Jasmina Milicevic

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INTERNATIONAL CONFERENCE ON FIRST AND SECOND LANGUAGES: EXPLORING THE RELATIONSHIP IN PEDAGOGY-RELATED CONTEXTS DEPARTMENT OF EDUCATION UNIVERSITY OF OXFORD 27-28 MARCH, 2009 Lexical Functions and Paraphrasing Rules as a Bridge between L1 and L2 Jasmina Milićević Dalhousie University Abstract Acquisition of lexical and paraphrastic relations enhances production capabilities of language learners, in L1 and L2 alike. This paper explains how the cross-linguistically valid formalisms proposed by Meaning-Text theory to model these relations, namely, lexical functions and paraphrasing rules, can be used in L2 teaching to help the learner get a better grasp of similarities and differences between L1 and L2 and thus achieve higher accuracy and idiomaticity in L2. A learner-friendly encoding of lexical functions and paraphrasing rules

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Transcript of Jasmina Milicevic

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INTERNATIONAL CONFERENCE ON FIRST AND SECOND LANGUAGES: EXPLORING THE RELATIONSHIP IN PEDAGOGY-RELATED CONTEXTS

DEPARTMENT OF EDUCATIONUNIVERSITY OF OXFORD

27-28 MARCH, 2009

Lexical Functions and Paraphrasing Rules

as a Bridge between L1 and L2

Jasmina Milićević

Dalhousie University

Abstract

Acquisition of lexical and paraphrastic relations enhances production capabilities of language learners, in

L1 and L2 alike. This paper explains how the cross-linguistically valid formalisms proposed by Meaning-

Text theory to model these relations, namely, lexical functions and paraphrasing rules, can be used in L2

teaching to help the learner get a better grasp of similarities and differences between L1 and L2 and thus

achieve higher accuracy and idiomaticity in L2. A learner-friendly encoding of lexical functions and

paraphrasing rules is presented, along with sample exercises intended to help an Anglophone learner of

French learn how to use these tools.

1 The Place of L1 and Linguistics in L2 Instruction

To explore the relationship between the first and the second languages (henceforth L1 and L2) in

the context of L2 lexicon acquisition and teaching, one may look into ways in which universal,

i.e., cross-linguistically valid, formalisms developed by linguistics can be used as a bridge

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between learners’ L1 and L2. Due to their universality, these formalisms provide a basis for

comparison of the two languages: by highlighting differences and similarities between L1 and

L2, they help L2 learner achieve higher accuracy and idiomaticity. This use of linguistic

formalisms is similar to Wierzbicka’s use of semantic primes for cross-cultural semantic analysis

(Wierzbicka, 1996).

The present proposal is based on two assumptions. First, the use of L1 can be beneficial in L2

instruction (Ringbom, 1978; Sparks et al., 2009). This, of course, is not a new insight:

contrastive analysis of, and translation between, L1 and L2 have long been used in L2 instruction

to mitigate the effects of negative transfer and enhance the positive one. Second, the use of

linguistic concepts and formalisms, adapted to learners’ needs, can greatly facilitate language

acquisition by making it more structured and systematic (Polguère, 2004: 1). While this view has

long-standing proponents among linguists (Halliday et al., 1964; Huot, 1981) and, more recently,

has been corroborated by research in applied linguistics turning up significant correlations

between meta-linguistic abilities of learners (in our terms, knowledge about language, as

opposed to knowledge of a language) and their proficiency levels (Elder et al., 1999; Roehr,

2006), it has not yet trickled down to schools and universities – at any rate, not in North

America, where language classroom is largely off limits for linguistics. This paper will hopefully

contribute to making a case for an explicit and systematic use of linguistics in language teaching

and learning.

We will be looking into two specific formalisms put forward by Meaning-Text linguistic

theory, in particular its Explanatory-Combinatorial Lexicology (Mel’čuk et al., 1995): lexical

functions, used to model lexical relations (semantic derivations and collocations), and

paraphrasing rules, modeling paraphrasing relations (i.e., relations of (near-)synonymy between

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sentences). It is widely recognized that fluent and idiomatic speech hinges upon Speaker’s

knowledge of lexical relations and his ability to manipulate these relations in order to produce,

when necessary, a varied output, i.e. to paraphrase (Žolkovskij & Mel’čuk, 1967; Russo & Pippa,

2004). It is therefore crucial that lexicon instruction in L2 be focused on the acquisition and

active use of these relations. This, in turn, explains the importance of the corresponding

formalisms.

