Teaching Languages in a Common Language

Christopher C. Adam

University of California, Los Angeles

Cynthia Eid

Antonine University, Baabda, Lebanon

Adam and Eid first set out to establish that the same approach used throughout the ages for successfully teaching mathematics, which is a common language of the world, can be applied to the instruction of SLs (i.e. subsequent languages like English, French, etc.). Once said feasibility is established, they proceed to argue for the benefits of doing so.

0. Introduction: A True Lingua Franca

Math is taught with the same symbols and standards of usage around the world. Even in countries where a local tradition once differed immensely in its use of symbols, today even those nations generally adhere to what is often known as the “Western” standard usage, despite it actually owing its origins to a combination of the Indian, Greek and Arabic traditions.

Figure (1). The Pythagorean Theorem

Furthermore, this common language is taught in terms on its timeless formulas which are passed down from generation to generation, as in the case of the Pythagorean Theorem, which students have learned for more than two thousand years:

(1) a² + b² = c²

No less familiar to students throughout the generations is the following formula:

(2) {a+b}{a-b} = a²-b²

In contrast to the timeless nature of formulas, specific instantiations of these formulas are used in practice and then forgotten as soon as they have lost their immediate relevance. For example, if we put specific numbers in the Pythagorean Theorem {a² + b² = c²} to serve a specific purpose, then we might have something like the following:

(3) 339² + 486² = 592.55²

And again referring to the familiar formulaic face of {a+b}{a-b} = a²-b², we might see a specific case such as the following which, unlike the abstract formula, is only useful for a specific instance:

(4) {339+486}{339-486} = 339²-486²

Similarly, in the instruction of language, specific rules of conduct for different grammatical structures often meet the guidelines of identical mathematical formulas (usually far simpler than those of first-year introductory algebra) which can be learned as a general means for determining the grammaticality of a given structure.

This being the case, the same approach used for teaching the universal language of mathematics successfully throughout the ages, when applied to the instruction of SLs (i.e. subsequent languages like English, French, etc.), cuts across differences in NL (native language or L1) and SL (subsequent language L2) to ensure similar success in mastery of basic mechanical elements.

I – How Mathematical Logic Can be Applied to Language and Style:

• Factorization

• Variable-Equation Substitutions

• The Existence of Zero

II – Why Mathematical Logic Should be Applied to Language and Style:

• A Past Precedent Developed

• A Reliable Touchstone/Standard

• Beyond Grammar

1a. Factorization

One area where mathematics is quite analogous to language (and can thus easily be applied to it) is grammatical parallelism, which like algebra operates within the norms of factorization. Analyzing sentences like algebraic formulas gives one the opportunity to check for parallel distribution of a value across the terms it applies to. For example, in the case of the following erroneous sentence (marked with an asterisk *), the word to does not appear equally distributed:

(5)	*I like to bathe, __ jog, to eat and __ swimming.

One solution to this lack of parallel distribution of the value to involves including it in every term:

(6)	I like to bathe, to jog, to eat and to swim.

However, it is generally the case that we can factor out a value which is common to all terms, as with this algebra equation:

(7)	9A–AB+11AC => A {9__–__B+11__C}.

Like algebraic factorization, then, the sentences can also factor out a value common to all the terms. If we factor out the word to, we do not have to repeat it every time:

(8)	I like to {__ bathe, __ jog, __ eat and __ swim}.

It is often the case that we can factor out multiple values from a set of terms. Since language has syntax, in addition to values, the side to which a value is factored out is not arbitrary but in accordance with a given language’s syntactic demands. For example, that which characteristically appears at the left of a term, like infinitival to in relation to its verb, will always be factored out to the left of the brackets, while that which characteristically appears at right end of its term will be factored out to the right:

(9)	We like to speak about them, to think about them, and to dream about them.

becomes

(10)	We like to {__ speak __, __ think __, and __ dream __} about them.

Since syntax dictates that infinitival-to precede the verb while the prepositional phrase about them be factored out to the right of the verb, an additional linear characteristic must be recognized. However, this does not invalidate the presence of factorization; rather it simply adds another element to it.

