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FOREWORD

This book advocates a novel approach to the study of natural language. Rather than pleading for it, however, it applies it to a substantial fragment of the English language and presents a grammar for it. The grammar takes the form of a system of fully articulated rules which generate infinitely many English expressions. The methodology can thus be judged not just by arguments proffered in its favour but by the results it yields.

Here are several aspects in which the present approach contrasts with that underlying most of current linguistic research.

a) It does not take the task of the grammarian to be merely that of circumscribing the class of expressions which English endows with meaning. It assumes rather that the grammarian must also determine what those meanings are. In other words, it takes seriously the definition of grammar as a generator of expression/meaning pairs.

b) It is based on the view that any separation of the study of ways in which expressions are combined into compounds from the study of what those expressions mean is methodologically unsound. One obvious objection to such a procedure is that it is wasteful. If sentences are first generated as mere strings in a separate 'syntactic module' of a grammar, then (if a compositionality hypothesis, however weak, is correct) the same generation process has to be run through over again in the 'semantic module', this time generating the meanings borne by the strings as well as the strings themselves. There is a deeper objection, however. A purely syntactic generator of well-formed expressions is in principle impossible because the well-formedness of a compound expression often depends not only on whether it's components are well-formed but also on what they mean. Syntax and semantics must go hand in hand.

c) It resists the temptation to generate sentence-meaning pairs indirectly by associating English sentences with their translations into another language, be it a language of 'deep structures' invented by linguists or a symbolic notation invented by logicians. This is not only because the other language is itself in need of interpretation, so that the semantic problem is passed on rather than solved. The main reason is that on this approach the peculiar method in which a natural language associates expressions with their meanings is left unexamined. A natural language is a code and an important part of the grammarian's task is to decipher that code. This task is not discharged by translating vernacular expressions into an invented 'ideal' notation which is based on substantially different coding principles.

d) The notion of a code presupposes that prior to, and independently of, the code itself there is a range of items to be encoded in it. Hence if the grammarian is to be seen as someone attempting to crack the code of a language, meanings cannot be conceived of as products of the language itself. They must be seen as logical rather than linguistic structures, amenable to investigation quite apart from their verbal embodiments in any particular language. To investigate logical construction in this

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way is the task of logic. The linguist's brief is to investigate how logical constructions are encoded in various vernaculars.

The aim of the present book is to decipher the coding system operative in the English language. The result, a 'meaning driven grammar', is a set of rules which are entirely free of the sort of corner-cutting that is taken for granted in conventional styles of linguistic description. Some of these rules are considerably intricate, but hopefully not more so than they need to be. Thus we get a first glimpse of the actual degree of combinational complexity - or, as Frege used to call it, 'mathematical multiplicity' - involved in natural codes. It transpires, inter alia, why all attempts at producing mechanical translators have so far been doomed to failure: they have all been based on gross oversimplifications of the system of rules which (unreflectively) guide speaker/hearers in generating and interpreting natural-language expressions. But in the age of the microchip complexity is fortunately no object.

CHAPTER ONE: THE STATE OF THE ART

1.1 'AUTONOMOUS SYNTAX'

A linguistic expression is a means of conveying information. To understand it is to know what particular piece of information it conveys. Hence the linguist's task vis-a-vis a vernacular sentence is that of establishing what particular message it carries and how it encodes this message. His task vis-a-vis the vernacular as a whole is that of showing how the encoding works in general to generate an infinite number of message-bearing sentences. In other words, his task is that of construction a grammar, a system for generating sentence-meaning pairs.

It is an intriguing fact that although this definition of the linguist's task is almost universally accepted, no sentence-meaning pairs, or system for producing such, are to be found in the linguistic literature.

One reason for this curious situation is the endemic view that linguistic research can be compartmentalised into two separate domains, syntax and semantics, and that the former can be pursued quite independently of the latter. Noam Chomsky, who has coined the definition of grammar as a generator of sentence-meaning pairs, is a proponent of so-called autonomous syntax, based on 'the thesis ... that the question of what the syntactic structure (underlying or superficial) of a sentence is entirely independent of the question of what its semantic structure may be.'⁽²⁾

It is not easy to see what advantage there can be in taking Chomsky's advice and deliberately ignoring meaning when studying syntax. The constitution of any artefact is dictated by the purpose it is to serve. The shape of a piece of clothing, for example, is dictated by the fact that it is to cover a certain part of the human body. Clearly no theory of clothing can explain why a shirt, say, takes the geometric shape it does without reference to that fact. Analogously, one would have thought, the syntactical shape of a sentence is dictated by the fact that it is to encode a definite message. No

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theory of language can explain why a sentence like 'Ali works' takes the shape is does without reference to the fact that it bears a message which ascribes a definite activity, that of working, to a definite individual, Muhammad Ali.

