Adrien Revault D'allonnes

Marie-Jeanne Lesot
email: marie-jeanne.lesot@lip6.fr

If I don't know, should I infer? Reasoning around ignorance in a many-valued framework

Many-valued logic allows to reason with partial truth measured by degrees on a discrete scale, but it suffers from an ambiguous interpretation of the middle truth level, considered as intermediate truth or as ignorance, i.e. inability to assess truth. The L e M extension introduces an additional value, outside the truth scale, to distinguish between these two notions. This paper studies L e M from a logical perspective, examining how to reason in this framework: it discusses the definition of appropriate semantics for the logical connectives and it considers an inference task, proposing a Modus Ponens variant for L e M .

I. INTRODUCTION

The many-valued logic paradigm extends binary logic to represent and manipulate partial truth: it is based on a symbolic, discrete truth scale and offers a formal framework to combine and reason with these degrees. Many variants have been proposed, differing in their interpretation, as well as their theoretical properties, see e.g. [START_REF] Rescher | Many-valued Logic[END_REF]- [START_REF]Many-valued logics[END_REF].

One of the reasons for the diversity of these variants in particular relates to the interpretation of the truth degree located at the middle position on the scale (usually, an odd number of intermediate truth levels is considered, so that the middle value is forced into the scale): as described and discussed in details in [START_REF] Dubois | On ignorance and contradiction considered as truth-values[END_REF], it bears an intrinsic ambiguity. Indeed, on the one hand, it can be understood as a neutral truth value, neither true, nor false; on the other hand, it can be interpreted as expressing ignorance, i.e. the inability to assess the truth degree. As such, it corresponds to an information state of the agent, and not to a truth degree. It can thus be considered as representing partial truth or incomplete knowledge, although the two notions clearly have distinct nature [START_REF] Dubois | On ignorance and contradiction considered as truth-values[END_REF].

The interpretation as ignorance opens the way to an epistemic reading, where the truth degrees should actually be understood as belief degrees. Therefore, it has been proposed to exploit modal logic to formulate Kleene three-valued logic and Belnap four-valued logic with improved characteristics [START_REF]Reasoning about ignorance and contradiction: many-valued logics versus epistemic logic[END_REF].

This paper studies a different approach, based on an extension of many-valued logic, called L e M [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF]: in this formalism, the distinction between neutral truth and ignorance is made through the introduction of an additional assessment value, that does not belong to the truth degree scale, as detailed in Section II-B: being outside the scale is the crucial feature that captures the essential difference between gradual truth and ignorance. This extended many-valued framework has been proposed to combine multi-criteria evaluations, so as to manipulate incomplete assessments of pieces of information [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF].

In this paper a logical perspective is considered, to examine the properties of L e M as a reasoning framework, in a propositional logic case: the issue is to define appropriate operators to implement the semantics of the connectives, conjunction, disjunction and implication. A second task addresses the definition of an inference process, to allow to establish new pieces of knowledge together with their evaluation in L e M , using an appropriate Modus Ponens variant. In both cases, the main question is how to manipulate ignorance.

The paper is structured as follows: after giving in Section II some preliminaries, regarding classic many-valued logic, L M , and its extension L e M to distinctly take into account ignorance, it studies in Section III the question of defining the semantics of the logical connectives to manipulate this ignorance. Section IV examines the issue of inference task in this framework. Section V concludes the paper. II. PRELIMINARIES: L M AND L e M This section briefly presents some of the main many-valued logic systems and discusses differences in their interpretations and their consequences on their definitions (see e.g. [START_REF] Rescher | Many-valued Logic[END_REF]- [START_REF]Many-valued logics[END_REF] for complete surveys and discussions). It then describes the extended many-valued framework on which the proposed tools to reason around ignorance rely.

A. Many-Valued Logic: L M Many-valued logic models reasoning using a discrete truth scale allowing to represent and manipulate partial truth: it uses M > 2 truth degrees of a totally ordered set L M = {τ 0 , . . . , τ M -1 }, where τ α ≤ τ β ⇔ α ≤ β. The ordering property guarantees all degrees in L M are comparable. They span, at a granularity varying with M , different levels of veracity from τ 0 , meaning 'false', to τ M -1 , for 'true'. The switch between the two appears around the middle value τ M -1 2 , usually forced into the chosen scale by choosing an odd M .

This section briefly discusses some classic three-valued logics and then turns to their generalisations for higher M values. Here as in the following sections, the logical symbols are abusively applied to truth degrees, and not only to logical formulae, to express the truth value of the compound formula for which the individual truth values are considered.

¬ τ 0 τ 1 τ 2 ¬ τ 1 τ 2 τ 1 ¬ τ 2 τ 0 τ 0 Łukasiewicz Ł 3 ∨ ∧ → 0 τ M -1 2 1 0 τ M -1 2 1 0 τ M -1 2 1 τ 0 τ 1 τ 2 τ 0 τ 1 τ 2 τ 0 τ 1 τ 2 τ 0 τ 0 τ 1 τ 2 τ 0 τ 0 τ 0 τ 2 τ 2 τ 2 τ 1 τ 1 τ 1 τ 2 τ 0 τ 1 τ 1 τ 1 τ 2 τ 2 τ 2 τ 2 τ 2 τ 2 τ 0 τ 1 τ 2 τ 0 τ 1 τ 2 Bochvar B 3 τ 0 τ 0 τ 1 τ 2 τ 0 τ 1 τ 0 τ 2 τ 1 τ 2 τ 1 τ 1 τ 1 τ 1 τ 1 τ 1 τ 1 τ 1 τ 1 τ 1 τ 2 τ 2 τ 1 τ 2 τ 0 τ 1 τ 2 τ 0 τ 1 τ 2 Kleene K 3 τ 0 τ 0 τ 1 τ 2 τ 0 τ 0 τ 0 τ 2 τ 2 τ 2 τ 1 τ 1 τ 1 τ 2 τ 0 τ 1 τ 1 τ 1 τ 1 τ 2 τ 2 τ 2 τ 2 τ 2 τ 0 τ 1 τ 2 τ 0 τ 1 τ 2 Post P 3 τ 0 τ 0 τ 1 τ 2 τ 1 τ 1 τ 1 τ 2 τ 2 τ 2 τ 1 τ 1 τ 1 τ 2 τ 1 τ 2 τ 0 τ 0 τ 1 τ 2 τ 2 τ 2 τ 2 τ 2 τ 1 τ 0 τ 0 τ 1 τ 1 τ 2
1) Original Tri-valued Logics: The first proposed manyvalued logic systems considered the case of three truth degrees, often denoted 0, 1/2 and 1 and mainly differ in the interpretation of the middle value. Their definitions are usually given through the truth table for the negation and one connective among {∨, ∧, →}, the two other ones being derived from syntactical expressions. They also differ in the choice of the reference connective, its definition and the expressions to derive the others (see [START_REF] Rescher | Many-valued Logic[END_REF], [START_REF] Gottwald | A treatise on many-valued logics[END_REF]). Table I gives the resulting truth tables, using the notations introduced above i.e. denoting 0 as τ 0 , 1/2 as τ 1 and 1 as τ 2 .

Post system is based on the definition of the truth tables for disjunction and negation, from which the other connectives are then derived. It is based on a very specific, cyclic, negation; as a consequence, it does not satisfy the normal conditions, i.e. it does not coincide with the classic binary logic case when restricted to the extreme degrees {0, 1}, which makes it difficult to interpret.

The three other systems satisfy the normal conditions. The Bochvar system considers the intermediate value τ 1 as an absorbing element, which makes any evaluation τ α τ β = τ 1 , for all operators ∈ {∨, ∧, →} if τ α or τ β equals τ 1 . Indeed, Bochvar considers that τ 1 corresponds to ignorance, which "contaminates" any evaluation. As a consequence, τ 0 ∧τ 1 = τ 0 and τ 2 ∨ τ 0 = τ 2 .

Łukasiewicz and Kleene systems return the intuitive values in these cases, they only differ in the evaluation of τ 1 → τ 1 : it is a tautology for Ł 3 but not for K 3 . Indeed, K 3 exploits a partial truth interpretation of the intermediate degree τ 1 and considers that if both premise and conclusion are half true, so is the implication.