1b. Variable-Equation Substitutions: HE » WHO

Another area where the mathematical metaphor can be extended is that of substituting equal-value variables. Each relative pronoun (wh-word) represents the same value as a type of personal pronoun, and such characteristics as final vowels, m-sounds and s-sounds clue us in on this. Since these two word types represent the same value, only one at a time is used in English.

(that)	WHO	»	HE, THEY
(that, Ø)	WHO(M)	»	HIM, THEM
	WHOSE	»	HIS, THEIR(S)

Table (1)

Relative structures are used to combine two simple sentences into one complex sentence. Note the following structures, in which each word of the first sentence pair (with personal pronouns) has a one-to-one logical correspondence with the words of the equivalent complex sentence (with wh-relative pronouns):

(11)

{

You saw the man. Hex ate my chicken. You saw the man whox ate my chicken. (that)x

Note that the restrictive uses of both relative subject-who and relative object-who(m) can be substituted with that, while only the latter can be substituted with a zero-form Ø.

(12)

{

I saw the man. You know himx. I saw the man who(m)x you know. (that)x (Ø)x

Relative object-who(m) is moved to the beginning of its clause regardless of whether it is a direct object as in (12) or a prepositional object as in (13). What is important is that the one-to-one correspondence is intact, i.e. with no redundant repetition of the value x:

(13)

{

We saw the man. You talked about himx. We saw the man who(m)x you talked about. (that)x (Ø)x

spoken and written

Sometimes, relative object-who(m) and the preposition that governs it move to the beginning of their clause together as if they shared an inescapable box (or set of brackets). This is especially common in written English, and when this is done, whom cannot be substituted with that or Ø despite the former being restrictive:

(14)

{

We saw the man. You talked {about him}x. We saw the man {about whom}x you talked. ({about that})x ({about Ø})x

mostly written

Finally, relative possessive-whose and the noun it governs move together to the beginning of their clause when two simple sentences are combined into a complex one:

(15)

{

You saw the man. We ate {his chicken}x. You saw the man {whose chicken}x we ate.[1]

Again, since relative pronouns represent the same value as personal pronouns, the two never occur together. Consider the following:

3·3=9 and 9=x, so 3·3=x, but 3·3≠9x

Since 9 and x represent the same value and are both equal to 3·3, only one is necessary in the equation while using both together is erroneous. Similarly, a structure like the following does not occur in standard spoken or written English because the same value is repeated twice thus incurring a grammatical error:

(16)

{

We saw the man. You talked about himx. We saw the man who(m)x you talked about himx. (that)x (Ø)x

One need not appeal simply to principles of usage when calling this an error. Rather, it can simply be stated that English adheres to a principle which some languages simply do not adhere to, that of prohibiting redundancy or a value.

Note for example that, for a sentence with the same elements in Arabic, repetition of the value representing him is not only non-erroneous but in some cases required:[2]

(17)	xعنــه	تكلمتَ	.	الرجل	رأينا
	him‑about	you:spoke		man‑the	we:saw

(18)	xعنــه	تكلمتَ	xالذي	الرجل	رأينا
	himx‑about	you:spoke	thatx	man‑the	we:saw

(19)	xعنــه	تكلمتَ	.	رجلا	رأينا
	himx‑about	you:spoke		a‑man	we:saw

(20)	xعنــه	تكلمتَ	xØ	رجلا	رأينا
	himx‑about	you:spoke	thatx	a‑man	we:saw

Since Arabic requires the same resumptive structure that English prohibits, i.e. where the same value is repeated twice, it is vital that students with Arabic as their NL approach English structures such as relative clauses with a logical approach that does not rely on prior spoken language use but rather appeals to the same universal logic that is represented in mathematical structures.

(21)	xكلبــه	سرقتُ	.	الرجل	رأينا
	{his‑dog}x	I:stole		man‑the	we:saw

(22)	xكلبــه	سرقتُ	xالذي	الرجل	رأينا
	{his‑dog}x	I:stole	thatx	man‑the	we:saw

(23)

{

We saw the man. I stole {his dog}x. We saw the man {whose dog}x I stole. not We saw the man thatx I stole hisx dog.

