A set of semantic and pragmatic criteria for descriptive vs. metalinguistic negation

This article is about metalinguistic negation and the types of criteria making a straightforward distinction between descriptive (DN) and metalinguistic negation (MN). First, I will distinguish three types of negation: one type of DN and two types of MN, where MN1 is upward metalinguistic negation and MN2 presuppositional metalinguistic negation. This distinction is based on two semantic criteria (entailment and scope of negation), and one discursive or pragmatic criterion (connectives). Then, the use of the connective criterion will imply that maisSN (Ger. sondern and Sp. sino) triggers DN, whereas maisPA (Ger. aber, Sp. pero) licenses MN1 and a causal connective like parce que ‘because’ or puisque ‘since’ generates MN2. The general philosophy of the analysis of negation is both monoguist (negation is semantically unambiguous, and its meaning is logical) and contextual (there is no reinterpretation of the negative clause when negation is metalinguistic). Finally, this article aims to lend theoretical support to the experimental work on negation reported in Blochowiak & Grisot (2018).


Introduction
The issue of negation has been radically changed by Horn's article on metalinguistic negation and pragmatic ambiguity (Horn 1985). The idea of an unambiguous meaning, challenging the classical approach to negation, was the direct consequence of a systematic application of Gricean program in linguistic pragmatics: linguistic meaning is not multiple, as required by the Modified Occam's Razor Principle that "Senses are not to be multiplied beyond necessity" (Grice 1978: 118-9). In effect, a pragmatic research program in linguistics can take advantage of the idea that differences in meaning originate from usage rather than semantics. For instance, the difference in meaning between the two usages of negation in (1) and (2) could thus be described as originating from a difference in scope, which does not imply a difference in meaning of the negation operator: (1) The king of France is not bald.
The king of France is not bald, because there is no king of France.
As I assume a monoguist approach to meaning, the main challenge is to explain how different meanings in use can be associated with a single linguistic meaning. Regarding negation, I follow here the pragmatic trend, which claims that negation is not semantically ambiguous (Horn 1985). However, I also defend a more radical pragmatic analysis, which refuses to see the descriptive vs. metalinguistic uses of negation as cases of pragmatic ambiguity (Horn 1985;. The classical Gricean and Hornian approach would predict that (2) is processed in two steps (see also Gazdar 1979 for a similar analysis). For instance, (2) would give rise to the analysis given in (3): (3) First step: the king of France is not-bald.

Second step: not [the king of France is bald]
In this analysis (as reported in Carston 1996), the first step is narrow scope, where negation scopes over the predicate bald; the second step is wide scope, including in its domain the existential presupposition, since the follow-up clause (there is no king of France) entails the negation of the presupposition ('there is a king of France').
Our paper advocates an alternative to the pragmatic ambiguity analysis. The aim of the alternative analysis is to argue for a contextualist approach, where the issue of the scope of negation is a question of context. According to the contextualist approach, the semantics of negation has wide scope, and pragmatic processes explain why negation can have either narrow scope (descriptive negation) or wide scope (metalinguistic negation).
One of our article's main topics is therefore the truth-conditional use of negation, or descriptive negation, whose semantic description can be represented roughly as in (4): (4) It is not the case that P.
Alternatively, what is described by metalinguistic negation is generally defined as a nontruth-conditional use of negation. The metalinguistic negation exhibits a strong use of negation: negation can scope over the very act of saying a sentence or a proposition, and not just over a proposition. In that case, the common paraphrase of metalinguistic negation would not be (4) but (5), and the difference in meaning between (1) and (2) is made explicit in (6) and (7): (5) I cannot affirm P.
It is not the case that the king of France is bald.
I cannot affirm that the king of France is bald.
Several issues are linked to the descriptive vs. metalinguistic distinction. The first is the status of utterances containing metalinguistic negation; the second is the nature of the criteria which can make this distinction operative. In terms of the first issue, metalinguistic negation is generally associated with metarepresentation. As defined by Wilson, "a metarepresentation is a representation of a representation: a higher order representation with a lower representation embedded within it" (Wilson 2012: 230). An embedded representation can be an utterance, a thought or a proposition/sentence: a public representation, a mental representation or an abstract representation, respectively. MN examples are those where embedded representations are public representations, or utterances. The classical examples of MN (Horn 1989;Carston 1996) are clear enough on this point: Around here we don't eat tom [eiDouz] and we don't get stressed out. We eat tom[a:touz] and we get a little tense now and then.
Mozart's sonatas weren't for violin and piano, they were for piano and violin.
I didn't manage to trap two mongeese: I managed to trap two mongooses.
In other words, if uses of metalinguistic negation are metarepresentations, in the sense given by Wilson, then negation is a higher-order representation, and the utterance under the scope of negation is a lower-order representation.
This analysis is uncontroversial, but it only provides a narrow description of the uses of metalinguistic negation. In this paper, however, I aim to show that metalinguistic negation has truth-conditional effects. In order to develop a solid argument, I need to use clear-cut criteria to disentangle the two uses of negation. Three tests will be used: two semantic ones, entailment and scope, and a pragmatic one, the French connective mais 'but', as it appears in the contrast between (11) and (12): 1 (11) Abi n' est pas belle, mais ordinaire. Abi neg is neg beautiful, but ordinary 'Abi is not beautiful, but ordinary.' (12) Abi n' est pas belle, mais extraordinaire. Abi neg is neg beautiful, but gorgeous 'Abi is not beautiful, but gorgeous.' First,DN in (11) and MN in (12) entail different things: the corrective clause in (11) entails 2 the negative clause, whereas it entails the positive counterpart in (12). Second, these examples differ in scope: while negation scopes over the positive proposition (POS) in (11) ('Abi is beautiful'), it scopes over its implicature in (12) ('Abi is not gorgeous'). Third, the connective relating the negative clause (NEG) and the corrective clause (COR) is not the same mais 'but'. Referring to Anscombre & Ducrot's (1977) distinction between mais SN and mais PA , I will argue that DN triggers SN, whereas MN licenses PA.
The article is organized as follows. Section 2 introduces the semantics and pragmatics of DN, while in section 3 I discuss the different usages of MN. Section 4 summarizes the criteria distinguishing these two types of negation and their relations, while section 5 develops the mais-test. The final concluding section describes the main consequences of this pragmatic analysis of negation.