The rest of the paper is organized as follows. Section 2 presents the formalisms of lexical

functions and paraphrasing rules and proposes learner-friendly versions thereof with

intermediate-to-advanced Anglophone learners of French as target audience. Section 3 illustrates

the use of these formalisms in language instruction, and Section 4 serves as a conclusion.

2 Universal Linguistic Formalisms for L2 Lexicon Acquisition and Teaching

2.1 Lexical Functions and Paraphrasing Rules: The Linguist’s Version

2.1.1 Lexical functions

Lexical functions [= LFs] (Wanner, ed., 1996) cover both semantic-derivation relations

(synonym, antonym, conversive, nominalization, verbalization, result name, actant names,

adjectives characterizing actants, etc.) and collocational relations (intensifier, positive evaluator,

light verb, realization verb, etc.).

A LF link is encoded as shown in (1a); f is the name of LF, lexeme L is the keyword of f, and

lexemes {L1, L2, …, Ln} are elements of the value (of f for L). The application of an LF – the

intensifying attribute Magn – to an L is illustrated in (1b).

(1) a. f(L) = {L1, L2, …, Ln}

b. Magn(amour) = grand | preposed, fou

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Some 60 standard simple LFs form the core of the LF system; they can combine into complex

LFs, like the last two LFs in Figure 1 below. These LFs are valid cross-linguistically (their actual

applicability depends of course on the available lexical material in each individual language).

Examples of the most common standard LFs applied to some English lexemes and their French

equivalents follow. Note how widely different the elements of LF values are even in such closely

related languages as English and French.

Semantic derivations

Syn(WISH) = desire Syn(DÉSIR) = souhait

Conv21(FEAR(V)) = frighten Conv21(CRAINDRE) = effrayer

S0(STEAL) = theft S0(VOLER) = vol

S1(STEAL) = thief S1(VOLER) = voleur

S2(PLUNDER) = loot S2(PILLER) = butin

A1(KNOW) = aware A1(SAVOIR(V)) = _au courant_

A2(KNOW) = known <familiar> A2(SAVOIR(V)) = connu

Collocations

Magn (APPLAUSE) = heavy < thunderous Magn(APPLAUDISSEMENTS) = nourris < frénétiques

Bon(ANALYSIS) = fruitful Bon (ANALYSE) = lucide

Ver(TRUTH) = plain Ver(VÉRITÉ) = stricte | antepos.

Oper1(DECISION) = make [ART ~] Oper1(DÉCISION) = prendre [ART ~]

Real2(JOKE) = get [ART ~] Real2(BLAGUE) = comprendre [ART ~]

AntiMagn(APPLAUSE) = scattered AntiMagn (APPLAUDISSEMENTS) = clairsemés, rares

CausOper3(COURSE) = enroll [NZ in ART ~] CausOper3(COURS) = inscrire [NZ au ~]

Figure 1: Some standard LF’s applied to English and French lexemes

The name of an LF encodes the semantic and the syntactic role the LF value plays with respect to

the keyword L. Thus, Magn is semantically an intensifier and syntactically an attributive

modifier of L, Operi is a semantically empty verb taking L as its main object and the i-th actant

of L as it subject, etc.

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2.1.2 Paraphrasing rules

A paraphrasing rule is an equivalence rule; it specifies a substitution of linguistic material at a

given level of representation (semantic, deep-syntactic, etc.) such that it does not (significantly)

alter the initial meaning the Speaker intended to convey. There are two major types of

paraphrasing rules.

Lexical-syntactic paraphrasing rules (Žolkovskij & Mel’čuk, 1967; Mel’čuk, 1992) operate at

the level of deep-syntactic structure; they describe possible lexical substitutions and concomitant

syntactic restructurings in a deep-syntactic dependency tree. Since these rules are formulated in

terms of lexical functions, they too are universal, i.e., in principle applicable within any

language, as well as between languages; thus, they can be used in translation, which can be

viewed as interlingual paraphrasing. Two lexical-syntactic paraphrasing rules follow: a

synonymic substitution (in a broad sense) and a derivative substitution triggering head switching.

For each rule, the illustration of its application is given within English and French and between

the two languages.