Since that and his both represent the same third person value here, standard English does not repeat this value. Again, however, since Arabic requires it, NL Arabic students can be referred to a universal standard to understand how to approach the English structure, rather than simply learning that English is different but without learning how to approach the problem systematically.

1c. The Existence of Zero

The application of zero in modern mathematics has infinitely expanded our understanding of many processes, to the extent that the great many higher forms of math are indebted to its usage. Apparently more ancient than zero’s use in the ancient mathematic tradition of India[3] is its use in India’s ancient linguistics tradition, started by Pāṇini,[4] who saw that language has a paradigmatic zero which explains a diverse many phenomena, otherwise obscured by our tendency to not view the presence of absence.

i. Proof of the Unseen

Like the Invisible Man of H.G. Wells’s science fiction novel of the same name, there are things which, although not visible, do exist and affect movement on entities which we can see. Such entities as gravity, air and radiation commonly affect objects in our environs, and we do not question its existence because we see the outcome on a normal basis.

Sometimes, like Wells’s Invisible Man, what is now invisible used to be visible, but has since changed. Note for example, the direct objects in Modern Standard Arabic:

(24)	يسرق كتابا	(25)	يسرقــه	(26)	يسرقــهم
	yásruqu kitāban		yásruqu-hu		yásruqu-hum
	he:steals-a:book		he:steals-it		he:steals-them

Spoken Arabic dialects have generally lost the h-sound of the third-person masculine singular object ‘it’, and Lebanese Arabic has gone a step further in losing all third-person object h-sounds in some environments:

(27)	byísroq ktēb	(28)	byísirq-o	(29)	byisríq-on
	he:steals a:book		he:steals-it		he:steals-them

In some cases, this has resulted in the third-person masculine object pronoun being inaudible as a suffix. However, it still affects the verb’s stress pattern, by drawing the stress towards the end of a word as do audible suffixes:

(30)	byísirqu ktēb	(31)	byisirqū^´-Ø	(32)	byisirqū^´-on
	they:steal a:book		they:steal-it		they:steal-them

(33)	btísirqi ktēb	(34)	btisirqī^´-Ø	(35)	btisirqī^´-on
	you(f):steal a:book		you(f):steal-it		you(f):steal-them

As such, the third-person masculine object pronoun, though not audible (or, if you will, “visible”) in the sense that other pronouns are, still makes its presence felt, crucially proving that it does exist.

ii. Zero-Form of to be

Once we have accepted the existence of zero-forms, we have a valuable tool for explaining many language phenomena. For example, Small clauses in English, such as those following the causative use of the verb to make, use a non-conjugated form of the verb very similar to the English subjunctive, except for the verb to be which is replaced by zero:[5]

	Main Clause	Small Clause
		Subject	Verb	Complement
(36)	I make	a student	eat	burgers
(37)	I made	a student	give	us money
(38)	*I make	students	be	angry
(39)	I make	a student	Ø	angry

The substitution of to be for zero is similar to what happens to the Arabic verb for ‘to be’ in present-tense main clauses:

	Complement	Verb	Subject
(40)	معلمة	كانت	مريم
	a:teacher	was	Maryam

(41)	معلمة	Ø	مريم
	a:teacher	is	Maryam

The Ø, which means ‘to be’, is certainly not meaningless, and analyzing that meaning as being encoded in the zero-form of كان levels the paradigm more than does saying that Maryam or teacher is the verb, or claiming that some sentences do not need verbs while others do, when it is clear that, in languages as distinct as Arabic and English, the verb ‘to be’ appears to be unique in this tendency.

iii. Zero-Forms in Relative Constructions

Some of the complications of SL students mastering relative-clause structure have been discussed in the previous sections. It might be noted from above that wh-relative pronouns representing objects, such as the restrictive uses of whom and which, can be substituted with either that or Ø, the two latter forms being indistinguishable in usage:

(42)	We saw the man who(m) you talked about. We saw the man that you talked about. We saw the man Ø you talked about.