The issue of tautologies in tri-valent and general manyvalued logics is discussed in details in [START_REF] Rescher | Many-valued Logic[END_REF]. It has been proposed as a counter-argument to a truth table approach [START_REF] Dubois | On ignorance and contradiction considered as truth-values[END_REF]. However it may be considered that the use of a general many-valued framework, that allows for several intermediary truth degrees between true and false, i.e. in the case where M > 3, usually leads to abandon the principle of excluded middle (see e.g. [START_REF] Rescher | Many-valued Logic[END_REF] for a detailed discussion): it can thus be considered as relevant that some tautologies are not recovered.

2) General Case: Table II summarises six main manyvalued logics, as can be found for instance in [START_REF] Rescher | Many-valued Logic[END_REF], using the notations defined in the previous section. They are usually based on the definition of negation and a binary connective, as well as some syntactic equivalences which make it possible to derive the other binary connectives. The table omits a common alternative for implication, denoted τ α ⊃ τ β = ¬τ α ∨ τ β , that is considered in the systems U M , S M and P M and that leads to define τ α ⊃ τ β = max(¬τ α , τ β ) in all three cases.

In the case of B M , two cases must actually be distinguished, depending on whether M is odd or even, which determines whether τ M -1 2 belongs to the scale or not. In the latter case, τ M -1 2 must be replaced by τ M -1 2 , i.e. the degree immediately below the intermediate level that does not belong to the scale. In Table II, both are denoted τ M -1 2 to simplify the notations. The U M system only considers the case of odd M .

Ł M generalises the trivalent Łukasiewicz logic, B M the Bochvar logic, S M using the ⊃ implication the Kleene logic, G M the Gödel logic and P M the Post logic [START_REF] Rescher | Many-valued Logic[END_REF].

Except G M and P M , all systems agree on the negation definition. G M negation only allows two values as result, falling back to a binary logic. P M considers an atypical cyclic definition, generalising the one of P 3 indicated in Table I. It can be underlined that for all systems except G M and P M , ¬τ M -1

2 = τ M -1 2
, for G M , it equals τ 0 , i.e. false, and for P M , τ M -1 2 -1 .

Except B M , all systems agree on the definition of disjunction and conjunction, whether they are directly defined or derived from other connectives. In the case of Ł 3 , a so-called strong variant of these connectives is also proposed, according to which

τ α ∨ τ β = ¬τ α → τ β = min(τ α+β , τ M -1 ) and then τ α ∧ τ β = max(τ α+β-(M -1) , τ 0 ). With this variant, τ α ∨ τ α = τ α , except for τ M -1 2
, whereas it is the case for the consensus definition as max(τ α , τ β ). Regarding B M , the disjunction has value τ M -1 2 except in the cases requiring compatibility with binary logic, i.e. when both τ α , τ β ∈ {τ 0 , τ M -1 }: even more than in the tri-valued case, τ M -1 2 plays a pervasive role, B M actually reduces to B 3 after a few degree combination steps.

More variations are observed for the definition of the implication connective: all systems agrees on value τ M -1 if τ α ≤ τ β , but otherwise, the resulting value ranges from τ 0 (for S M ) to τ β (for G M ), or greater values depending on the difference between τ α and τ β .

B. L e

M : Introduction of an Ignorance Degree, τ ? It has been proposed to extend the previous many-valued framework, in order to distinguish explicitly between neutral

Ł M U M B M ¬τα τ M -1-α τ M -1-α τ M -1-α τα → τ β τ M -1 if τα ≤ τ β τ M -1-(α-β) otherwise        τ M -1 if τα ≤ τ β τ M -1 2 if τα ≥ τ β and α -β ≤ M -1 τ 0 otherwise    τ M -1 if τα = τ 0 and τ β ∈ {τ 0 , τ M -1 } τ β if τα = τ M -1 and τ β ∈ {τ 0 , τ M -1 } τ M -1 2 otherwise τα ∨ τ β (τα → τ β ) → τ β ¬(¬τα ∧ ¬τ β ) ¬(¬τα ∧ ¬τ β ) max(τα, τ β ) max(τα, τ β ) max(τα, τ β ) if τα, τ β ∈ {τ 0 , τ M -1 } τ M -1 2 otherwise τα ∧ τ β ¬(¬τα ∨ ¬τ β ) min(τα, τ β ) min(τα, τ β ) min(τα, τ β ) if τα, τ β ∈ {τ 0 , τ M -1 } τ M -1 2 otherwise S M G M P M ¬τα τ M -1-α τ M -1 if τα = τ 0 τ 0 otherwise τ M -1 if τα = τ 0 τ α-1 otherwise τα → τ β τ M -1 if τα ≤ τ β τ 0 otherwise τ M -1 if τα ≤ τ β τ β otherwise τ M -1 if τα = τ 0 max(τ α-1 , τ β ) otherwise τα ∨ τ β max(τα, τ β ) max(τα, τ β ) max(τα, τ β ) τα ∧ τ β min(τα, τ β ) min(τα, τ β ) min(τα, τ β )
truth value and ignorance, and thus to remove the ambiguity between these two notions [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF]: L e M introduces an additional value to assess a given formula, denoted τ ? . L e M is not proposed as a reasoning framework, but as a framework to perform symbolic multi-criteria evaluation, more precisely for a task of information scoring: τ ? is primarily introduced to allow abstaining from evaluating some of the criteria and to make an explicit difference with the case where they have a neutral assessment.

As a consequence, τ ? is proposed to satisfy the following properties, where denotes a computational multivalued binary operator:

(P 1 ) τ ? ∈ L M (P 2 ) τ ? τ α = τ α τ ? = τ α forall τ α ∈ L M (P 3 ) ¬τ ? = τ ? (P 4 ) τ ? τ ? = τ ?
(P 1 ) sets τ ? apart from other degrees, not submitting it to the same ordering constraints as τ α ∈ L M : an element which cannot be evaluated cannot be compared to an element whose evaluation is known, no matter its value. Therefore, τ ? is defined as an exception to the total order rule and of a different nature than other degrees. (P 2 ) defines τ ? as a neutral element for all considered manipulation operators: it applies a principle according to which what is not known should not be taken into account when combining the evaluation of several criteria. This can be interpreted as a neutral posture, for instance as opposed to an optimistic one that would replace the unknown by the highest possible degree. The aim of this paper is to study this principle in the case where logical operations are considered, i.e. in the case ∈ {∧, ∨, →}.

The last two properties (P 3 ) and (P 4 ) define the behaviour of τ ? when combined with itself.

The evaluation is then performed on L M extended with τ ? , denoted by L e M = L M ∪{τ ? }. It is shown to offer an essential gain in expressiveness, illustrated in an information scoring context.

III. LOGICAL CONNECTIVES IN L e M

The additional truth value τ ? has been introduced to manipulate the degrees at an arithmetical level [START_REF] Revault D'allonnes | Formalising information scoring in a multivalued logic framework[END_REF], its reasoning properties have not been studied. This section examines extended semantics for the logical connectives, to study how they can process τ ? . After discussing the cases of negation, disjunction and conjunction, the section details the case of implication, for which many choices can be considered. Of course, for the combination of τ α and τ β in L M , i.e. distinct from τ ? , classical operators as summarised in Table II can be used. The issue is to define their combination with τ ? , as well as its combination with itself.

A. Negation, Disjunction and Conjunction Semantics in L e

M

This section discusses the cases of the negation, disjunction and conjunction connectives in turn.

1) Negation Connective: We propose to set

¬τ α = τ M -1-α if τ α ∈ L M ¬τ ? = τ ? (1)
The first line agrees with most classical negation (see Table II), the second one is identical to property (P 3 ). They correspond to a classic choice for L M and an arguably natural one for τ ? : there does not seem to be a reason why the negation of ignorance should be made to a known value, nor how one should be selected. This definition leads to a similarity in behaviour between τ ? and τ M -1 2 : they are both equal to their own negation.

2) Disjunction Connective: As for instance discussed in [START_REF] Gottwald | A treatise on many-valued logics[END_REF], and using the notations introduced in Section II, a disjunction operator is required to satisfy the properties of associativity, commutativity, non-decreasing monotonicity in both arguments and to have τ 0 as neutral element.