Semantic properties of DN
Descriptive negation (DN) corresponds to ordinary -that is, truth-conditionalnegation. As such, it is not enough to refer to the logical meaning of negation, simply because logical negation is propositional (it is not the case that P), whereas linguistic negation is syntactically either the negation of a sentence (13) or the negation of a constituent (14) (Klima 1964). For Klima, sentence negation can be tested by the addition of neither or not even, whereas constituent negation does not license not even, as (13) and (14) show: Peter did not come, {neither Paul, not even on weekends}.
There was some rain not long ago, {* not even, even} in the desert.
Even if the semantics of (15) is given by (16), and corresponds syntactically to a sentence negation, its complete interpretation must specify the semantic focus of negation: (15) Abi is not married.
It is not the case that Abi is married. 1 The uses of French examples is principally motivated by lexical contrasts and the uses of connectives. 2 I use entailment in its classical logical meaning: P entails Q if all situations that make P true make Q true.
Moeschler: A set of semantic and pragmatic criteria for descriptive vs. metalinguistic negation Art. 58,page 4 of 30 If the interpretation of the proposition 'Abi is married' in (15) is given as false by the negation operator, the question is why this proposition is false. Two possible answers can be given here. First, if the question under discussion (QUD, Roberts 2004) is (17): (17) Who is not married?
the propositional function 'x is married' is false of Abi if Abi does not belong to the extension of married. However, in that case, a better answer would simply be Not Abi rather than Abi is not married. Second, if Abi is the topic of the discourse, what is expected is that Abi is defined by a set of properties, in which 'being married' is excluded. So, the complete interpretation of a negative sentence can be expected to contain a corrective clause, stating which property (or predicate) is not only true of Abi, but also relevant in the context of the conversation. For instance, a possible corrective clause might be (18): (18) Abi is not married; she is engaged.
So, the semantic meaning of (18) is inferred from NEG and COR, where NEG stands for the negative clause and COR for the corrective clause. In other words, I assume here that the complete interpretation of a negative clause must contain an alternative to the positive counterpart of the negative clause (POS), either explicitly or implicitly. Hence, if COR is overt, COR belongs to the set of alternatives to POS; if COR is covert, a complete interpretation of NEG must license a set of alternatives to POS, as represented in (19): As an initial conclusion, it can be assumed that the investigation of negative utterances implies a relation between three propositions: NEG, COR and POS. The next sections will show that these relations are key to understanding the difference between descriptive and metalinguistic negation. Turning to the relationship between POS and COR, three properties can be usefully implemented, following Moeschler's (2013a) proposal: COR entailments, scope of negation, and discourse connectives linking NEG and COR.

Entailment
The entailment relation between NEG and COR is that COR entails NEG, as (20) shows: This entailment property is semantic. With scalar predicates, the strong predicate entails the weak one, whereas with antonyms, the affirmation of one predicate entails the negation of the other, the converse only being the case for complementary antonyms. As I will discuss in section 3, this entailment property exhibits a strong contrast between upward metalinguistic negation and downward descriptive negation.

Scope of negation
When it comes to the scope of negation, the inferential property again contrasts upward MN with DN, as in (22) and (23): Abi is not beautiful; she is gorgeous.
What is interesting here is that a DN use gives rise to a non-ambiguous implication. In other words, we are invited to infer from (22) that Abi is less than beautiful, which is made explicit in (24): (24) Abi is not beautiful; she is ugly.
Therefore, the issue is how COR entailments match the scope of negation (a proposition). I assume that negation semantically scopes over a proposition -here, 'Abi is beautiful' -rather than over an utterance or over a sentence. 3 Thus, this minimal meaning is a consequence of the asymmetry between the affirmation and the negation of antonyms: if 'x is beautiful' entails 'x is not ugly', then it is possible that 'x is not beautiful' simply describes a degree of beauty less than that which would qualify as beautiful, without having a precise target. For instance, if it is not the case that Abi is beautiful has a descriptive meaning, then the set of alternatives, made explicit in COR, must entail NEG, that is, 'Abi is not beautiful', as shown in (25): (25) Abi is not beautiful, she is {ordinary, ugly}.
On the other hand, if (22) is true, the upward alternative to beautiful is not expected, because, as in (23), COR entails POS, and not NEG. I hypothesize that the convergence between COR entailments and the scope of negation (i.e. a proposition) makes COR covert. This analysis will only be complete when metalinguistic negation will be taken into account, because, in this case, the presence of COR ensures the expected wide scope interpretation -that is, negation scoping on a scalar implicature. However, this does not mean that DN is obtained by default (see Blochowiak & Grisot 2018 for experimental confirmation). My argument is that negation is semantically unambiguous, and pragmatically specified as DN or MN.

Connectives
The third criterion is the connective used when NEG is followed by COR. COR introduces a correction of POS, which is explicit in dialogue, but implicit in a response to an initial move (26A). (26) is a dialogical illustration, where the prototypical connective in French is au contraire 'on the contrary', whereas in (27) au contraire follows NEG (27a), and not POS (27b): A: Abi est belle. Abi is beautiful 'Abi is beautiful.' B: Au contraire, elle est ordinaire. on the contrary she is ordinary 'On the contrary, she is ordinary.' (27) a. Abi n' est pas belle, au contraire elle est ordinaire. Abi neg is neg beautiful on the contrary she is ordinary 'Abi is not beautiful; on the contrary she is ordinary.' b. #Abi est belle, au contraire elle est ordinaire. 4 Abi is beautiful on the contrary she is ordinary 'Abi is beautiful; on the contrary she is ordinary.' On the other hand, mais 'but', which I will return to discuss in section 5, is only possible in monologic utterances -see the contrast between (28) and (29): (28) A: Abi est belle. Abi is beautiful 'Abi is beautiful.' B: #Mais elle est ordinaire. but she is ordinary 'But she is ordinary.' Abi n' est pas belle, mais ordinaire. Abi neg is neg beautiful but ordinary 'Abi is not beautiful, but ordinary.' Essentially, in DN, COR entails NEG, the scope of negation is a proposition, and the prototypical connectives relating NEG and COR are au contraire and mais in French -three properties which, as we will see, are different from those that define MN.