Similarly, while definite nouns are relativized with a pronounced الذي in Arabic, indefinite nouns are relativized with an unpronounced Ø:

(43)	عنــه	تكلمتَ	الذي	الرجل	رأينا
	him‑about	you:spoke	that	the‑man	we:saw

(44)	عنــه	تكلمتَ	Ø	رجلا	رأينا
	him‑about	you:spoke	that	a‑man	we:saw

Here, as in the spoken Arabic dialects, knowledge of zero-forms gives students a valuable tool to understand the existence of forms which do exist although they are not immediately visible.

iv. Subject-be Zero-Substitution

Such obstacles as dangling modifiers can also be overcome with an understanding of the zero-forms which they contain. Adverbial relative clauses can be shortened by substituting zeroes for both the subject and the verb to be. And subject-be zero-substitution is necessary before a relative clause can be fronted to the beginning of the sentence:

(45)	The singer {whox wasy unhappy with his life} attacked my cat. becomes The singer { Øx Øy unhappy with his life} attacked my cat. becomes { Øx Øy unhappy with his life}, the singer attacked my cat.

Understanding how fronted relative clauses work allows one to trace them back to the relative clauses from which they originate, thus making clear how to avoid creating sentences with dangling modifiers:

(46)	*{ Øx Øy meowing and purring}, the singer attacked my cat. comes from *The singer {whox wasy meowing and purring} attacked my cat.

The same subject-be zero-substitution can be applied to adverbial relative clauses and, though not necessary for fronting, can also accompany such movement:

(47)	The singer {when hex wasy singing his favorite song} enjoyed the audience’s attention. becomes The singer {when Øx Øy singing his favorite song} enjoyed the audience’s attention. becomes {When Øx Øy singing his favorite song}, the singer enjoyed the audience’s attention.

As with fronted adjectival clauses, dangling modifiers can be avoided by tracing fronted adverbial clauses to the relative clauses from which they originate:

(48)	*{While Øx Øy meowing and purring}, the singer attacked my cat. comes from *The singer {while hex wasy meowing and purring} attacked my cat.

Typically, the problem of dangling modifiers is addressed as a logical error, which it also is. However, diagnosing the structure’s problem and tracing it back to a flaw in the original relative clause gives one a mechanical basis for repairing the construction.

v. Zero-Form of Prepositions

An understanding of zero-forms gives one the ability to work with another complex form which is the source of much confusion for SL learners of English, the infinitive. The word to is not always a preposition, as it can form part of the infinitive. Like the gerund, with -ing, the infinitive is a verbal noun, like the Arabic maṣdar (مصدر). For this reason, both infinitives with to and gerunds with -ing are generally used in the same parts of sentences where nouns are used, e.g. as subjects and direct objects:

	subject	verb	adjective	subject	verb	direct object
(49)	school	is	useful	we	like	school
(50)	studying	is	useful	we	like	studying
(51)	to-study	is	useful	we	like	to-study

This seems simple enough, but there are complications. Because infinitival-to looks like the preposition to, prepositions generally cannot precede it lest it sound as if two prepositions were together:

	subject	verb	preposition	prepositional object
(52)	we	think	about	school
(53)	we	think	about	studying
(54)	we	think	about	to-study

Infinitival-to sound especially awkward when it follows the preposition to:

	subject	verb	adverb	preposition	prepositional object
(55)	we	look	forward	to	school
(56)	we	look	forward	to	studying
(57)	we	look	forward	to	to-study

Only the preposition for can precede infinitival-to, and then there is one complication, namely that for is typically substituted with a zero-form:

	subject	verb	preposition	prepositional object
(58)	we	pay	for	school
(59)	we	pay	for	studying
(60)	we	pay	for à Ø	to-study

In older varieties of English, one could say we pay for to study, and many older British and American songs say for to, for example in older (yet still well-known) songs like “Oh! Susanna”:

I came from Alabama

With my banjo on my knee,

I’m going to Louisiana

My true love for to see...

And, of course, Shakespeare uses this construction three times in the first seven lines of the following passage of King Leir:

Should you enjoin me for to tie a millstone

About my neck, and leap into the sea,

At your command I willingly would do it: [3.45]

Yea, for to do you good, I would ascend

The highest turret in all Brittany,

And from the top leap headlong to the ground:

Nay, more should you appoint me for to marry

The meanest vassal in the spacious world, [3.50]

Without reply I would accomplish it. . .