In the case of L e M however, two of these properties need to be questioned: first, as the set of truth values is not ordered, by definition (property (P 1 )), the monotonicity requirement cannot be defined and thus will not apply. Second, the definition of τ 0 as neutral element is also reconsidered.

Indeed, as mentioned in Section II-B, an underlying principle of L e M states that what is not known should not be taken into account. It leads to set, for all

τ α ∈ L e M , τ ? ∨ τ α = τ α (2)
considering that, in absence of knowledge, the disjunction is as true as its known component, in agreement with property (P 2 ). This equation leads to τ ? ∨ τ ? = τ ? which seems to be a natural choice. Note that as a consequence of the negation definition, it also leads to consider that ¬τ ? ∨ τ ? = τ ? , which may be debatable [START_REF] Dubois | On ignorance and contradiction considered as truth-values[END_REF]. However, as mentioned before, manyvalued logic does not require the excluded middle to hold, so it can be argued that it should not be required for the specific case of τ ? either.

Another consequence of this choice is that τ 0 ∨ τ ? = τ 0 , meaning that τ 0 is not a neutral element. However, it still holds that τ M -1 is an absorbing element: for τ α ∈ L M , the proposed definition does not modify the behaviour, therefore τ M -1 ∨ τ α = τ M -1 , and for τ ? , it holds that τ M -1 ∨ τ ? = τ M -1 .

This behaviour defined for τ ? differs from the one for τ M -1 2 : in most cases (see Table II), it holds that τ M -1

2 ∨ τ α = τ α only for τ α ≥ τ M -1 2 , otherwise τ M -1 2 ∨ τ α = τ M -1 2
: τ ? is a neutral element for disjunction, whereas τ M -1 2 only is for greater values, it is absorbing otherwise.

3) Conjunction Connective: The definition of conjunction semantics can be derived from that of the disjunction and negation (Eq. ( 1) and ( 2)), based on the De Morgan laws:

τ α ∧ τ β = ¬(¬τ α ∨ ¬τ β ) and, therefore, τ α ∧ τ ? = ¬(¬τ α ∨ τ ? ) = ¬(¬τ α ) = τ α : as a consequence τ ? ∧ τ α = τ α (3)
Another approach to define conjunction can be based on a discussion similar to the one conducted for the disjunction in the previous subsection. It leads to an identical result as the one derived using the De Morgan laws.

Usually, a conjunction operator is rquired to satisfy the properties of associativity, commutativity, non-decreasing monotonicity and to have τ M -1 as neutral element (see e.g. [START_REF] Gottwald | A treatise on many-valued logics[END_REF]): as for disjunction, monotonicity and neutral element do not hold in L e M , however, τ 0 remains an absorbing element.

B. Implication Semantics in L e

M

The implication definition opens the way to more discussions: there is no general agreement on the properties an implication operator should satisfy (see e.g. [START_REF] Gottwald | A treatise on many-valued logics[END_REF] and [START_REF] Yager | On some new classes of implication operators and their role in approximate reasoning[END_REF]) and numerous derivations from disjunction, conjunction and negation can be considered (see for instance [START_REF] Gottwald | A treatise on many-valued logics[END_REF]). They are discussed in turn below.

1) Residuated Implications: R-implications are derived from conjunction and defined as τ α → τ β = sup{τ |τ α ∧ τ ≤ τ β }. As a consequence, this approach cannot be applied in the L e M case: it requires a total ordering on the considered truth values, whereas the essential characteristic of L e M precisely is its absence of total order, that makes it possible to introduce the crucial difference between τ ? and any truth degree τ α .

More precisely, due to the conjunction operator (Eq. 3), if τ α = τ ? , the r-implication definition can lead to set τ ? → τ β = sup{τ |τ ? ∧ τ ≤ τ β } = sup{τ |τ ≤ τ β } = τ β . However, the value for τ α → τ ? cannot be established using this definition, nor that of τ ? → τ ? .

2) Strong Implications: S-implications are derived from negation and disjunction as τ α → τ β = ¬τ α ∨ τ β . Using the definitions of Eq. ( 1) and ( 2), if τ α and τ β denote values different from τ ? (i.e. τ α and τ β ∈ L M ), it leads to

τ ? → τ β = τ β τ α → τ ? = ¬τ α τ ? → τ ? = τ ? (4)
For τ ? → τ β , this definition leads to the same value as the one suggested by r-implications. However, it also allows to process the other cases. Setting τ ? for the case τ ? → τ ? is a natural choice, and is in agreement with property (P 4 ). The specificity comes from τ α → τ ? = ¬τ α ∨ τ ? = ¬τ α because of the disjunction definition given in Eq. ( 2).

This choice allows to make the difference between τ ? and τ M -1 2 : depending on the values of τ α and τ β , neither does τ M -1 2 → τ β always equal τ β , nor does τ α → τ M -1 2 always equal ¬τ α . Thus there are cases where the result differs between τ ? and τ M -1 2 .

3) Quantum Logic Implications: QL-implications are derived from conjunction and negation as τ α → τ β = ¬(τ α ∧ ¬(τ α ∧ τ β )) Using the definitions given in Eq. ( 1) and (3), if τ α and τ β denote values different from τ ? , they lead to

τ ? → τ β = τ β τ α → τ ? = ¬τ α ∨ τ α τ ? → τ ? = τ ? (5)
This approach agrees with the previous ones for τ ? → τ β and τ ? → τ ? , the difference comes from the τ α → τ ? , whose result depends on the choice of the disjunction for values in L M : it can be a tautology, i.e. τ M -1 , if the law of excluded middle is imposed, or τ max(α,M -1-α) using classic variants (see Table II). As s-implications, it leads to different behaviours for τ M -1 2 and τ ? . 4) Proposed Implication: We propose another variant for implication with ignorance, which can be considered as a radical choice that always produces τ ? , in agreement with (P 2 ):

τ ? → τ β = τ ? τ α → τ ? = τ ? τ ? → τ ? = τ ? (6)
This choice relies on the principle that if the premise or the conclusion is unknown, it is not relevant to try to make a guess, and to try for instance a cautious or, on the contrary, an optimistic choice for the unknown value: the strong implication can be interpreted as using the minimal value that can be established, in a cautious approach. Indeed, for instance for the case τ α → τ ? , if the conclusion turns out to be τ 0 , the truth value of the implication would be τ α , whereas it would be τ M -1 if the conclusion turns out to be τ M -1 . Setting the result as τ α thus corresponds to the most cautious choice. The same interpretation applies to the case τ ? → τ β .

The proposed implication rather keeps the unknown state and propagates it: if either the premise or the conclusion cannot be evaluated, it is considered that the truth value of the implication cannot be evaluated either. The next section examines the consequence of this choice on the inference process to derive new pieces of information.

Obviously, this choice it leads to different behaviours for τ M -1 2 and τ ? .

C. Consistency Issue

The consistency issue of a logical system is related to its impossibility to infer a contradiction. In the case of manyvalued logic, this property is already relaxed since the laws of excluded middle and noncontradiction usually do not hold. Now adding τ ? does not introduce inconsistency: for disjunction and conjunction, τ ? is defined as a neutral element, for the proposed implication it is a absorbing element. As a consequence, any inconsistency proved after its inclusion will persist should it be removed.

IV. LOGICAL INFERENCE IN L e M

The inference task aims at establishing new pieces of knowledge, i.e. new formulae that can be derived from given formulae, together with their truth values. After discussing the considered issue, this section describes the proposed inference scheme based on an extended Modus Ponens rule and illustrates it in the case of L e 5 .

A. Principle and Discussion

Inference in many-valued logic (see e.g. [START_REF] Gottwald | A treatise on many-valued logics[END_REF]) usually is a binary question, based on the notion of designated truth values, i.e. a set of truth degrees that are interpreted as degrees of truth, as opposed to anti-designated values, interpreted as degrees of falsity: inference requires that if a set of formulae has designated truth values, then a derived formula also has a designated truth value.