Pragmatic properties of MN
MN was first discussed in examples of metarepresentation, contrasting other cases of direct discourse, indirect discourse, free indirect style, hearsay adverbs, echoic utterances with dissociating attitudes, pragmatic markers and tag-questions, among others (Wilson 2012). Manifestly, MN examples are those in which embedded representations are public representations, which is to say utterances -see examples (8) to (10), repeated here for convenience: (8) Around here we don't eat tom[eiDouz] and we don't get stressed out. We eat tom[a:touz] and we get a little tense now and then.
Mozart's sonatas weren't for violin and piano, they were for piano and violin. (10) I didn't manage to trap two mongeese: I managed to trap two mongooses.
If a metarepresentation is a representation of a representation -that is, a higher-order representation R1 embedding a lower-level representation R2, as in (30)  In this case, the COR clause (I managed to trap two mongooses) corresponds to what the speaker takes responsibility for saying. However, the Hornian and Relevance-based interpretations of MN are not identical: in the Relevance-based interpretation of MN as metarepresentation, negation is semanticallyas well as pragmatically -unambiguous. What makes the difference between DN and MN is a question of complexity: DN is a simple and single representation, where negation scopes over a sentence, whereas MN scopes over another representation, which is to say an utterance.
The question of the nature of the embedded representation (utterance, thought, sentence or proposition) is a question of context, as will be seen later. 6 From a formal point of view, metalinguistic negation can scope over a mental representation like a thought, when the context of use refers to an attributed thought -as, for instance, in the case of irony -or an abstract proposition, especially when some entailment relations hold. Below, I will mainly discuss cases where the embedded representation is a sentence whose semantic format is propositional. I will capitalize on formal and logical properties of propositions to differentiate straightforwardly between DN and MN. Horn (1989: 384)  (38) Quelques étudiants n' ont pas réussi l' examen de pragmatique, some students neg have neg passed the exam of pragmatics tous ont réussi. all have passed 'Some students didn't pass the pragmatics exam; all of them passed.' 7 5 It is worth noting that what is under the scope of negation is an utterance -in this case, a public representation.
The question of whether it should be a sentence, or an utterance, is open. In Horn's interpretation of MN, the embedded representation is an utterance. I will give a more abstract definition of MN, in which a sentence is embedded. The argument will be that some property of sentence meaning, such as an implicature or a presupposition, is in the scope of MN. 6 One of the reviewers noticed that this is also a question of ontology. As I understand it, the nature of what is modified by a metalinguistic use of negation is crucial not only for theory of grammar, but also for theory of cognition. I suspect that, in a standard Relevance Theory perspective, DN would scope over sentences, and MN mainly over utterances. My suggestion is that there is something more abstract (such as propositions) in between these two solutions. 7 There seems to be a major difference between French and English. The English translation of (38)  If R2 is a proposition, it must be explained why, while scoping over a positive particular, negation does not trigger the classical pragmatic interpretation -that is, the negative particular implicature. In other words, why does (38) not give rise to the classical DN interpretation, made explicit in (40)?
(40) Quelques étudiants n' ont pas réussi l' examen de pragmatique, some students neg have neg passed the exam of apragmatics mais quelques-uns l' ont réussi. but some pro-3pl have passed 'Some students didn't pass the pragmatic exam, but some of them passed.' If the DN interpretation is not licensed in (38), other semantic scales should give the same type of result. However, this does not happen: the <certain, possible> scale gives rise to a contradiction, as does the <and, or> scale: #It isn't possible she'll win, it's downright possible she will. (42) #John isn't patriotic or quixotic, he's patriotic or quixotic.
Why does this happen, though? Some and some not are compatible (they are subcontraries), but possible P and not possible P are not logically connected in a similar fashion: 8 what is logically connected is the pair possible P and possible not P. As far as the <and, or> scale is concerned, not (P or Q) and (P or Q) are contradictory. So, how is it possible that some not and all do not give rise to a contradiction? From a logical perspective, a positive universal (all) and a negative particular (not all, some not) are contradictory. So, what should be expected is that some not and all are incompatible (they are contradictories), as elucidated by the logical square in Figure 1: Contradictories some some not Particulars I Subcontraries O Figure 1: The logical square.
metalinguistic reading is fine, if some is pronounced contrastively with a full vowel. It is, however, quite ungrammatical with a weak indefinite sm [unstressed some] NP as subject. And it is also impossible to have the DN reading with wide-scope negation -that is, some students didn't pass cannot mean 'no students pass'" (see Israel 1999) -exactly what happens in French. In French, the semantic scope is Neg-Q, whereas the syntactic order is Q-Neg. For instance, the usual interpretation of tous les étudiants n'ont pas réussi 'not all students passed' is ¬∀, and not ∀¬, even though the latter universal negation reading (tous…pas 'all not' meaning aucun 'no') is possible (with a stressed tous).
So why can some not and all be compatible in MN? The evident answer is that negation is used not descriptively but metalinguistically. Hence, pragmatic analysis may provide an initial answer, either in terms of metarepresentation (Carston, Wilson), or in terms of speech act (Horn). In the metarepresentational analysis, negation scopes over the implicature of some, or 'not all', which makes the 'not all' and 'all' meanings compatible. In the speech act analysis, the contrast is between what the speaker can and cannot affirm: From a logical perspective, both (43) and (44) are consistent. So, it seems that MN receives a non-contradictory interpretation, on condition that MN is analyzed pragmatically. In spite of this, I propose a simpler analysis, which is neither metarepresentational nor speech act-based; I intend to show that MN also has representational effects, which can be made explicit by resorting to COR entailments, scope of negation and connectives. On this basis, a pragmatic analysis will be proposed.

Entailment relations
I will first examine entailment relations. Gazdar (1979) provided one of the first semantic and pragmatic analyses of scalar implicature and presupposition. The notion at stake is potential vs. actual inferences -that is, potential implicatures (im-plicatures) and potential presuppositions (pre-suppositions), as opposed to actual implicatures and presuppositions. The crucial point is that both generalized conversational implicature (as defined by Grice 1975) 9 and lexical presupposition, shown in (45) and (46) respectively, can be defeated: Some of the students, in fact all, came to the party.
Abi does not regret having failed, because she passed. 10 In Gazdar's terms, potential quantitative implicatures are "all the implicatures which the sentence could possibly have prior to contextual cancellation" (Gazdar 1979: 55). Similarly, potential presuppositions, or pre-suppositions, are "what the presupposition would be if there was no "projection problem," no "ambiguity" in negative sentences, and no context sensitivity" (Gazdar 1979: 124). In other words, the scalar implicature 'not all students' is a potential implicature of some of the students, cancelled by the parenthetical in fact all; equally, 'Abi failed' is a potential presupposition, cancelled by the corrective clause she passed. So, these cases are explained by the cancellation of a potential inference, valid before contextualization, and thus neither added to nor retrieved from the common ground. 11 Gazdar's proposal, while incomplete, has one substantial positive point: it shows that entailments are stronger than implicatures and presuppositions. In order words, semantic meaning is stronger than pragmatic meaning. 12 I will follow Gazdar's suggestion and make use of Moeschler's proposal (2013b) on the relation between semantic and pragmatic meanings.
First, I will draw a distinction between two types of MN: MN1 scoping over a potential scalar implicature, and MN2 scoping over a potential presupposition.

MN1 entailments
In MN1, COR entails POS, and negation scopes over POS scalar implicature. First consider the entailment relation: a. Abi is not beautiful, she is gorgeous. b. Abi is gorgeous → Abi is beautiful (48) a. Some students did not pass, all passed. b. all students passed → some students passed In these examples, COR contains an upper-bound predicate or quantifier entailing its lower-bound correlate, the relation being one of entailment. The reason is that scalar predicates and quantifiers belong to quantitative scales: 13 <all, some>, <gorgeous, beautiful>. In other words, there is a strong contrast between MN1 and DN: in DN, COR entails NEG, whereas in MN1, COR entails POS.