RuleEQUIV.LEX-SYN-1 : L(V) ≡ Oper1IIS0(L(V))

(2) a. English ~ English

He fell [= L]. ≡ He took [= Oper1] a fall [= S0(L)].

b. French ~ French

Il est tombé [= L]. ≡ Il a fait [= Oper1] une chute [= S0(L)].

c. English ~ French

I decided to make [= Oper1] a trip [= S0(L)] to Europe. ≡

J’ai décidé de voyager [= L] en Europe.

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RuleEQUIV.LEX-SYN-2 : L1IIL2 ≡ L2´ATTRAdv1(L1)

(3) a. English ~ English

The rain contined [= L1] to fall [= L2]. ≡

The rain was falling [= L2´] still [= Adv1(L1)].

b. French ~ French

La pluie continuait [= L1] à tomber [= L2]. ≡

La pluie tombait [= L2] toujours [= Adv1(L1)].

c. English ~ French

The crisis caused [= L1] an incrase [= L2] in prices. ≡

Les prix ont augmenté [= L2´] suite à [= Adv1(L1)] la crise.

Semantic paraphrasing rules (Milićević, 2007a: 181-245) operate at the level of semantic

representation; essentially, they exploit semantic decompositions (≈ definitions) and alternative

lexicalizations they make possible. For instance, paraphrases (4a) and (4b) are linked via the

following rule, which gives the semantic decomposition of the lexeme APPROVE.

RuleEQUIV.SEM-1 ‘X approves Z’s Y’ ≈ ‘X, who is convinced that Z has done or will do Y,

believes (and says) that Y is a right thing to do.’

(4) a. I approve your decision to leave.

b. I think that your decision to leave is correct.

Semantic paraphrasing rules are universal, as well, although in a different sense than lexical-

syntactic rules: what makes them universal is the underlying mechanism of semantic

decomposition.

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2.2 Lexical Functions and Paraphrasing Rules: The Learner’s Version

2.2.1 Lexical Functions for L2 Learners

Two learner-friendly ways of encoding lexical-functional links will be presented.

The first way, consists in using as an LF name a literal expressions of its meaning; such an

expression, called an LF glose, is in fact the default, or the most neutral, value of the

corresponding LF (Polguère, 2000, Popovic, 2003). Thus, a real linguistic expression is used,

rather than an artificial name, such as Magn, Oper1, etc. This way of coding is adopted in French

monolingual learners’ dictionaries or dictionary projects based on Explanatory-Combinatorial

Lexicology: DiCo (Polguère, 2000), LAF (Mel’čuk & Polguère, 2007) and DA (Milićević &

Hamel, 2007); it is illustrated in Figure 2, presenting a fragment of the LF zone in the entry for

CONSEIL#I.1 ‘advice’ in the DA dictionary:1

Lexical functionsElements of value

Hard Coding Soft Coding

Oper1 [X] donner un C. ‘give A.’ donner ‘give’ [(ART) ~ à NZ],

fournir ‘fournish’, offrir ‘offer’ [ART ~ à NZ]

beaucoup de C.+ Oper1

[X] donner beaucoup de C. ‘give

much A.’

prodiguer ‘pour out’ [ART ~s à NZ]

Oper3 [Z] recevoir un C. ‘receive some A.’ demander ‘ask’ [(ART) ~ à NX]

solliciter ‘solicit’ [ART ~ à NX],

Real3 [Z] réagir au C. de la façon

attendue ‘react to A. in the expected

way’

accepter ‘accept’, écouter [ART ~] ‘listen to’,

obéir [à ART ~] ‘obey’;

< suivre ‘follow’ [ART ~]

AntiReal3 [Z] réagir au C. contrairement à la

façon attendue ‘react to A. contrary to

rejeter ‘reject’ [ART ~]

1 Cf. the actantial structure of CONSEIL#I.1: ~ de X à Z de faire Y / concernant W ‘advice by X to Z to do Y /

concerning W’ (e.g., monX conseil à JeanZ de partirY ‘my advice to Jean to leave’ / monX conseil à JeanZ

concernant son départW. ‘my advice to Jean concerning his departure’).