The issue considered here is closer to the inference question in fuzzy logic, where the aim is to precisely identify the truth value of the derived formula: formally, given a set of propositional formulae associated with their truth values F = {(F i , τ i ), i = 1..n}, where F i are formulae and τ i ∈ L e M , the question is to examine what can be established for a given formula G regarding its truth value. It is thus related to the issue of tableau calculus for many-valued logic (see e.g. [START_REF] Gottwald | A treatise on many-valued logics[END_REF] for an overview).

The next section discusses the case of Modus Ponens, also called rule of detachment, i.e. it considers F = {(F, τ F ), (F → G, τ I )}, where F and G can be compound complex formulae, and when τ F , τ I ∈ L e M : the question is to possibly infer (G, τ G ), establishing the value of τ G .

B. Extended Modus Ponens

In binary logic, the Modus Ponens inference rule allows to infer G from F = {F, F → G} as the single interpretation that makes both the observation F and the rule F → G true also makes G true. In fuzzy logic, for instance considering the Łukasiewicz operators, F = {(F, u), (F → G, v)} where u, v ∈ [0, 1], the Generalised Modus Ponens derives (G, max(0, u + v -1)).

In the case of propositional L e M , we propose to examine all interpretations compatible with F i.e. the set T = {τ | i(τ F , τ ) = τ I } where i is the proposed implication operator discussed in the previous section. The definition of τ G depends on the cardinality of T and distinguishes between three cases:

1) if |T | = 1: i.e. if a single interpretation is compatible, then the truth value of G naturally equals this value.

2) if |T | > 1: i.e. if several interpretations are possible, several approaches can be considered: one consists in abstaining from taking a decision and allowing truth values as subsets of L e M . This approach can be related to the manyvalued tableau calculus, but may be difficult to interpret.

Another approach consists in aggregating the obtained values: a cautious choice can for instance select the minimal value, i.e. setting τ G = min{τ ∈ T }, indicating that at least this degree can be achieved. This definition makes sense under the assumption that T does not contain both τ ? and values from L M , otherwise the min operator cannot be applied. Now as illustrated in the next section, when using the proposed implication defined in Eq. ( 6), the only problematic case is obtained if τ F = τ I = τ ? : in such a case, it seems reasonable to set that τ G = τ ?

A third approach can also consider that if several values can be obtained, it means that the available pieces of information are not conclusive enough, i.e. that the rule together with the observation does not allow to conclude: this principle leads to set τ G = τ ? in such a case.

3) if T = ∅: i.e. no interpretation is compatible with F, then we propose to set τ G = τ ? , meaning that the rule does not allow to infer any new piece of information.

It may be argued that this case corresponds to a contradiction and requires to be processed as such. However the expressiveness allowed by the introduction of τ ? allows to take a less definite decision. Indeed, if τ G = τ ? , using the proposed implication defined in Eq. ( 6), the truth value of the implication F → G gets value τ → τ ? = τ ? which is not contradictory with the available value τ I as τ I ∧ τ ? = τ I due to the conjunction operator defined in Eq. (3).

C. Illustration in L e

5

This section illustrates the proposed extended Modus Ponens in the case of L e 5 , considering for implication operator

τ I τ 0 τ 1 τ 2 τ 3 τ 4 τ 0 τ 0 τ 0 τ 0 τ 0 τ 0 τ 1 τ 0 τ 0 τ 0 τ 0 τ 1 τ F τ 2 τ 0 τ 0 τ 0 τ 1 τ 2 τ 3 τ 0 τ 0 τ 1 τ 2 τ 3 τ 4 τ 0 τ 1 τ 2 τ 3 τ 4
the proposed operator defined in Eq. ( 6) combined with the Łukasiewicz operator for values other than τ ? (see first column of Table II). Table III gives the obtained results, where the colours give details about the computations: blue values correspond to the case where T = ∅ and green ones to the case where |T | > 1, using the min operator as aggregation.

Table IV gives the result obtained when applying a fuzzylike Generalized Modus Ponens (as mentioned in the previous section) to the discrete case of L 5 , as a comparison.

In the case where it is not known whether the rule F → G holds, i.e. when τ I = τ ? (see last column in Table III), it seems reasonable that the extended Modus Ponens leads to the conclusion that τ G = τ ? , whatever the truth value of F is: indeed, no information can be inferred from an unknown rule. This desired result is recovered due to the fact that if τ F = τ ? , the only way to obtain τ ? for the implication F → G comes from the case where τ G = τ ? . If τ F = τ ? , actually T = L e 5 and thus contains both contains τ ? and truth degrees. However it seems reasonable to set τ G = τ ? in this case.

When it is not known whether the observation F holds, i.e. when τ F = τ ? (see last row in Table III), then whatever the truth value of the rule, again, as can be desired, the truth value of the conclusion is τ ? . However, the justification for these values differs from the previous ones: they come from the fact that then T = ∅.

All the previous cases correspond to cases where, indeed, F and F → G alone do not justify deriving the conclusion G, making the resulting value τ ? relevant. Note that it may be the case that G holds, due to other inferences or as initial piece of knowledge, but the derivation through F and F → G is not conclusive. However, if a known truth degree τ can be established from a different inference, the combination will lead to τ ∨ τ ? = τ , preserving the identified piece of information. This also justifies the choice of the semantics for the disjunction connective discussed in the previous section.

Other cases where the result is τ ? can be observed in the upper left part of Table III: again, they can be interpreted as cases where the observation together with the rule are not conclusive, because either one or the other have too low truth degrees. This behaviour differs from the GMP one (see Table IV), where the conclusion is considered to be false, as τ G = τ 0 . The choice of not triggering the rule and getting τ ? as a result can be considered as relevant, even if debatable. The rest of the two tables is identical, in particular thanks to the considered min aggregation for the blue values. It can be argued that the fact that a conclusion is considered false (τ G = τ 0 ) if both the rule and the observation are half true (τ M -1 2 = τ 2 ) is not intuitive, but this is a result of building here the inference scheme on the Ł 5 implication operator, for which τ 2 → τ 0 = τ 2 : other implication choices can lead to other results, although the whole table needs to be recomputed and interpreted.

If the observation is true (τ F = τ 4 ) and the rule is half true (τ I = τ 2 ), then the conclusion is considered to be half true, whereas it is unknown, τ ? , if the rule also is: beside allowing to avoid triggering the rule in inappropriate cases, as discussed above, the introduction of τ ? also allows to distinguish between ignorance and neutral value, in agreement with its initial motivations.

V. CONCLUSION AND FUTURE WORKS

This paper discussed, from a logical point of view, the introduction of an additional truth value, distinct from a truth degree, in the framework of many-valued logic, so as to enable to explicitly distinguish between neutral truth and ignorance: τ ? places the comparison between a 'half true' fact and another back in its rightful place in L M 's semantic hierarchy. The definition of appropriate connective semantics has been discussed, illustrating alternative possibilities in particular for implication. For a specific choice, the issue of inference through an extended Modus Ponens has also been discussed, highlighting how τ ? allows to increase the legibility and expressiveness of the induced reasoning framework through the allowed disambiguisation. Ongoing works aim at further studying L e M at a theoretical level, in particular examining other inference schemes, as well as at an applicative level, to assess the practical relevance of the proposed tools for a real-life approximate reasoning framework.

TABLE I TRUTH

 I TABLES FOR SOME CLASSIC TRI-VALENT LOGICS

	Negation Post Others

TABLE II OVERVIEW

 II OF SIX CLASSIC MANY-VALUED LOGICS, AS CAN BE FOUND E.G. IN[START_REF] Rescher | Many-valued Logic[END_REF]

TABLE III TRUTH

 III TABLE OF THE PROPOSED INFERENCE SCHEME, BASED ON EQ. (6) AND THE ŁUKASIEWICZ IMPLICATION.

					τ I
			τ 0	τ 1	τ 2	τ 3	τ 4	τ ?
		τ 0	τ ?	τ ?	τ ?	τ ?	τ 0	τ ?
		τ 1	τ ?	τ ?	τ ?	τ 0	τ 1	τ ?
	τ F	τ 2	τ ?	τ ?	τ 0	τ 1	τ 2	τ ?
		τ 3	τ ?	τ 0	τ 1	τ 2	τ 3	τ ?
		τ 4	τ 0	τ 1	τ 2	τ 3	τ 4	τ ?
		τ ?	τ ?	τ ?	τ ?	τ ?	τ ?	τ ?