(King Leir, 1.3.43-49)

Of course, there are still modern dialectal uses of this construction as seen in the work of popular Irish musicians like U2 in their somewhat recent song entitled “Mofo” (1997):

Looking for to save my save my soul

Looking in the places where no flowers grow

Looking for to fill that God-shaped hole

In standard modern English, however, the expression is shortened to we pay for to study, possibly because, according to speakers, for and to already have similar meanings.[6]

When a new subject of the infinitive comes between for and the infinitive, however, the preposition for is not substituted with zero. For example, in the sentence, we paid for Joseph to study the name Joseph comes between for and to study.

	subject	verb	preposition	prepositional object
(61)	we	pay	for	[Joseph to-study]
(62)	we	pay	for à Ø	[   Ø to-study]

This seems to happen because, if a new subject like Joseph comes between for and to, then they are kept far enough apart for it not to seem as though two prepositions were adjacent. However, if there is no new subject, then for and to become too close and for must take a zero-form.

2a. A Past Precedent Developed

Since variables within a formula allow variation only in the variable itself, not in the formula, a clear distinction is drawn between what can and cannot be changed. This eliminates the potential for ambiguous writing.

Indeed, the attempt to weed out ambiguity from the language has already been applied in the shaping of the English language education in the British and American education systems throughout the centuries. Note the famous example of the double negative in English sentences, like the following, which follow the same principle as the mathematical double negative. In English, in order to negate a sentence, only one negative word is used, just as in mathematics only one negative is used:

(63) I can’t do a thing or I can do nothing

(64) (-x)·(x) = (-x²) or (x)·(-x) = (-x²)

However, two negatives in English make an emphatic positive:

(65) I can’t do nothing = I can do something!

(66) (-x)·(-x) = (x²)

This is important in the case of students educated in French (a not uncommon language of education in the Middle East), which like other Romance languages does not apply the same standard:

(67) French: Je ne peux rien faire. ‘I cannot do anything’/‘I can do nothing’

(68) Spanish: no puedo hacer nada. ‘I cannot do anything’/‘I can do nothing’

In the case of languages like French and Spanish, two negatives maintain a negative meaning. Instructors of English throughout the ages who have been aware of this trait of languages like French have been effective in applying the English standard of “two negatives make a positive”: (‑x)·(-x) = (x²).

Of course, this paradigm could be expanded to cover structures which non-native speakers of a language more often than not fail to master. One such difficult-to-master construction is that of the inherently negative irrealis (unreal) subjunctive, in such sentences as if I were…, then….

Since the irrealis (unreal) subjunctive is the opposite of what is true, it is inherently negative in truth value. In other words, a positive irrealis subjunctive, which is internally negative (‑x), means the same thing as a negated indicative, which is externally negative -(x):

(69)	irrealis (aff)	If	Mary	were(‑x)	pretty,	then I	would be(‑x)	happy.
	=
	indicative (neg)	Ø	Mary	is not‑(x)	pretty,	so I	am not‑(x)	happy.

Since this is a universal tendency, it is not surprising that both languages related to English, like French, and languages unrelated to English, like Lebanese Arabic, have identical structures, i.e. the use of a verb with past-tense morphology that has present meaning:

(70)	irr. (aff)	Si	Marie	était(‑x)	jolie,	donc je	serais(‑x)	heureux.
	=
	ind. (neg)	Ø	Marie	n’est pas‑(x)	jolie,	donc je	ne suis pas‑(x)	heureux.

and[7]

(71)	irrealis (aff)	مبسوط.	(‑x)كنت	لـ	حلوة ،	(‑x)كانت	مريم	لو
	=
	indicative (neg)	مبسوط.	‑(x)مش	أنا	حلوة .	‑(x)مش	مريم	Ø

The irrealis subjunctive encodes negativity at its core (-x), while an indicative verb takes its negation from outside the core -(x). Nevertheless, both end up with the same negative value: since either (-x) or -(x), once the parentheses are removed is -x.