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the expected way’

Adv1Real3 [Z faire qqch] en suivant un C. ‘[Z

do sth] following A.’

sur ‘on’ [ART ~]

Figure 2: six LFs of CONSEIL#I.1 ‘advice’ in hard and soft coding in the style of Polguère (2000)

The LF glose can be used as a “scaffold” if the learner cannot recall an element of the value of

the LF (which would be more idiomatic). With respect to the hard formalism, this is a more

transparent but less general way of encoding lexical relations: often, for a single hard-coded LF,

there are several soft-coded ones, as a function of the argument of the LF.2

The second learner-friendly way of coding LFs, more general than (and radically different

from) the first one, would consist in keeping artificial names for the LFs while making them

more transparent (the problem of encoding the actantial role with verbal functions – Oper1 vs.

Oper2, etc. – would have to be figured out):

FLs describing derivations FLs describing collocations

Hard coding Soft coding Hard coding Soft coding

Syn Synonym Magn/AntiMagn Intensifier/Attenuator

Anti Antonym Bon/AntiBon SubjectiveEvaluatorPos/Neg

Convijkl Conversive Ver/AntiVe ObjectiveEvaluatorPos/Neg

S0 Action Name Operi SupportVL=principalCO

S1 Agent Name Funci SupportVL=Subjet

S2 Patient Name Reali RealizationVL=principalCO

A1 Agent’s Characteristics IncepOperi PhaseStartSupportVL=principalCO

A2 Patient’s Characteristics FinFunci PhaseEndSupportVL=Subjet

Figure 3: Another possible soft coding of LFs

2 Here are some soft LFs taken from the DA database which correspond to the hard LF Magn: intense ’intense’,

grand ’big’, important ’important’, beaucoup ’much’, intensément ’intensely’, de façon marquée ’in a

marked way’, etc. Light LFs adapt to the part of speech of their keyword, which is not the case with classical

LFs.

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For the time being, I prefer not to choose between the two possible formalizations. Note,

though, that either way learner-friendly encoding of LFs becomes quite involved at some point,

and, consequently, that of paraphrasing rules.

2.2.2 Paraphrasing Rules for L2 Learners

Here again, we have two encoding options, depending on what soft LFs are used (Milićević,

2007b). Two soft versions of the RuleEQUIV.LEX-SYN-1, cited in Section 2.1.2, follow.

Soft version-1

Generalizations are possible from the semantic label of a given lexeme (Polguère, 2003).

Semantic labels are taxonomic characterizers (object, person, event, act, action, process, etc.)

used to describe in an approximate way the meaning of lexemes and semantic type of their

actants. A lexeme L inherits from its label its semantic and, to some extent, co-occurrence

properties: the Ls with the same semantic label tend to admit the same LFs and to have the same

values for these LFs. For instance, the default value for Oper1(Lact) is do or make, that for

Oper1(Lnegative≥act) is commit; the value of Oper1(Lfeeling) is feel, that of Oper1(Linformation) is give, etc.

This information can be incorporated into paraphrasing rules.

V[X ~ Y]action ≈ NX peform <carry out> N action of NY

E.g.: (to) analyse ≈ perform <carry out> an analysis;

(to) experiment ≈ perform <carry out> experiments

V[X ~ Y]feeling ≈ NX feel Nfeeling for NY

E.g.: (to) respect ≈ (to) feel respect; (to) love ≈ (to) feel love

V[X ~ Y à Z]information ≈ NX give to NZ Ninformation about NY

E.g.: (to) advise ≈ (to) give a piece of advice; (to) hint ≈ (to) give a hint

For a single hard-coded paraphrasing rule, we have several soft rules.

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This coding technique is limited in two respects. First, not all LFs and not all values of LFs

for Ls with the same semantic label coincide; second, not all rules can be rewritten in this way

(namely, the more syntactic ones, like the rule RuleEQUIV.LEX-SYN-2, Subsection 2.1.2).

Soft version-2

V[X ~ Y] ≈ NX support V + NAction.Name(V) + (PREPOSITION) + NY.

This notation is more general but less transparent than the one suggested above, just as are the

LFs in terms of which the rule is written.

3 Learner-friendly Formalisms in Use: An Illustration

3.1 Lexical functions and paraphrasing rules in a sample entry of a learners’

dictionary

Below I show how a part of an entry in a learners’ bilingual English ~ French dictionary could

look like, with learner-friendly LFs and paraphrasing rules as used in the DA dictionary.