TABLE IV TRUTH

 IV TABLE OF THE ŁUKASIEWICZ GMP INFERENCE SCHEME.


Iñaki Etaio
email: inaki.etaio@ehu.eus

Sophie Meillon

Francisco J Pérez-Elortondo

Pascal Schlich

Dynamic sensory description of Rioja Alavesa red wines made by different winemaking practices by using Temporal Dominance of Sensations

Keywords: TDS, Rioja wine, carbonic maceration, sensory analysis, Tempranillo

BACKGROUND: Although sensory description of wines in scientific literature is very large, there is an evident lack of studies describing wines from a dynamic approach. The objective of this study was to describe the evolution of the sensations perceived in red wines from Rioja Alavesa by using Temporal Dominance of Sensations (TDS) and also to compare wines made with the two winemaking procedures used in Rioja Alavesa: carbonic maceration (CM) and destemming (DS). RESULTS: Ten sensory attributes were evaluated in eight wines (four CM and four DS wines) in triplicate by a panel of 16 trained assessors. Red/black berry and woody aromas were dominant firstly, whereas heat, astringent, bitter and pungent sensations were dominant later. CM wines showed higher dominance for woody, spicy, pungent and acid sensations and lower dominance for red/black berry aroma and astringency than DS wines.

CONCLUSION: This study is the first to describe Rioja wines from a dynamic approach and it also provides information about the dynamic sensory differences between wines made by CM or by DS. In this sense, this work shows the usefulness of TDS to describe and differentiate wines and to provide additional information to the conventional static descriptive analysis.

INTRODUCTION

Rioja Alavesa (RA) is one of the three sub-zones in the Rioja wine region included in the Protected Designation of Origin (PDO) Rioja. Among the grape varieties accepted for winemaking, Tempranillo is the predominant one in red wines (96.36% of the surface cultivated with red grape varieties). [START_REF]Estadísticas 2013[END_REF] Most of the red wines are made exclusively with the Tempranillo grape variety or by mixing Tempranillo with other grape varieties in a low proportion (usually lower than 10%). These minor grape varieties also include some white grape varieties such as Viura (also called Macabeo) which is the main one.

Young wine is the most typical wine in RA and it has been traditionally made by the carbonic maceration (CM) technique. CM is also used in different European regions to make traditional wines, such as Médoc, Bordeaux, Beaujolais or Provence. [START_REF] Ribéreau-Gayon | Sciences et Techniques du Vin. Tome III. Vinifications[END_REF][START_REF] Carnacini | Caratteristiche di vini ottenuti mediante macerazione carbonica[END_REF] The maintenance of many small, family wineries in RA has perpetuated this practice although the use of destemming (DS), i.e. winemaking by crushing the grapes before fermentation, is progressively displacing CM.

CM was described for the first time by Flanzy [START_REF] Flanzy | Nouvelle méthode de vinification[END_REF] and it refers to placing the entire grapes in a vat within a CO 2 atmosphere (anaerobiosis) for several days so that several enzymatic reactions take place without yeast participation: ethanol production (until 20-30 mL L -1 ), malic acid consumption and polyphenol transfer from skin to the pulp. Then, the grapes break down because of the internal pressure and the fermentation by yeast starts.

Actually, CM practised in RA is not a strict CM since some grapes break when they are placed in the vat (thus initialising the fermentation by yeast, so both phenomena take place simultaneously), SO 2 is added and CO 2 is not. [START_REF] Jaime | Experiencias a escala semi-industrial de vinificación en tinto con maceración carbónica no estricta[END_REF] Next, free run must from the grapes and must from pressed clusters are mixed to continue the alcoholic fermentation, usually without the skins. In the DS process, grapes are crushed and fermentation by yeast occurs immediately in contact with skins.

Although RA wines are recognised as high quality wines, scientific reports describing their organoleptic characteristics using sensory panels are very scarce. [START_REF] Etaio | Development of a quantitative sensory method for description of young red wines from Rioja Alavesa[END_REF] Regarding sensory characteristics that differentiate CM wines from DS wines, the scientific literature is quite general and ambiguous. In addition, the sensory practices used do not always meet the minimum required in scientific sensory analysis. Odour of CM wines has been described as 'distinctive' and 'rich'. [START_REF] Carnacini | Caratteristiche di vini ottenuti mediante macerazione carbonica[END_REF][START_REF] Lõrincz | Examination of volatile components in carbonic macerated red wines[END_REF][START_REF] Flanzy | Vinificación por maceración carbónica, in Enología: Fundamentos Científicos y Tecnológicos[END_REF] Sensory attributes frequently used to describe the aromas of these wines are related to fruits, i.e. generic fruit, [START_REF] Jaime | Experiencias a escala semi-industrial de vinificación en tinto con maceración carbónica no estricta[END_REF][START_REF] Lõrincz | Examination of volatile components in carbonic macerated red wines[END_REF][START_REF] Alvarez | Experiencia de maceración carbónica con la variedad Tempranillo en Tierra de Barros. Campañas 96-97[END_REF] or specific fruit attributes such as strawberry and raspberry [START_REF] Versini | Confroto tra i componenti volatile dei vini rossi ottenutti con macerazione tradizionale e macerazione carbonica. Importanza differenziante del cinnamato di etile[END_REF] or cherry and kirsch. [START_REF] Ducruet | Comparison of the headspace volatiles of carbonic maceration and traditional wine[END_REF] Other aromas reported are floral [START_REF] Lõrincz | Examination of volatile components in carbonic macerated red wines[END_REF][START_REF] Alvarez | Experiencia de maceración carbónica con la variedad Tempranillo en Tierra de Barros. Campañas 96-97[END_REF] and spicy. [START_REF] Carroll | Effects of carbonic maceration on chemical, physical and sensory characteristics of muscadine wines[END_REF] Regarding sensations in mouth, CM wines have been described as soft and velvety [START_REF] Ribéreau-Gayon | Sciences et Techniques du Vin. Tome III. Vinifications[END_REF][START_REF] Jaime | Experiencias a escala semi-industrial de vinificación en tinto con maceración carbónica no estricta[END_REF][START_REF] Lõrincz | Examination of volatile components in carbonic macerated red wines[END_REF][START_REF] Carroll | Effects of carbonic maceration on chemical, physical and sensory characteristics of muscadine wines[END_REF][START_REF] Fuleki | Application of carbonic maceration to change the bouquet and flavor characteristics of red table wines made from Concord grapes[END_REF] but without comparing them directly with wines made by DS. Among the very few studies comparing CM and DS wines by using sensory panels, Fuleki [START_REF] Fuleki | Application of carbonic maceration to change the bouquet and flavor characteristics of red table wines made from Concord grapes[END_REF] reported that CM wines from Concord grapes (Vitis labrusca) presented a softer taste and better flavor (lower Concord character) than DS wines made with the same grapes. In the case of Tempranillo wines from RA, Etaio et al. [START_REF] Etaio | Effect of winemaking process and addition of white grapes on sensory and physicochemical characteristics of young red wines[END_REF] described CM wines as stronger in red berry odour and aroma, alcoholic aroma and acidity, and lower in licorice and tree fruit odour than DS wines.

In all the works found in the scientific literature sensory description of CM wines has been made by the traditional approach, that is, by evaluating the product in a static way, without considering the evolution of the sensations in the mouth throughout the time. The Temporal Dominance of Sensations (TDS) technique [START_REF] Pineau | Temporal Dominance of Sensations: Construction of the TDS curves and comparison with time-intensity[END_REF] -17 allows the sensory characteristics elicited by food products to be described by taking into account the temporal dimension. TDS consists of identifying and rating sensations perceived as dominant throughout the tasting and enables the different sensations to be evaluated simultaneously. After introducing the food or drink into the mouth, each assessor has to indicate all the sensory descriptors (attributes) perceived as dominant among a list of attributes displayed on the computer screen throughout the evaluation, and also she/he has to rate their intensity on a continuous scale. Data collected from all the assessors are then analysed as described in Labbe et al. [START_REF] Labbe | Temporal dominance of sensations and sensory profiling: A comparative study[END_REF] and Pineau et al. [START_REF] Pineau | Temporal Dominance of Sensations: Construction of the TDS curves and comparison with time-intensity[END_REF] to calculate dominant rates for each attribute.