MN2 entailments
In MN2, COR entails NEG and the negation of the presupposition of POS (PP POS ), as (49) and (50)  So, the corrective clause COR entails both the negation of the implicature and the presupposition. In this regard, whether COR is necessary for such a cancellation is an without negation, because of the contradiction between the potential implicature and the corrective clause. However, as (i) and (ii) show, this is only the case for implicatures: (i) Some of the students, in fact all, came to the party.
(ii) #Abi regrets having failed since she passed. 12 It is worth noting that the difference between a potential and an actual presupposition leads to the pragmatic definition of presupposition put forward by Gazdar (1979: 103-108), on the basis of Stalnaker (1977). 13 To simplify, semantic scales are defined as lexical here. 14 The question of whether COR entails NEG is under discussion. In Moeschler (2013a), I defend this assumption. One objection might be that, when a presupposition is objected to and cancelled by COR, this entailment affects the truth of the matrix clause in (49) and the assertion in (50). A moderate version would predict that this is typically an implicature, not an entailment: it could be true or false. For instance, if Abi does not know that she passed, but believes that she failed, she might have regrets. In my analysis, the cancellation of POS is a consequence of the cancellation of its presupposition. In cases where MN does not scope over a linguistic form but over pragmatic content (implicature or presupposition), the question remains open. In section 4.1, I will argue that the interpretation as metalinguistic is the result of a process of assumption formation and confirmation, not giving rise to a two-step process of MN. In other words, the wide scope reading for MN is anticipated, and COR is simply expected to be a confirmation of this move. However, what is striking is that MN1 and MN2 have different semantic effects: in MN1, POS is conserved, whereas MN2 cancels both POS and its presupposition (PP POS ). Table 1 makes these differences explicit in terms of the representational effects of DN, MN1 and MN2:  Table 1 shows, entailment is a good diagnostic tool for the distinction between DN and MN, and between two types of MN. The entailment test also provides a reason why examples of MN relating to form, such as (10), do not trigger any entailment relation. To make this point clearer, no entailment holds between COR, POS or NEG in (10), repeated in (52a) below:

Types of negation Entailments
(52) a. I didn't manage to trap two mongeese: I managed to trap two mongooses. b. I managed to trap two mongooses ⇸ it is not the case that I managed to trap two mongeese Here, the pragmatic analysis, either by way of an echoic or a speech act interpretation, is the only relevant analysis, as shown in (53) and (54) This explains why pragmatic approaches to MN were initially interested in those cases which had no specific meaning issues (either semantic or pragmatic). This is also the main reason why I will not propose any particular account of these examples. 15 descriptive vs. metalinguistic negation Art. 58, page 12 of 30

Scope of negation in MN2
The scope of negation in MN2 is a direct consequence of COR entailments. Traditionally, as in the Russellian logical analysis of the king of France example, an MN2 reading is wide scope. So, (55), in which there is no king of France, receives the logical form in (56) (see Moeschler 2010 for a more complete analysis): (55) The king of France is not bald.
'It is not the case that there is an x such as x is a king and there is no y such that y is a king and y is different from x and x is bald'.
This wide scope reading explains why the assertion (POS) and its presupposition (PP POS ) are false, as the logical form (57), equivalent to (56), shows: For all x, x is not a king or there is a y such that y is a king and y is different from x or x is not bald'.
What (57) shows is that, for every x, either there is no king of France -¬King(x) -or he is not unique -[King(y) ∧ (y≠x)] -or he is not bald -¬bald(x). Therefore, the negation of the presupposition $x[King(x)] or the assertion [bald(x)] is a consequence of the wide scope of negation.

Scope of negation in MN1
However, what about upward negation -that is, MN1? How can we predict that negation scopes over the POS scalar implicature (SI POS )? As we have seen, COR entails POS, which means that if negation were not present (as in the structure POS, COR), SI POS would have been triggered by POS, even if defeated by COR, as in (58) (59B) can certainly constitute a possible refutation of A's SI: if A compliments Abigaël's father, A may convey that she is nothing but beautiful. So, an answer such as (59B) is possible, but may be considered risky, mainly because it is difficult to interpret without a tone of aggressiveness or irony, as in (60): (60) B: Do you mean 'nothing but beautiful'?
In addition, how does one explain the scope of negation in MN1, as, for instance, in (61)?
(61) Abi is not beautiful, she is gorgeous.
The main point of my analysis is that negation cannot be descriptive -that is, scoping over the proposition 'Abi is beautiful'. The reason for this is simple: if this is the case, the speaker would have produced a contradiction, saying both P and not-P, since gorgeous entails beautiful: (62) Abi is gorgeous → Abi is beautiful (63) It is not the case that Abi is beautiful, and Abi is beautiful.
However, note that this contradictory step is exactly what is predicted by a speech act analysis: first, negation is taken to be descriptive, and then it is reinterpreted as metalinguistic, since the ambiguity interpretation at the semantic level (Burton-Roberts 1989) first predicts a narrow scope (descriptive) interpretation, and it is only by way of a second step that a wide scope interpretation involving SI cancellation arises. 16 In my proposal, interpretation only includes a one-step wide scope, mainly because it is contextually accessible (see Blochowiak & Grisot 2018 for an experimental confirmation of this). Therefore, the one-step interpretation makes POS available in the context, which is confirmed by the entailment from COR to POS. In other words, the only possible target for negation is something beyond semantic content, as it must be defeasible: since POS implicates not-COR, negation scopes over SI POS -that is, not-COR. The full process is described in (64) below: (64) a. Scope of negation: Abi is not more than beautiful b. COR entailments: i. Abi is gorgeous → Abi is more than beautiful ii. Abi is gorgeous → Abi is beautiful It is possible to give a more precise explanation as to why COR is expected to be presentalthough not mandatory -in MN1. The explanation is a side effect of a weak constraint on DN: since scope and entailments are identical in DN, COR is not informative, and when it occurs, it allows for the specification of NEG at the level of the explicature. On the other hand, with MN1, scope and entailments are not identical. As far as quantitative scales are concerned, while negation scopes over a specific degree d of a property P -here d(beautiful) -with an 'only' or 'no more than' interpretation, entailments include all lowerbound degrees of P.
The complete division of labor between scope and entailments predicts how negation is processed in different contexts. In Moeschler (2013a), these semantic properties are confirmed by the contextual effects of negative utterances, in the sense of Relevance descriptive vs. metalinguistic negation Art. 58,page 14 of 30 Theory (Sperber & Wilson 1995;Wilson & Sperber 2012), like adding new information, and strengthening or suppressing old information. 17 Table 2 summarizes the different scopes of DN, MN1 and MN2:

Connectives in MN
What about connectives -that is, instances of explicit linguistic material introducing COR? I have shown that the prototypical connectives following DN in French are au contraire 'on the contrary' and mais 'but'. But what about MN1 and MN2? Metalinguistic negation is followed neither by the same type of corrective clause, nor by the same connective. Whereas the relation between NEG and COR is a contrast relation with MN1, it is an explanation relation with MN2. Contrast relations are typically introduced by but in English, and mais in French. So, based on the observation that mais introduces COR both in DN and in MN1, the question is how and why the same connective can be associated with different entailments and scope relations in DN and MN1, an issue which will be extensively discussed in section 5.
As regards MN2, the data clearly show that neither of the two connectives introducing COR in DN are acceptable in MN2, as (65) shows. The only possible connective is a causal one, having an explanation function, like because or since in (66) In other words, since MN2 suppresses both an assertion and its presupposition, an Explanation discourse relation is required (Moeschler 2013a). This not the case when the order is not NEG, connective COR but COR, connective NEG, as illustrated by Carston (2002: 310): (67) There is no king of France: so, the king of France is not bald.
A: When did you give up smoking? B: I've never smoked in my life; so, I haven't given up smoking.
In (67) and (68), the relation between COR and NEG is a cause-consequence relation made explicit by so, which suggests that MN is not the result of a two-step process, since NEG can follow COR. Moreover, COR is here the cause of defeating PP POS . What, then, is the prototypical connective for MN1? As (69) shows, neither the DN nor the MN2 connectives are appropriate: Abi is not beautiful, {# on the contrary, # because 18 } she is gorgeous. 17 These respective contextual effects are the suppression of POS for DN, the strengthening of POS for MN1, and the suppression of POS and PP POS for MN2. 18 See gloss (72) for a possible use of because in (69).  With on the contrary, the NEG-COR relation would mean that the discourse relation is Correction. But the speaker is not correcting -which implies a single speech act -as happens when POS is corrected by COR in DN. On the contrary, in MN1, there is a Contrast discourse relation, and the connective used can have different discourse functions. In particular, the NEG-COR relation in DN is a specification of the explicature of NEG, which is to say the degree of property P below the predicate under negation. For instance, being not beautiful could be an intermediate predicate between antonyms, or a full antonym, as (70) shows:

Types of negation Scope of negation
(70) a. Abi is not beautiful, but ordinary. b. Abi is not beautiful, but ugly.
As regards causal connectives, such as because or since, their presence can be acceptable if the speaker aims to explain why she cannot assert POS, in which case the reading of (71) is upward, as in (72): (71) Abi is not beautiful, because she is gorgeous.
Abi is not beautiful, and the reason why I cannot affirm that she is beautiful is because she is more than beautiful -that is, gorgeous.
Moreover, because has a speech act use in (71) and not a content use, as with DN in (73) (Sweetser 1990): 19 (73) Abi is not beautiful because she has a very sad smile.
Apart from the possible explanation or causal connective with a speech act use in MN1, the most predictable connective is but, as shown in (74), and mais in the French example (75): (74) Abi is not beautiful, but gorgeous.
Abi n' est pas belle, mais extraordinaire. 20 Abi neg is neg beautiful, but gorgeous 'Abi is not beautiful, but gorgeous.' As such, the principal question is why the same connective mais 'but' is possible for both DN and MN1. This is a question to which I will return in section 5. However, before giving an exhaustive summary of properties distinguishing DN and the two types of MN, I would like to clarify one point about the interpretation of connectives in the NEG-COR relation. First, and perhaps least importantly, it could be argued that the presence of the same connective in DN and MN1 shows a genuine ambiguity of negation, letting COR disambiguate it. Second, as will be seen in section 5, two connectives are at stake for each type of negation: SN for DN, and PA for MN1. In a nutshell, four connectives can introduce three discourse relations, one for each use of negation (Moeschler 2013a), as shown in Table 3: 19 It is worth recalling that Sweetser (1990) makes a seminal distinction in connective uses based on their domains: content (because connects two propositions causally related), epistemic (because connects an argument justifying a belief) and speech act (because introduces the reason why a speech act has been uttered), as in (i) to (iii) respectively: (i) He came back because he loves her.
(ii) He loves her, because he came back.
(iii) Are you ready? Because we are late. 20 As I will show, a complete clause following mais is grammatically infelicitous here, as noticed by Horn (1989). I will return to this property of mais later.

Criteria for descriptive and metalinguistic negation
At this point, we have three robust semantic and pragmatic criteria for distinguishing DN, MN1 and MN2. Table 4 is a synthesis of Tables 1 to 3: I intend to show that these criteria are not only compatible (a possible outcome), but practically inevitable (a necessary outcome).

Criteria for DN
First, if COR entails NEG in cases of DN, this implies that negation scopes over POS. These two criteria are therefore entrenched. So, what about connectives? Au contraire is the connective stating a contrary relation. COR and POS are not in a contradiction relation (which is signaled by not in English, ne…pas in French). On the contrary, following Horn (1989), the semantic relation between antonyms is a contrary relation, as illustrated by the logical square in Figure  So, by saying Abi is not beautiful, NEG contains a lower bound negative predicate, which is entailed by its upper bound correspondence (she is ugly) -that is, the contrary of beautiful. Note that this can be inferred by implicature, which is not scalar, but rather an R-implicature in Horn's (1984) terminology (R stands for the R-Principle, derived from the Gricean maxim of relation or relevance). By saying Abi is not beautiful, the speaker says what is necessary to allow a lower bound implicature to be inferable ('Abi is ugly'). This implicature is conversational, since it can be cancelled, as shown in (76): (76) Abi is not beautiful, but she is not ugly either.
The two negative predicates (not beautiful, not ugly) are compatible, as they are subcontraries, and subcontraries can be true together -unlike contraries, which can only be false together, but not true together.

Criteria for MN1
Turning to MN1, the prediction is that the possible semantic and pragmatic relations are different: since COR entails POS, and POS is true, negation cannot scope over POS, and the only possible candidate is one aspect of POS meaning, which is to say SI POS . In this case, the logical square is again useful: the scale <gorgeous, beautiful> is a positive quantitative scale, and gorgeous and not beautiful are contrary, as shown in Figure 3:

