPOSITION 4 . 3 .

 43 Let >. be strictly less than 1. Assume that the initial data and the right-hand side are discretized according to ( 4.8) -( 4.10). Then there exists a constant C which depends only on luol, llutll, lfi£2(QT) and T such that ll ukll + Sh uh -uh uh -uh all k less than or equal to T I !:::.t. Proof. With exactly the same technique as in the proof of stability in Theorem 3.2, we consider the expression ( v-u v-u) R'(u,v)=a(u,v)+sh Tt'"'Tt'".

From

dx

  In the smooth case, the initial position uo is defined by{ -[(xx 0 ) 2 -o: 2 ]2 o:-4 on [xoo:, xo + o:], uo(x) = 0elsewhere.In our experiments, o: = 0.2, xo = 0.6.

  FIGURE 2

  FIGURE 3

FIGURE 4

 4 FIGURE 4

FIGURE 5

 5 FIGURE 5

  FIGURE 6

  t ~ T-' Yf, and for lx -Ll + t :::; e. This space is tailored so as to avoid any (possible) difficulty with Ut and Ux in the corner (O,L).

	By integration by parts and density,
	loT ux(L, t)(v(L, t)-u(L, t)) dt ~ 0, '1:/v E% such that v-uE 2;_E.

This makes sense, in view of Corollary 2.4. A classical convexity argument and a number of passages to the limit conclude this part of the proof.

Conversely, let u satisfy (1.1)-(1.7). As u belongs to 22 and f to L 2 (Qr ), Ux has a trace over {L} x [O,T], and (1.3)-(1.5) make sense. One has to check only (2.10), the other relations being clear. The main step is to show that ux(-, t) belongs to L 2 (0, T-' Y/) for all positive 'Y/; the difficulty is with the corner (L, 0). Thus, we have a lemma: LEMMA 2.6. Let u satisfy (1.1)-(1.7). Then for all positive 'Y/, u belongs to

  Kh, for all t, and for h small enough. The rest of the proof is standard.

	3.7) is
	vh(t) = uh(t) + Qh(v 11 (t)-u(t));
	by a continuity argument, and the previous construction of v'~, Vh (t) is an element
	of

D 3.2. Convergence of Fully Discretized Finite Element Schemes.

  Consider the quadratic form over vh X vh defined by

	2 R(u,v)=(Ahu,v)+ ---;5;t , u-v
	I	1
	which appears twice in (3.21). We shall show that it is positive definite if condition
	(3.20) holds. We have the identity
	1	1
		converges to the unique
	solution of (2.8) -(2.10) as h and flt converge to zero.
	Proof. Let us first prove the stability: if we let v = u~-1 in (3.16), we can write
	(3.16) as
	(3.21)

  4 11u~+ 1 +u~ll+o: h tl.~ h ~C T,R(u~,ujJ, L lfkl 2 tl.t .

				2
	(3.23)	1	luk+1 ukl	(	)
					1:::;1:s;M
	Define an interpolation uh by
	( ) uh x, t = uh x k( )(k+1)f:l.t-t tl.t	k+1( )t-kf:l.t £ kA + uh x tl.t or u.t ~ t ~ + 1 u.t. (k )A
	Relation (3.23) implies that we can extract from the sequence ( uh)h a subsequence
	(again denoted by (uh)h) such that
		uh ~ u in L 00 (0,T;V) weak *•
		duh

11 2 ~ R(uo, u1) + ~ 1!112 tl.t + 2 ~ u tl.~ u tl.t. This is a discrete Gronwall inequality, which can be integrated readily [5, Lemme 4.1, p. 76], and yields, if M is the largest integer such that Mh ~ T,

  Assume that the initial data and the right-hand side are discretized according to (4.8)-(4.10), and that .A is strictly less than 1. Then, the sequence uh converges to the unique solution u of (2.8}--(2.10) for the strong topology of f!t2.

	2. D
	We interpolate u~ by letting
	( ) Uh X, t = Uh X k( )(k+1)~t-y ~t	k+l( )t-~t + Uh X ----;s:;:-•
	We can now state a convergence result.
	Proof. From Proposition 4.2, we can see that we can extract a converging sub-
	sequence, again denoted by uh, such that
	Uh ~ u in L 00 (0,T;V) weak*,
	duh dt __.. u in L 00 (0, T; H) weak*,
		1 'V(3 < 2'

THEOREM 4. 4.

If m is an integer such that m.6.t :::; "1 < (m + 1).6-t, we have finally the inequality For the last two terms, we have the easy estimates

Let Uj be defined by (4.1), with data (4.8)-(4.10). The interpolation u~ of Uj is given by ( 4.3). Then R' ( u~+ 1 , u~) is bounded for kh :::; T by a constant depending only the data uo, u1, and f and on T.

Proof. If ~t = ~x, the quadratic form R' defined above is no longer coercive on vh X vh, uniformly in h; nevertheless, it is still positive definite. Its explicit expression is indeed equal to R'(

Elementary manipulations give a form which is easier to handle, namely

R'(u~,u~+l):::; R'(u~-1 ,u~) + sh(!k, uh ;;.xuh )~x.

But we have the inequalities

Sh fk, h ~X h

and Therefore,

From a discrete Gronwall estimate we can deduce that Let us now find a variational inequality satisfied by the uim). We go back to (4.7), and we choose a v = vk inK such that

From now on, we assume that n is even, and the technical modifications which must be made in the case n odd are left to the reader. We add twice (4.7) at discrete time k to ( 4. 7) at discrete time k -1 and at discrete time k + 1. We obtain

Let Vh be the subspace of all functions of Vh which are affine on We define a mapping from Vh to Vh by

We define a scalar product on Vh by

j=1

With these notations, we can write ( 4.21) as follows:

-

Let Qh be the orthogonal projection on Vh with respect to the scalar product (•, •). We define v~ by

The next set of computations (Figure 7) presents the evolution in time of the solution with initial data 0 if 0:::; X:::; 0.5, 50(x-0.50) if 0.50 :::; X :::; 0.51, 0.5 if 0.51 :::; X :::; 0.60, uo(x) = 50(0.61-x) if 0.60 :::; X :::; 0.62, -0.5 if 0.62 :::; X :::; 0.72, 50(x-0.73) if 0.72:::; X:::; 0.73, 0 if0.73:::; X:::; 1. This initial condition has quite steep spatial derivatives; with a CFL of 0.08, and a space step of 1/1000, the result has the normally expected oscillations, but is acceptable.

Finally in the last set of computations (Figure 8) we display the evolution of the solution with data { 0 if 0 :::; X :::; 0.25, uo(x) = cos(201r(x-0.25)) if 0.25:::; x:::; 0.75, 0 if0.75:::;x:::;l.

We take a space step of 1/200 and a characteristic scheme. The humps are returned one by one by the unilateral constraint. The choice of a scheme can be governed by the following principles: if the initial data are smooth, the characteristic scheme gives very good results; if the initial data are not smooth, the nonlinear boundary conditions make the computation harder, and create a first-order notch. However, the subcharacteristic scheme gives worse results, even if they look smoother.