The TDS technique has been successfully applied to evaluate several food products and drinks, such as gels containing odorants, [START_REF] Labbe | Temporal dominance of sensations and sensory profiling: A comparative study[END_REF] alcoholic beverages, [START_REF] Deleris | Impact of swallowing on the dynamics of aroma release and perception during the consumption of alcoholic beverages[END_REF] candies, [START_REF] Saint-Eve | How texture influences aroma and taste perception over time in candies[END_REF] extra-virgin olive oil with pureed bean and tomatoes, [START_REF] Dinnella | Sensory functionality of extra-virgin olive oil in vegetable foods assessed by Temporal Dominance of Sensations and descriptive analysis[END_REF] blackcurrant squashes, [START_REF] Ng | Using quantitative descriptive analysis and temporal dominance of sensations analysis as complementary methods for profiling commercial blackcurrant squashes[END_REF] coffee, [START_REF] Barron | Impact of crema on the aroma release and the in-mouth sensory perception of espresso coffee[END_REF] fish sticks, [START_REF] Albert | Comparison between temporal dominance of sensations (TDS) and key-attribute sensory profiling for evaluating solid food with contrasting textural layers: Fish sticks[END_REF] beer, [START_REF] Vázquez-Araújo | Comparison of temporal-sensory methods for beer flavor evaluation[END_REF] ice-creams [START_REF] Varela | How hydrocolloids affect the temporal oral perception of ice cream[END_REF] or wine. [START_REF] Sokolowsky | Evaluation of bitterness in white wine applying descriptive analysis, time-intensity analysis, and tem-poral dominance of sensations analysis[END_REF] -30 For a more detailed list of studies applying TDS see Di Monaco et al. [START_REF] Monaco | Temporal Dominance of Sensations: A review[END_REF] The main objective of the present work was to describe the temporal sensory perception in mouth of eight red wines from Rioja Alavesa using the TDS method and to study the sensory differences related to the winemaking process, i.e. carbonic maceration versus destemming.

EXPERIMENTAL Wines

Eight young red wines from different wineries from RA were analysed in the study. Four wines were made by CM and four wines by DS. In each group of wines (CM and DS), three wines were made exclusively with the Tempranillo variety and the other wine was made with Tempranillo but also adding Viura variety grapes (approximately 10% of the total grape weight). This wine with Viura grapes was included in each group since adding a small percentage of white grape varieties to make red wine is a usual practice carried out by many wineries from RA and it is expressly accepted in the regulations of PDO Rioja. 31 As wines were produced to be commercialised as PDO Rioja, enological practices followed by the wineries fitted the rules of this PDO. All the wines underwent malolactic fermentation. Bottles (750 mL) were picked up directly from the wineries and kept laid down in a cellar (12 ± 3 ∘ C and 90 ± 5% of relative humidity) until being analysed. At the laboratory, wines were stored at 10 ∘ C. Twenty-four hours before being served, wines were put at room temperature in order to reach the serving temperature of 21 ± 1 ∘ C. Wine bottles were opened 1 h before the evaluation. The main chemical characteristics of the wines are shown in Table 1.

Assessors

Sixteen French assessors (nine women and seven men) aged 29-65 years (mean age 45 years) participated in this study which took place at the Centre des Sciences du Goût et de l'Alimentation (CSGA) in Dijon (France). They were selected according to their results in previous sensory discrimination, ranking and recognition tests and had recently participated in a TDS panel training and red wine sensory profile (sixteen 1-h sessions) for 3 months. They were paid for their involvement in the study.

Training sessions

Assessors were trained during three 1-h sessions to TDS evaluation of the RA wines. In the first session descriptive sensory attributes were generated for the eight wines of the study. Assessors were instructed about the tasting protocol: [START_REF] Meillon | Contribution of the Temporal Dominance of Sensations method to the sensory description of subtle differences in partially dealcoholized red wines[END_REF] at zero seconds, put the wine in mouth; at 12 s, inspire three times drawing air through the wine; at 20 s, click the tongue three times and wait until no more sensations are perceived. Throughout the evaluation of each sample, assessors had to write down all the mouth sensations they perceived as dominant. The 10 more cited terms were selected and kept for further TDS evaluations. The two following sessions were dedicated to check the appropriateness of the attribute list, to define the sensory reference corresponding to each attribute and to train the panel in sample evaluation by using TDS method. In both sessions sensory references and wine samples were evaluated and discussed in order to reach a consensus about the attributes perceived in the samples and the references that best reproduce them. The final 10 attributes selected for the study (four aromas and six gustative and mouth-feel sensations) and the composition of the sensory references are shown in Table 2.

Wine evaluation by TDS

Wine evaluations by TDS took place in isolated sensory booths at standardised temperature (20 ± 1 ∘ C) and under red light to mask wine colour. Wine samples were presented in plastic opaque cups (10 mL) coded randomly with three digits.

Evaluations were carried out in three 1-h sessions in order to have three replications. In each session, the eight wines of the study were monadically presented according to a Williams Latin Square design balanced for order and carry-over effects.

Data acquisition was achieved with Fizz Software (BIOSYSTEM, Couternon, France) with a period of 500 ms. The list of the 10 attributes was displayed on a computer screen associated with unstructured scales (30 cm), ranging from weak (left) to strong (right). The order of the attributes in the list was randomised across assessors to balance the fact that first attributes of the list could be more often cited. [START_REF] Pineau | Temporal Dominance of Sensations: What is a good attribute list?[END_REF] Assessors put all the cup content in mouth and clicked the 'start' button to start the temporal sensory evaluation. Once the chronometer started, they identified and rated the intensity of the sensations they perceived as dominant while performing the tasting protocol. Score recording extended throughout 120 s although when not perceiving sensations anymore, assessors had to click 'stop' button. A 2 min break was established between samples to eliminate residual sensations by rinsing the mouth with unsalted crackers and Evian water.

Data analysis

Data analysis was carried out as described by Meillon et al. [START_REF] Meillon | Contribution of the Temporal Dominance of Sensations method to the sensory description of subtle differences in partially dealcoholized red wines[END_REF] for producing the TDS curves for each wine and the TDS curves of differences between CM and DS wines. Curves were produced by importing the Fizz data into the TimeSens software (www.timesens.com).

The dominant rates of the different attributes are calculated for each wine by dividing the number of citations of each attribute (over all assessors and replications) by the total number of evaluations (number assessors × number replications). Since an assessor can have only a single dominant attribute at each time, the sum of the dominance rates over attributes is equal to 1 at each time. The higher the dominant rate, the better is the agreement among assessors (i.e. if 10 of the 16 assessors had marked an attribute as dominant at the same time, the dominance rate of this attribute is 10/16, which indicates a better agreement than, for instance, five assessors coinciding in an attribute, so with a dominance rate of 5/16).

The dominance rates of each attribute are then smoothed along time to produce a time-dominance curve. The different curves for a given wine are then displayed on a graph that also includes 'chance level' and 'significance level' curves. The 'chance level' line represents the dominance rate that an attribute can obtain by chance (1/number of attributes). The 'significance level' line, based on a binomial test, expresses the smallest value of the proportion being significantly (P ≤ 0.10) higher than the 'chance level'. When the curve of an attribute overcomes this significance level, it is considered as significantly dominant. In this manner, the sequence of the dominant sensations perceived in a product can be described.

In addition, TDS difference curves showing the attributes that are different between CM and DS wines were produced as described by Pineau et al. [START_REF] Pineau | Temporal Dominance of Sensations: Construction of the TDS curves and comparison with time-intensity[END_REF] These curves are drawn by subtracting time by time dominance rates from one product to another one. TDS difference curves were plotted only when a significant (P ≤ 0.10) difference was observed between the average dominance rate of the four CM wines and the average dominance rate of the four DS wines.

The table composed of dominance durations of products by subjects as observations and attributes as variables was analysed by Canonical Variate Analysis (CVA). The first two canonical variates were displayed as a biplot with superimposition of the 90% confidence ellipses of product means. CVA was complemented by T 2 Hotelling tests to compare the products two by two multi-variately based on their dominance durations. CVA was also carried out by using TimeSens software.