X’s ATTENTION to Y ATTENTION de X pour Y

Lexical functions Element(s) of value

Oper1 = [X] have A. for Y

devote, pay [~ to NY] accorder, faire, porter [~ à NY]

Oper2 = [Y] have A. from X

get, receive [~ from NX] N/A

Caus2Func =

[Y] cause A. to be on Y attract, catch; arrest [~] capturer, retenir [~]

MagnCaus2Func2 =

[Y] cause intense A. to be on Y grab [N’sX ~] N/A

CausOper2 =

[someone] cause X to have A. for Y bring [NY ~ to N’sX ~] porter [à ART ~ de NX];

SIGNALER3

3 Although signaler can be described as a fused value the LF CausOper2 (i.e., a value expressing together the

meaning of the LF and its keyword), the equivalence bring Y to X’s attention ≈ signaler Y to X has a distinctly

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CausFunc2 =

[someone/sth] cause A. to be on Y

call, draw [N’sX ~] attirer [~ sur NY]

Magn+Caus1Func2 =

[X] cause X’s intense A. to be on Y N/A fixer [ART ~ sur NY]

Magn =

Intense

Intense-focused on details

Intense-not shared

Intense-continuous

intense, heavy, heightened;

careful, close, meticulous;

undivided

N/A

N/A

minutieuse

N/A

soutenue

Figure 4: LF zone in the entry for ATTENTION in a learner’s dictionary (fragment)

Paraphrasing rules Exemples

Synonymic substitution based on the

decomposition of SIGNALER

W brings Y to X’s attention ≈

W signale Y à X

The problem was brought to my attention by Mr. Who. ≈

Ce problème m’a été signalé par M. Untel.

Synonymic substitution & addition of Intense

[Y] cause intense A. to be on Y ≈

[s’one/sth] cause A. to be on Y+Intense A.

This case grabbed the intention of the nation. ≈

Ce cas a attiré beaucoup d’attention partout au pays.

Conversive substition

[Y] have A. from X ≈

[X] have A. for Y

She was not getting enough attention from her husband. ≈

Son mari ne lui accordait pas assez d’attention.

Implicative substitution & addition of Intense

(applied from right to left)

[X] cause X’s intense A. to be on Y ≈

[X] begin to have A. for Y+Intense A.

He started paying close attention to this new problem. ≈

Il fixa son attention sur ce nouveau problème.

Figure 5: Paraphrasing rules in the entry for ATTENTION in a learner’s dictionary (fragment)

This lexicographic information can be exploited in various language production tasks:

writing, revising, reformulating, abstracting, translating, etc., some of which will be illustrated

below.

3.2 Sample exercises with lexical functions and paraphrasing rules

semantic feel about it and is thus better described via a semantic paraphrasing rule: see Figure 5.

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Two exercises with each of the formalisms will be presented.

3.2.1 Exercises with Lexical Functions

Exercise-1

Describe the following English collocations in terms of lexical functions and translate them

into French. What do you observe?

(5) eat dinner, cook dinner, make breakfast, ask a question, give a cry, make a gesture

Goals: 1) Familiarize students with a highly idiomatic character of collocations; 2) make them

aware of the negative transfer from L1 they are likely to experience with collocations.4

Responses:

[X] ingest L = Real1 eat dinner prendre son dîner

[X] get ready to ingest L = PreparReal1 cook dinner préparer le dîner

[X] get ready to ingest L = PreparReal1 make breakfast préparer le petit déjeuner

[X] cause L to exist = CausFunc0 ask a question poser une question

[X] produce L = Oper1 give a cry pousser un cri

[X] do L = Oper1 make a gesture poser un geste

Figure 6: Key to Exercise 1

Exercise-2

Supply an element of the value for the LFs in perentheses.

(6) a. Quelle conclusion peut-on tirer de l’ _______ (Fact.of = S0(ESSAYER#1)) de ce produit?

‘What conclusion can one draw from the _______ of this product?’

b. J’ai enfin rendez-vous pour le premier _______ (Fact.of = S0(ESSAYER#2)) de ma robe.

‘I finally have an appointment for the first _______ of my dress.’

(7) a. À quel _______ (Name.for.W = S4(ACHETER#1)) a-t-il acheté#1 cette voiture?

4 Cf. the following data from a learners’ corpus (Milićević & Hamel, 2007): *manger le dîner, *demander une

question, *donner un cri., etc.