Contraries gorgeous not beautiful
Contradictories beautiful not gorgeous Subcontraries So, saying not beautiful, from a purely semantic point of view, is saying something contrary to gorgeous. However, as gorgeous entails beautiful, to avoid contradiction, negation can only scope over its implicature, which is its subcontrary not gorgeous. As the implicature is defeated by MN1, a R-implicature from not gorgeous to not beautiful is not obtained. (77) summarizes this process (→ for 'entails', +> for 'implicates'): (77) a. Abi is not beautiful → Abi is not gorgeous b. Abi is gorgeous → Abi is beautiful c. Abi is beautiful +> Abi is not gorgeous d. Abi is gorgeous → it is not the case that Abi is not gorgeous Still, (77) is the description not of a cognitive process, but only of a logical reconstruction. From a pragmatic cognitive perspective, what takes place is a process of assumption formation and confirmation. As stated by Sperber & Wilson, "the calculation of implicatures is a matter of non-demonstrative inference. It involves a partly non-logical process of assumption formation and confirmation" (Sperber & Wilson 1995: 201): 21 (78) a. Assumption formation (from NEG): Abi is not beautiful +> it is not the case that Abi is not gorgeous b. Assumption confirmation (from COR): Abi is gorgeous → it is not the case that Abi is not gorgeous descriptive vs. metalinguistic negation Art. 58, page 18 of 30 As we see, COR confirms what is expected by treating negation as wide scope in (78a)that is, as negating POS implicature ('Abi is not gorgeous'). As entailments are stronger than implicatures, the assumption formed in (78a) is confirmed by COR in (78b). This leads to a very important conclusion: COR is not defeating an initial comprehension process, where negation is descriptive; on the contrary, it confirms what is assumed, which is to say the cancellation of SI POS . However, one can imagine a situation where the narrow scope interpretation of NEG is canceled. I hypothesize that this is what happens in humor. As an example of such a process, consider the famous joke by the French humorist Pierre Desproges: 22 (79) Marguerite Duras, qui n' a pas écrit que des conneries. Marguerite Duras who neg has neg written just bulshitt Elle en a aussi filmé. she pro has also filmed 'Marguerite Duras, who has not just written bullshit. She has filmed some, too.' In (79), there is a real trap by Desproges. He wants the audience to infer (80), but his corrective clause defeats it, as shown in (81) The effect is a manipulation, since it is unexpected, and, of course, gives rise to nonpropositional effects such as laughter.
The last point concerns the compatibility of connectives: what about the contrastive connective mais 'but' and parce que 'because'? The causal connective explains why the speaker cannot affirm POS, and so receives a speech act interpretation. In other words, when parce que occurs, as in (84), the only possible reading is a speech act reading: Abi is not beautiful, because she is gorgeous.
So, the explanation (because she is gorgeous) looks like a justification for not saying Abi is beautiful. Indeed, the entailment between gorgeous and beautiful is sufficient. So, the processing of (84) is restricted to (85): (85) justification [Abi is gorgeous, the speaker cannot say "Abi is beautiful"] 22 Marguerite Duras was a very active intellectual and famous writer (nouveau roman) during the eighties. The story becomes more complex with mais, as in (86): (86) Abi n' est pas belle, mais extraordinaire. Abi neg is neg beautiful but gorgeous 'Abi is not beautiful, but gorgeous.' What is required here is that the connective mais 'but' be compatible with the set of entailments of COR and the scope of negation. As mais is a contrastive connective, the conclusion drawn from NEG and the conclusion drawn for COR should be contradictory. This is easily demonstrable. Suppose that Abi is applying for the role of Juliet in a theatre company. Speaker A claims that Abi is beautiful, which should be a positive argument for hiring her. However, speaker B thinks that this argument is not enough, and then he wants to use a stronger argument (Abi is gorgeous): (87) a. Abi is not beautiful +> Abi cannot play Juliet b. Abi is gorgeous +> Abi can play Juliet From the contribution of mais, both entailments and scope can be explained. For one, as the second conjunct is stronger than the first, the entailment relation between COR and POS is explained on the grounds that gorgeous semantically entails 'beautiful'. Then, COR contradicts the POS scalar implicature, and, as COR is introduced by mais, it is stronger than NEG.

Criteria for MN2
I will now turn to the relations between COR entailments, the scope of negation and connectives in MN2. COR entails both NEG and POS presupposition, which means that negation scopes over POS and its presupposition. As regards the causal connective, what must be stressed is that a metalinguistic reading hardly be obtained alone. For instance, the reply in (88B) is highly implausible, because the presupposition belongs to the common ground, and cannot be removed without a negative utterance: (88) A: Abi does not regret having failed. B: #Do you mean she passed?
So, why do we have a causal or explanation connective introducing COR? As shown in 4.2, a causal connective introduces a justification of a negative utterance with a wide scope negation. As such, a wide scope interpretation means that what is proposed to belong to the common ground or expected already to be in the common ground must be rejected or retrieved. The simplest way to do so is to negate both the assertion and its presupposition (89). However, this move is not enough, simply because what NEG can do is refute the addition of POS to the common ground. So, the best way to do this is by either presenting it as new information, introduced by parce que 'because', or old information with puisque 'since', which contradicts what is believed to belong to the common ground (Zufferey 2012). This could not be accomplished by a corrective clause nor by a contrastive connective, such as on the contrary or but, because COR in MN2 introduces the negation of POS presupposition (89), while mais can either introduce the negation of POS implicature in MN1, or indirectly negate POS via its entailment (NEG) in DN (90):

The big picture
As a conclusion to this section, there are strong arguments for putting forward a semantic and pragmatic distinction between three types of negation: DN, MN1 and MN2. Table 5 gives a more precise picture of the different types of propositional effects involved in the three uses of negation: However, some cells in Table 5 still require further comment. First, the entailment of not-P can be positive or negative: when there is a categorical relation between P and Qas CHOW(x) → DOG(x) -, Q is not always the case: (91) a. Nath did not buy a Chow, he bought a Labrador. not-P → Q b. Nath did not buy a Chow, he bought a Siamese.
not-P → not-Q Second, in MN1, not-P is erased (not-P), because MN1 entails both P and the negation of its implicature (not-Q), while its implicature is also erased (Q). Third, the simplest case is MN2, where not-P erases its presupposition Q (Q).
One key point, observed in Table 5, is that MN2 requires that COR equals the overt negation of POS presupposition. The presence of a positive clause in Abi does not regret having failed, since she passed is simply due to the entailment of COR: Abi passed entails Abi did not fail.

The mais-test
In Horn (1989), three criteria for distinguishing between DN and MN are proposed. The first two are uncontroversial, but, as I will show, the third one can be challenged. Since it relates to the French mais 'but', it is crucial for distinguishing between DN and MN1. However, the two uncontroversial criteria will be the first to be examined.

MN and incorporation
First, metalinguistic negation is not incorporated. This means that the morphological negator (negative prefix) un-/in-cannot be used for MN, as shown in (92) (Horn 1989: 392): a. #It is impossible for you to leave, it is necessary. b. It is not possible for you to leave, it is necessary. descriptive vs. metalinguistic negation Art. 58,page 21 of 30 Note that when negation is lexically incorporated, the same impossibility occurs: (93) a. #I hate you, I adore you. b. I don't love you, I adore you.
This being highly uncontroversial, one legitimate question is to ask why it is the case. Remember that DN and MN are indicated by normal sentence negation, like (do) not in English, and ne…pas in French. The necessity of sentence negation shows why negation can trigger a wide scope reading. Being incorporated into a predicate makes the wide scope reading impossible. In other words, negation prefixes, as well as constituent negation constructions, cannot have MN uses because they cannot have wide scope. Moreover, the relation between a negative-marked predicate and its positive counterpart generally gives rise to a contrary relation, and not to a contradictory one. However, both DN and MN trigger contradiction relations, rather than contrary ones.