RESULTS AND DISCUSSION

Dynamic descriptions of the wines

The attributes selected to be evaluated in the RA wines of the present study (Table 2) did not fit, to a certain extent, the attributes previously reported for young red wines from Rioja Alavesa. [START_REF] Etaio | Development of a quantitative sensory method for description of young red wines from Rioja Alavesa[END_REF] It could be explained by the fact that TDS is a sensory approach different to the conventional descriptive analysis, which does not consider expressly the evolution of sensations. But it could also be explained in some sense by the cultural differences between panels from different regions (Basque Country and Burgundy in this case), as reported by several authors. In fact, Drake et al. [START_REF] Drake | Comparison of differences between lexicons for descriptive analysis of Cheddar cheese flavour in Ireland, New Zealand, and the United States of America[END_REF] reported differences in the usage of several terms to describe Cheddar cheeses among panels from Ireland, New Zealand and USA, and found some attributes that likely displayed distinct cultural influences. Also, Strauss [START_REF] Strauss | The linguistic aestheticization of food: A cross-cultural look at food commercials in Japan, Korea, and the United States[END_REF] reported cultural variations in the taste patterns used in Japan, Korea and USA, and Tu et al. [START_REF] Tu | Cultural differences in food description and preference: Contrasting Vietnamese and French panellists on soy yogurts[END_REF] found cultural specificities among Vietnamese and French panels when describing the aroma of yogurts.

TDS curves for the eight wines analysed are shown in Fig. 1. The attributes more consistent at panel level (curves over the significant level line; P = 0.10), so valid to describe the wines in a general manner, were heat, red/black berries aroma and astringent (significantly dominant in eight, seven and seven wines, respectively). Bitter was significantly dominant in five wines, woody aroma and pungent in four wines and acid in three wines. Spicy and banana aromas and sweet were not significantly dominant in any wine. To summarise, three attribute groups can be clearly differentiated regarding the evolution of the sensations perceived in the mouth: attributes with dominance in all or almost all the wines (heat, red/black berries aroma and astringent), attributes with dominance in approximately half of the wines (bitter, woody aroma, pungent and acid), and attributes without any dominance (spicy, banana aroma and sweet).

Regarding the sequences of dominant sensations in the wines as a whole, aromas (red/black berries and woody) were dominant at the beginning of the tasting experience while gustatory and trigeminal sensations (bitter, heat, astringent, pungent and acid) were dominant later. This clear separated perception order has not been observed in other studies describing wines by TDS method, where the maximum dominance peaks of these different sensations appeared mixed and also, in some cases, with taste and trigeminal sensations perceived as dominant sooner than aromas (Meillon et al. in Syrah and Merlot wines [START_REF] Meillon | Contribution of the Temporal Dominance of Sensations method to the sensory description of subtle differences in partially dealcoholized red wines[END_REF][START_REF] Meillon | Impact of partial alcohol reduction in Syrah wine on perceived complexity and temporality of sensations and link with preference[END_REF] and Pessina et al. in Falanghina and Falanghina and Muscat mixed wine [START_REF] Pessina | Application of Temporal Dominance of Sensations to taste and flavor in wine[END_REF] ).

The time when the dominance peak was highest was quite similar for red/black berry and for woody aromas, ranging from 8 to 14 s and from 7 to 11 s, respectively (with the exception of red/black berry aroma in wines DS1 and DS3, with maximum peaks at 26 and 20 s). Curves above the significant level for taste and trigeminal sensations appeared later than aromas and were broader, remaining significantly dominant for a longer time. With the exception of pungent (with a maximum dominance peak ranging from 29 to 35 s), there was a quite high dispersion among wines regarding the times when the dominance was maximum (heat from 21 to 46 s, astringent from 14 to 38 s and bitter from 16 to 30 s). Also, the order of the maximum peaks of these sensations throughout the time varied according to each wine. Despite this, some general trends were observed: pungent presented a dominance peak quite regular around 29-35 s; bitter, when significantly dominant, was the first or second maximum peak; and heat was the last maximum peak for majority of the wines (in seven of the eight wines).

Dynamic sensory differences between carbonic maceration wines and destemming wines

When comparing DS and CM wines (Fig. 2), significant differences (P ≤ 0.10) were found for several attributes. Regarding the aromas, DS wines presented a significantly higher dominance of red/black berry aroma compared with CM wines in the time interval from 15 to 30 s. On the other hand, CM wines presented a higher dominance for woody and spicy aromas at 5-8 and 15-22 s, respectively. Although spicy aroma did not present a significant dominance in any wine, when comparing both kinds of wines the differences were significant. The higher dominance of red/black berry aroma in DS wines in comparison with CM wines was unexpected. Although there are no other similar studies of TDS application to compare with, the very limited number of scientific reports comparing CM and DS wines by using conventional descriptive analysis indicate that CM wines are characterised by higher fruity and, specifically, red berry aromas. Indeed, Etaio et al. [START_REF] Etaio | Effect of winemaking process and addition of white grapes on sensory and physicochemical characteristics of young red wines[END_REF] found that CM wines presented a significantly higher red berry aroma than DS wines (there were not significant differences for black berry aroma). Versini and Tomasi 10 described CM wines as higher in strawberry and raspberry in comparison with non-CM wines as well. Also, the higher fruity character of CM wines reported by several authors [START_REF] Jaime | Experiencias a escala semi-industrial de vinificación en tinto con maceración carbónica no estricta[END_REF][START_REF] Lõrincz | Examination of volatile components in carbonic macerated red wines[END_REF][START_REF] Alvarez | Experiencia de maceración carbónica con la variedad Tempranillo en Tierra de Barros. Campañas 96-97[END_REF] has not been observed in the present study. The higher dominance for red/black berry aroma in DS wines could be explained for later and broader peaks over the significance line for this attribute in comparison with CM wines (Fig. 1). In this sense, the use of a dynamic approach seems to reveal a small time difference in the perception of this sensation as dominant, which could not be noticed when using the conventional descriptive analysis. This observation would support the interest of using both kind of analysis to better understand the sensations perceived in mouth when evaluating a food product.

Regarding taste and trigeminal sensations, heat, the unique attribute that resulted in being significantly dominant in the eight wines, did not present differences among DS and CM wines. Thus, this attribute contributed to describing the sensations elicited in mouth by these wines but not to discriminating among these two winemaking practices. In relation to heat sensation, Etaio et al. [START_REF] Etaio | Effect of winemaking process and addition of white grapes on sensory and physicochemical characteristics of young red wines[END_REF] reported higher intensity for alcoholic sensation in CM wines in conventional descriptive analysis. No additional references about possible differences between CM and DS wines for this sensation (heat, alcoholic, burning) have been found.

In relation to astringency, DS wines presented an evident higher dominance, which extended throughout 25 s. There is no conclusive data about the differences in astringency between CM and DS wines, mainly due to the scarce scientific reports published (all of them using conventional descriptive analysis). In a study assessing wines made with Muscadine grape variety (Vitis rotundifolia Michaux) Carroll [START_REF] Carroll | Effects of carbonic maceration on chemical, physical and sensory characteristics of muscadine wines[END_REF] reported lower astringency for CM wines in comparison with DS wines, whereas Etaio et al. [START_REF] Etaio | Effect of winemaking process and addition of white grapes on sensory and physicochemical characteristics of young red wines[END_REF] did not find differences in Tempranillo wines from Rioja Alavesa regarding these two winemaking practices.

CM wines presented a slight higher dominance for pungent and for acid sensations during a brief period of time (6 and 4 s, Shaded cells mean that there is significant difference between products (P ≤ 0.10) and cells without shading mean that there is no significant difference between products (P > 0.10).

respectively). References of other authors about differences in pungent sensation between DS and CM wines have not been found. Regarding acid sensation, the mentioned slight higher dominance in CM wines agreed with the results of Etaio et al. [START_REF] Etaio | Effect of winemaking process and addition of white grapes on sensory and physicochemical characteristics of young red wines[END_REF] Bitter attribute presented a singular performance, since its dominance was slightly higher for DS (between seconds 8 and 11) and later, from seconds 30 to 40, higher for CM. No clear explanation of this observation can be formulated but it could be linked to the dominance/lack of dominance of other sensations instead of being a very different perception of bitterness in CM and DS wines.