This being the case, a negative irrealis subjunctive -(-x) has the same meaning as a positive indicative (x):

(72)	irrealis (neg)	If	Mary	were not‑(‑x)	pretty,	then I	would not be‑(‑x)	happy.
	=
	indicative (aff)	Ø	Mary	is(x)	pretty,	So I	am(x)	happy.

Again French and Lebanese Arabic have identical structures for sentences with the same meaning:

(73)	irrealis (neg)	Si	Marie	n’était pas‑(‑x)	jolie,	…
	=
	indicative (aff)	Ø	Marie	est(x)	jolie,	…

irrealis (aff)	…	donc je	ne serais pas‑(‑x)	heureux.
=
indicative (neg)	…	donc je	suis(x)	heureux.

and

(74)	irrealis (neg)	مبسوط.	‑(‑x)ما كنت	لـ	حلوة ،	‑(‑x)ما كانت	مريم	لو
	=
	indicative (aff)	مبسوط.	(x)Ø	أنا	حلوة .	(x)Ø	مريم	Ø

Since many aspects of this construction apply across languages, pure comparison should not be ruled out for purposes of learning the construction. However, a mathematical principle operates within all of them, a principle which provides the logical basis for understanding the concepts.

2b. A Reliable Touchstone/Standard

Formulas do not lie. As such, in a world where many style books may disagree on what is preferable usage, mathematical logic can provide a touchstone by weeding out arbitrary preference, adhering to a logical standard corroborated by formulaic logic. For example, one style book might deem the following two sentences equally unacceptable:[8]

(75) *He likes to jog, he vandalizes and to wash cars.

(76) He enjoys jogging, vandalism and washing cars.

Stylists who believe that both (75) and (76) are both equally ungrammatical suggest “corrections” such as the following:[9]

(75´).	He likes to {__ jog, __ vandalize and __ wash cars}.

(76´). He enjoys {jogging, vandalizing and car washing}.

It is worth pointing out, however, that only (75) is truly wrong and in need of the repairs suggested in (75´), since only here can we not apply the left-factored he likes to all terms equally and find that without the suggested repairs of (75´) the sentence lacks grammatical parallelism:

(75´´). *He likes {to jog, he __ vandalizes and to wash cars}.

However, the so-called “repairs” of (76´) are not needed. Indeed, (76) is not really an error since the three terms (conjoined with and) are all noun-phrase (np) complements of the phrase he enjoys, as seen in (76´´).

(76´´). He enjoys {jogging_(NP), vandalism_(NP) and <washing cars>_(NP)}.

Writers on style may have strong opinions about what looks more or less even and symmetrical, but there is a sharp contrast between (75) that which truly violates grammatical principles and (76) that which simply may not meet up to an individual stylist’s ideals of evenness for the sake of elegance.

Although factorization formulas show the greatest connection between mathematical logic and language, the appeal to a scientific standard can be taken further. An area where stylists often disagree on which kind of structures they would prefer to see is in the use of modifiers. Note, for example, the basic English word order of the following sentence:[10]

(77)	Subject	Aux Verb	Manner Adjunct	Main Verb	Direct Object	Place Adjunct	Instrument Adjunct
	John	will	secretly	drive	the car	here	with gloves.

Out of fear that the reader will assume it is the car (and not John) which is with gloves, many stylists highly recommend moving the adjunct phrase with gloves to the beginning of the sentence, working under the assumption that the natural place of with gloves is alongside the subject (not closer to the object):[11]

(78)	Instrument Adjunct	Subject	Aux Verb	Manner Adjunct	Main Verb	Direct Object	Place Adjunct
	With gloves,	John	will	secretly	drive	the car	here	__.

Some confusion seems to arise to the effect of considering that the first example is erroneous while the second example is correct. In actuality, both are correct, but the first example (77) is more natural to the language, while the second example is merely an emphatic focus form (78). How do we know this?