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‘Should one pay _______ for a quality perfume?’

b. Ce cadeau ressemble à un _______ (Name.for.W = S4(ACHETER#2)) pour acheter#2 sa

patience.

‘This present looks like a _______ to buy his patience.’

Goals: 1) Familiarize students with polysemy and the fact that different word-senses of a

polysemous word may admit different LFs and have different values for the LFs that they share;

2) make them compare patterns of polysemy in L1 and L2.

Responses: (6) essai ‘trial’, essayage ≈ ‘fitting’; (7) prix ‘price’, pot-de-vin ‘bribe’.

3.2.2 Exercises with Paraphrasing Rules

Exercise-1

The following sentence contains problematic lexical material (in bold).5 Formulate the

problem and propose a solution.

(8) Le choix des patins doit recevoir une attention particulière.

‘The choice of skates should receive special attention’.

Goal: Enable students to analyze their lexical errors and correct them by using lexical links and

paraphrasing rules.

Response: In French, LF Oper2 does not have an idiomatic element of value with ATTENTION:

Oper2(ATTENTION) = ?mériter, ?recevoir. In English, it does: it is, exactly, receive, get

[attention]; cf. Figure 5 above. Here is a possible reformulation:

(9) Il faut accorder une attention particulière au choix des patins.

‘One should pay special attention to the choice of skates’.

5 This is an actual sentence, taken from the learners’ corpus mentioned previously (footnote 4).

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It makes use of two paraphrasing rules: a synonymic substitution involving quasi-synonyms

DEVOIRX doit faire Y and FALOIRIl faut que X fasse Y, and a conversive substitution, cited above, in the entry

for ATTENTION, which allows to pass from ?recevoir ATTENTION [= Oper2] to accorder

ATTENTION [= Oper1].6

Exercise-2

Translate (10) into French. Cite the paraphrasing rule(s) you used in the process.

(10)This kind of involvement benefits the society as a whole.

Goal: Make student aware of mismatches between L2 and L1 and show them how paraphrasing

rules can help them get around the problem.

A possible response:

(11)Ce genre d’engagement est bénéfique pour l’ensemble de la société.

‘This kind of involvement is beneficial for the society as a whole’.

English (to) benefit and French bénéficier are conversives: X benefits Y vs. Fr. Y bénéficie de

X [*X bénéficie Y]; plus, the English verb is transitive while the French one is not. The

substitution would change the sentence information structure too radically (by putting société in

the role of Theme). A possible reformulation that will preserve the information structure makes

use of the adjective for X: X is beneficial for Y X est bénéfique pour Y (this is the A1 of the

English verb, but the A2 of the French one). This reformulation uses two rules: a synonymic

substitution relating a verb with the adjective denoting the verb’s Agent characteristics, and a

conversion rule relating this adjective with the one denoting the verb’s Patient characteristics:

6 Among many other possible reformulations that I cannot describe due to lack of space, one can cite Les patins

doivent être choisis avec (beaucop de) soin ‘Skates should be chosen with (a lot of) care’ and Le choix des patins

est particulièrment important ‘The choice of skates is especially important.’

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V[X ~ Y] ≈ NX is + AdjAgent’s Charateristics(V) + PREPOSITION + NY;

Conversive(AdjAgent’s Characteristics) = AdjPatients’s Characteristics7

4 Conclusion

In lieu of conclusion, I will sum up the main points of the paper.

Language students, in particular students of L2, need to be systematically taught lexical and

paraphrastic relations as a way of enhancing their production skills. This requires introducing

some formalization into language instruction, borrowed from linguistics but adapted to suit the

specific needs of language learners. Thanks to their universality, linguistic formalisms allow the

learner to compare in an informed way his L2 and L1, which can ultimately ease the complex

task of lexicon acquisition.

Advanced L2 learners need to acquire quite a lot of linguistic concepts. In order to grasp the

notion of LF, for instance, however it is presented to them, they need to be familiar at least with

the notions of lexical unit, semantic and syntactic actant, free and set phrase. Thus, we need to

offer them support, in form of tutorials and exercises.

Acknowledgment

Many thanks to Igor Mel’čuk and Alain Polguère for commenting on a previous version of this

paper.

References

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7 Cf. the hard coding: L(V) ≡ Oper1IIA1(L(V)) and Conv21(A1) = A2.

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