With regards to the interval when dominance differences between DS and CM wines were significant, Fig. 2 shows that, in general terms, the main differences in attribute dominance took place from 15 to 44 s. So, it seems that it was necessary to have sufficient time after sipping the wine for the sensory differences to be clearly perceived.

A hotelling table from CVA and the corresponding bi-plot for wines and attributes are shown in Table 3 and in Fig. 3, respectively. The two axes represented in Fig. 3 explain 55.59 % of the variance. The main aspect observed in Fig. 3 and Table 3 is that CM wines were very close, overlapping all of them, with no significant (P > 0.10) wine-to-wine differences among them, and that, with some exceptions, they differed from DS wines. CM wines were related with attributes such as spicy aroma, bitter, banana aroma, sweet, pungent and, to a less extent, heat and acid. DS wines were more dispersed than CM wines in the bi-plot and there were also many significant wine-to-wine differences among DS wines (10 significant differences vs. two non-significant differences in Table 3). As shown in Fig. 3, wine DS1 was significantly different from the other six wines (with a very slight overlapping surface with CM3), wine DS3 was different from other six wines (the four CM wines and two DS wines), wine DS4 was different from wines DS1 and DS3, and wine DS2 only differed clearly from wine DS1. So, DS wines were more heterogeneous than CM wines regarding their dynamic sensory characteristics. Among DS wines, wine DS2 and wine DS3 were the closest ones, mainly described by astringent, red/black berry aroma and woody aroma. On the projection over these first two axes, wine DS1, the most different from the others, was not clearly differentiated by any attribute dominance (just in a slightly manner by acid sensation) and wine DS4 was mainly related to heat, pungent and, to a certain extent, to sweet sensation.

CONCLUSIONS

Using TDS to describe red wines from Rioja Alavesa has provided new information about the evolution of the sensations perceived in these wines, which cannot be achieved by static descriptive analysis. Results of the application of this method have shown that heat, red/black berries aroma, astringent, bitter, woody aroma, pungent and acid are the main attributes to describe the evolution of the sensations perceived in mouth, although attributes with significant dominance varied to a great extent depending on the wine. It has been also observed that there is a tendency for the aromas to dominate first, and the gustative and trigeminal sensations to dominate later, which constitutes an interesting aspect to be investigated in the future in relation to this method.

Analysis of wines by TDS has enabled the observation of significant differences in the evolution of the sensations in the mouth according to the winemaking used (CM vs. DS). CM wines showed higher dominance for woody and spicy aromas, and pungent and acid sensations, while DS wines showed higher dominance for red/black berry aroma and astringency. This information complements the scarce information available about the differences between wines made by these two winemaking processes, providing the wineries with additional data about the implications of using one winemaking process or the other one. Despite these results, additional studies are still necessary to confirm the observed dynamic differences due to winemaking process in these products. Besides, it would be interesting for further works to consider the cultural effect on the use of attributes to describe the sensations perceived in these kinds of wines.

Figure 1 .

 1 Figure 1. Evolution of dominance of sensory perceptions resulting from TDS for carbonic maceration (CM) and destemming (DS) wines. Curves over the significance level (P = 0.10) mean that the attribute is significantly dominant.

Figure 2 .

 2 Figure 2. Significant differences (P ≤ 0.10) between TDS curves of carbonic maceration (CM) and destemming (DS) wines (upper curves express higher dominance rates for DS wines, and lower curves higher dominance rates for CM wines).

Figure 3 .

 3 Figure 3. Plot from canonical variate analysis of the wines (CM, carbonic maceration; DS, destemming) and attributes (P < 0.0001; confidence ellipses at 90%).

Table 1 .

 1 Main chemical characteristics of the wines (mean values ± standard deviation) a Analysis carried out at the Official Laboratory of Araba County Council, one of the three official laboratories authorised to analyse wines protected by DOC Rioja.[START_REF] Eec | EEC) No 822/87 of 16 March 1987 on the common organization of the wine market[END_REF] In order to check accuracy internal standards and one repeated sample were used in the analysis of each parameter, according to the standardised laboratory protocol. b Determined by spectrophotometry of near infrared reflectance with Technicon InfraAnalyzer 400 (Technicon Instruments Corporation, Tarrytown, NY, USA) as described by Dumolin et al.33 c Determined by potentiometry. By potentiometry with H-plus equipment (Technicon), after calibration with pH standards. d Determined by acid-base titration. Potentiometric titration up to pH 7. 34 e Determined by spectrophotometry; absorbance at 280 nm.[START_REF] Eec | ECC Commission Regulation 2676/90, concerning the establishment of common analytical methods in the sector of wine (October 3, 1990)[END_REF]

						Total acidity
	Wine	Winemaking technique	Grape variety	Ethanol (cL L -1 ) b	pH c	(g tartaric acid L -1 ) d	Total polyphenol index e
	CM1	Carbonic maceration	Tempranillo	13.69 ± 0.13	3.75 ± 0.01	4.32 ± 0.13	55.25 ± 0.96
	CM2	Carbonic maceration	Tempranillo	14.48 ± 0.17	3.67 ± 0.02	4.45 ± 0.17	62.75 ± 1.26
	CM3	Carbonic maceration	Tempranillo	13.54 ± 0.13	3.80 ± 0.01	4.08 ± 0.13	59.25 ± 0.96
	CM4	Carbonic maceration	Tempranillo + Viura	13.96 ± 0.14	3.61 ± 0.01	4.27 ± 0.14	45.75 ± 0.50
	DS1	Destemming	Tempranillo	13.24 ± 0.18	3.75 ± 0.01	4.55 ± 0.18	63.50 ± 1.29
	DS2	Destemming	Tempranillo	14.13 ± 0.15	3.63 ± 0.01	4.67 ± 0.15	64.25 ± 0.96
	DS3	Destemming	Tempranillo	13.42 ± 0.06	3.89 ± 0.01	4.13 ± 0.06	70.75 ± 1.26
	DS4	Destemming	Tempranillo + Viura	14.03 ± 0.15	3.58 ± 0.01	4.83 ± 0.15	57.25 ± 0.96

a

Table 2 .

 2 Sensory attributes evaluated in the wines and reference composition

	Sensory attribute	Reference composition
	Aromas
	Woody/roasted	8 g oak chips maintained
		two days in 1 L of water
	Red/black berries	Strawberry (Monin), cherry
		(Monin) and
		blackcurrant syrups
		(Vedrenne) at a
		concentration of
		50 mL L -1 water (each
		one)
	Spicy	80 mL cinnamon syrup
		(Monin) L -1 water
	Banana	80 mL green banana syrup
		(Monin) L -1 water
	Tastes and mouthfeel sensations
	Acid	1 g citric acid L -1 water
	Bitter	0.08 g quinine
		chlorhydrate L -1 water
	Astringent	0.6 g tannic acid L -1 water
	Heat (alcohol)	70 mL ethanol L -1 water
	Pungent	-
	Sweet	8 g sucrose L -1 water

Table 3 .

 3 CVA Hotelling table for carbonic maceration (CM) and destemming (DS) wines

	Sample	CM1	CM2	CM3	CM4	DS1	DS2	DS3	DS4
	CM1	1	0.623	0.477	0.475	0.005	0.144	0.003	0.128
	CM2	0.623	1	0.222	0.102	0	0.035	0	0.311
	CM3	0.477	0.222	1	0.598	0.037	0.093	0.008	0.047
	CM4	0.475	0.102	0.598	1	0.025	0.143	0.001	0.052
	DS1	0.005	0	0.037	0.025	1	0.013	0.008	0.001
	DS2	0.144	0.035	0.093	0.143	0.013	1	0.471	0.055
	DS3	0.003	0	0.008	0.001	0.008	0.471	1	0.002
	DS4	0.128	0.311	0.047	0.052	0.001	0.055	0.002	1

ACKNOWLEDGEMENTS

The authors thank Christine Urbano, from Centre des Sciences du Goût et de l'Alimentation, for helping with the panel training and sample preparation.