Movement of elements from their place of origin is common in languages. For example, English moves the auxiliary verb before the subject in closed-ended (yes/no) questions:

(79)	Aux Verb	Subject		Manner Adjunct	Main Verb	Direct Object	Place Adjunct	Instrument Adjunct
	Will	John	__	secretly	drive	the car	here	with gloves?

And English additionally moves the interrogative wh-word to the very beginning of the sentence in open-ended questions:

(80)	Place Adjunct	Aux Verb	Subject		Manner Adjunct	Main Verb	Direct Object		Instrument Adjunct
	Where	will	John	__	secretly	drive	the car	__	with gloves?

In questions, special focus movement of instrument adjuncts does not bring them next to the subject. For this reason, both of the following attempts to move the instrument adjunct with gloves closer to the subject give ungrammatical results:

(81)	Place Adjunct	Aux Verb	Instrument Adjunct	Subject		Manner Adjunct	Main Verb	Direct Object
	*Where	will	with gloves	John	__	secretly	drive	the car	__	__?

(82)	Place Adjunct	Instrument Adjunct	Aux Verb	Subject		Manner Adjunct	Main Verb	Direct Object
	*Where	with gloves	will	John	__	secretly	drive	the car	__	__?

Instead, moving the instrument adjunct with gloves to the beginning of the sentence takes it outside of the entire structure, showing that this is truly a focus form, not the result of any natural tendency to be close to the subject:

(83)	Instrument Adjunct	Place Adjunct	Aux Verb	Subject		Manner Adjunct	Main Verb	Direct Object
	With gloves,	{where	will	John	__	secretly	drive	the car	__	__}.

Since its place of origin seems to be near the end of the sentence (after the object and place adjunct), moving the instrument adjunct to the beginning of the sentence can be understood as a means of stylistic emphasis, not as a more correct grammatical form.

2c. Beyond Grammar

Far from being useful in grammar alone, variables can also be applied to the systematic use of recurring main and supporting points in a formal writing structure. Furthermore, understanding how variables are used in formal writing, helps students determine what the variables of a standardized-test essay prompt are and how to how to create a thesis formula for them.

The first step is finding the main points implied in an essay prompt and labeling them X, Y & Z:

(84)

• • •

Discuss a misconception^X you once had Tell what you have since learned to be the truth^Y. Explain how^Z your attitude was changed by what you learned.

• • •

X Y Z

A thesis formula must then be constructed which can support the main points. Ideally, this will show causal relation with causal conjunctions (e.g. because, since, so, etc.), contrast with contrastive conjunctions (e.g. but, whereas, while, etc.), and equality with equation conjunctions (e.g. and, etc.):

• THESIS: I once wrongly believed X, but I have since learned the truth is Y, and the way this changed my attitude is Z.

The student can then proceed to supplement individual values for the variables X, Y & Z:

X = _________________________________________________________________

Y = _________________________________________________________________

Z = _________________________________________________________________

Finally, the variables can be plugged into the formula to create a complete thesis, upon which the structure of the essay will rest. Since the variables will be clearly labeled and individuated points, the student will better be able to understand that the x-paragraph is to contain x-related material, while the y-paragraph is reserved for y-material as is the z-paragraph for z-material:

• Paragraphs:

• Paragraph X with details: who, what, where, when, why & how[12]?

• Paragraph Y with details: “

• Paragraph Z with details: “

• Conclusion: Recap (Future Predictions?) on X, Y, & Z.

Sometimes finding X, Y & Z can be tricky since the information is not always broken up evenly, as in this example where X, Y & Z are divided between two bullets:

(85)

• •

Discuss a misconception^X you once had and what you have since learned to be the truth^Y. Explain how^Z your attitude was changed by what you learned.

• •

X Y Z

And here where X, Y & Z are all bunched up together on one line:

(86)

•

Discuss a misconception^X you once had, what you have since learned to be the truth^Y, and how your attitude was changed^Z by this.

•

X Y Z

Although, in a strict sense, the heavier mathematical principles underlying grammar are not present in writing style, the fact that formulas and variables are common throughout the world, while specific writing styles vary greatly from country to country and culture to culture, gives the SL student familiar elements of organization to work with. And since the more logical an argument style claims to be, the more it makes claims to being scientific (and thus mathematical) in nature.[13]

3. Conclusion

We have discussed the means by which and the reasons for which mathematical logic can be applied in language instruction.