MN and NPIs
Second, metalinguistic negation is generally incompatible with NPIs (negative polarity items): 24 (94) Chris didn't manage to solve {some, *any of the problems} -he managed to solve all of them.
Again, what could be the explanation? Manifestly, without COR, we find the inverse distribution of quantifiers: (95) Chris didn't manage to solve {*some, any of the problems}.
Interestingly enough, if we invert the order between the negation and quantifier, the order some not can give rise to both descriptive and metalinguistic uses: (96) a. Quelques étudiants n' ont pas réussi, quelques-uns ont réussi. some students neg have neg passed some have passed 'Some students did not pass, some of them passed.' DN b. Quelques étudiants n' ont pas réussi, tous ont réussi. some students neg have neg passed all have passed 'Some of the students did not pass, all of them passed.' MN So, apparently, the incompatibility with NPIs is more a syntactic constraint than a semantic one. This hypothesis can be tested with the French NPI ne pas lever le petit doigt 'not to lift a finger': In other words, these two tests (incorporation and NPI) are incompatible with MN. I conjecture that this is because of the meaning of negation in the negative clauses being either lexical or idiomatically descriptive. 25

SN and PA, DN and MN
The mais-test is based on Anscombre & Ducrot's (1977) distinction between two mais in French, labelled mais SN (for Ger. sondern and Sp. sino), and mais PA (for Ger. aber and Sp. pero). Horn's position is that MN1 licenses mais SN , whereas DN triggers mais PA : (100) Il n' est pas grand, mais PA petit. he neg is neg tall but small 'He is not tall, but small.' (101) Il n' est pas grand, mais SN très grand. he neg is neg tall but very tall 'He is not tall, but very tall.' His main argument is based on an observation by Anscombre & Ducrot (1977), according to which mais SN cannot be followed by a full corrective clause, as the contrast between (102) and (103) shows: (102) Il n' est pas grand, mais PA il est petit. he neg is neg tall but he is small 'He is not tall, but he is small.' (103) *Il n' est pas grand, mais SN il est très grand. he neg is neg tall but he is very tall 'He is not tall, but he is very tall.' However, there are three issues with this analysis. The first issue is linked to a unproved assumption by Anscombre and Ducrot: "We have grounded our description of PA on the assumption that when it follows negation, it can only be a descriptive negation, and not a polemic one" (Anscombre & Ducrot 1977: 32). 26 However, no argument is provided to support this assumption. Then, if we use genuine examples where the first conjunct is negative, the function and the meaning of mais is not the same: mais SN is corrective, whereas mais PA is contrastive, allowing for an argumentative function. Finally, related to this, and confirming the initial analysis of these connectives here, mais SN is preferably followed by au contraire 'on the contrary', whereas mais PA prefers pourtant 'however' or en revanche 'rather'. (104) and (105)  Remember that the test for DN is au contraire 'on the contrary'. So, my analysis, which is confirmed by (106) and (107), gives the opposite result of Horn's analysis: mais SN is associated with DN, and only mais PA can introduce COR in MN1. German, which makes a clear lexical distinction between sondern (SN) und aber (PA), further corroborates this point to a certain extent. Clearly, sondern is compatible with DN and MN1, as (108) shows: (108) a. Abi ist nicht schön, {sondern, *aber} gewöhnlich.
Abi is neg beautiful rather but gorgeous 'Abi is not beautiful, but gorgeous.' So, it appears that DN licenses SN, and MN1 licenses PA in some languages and SN in others, as Table 6 shows: In a nutshell, it seems that the mais-test is operative, but not in the sense predicted by Horn. That said, two points must be explained further: the first is linked to the syntactic constraint responsible for the difficulty of having a full COR clause following mais PA ; the other is the explanation of how and why mais PA can be compatible with MN1.

Syntactic or semantic constraint on PA?
How can it be explained that (109), unlike (110) and (111), is grammatically awkward?
(109) *Abi n' est pas belle, mais elle est extraordinaire. Abi neg is neg beautiful but she is gorgeous 'Abi is not beautiful, but she is gorgeous.' Abi n' est pas belle, elle est extraordinaire. Abi neg is neg beautiful she is gorgeous 'Abi is not beautiful, she is gorgeous.' Abi n' est pas belle, mais extraordinaire. Abi neg is neg beautiful but gorgeous 'Abi is not beautiful, but gorgeous.' One could argue as follows: what is contrasted is belle 'beautiful' and extraordinaire 'gorgeous', and this contrast is obtained by what is under the scope of negation, that is, the scalar implicature of belle 'beautiful'. In other words, since the scalar implicature of belle 'beautiful' is 'no more than beautiful' or 'not gorgeous', the lexical contrast allows for a syntactic short-cut, i.e. a contrast which may or may not be made explicit, as in (110)  What the speaker is doing in (112) is a correction, which is somehow less acceptable when it does not target the contrary of POS but an intermediary predicate, such as ordinaire 'ordinary': (115) ?Abi n' est pas belle, mais elle est ordinaire. Abi neg is neg beautiful but she is ordinary 'Abi is not beautiful, but she is ordinary.' A correction is a two-place relation, but a single speech act. This is explicitly assumed by Anscombre & Ducrot (1977: 40), who argue that mais SN must follow the complex negation non pas 'not', as (116) shows: (116) Pierre viendra non pas demain, *(mais après demain). Pierre come-fut neg neg tomorrow but after tomorrow 'Pierre is not coming tomorrow, but after tomorrow.' As they say, "It is therefore not surprising that [the correction] is introduced by mais = SN, if we admit that the utterance p SN q is the object of one unique act of enunciation. If mais = PA is, on the other hand, impossible in these conditions, it is because it would cause a succession of two acts of enunciation (which is blocked by non)." (Anscombre & Ducrot 1977: 40) 27 Now, we have a clear-cut prediction: if non pas + adj 'not + adj' is followed by a lowerbound adjective, then DN licenses SN, and if non pas + adj cannot be followed by an upper-bound adjective, then it confirms that MN1 licenses PA. This exactly what we find: (117) Abi est non pas belle, mais laide. Abi is neg neg beautiful but ugly 'Abi is not beautiful, but ugly.' (118) *Abi est non pas belle, mais extraordinaire. Abi is neg neg beautiful but gorgeous 'Abi is not beautiful, but gorgeous.' So, it seems that NEG + COR with DN realizes only one speech act, whereas NEG + COR with MN1 realizes two speech acts. The next section will provide evidence for this claim.