Factorization and the use of variables can be found to apply to a good many language phenomena often thought of as learnable only by principles of usage. The use of variables also provides scaffolding for writing structure, which could be facilitated by the use of universally understood terms.

As seen by past precedent, mathematical logic can provide a reliable means for determining the grammaticality of a form and for distinguishing the laws of good grammar from the opinions of good style. Since the goal of a good many instructors is to prepare students to write in error-free unambiguous language, the distinction should surely be made between logical and grammatical errors, on the one hand, and stylistic preferences, on the other.

What remains now is for these hypotheses to be tested on a wider range in order to determine whether they can contribute to how SLs are taught and if so within what contexts and limitations.

4. References

Bovée, Courtland L. and John Thill. 2004. Business Communication Today. 8th Edition. Prentice Hall, New York.

Brahmagupta. 628. Brahma Sputa Siddhanta.

Cowell, Mark W. 1964. A Reference Grammar of Syrian Arabic. Georgetown University Press, Washington, D.C.

Curtin, Elizabeth H. and Connie L. Richards. 2002. “Writing Across the Curriculum: Faculty Manual.” Revised edition. Salisbury University.

<http://www.salisbury.edu/wac/more/manual/wacmanual.doc>

Foster, Stephen. 1848. “Oh! Susanna” Song.

<http://www.bartleby.com/59/8/ohsusanna.html>

Langan, John. 2001. English Skills with Readings. 5th edition. McGraw-Hill Companies, New York.

Pāṇini. c. 400 BC Aṣṭādhyāyī.

Shakespeare, William, et al. 1997. “King Leir.” The Riverside Shakespeare. 2nd ed. Joseph Jay Tobin, et al. (ed.) Houghton Mifflin Company, New York.

Thackston, Wheeler M. 2004. An Introduction to Koranic and Classical Arabic. 2nd ed. Ibex Publishers, Bethesda, Maryland.

U2. 1997. “Mojo.” Pop. Audio CD. Island Records. Track 3.

Wright, William LL. D. 1896-98. A Grammar of the Arabic Language. Vol. 1-2. Cambridge University Press.

Antonine University

Hadath-Baabda, Lebanon

<eidcynthia@yahoo.fr>

UCLA

Department of Linguistics

3125 Campbell Hall

Los Angeles, CA 90095-1543

<cadam@ucla.edu>

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[1] In the last example, since [his chicken] is a phrase, in which “chicken” is governed by “his,” the whole phrase is moved as a block.

[2] Although the following examples are from MSA, Arabic dialects, such as those of the Levant, Gulf and Egypt, follow the same structural guidelines.

[3] Brahmagupta’s Brahma Sputa Siddhanta, written in 628.

[4] Pāṇini’s Aṣṭādhyāyī,written around 400 BC.

[5] The corresponding subjunctive sentences for the following two sentences would be “I demand that a student eat burgers,” “I demanded that a student give us money,” and “I demand that a student be angry.”

[6] The meanings of “for studying” and “to study” are not exactly the same, for the former generally signifies cause while the latter signifies purpose. This is because “for,” like Arabic لِـ/li-, can be used for cause, e.g.لذٰلك لم يذهب/li-ðālika lam yaðhab, or purpose, e.g. للحصول على الدكتوراه/li-al-ḥusūl ʕalā al-duktūrāh.

[7] The external negation of miš/muš is based on its being a compound of three morphemes: mā + huwa + šay’, the first of which is the same negative morpheme found in the irrealis mā kēnit of (74).

[8] One example is (Langan 2001).

[9] Bracketing and factorization in the following examples are our own.

[10] The term adjunct is being used in the linguistic sense as an umbrella term for, among other things, prepositional phrases and adverbs.

[11] One example is (Curtin and Richards 2002).

[12] Please note that while howcome counts as a wh-word, howdy does not.

[13] Bovée and Thill in Business English, for example, refer to one style of writing as the 2+2=4 method, since the various tiers of support for a main point must “add up.”