Why and how is PA compatible with MN?
So far, I have essentially made two claims: a) MN1 is followed by mais PA ; b) the sequence NEG, mais PA COR realizes two speech acts. So, what are the arguments in favor of these two claims? First, as was shown in the previous section, this analysis of MN1 is neither metarepresentational, nor speech act-based, but representational, i.e. having truth-conditional effects. So, the kind of effect obtained should be compatible with a representational analysis and a speech act analysis. As COR entails POS and MN1 scopes over POS implicature, what is expected is that the inferences obtained are compatible with the procedural meaning of mais PA . The semantics of this, given by Anscombre & Ducrot (1977: 28), is the following: "Let p and q two sentences: uttering p PA q is (1) Presenting p as a possible argument for a possible conclusion r.
(2) Presenting q as an argument against that conclusion, i.e.
[…] as an argument for ¬r. (3) Attributing to q more argumentative strength in favour of ¬r than in favour of r. The sequence p PA q, as a whole, is thus argumentatively oriented in favour of ¬r." 28 So, what could be the argumentative analysis of Abi n'est pas belle, mais extraordinaire 'Abi is not beautiful but gorgeous'? First, p, which corresponds to NEG, is opposed to p' Abi est belle 'Abi is beautiful' -that is, POS. As per my analysis, negation -having wide scope -would scope over p's implicature, that is 'Abi is not gorgeous'. In other words, p content (NEG) can be translated as 'Abi is not gorgeous'. This content is an argument for a conclusion: for instance, 'Abi cannot play Juliet', because the speaker is looking for a gorgeous young lady for Juliet's character. q can be reconstructed as 'Abi is gorgeous', which allows for the conclusion that 'Abi can play Juliet'. And finally, the whole sequence of two utterances gives rise to the same conclusion: 'Abi can play Juliet'. In this respect, my description of MN1 is compatible with Anscombre & Ducrot's analysis of mais PA . The second point to clarify is why we can say that the speaker realizes not one but two speech acts with MN1 followed by mais PA . The argumentative analysis provides an initial answer: p being an argument for r and q an argument for ¬r means that the speaker realizes two acts of argumentation. However, another argument, coming from truth-conditional pragmatics (Moeschler 2018a), is possible. As the complete utterance Abi n'est pas belle, mais extraordinaire gives rise to a conflict between two contradictory conclusions, these two conclusions cannot be assumed by the same speaker in the same speech act. Moreover, from an interpretative perspective, an inferential process of assumption formation and confirmation takes place, as shown in (78). This process is accomplished when the implicature 'it is not the case that Abi is not gorgeous', derived from Abi is not beautiful, is confirmed by the entailment of COR which has the same content representation. So, it makes sense to attribute to the process of assumption formation the status of one speech act, and to the process of assumption confirmation the status of a second speech act.
Finally, one might wonder what makes the difference in meaning between sequences with mais and without mais. This with or without issue (WWI) has recently been addressed in Moeschler (2018b), as follows. In the case of MN1, it is assumed that "with upward negation, mais introduces a contrastive focus, and further implicatures, following POS strengthening, are linked to the implicatures accessible from COR". In other words, COR receives via mais a focal position, whereas without mais, COR is simply a specification of the entailment domain of POS. As an illustration, the upward reading with and without mais PA receives the following descriptions: (119) Abi n' est pas belle, mais extraordinaire. Abi neg is neg beautiful but gorgeous 'Abi is not beautiful, but gorgeous.' (120) a. Contrastive focus: extraordinaire is selected from a set of alternatives {belle, très belle, extraordinaire, magnifique…}. b. COR implicature: Abi is eligible for Juliet's character.
(121) Abi n' est pas belle, elle est extraordinaire. Abi neg is neg beautiful she is gorgeous 'Abi is not beautiful, she is gorgeous.' (122) a. Specification (explicature): the highest degree of beauty is 'gorgeous'.
b. Implicature: It is not the case that Abi is only beautiful.
As expected, the same contrast is evident in DN cases, with or without a connective. The strategy with the connective is the following: "With downward negation, mais introduces a corrective focus, and all contextual implications are consequences of the suppressing effects of POS and the implicatures accessible from POS" (Moeschler 2018b). Consequently, the DN sequences with and without mais SN are as follows: (123) Abi n' est pas belle, mais ordinaire. Abi neg is neg beautiful but ordinary 'Abi is not beautiful, but ordinary.' (124) a. Corrective focus: ordinaire makes NEG true and gives the highest possible degree of beauty. b. COR implicature: Abi is not eligible for Juliet's character.
(125) Abi n' est pas belle, elle est ordinaire. Abi neg is neg beautiful she is ordinary 'Abi is not beautiful, she is ordinary.' (126) a. Specification (explicature): the right degree of beauty is below 'beautiful'. b. Implicature: it is not possible that Abi is at the lowest degree of beauty ('ugly').
Succinctly put, in Abi n'est pas belle, mais ordinaire 'Abi is not beautiful, but ordinary', the possible correction via an overt connective and a full clause can be explained by the same semantic status of both NEG and COR: COR simply replaces NEG. In MN1, COR does not replace NEG. So, there is no semantic equivalence, and mais 'but' cannot introduce a complete clause -see the unacceptability of (115).

Conclusion
In this paper, the challenges were multiple, and not without risks. First, I wanted to argue for a one-step process in the interpretation of metalinguistic negation, at least as regarding MN1, or upward negation. Second, I advocated a truth-conditional approach to both descriptive and metalinguistic negation. To do so, I established three criteria to disentangle descriptive negation from metalinguistic negation: entailment, scope and connectives. And finally, the greatest challenge of the paper was certainly to argue for a mais-test that distinguishes between descriptive and metalinguistic negation. I argued in favor of the assumption that DN licenses mais SN , whereas MN1 (upward negation) licenses mais PA . I also tried briefly to address the WWI (with or without issue) -that is, the explanation of the effect of the presence or the absence of a connective (mais) between NEG and COR. The main effect of the presence of a connective between NEG and COR is to introduce a contrastive focus and a specific implicature in the COR interpretation, whereas without a connective, the implicature is restricted to the suppression of the POS implicature, and COR triggers only a more specific interpretation of NEG. This is a real finding, because it clearly shows that the presence of a connective between NEG and COR triggers a much more precise interpretation than when a connective is absent. The WWI issue is, for me, a linguistic opening which might allow one to explain what might seem to be superficial differences in forms, but which I believe reveal important differences in the underlying cognitive act. However, to demonstrate this would ultimately require something more elaborate than what is proposed in this paper. In brief, what we need is sound semantics for corrective and contrastive connectives, as well as uncontroversial analysis of pragmatic effects of negation. If my semantic and pragmatic intuitions are well-founded, these two types of semantic constraints should match. This is the next logical step for my future research.