The same applies even if one accepts Chomsky's view that a language is not entirely an artefact and that sentences are the products of a 'language organ' which has been fashioned by biological evolution. Any explanation of why the sentences produced by the language organ take the shape they do will clearly have to refer to their adaptive function, just as any explanation of the shape of a bird's wing will have to refer to the fact that it is to keep the bird aloft. The adaptive function of a sentence is clearly to serve as a vehicle of information exchange between members of the species. Chomsky rejects all this. His hypothesis seems to be that in spinning its linguistic webs the human mind is programmed to produce certain primordial patterns which have little to do with the logical structure of the messages linguistic expressions serve to convey. Hence in trying to uncover those arcane patterns the linguist is best advised to ignore not only the structure of those messages but the messages themselves. Syntax, according to Chomsky, 'can be essentially defined without reference to interpretation'.

All the autonomous syntactician has to say about a sentence like 'All slowly works' is that it is an S consisting of an NP ('Ali'), and a VP ('slowly') and a V ('works'). This analysis is usually presented in the form of a tree-like diagram called a 'phrase marker':

S

                                                             ÚÄÄÄÄÁÄÄÄÄż

                                                            NP                   VP

                                                             ł              ÚÄÁÄż

                                                             ł              ADV     V

                                                             ł                 ł         ł

                                                           Ali          slowly     works

or, more economically, thus:

[_S[_NPAli][_VP[_Advslowly][_Vworks]]].

If a node branches into several others then the former is spoken of as a 'mother' and the latter as 'daughters' of the former and 'sisters' to each other. A mother 'dominates' her daughters, the daughters (if any) of her daughters, and so on. A string of words in the original sentence is a 'constituent' or 'phrase' if there is a node which dominates all and only the words in the string. According to the above phrase marker, 'slowly works', for instance, is a constituent, while 'All slowly' is not.

But what exactly is the syntactician telling us by means of a phrase marker? What do the letters 'S', 'NP', and 'VP' etc. stand for? They seem to suggest the familiar grammatical terms 'sentence', 'noun phrase', and 'verb phrase'. But the autonomous syntactician can hardly say that they simply abbreviate these traditional terms. For traditionally a sentence was explained as a bit of language

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expressing a complete message or thought, a noun phrase as a name of an object, and a verb phrase as a name of a property or activity ascribable to objects. All these explanations invoke what those bits of language mean, thus stepping outside the domain of autonomous syntax. The impression one gets is in fact that 'S', 'NP' etc. are not supposed to mean anything, that the autonomous syntactician does not use them to tell us something about what they stand for, but rather mentions them to tell us something about themselves. His theory seems to amount to a proposal for an artificial and uninterpreted notation.

It would be likewise in vain to ask an autonomous syntactician what the term 'constituent' means. He certainly cannot say that a constituent is an expression which is complete in that it refers all by itself to a definite entity, in contrast to an incomplete expression which refers only in combination with some other expressions. For that too would amount to leaving the domain of autonomous syntax. The term 'constituent' (or 'phrase') is apparently not to be burdened with any pre-theoretical meaning at all: a constituent is simply whatever the grammarians's theory brands as such in any particular case.

Finally, it would be equally idle to ask what governs the distribution of the mother/daughter relation. Why is it, for example, that 'slowly' is a sister of 'works' but not of 'Fred'? The syntactician cannot explain it by pointing out the obvious fact that 'slowly' stands for an activity modifier, ie for a mapping which takes activities to activities, and that the activity named by the VP 'slowly works' is the value of that mapping at the argument named by 'works'. In brief, he cannot say that 'slowly' is a sister of 'works' because the entities they stand for are related as a mapping and its argument. For that too would be transgressing the boundaries of autonomous syntax. No pre-theoretical meaning seems to attach to the term 'sister' either.

All this makes the syntactician's constituency, dominance, and sisterhood judgments - and thus say any sentence analysis he cares to propose - in principle uncriticizable. It can be shown, however, that by wilfully shutting their eyes to meaning, autonomous syntactisians frequently end up with analyses which fly in the face of their own purely syntactic principles. What follows are a few conspicuous examples.

Constituency

In a standard textbook endorsed by Chomsky, Adrian Akmajian and Frank W. Heny introduce the novice to the notions of phrase-marker and constituency in the following way. They cite the fundamental phrase-structure rule

(1.1) S ® NP VP

and comment,

[Phrase structure] rules introduce nodes in such a way that the sentence is broken down into a hierarchy of ever smaller constituents. So the immediate constituents of S are NP and VP ...⁽³⁾

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Then they note that a sentence like

(1.2) Fortunately John saw Mary

cannot be generated by means of (1.1), for the word 'fortunately' does not seem to be part of either the noun phrase or the verb phrase of (1.2). So, the authors tell us, we have to modify our S-expanding rule to include an optional adverb:

(1.3) S ® (Adv) NP VP.

Gives this rule, the 'hierarchical structure' of (1.2) will be represented thus:

(1.4) [_S[_Advfortunately][_NPJohn][_VPsaw Mary]].

The striking thing about this phrase marker is that it contains no node corresponding to the form of words 'John saw Mary'. Hence if the phrase marker correctly parses the sentence, 'John Saw Mary' is no constituent of (1.2).

It is hard to believe that 'the many students who used earlier drafts of [Akmajian's and Heny's] text' all accepted this conclusion without demur. At least some of them must have given vent to an intuition that in (1.2) 'fortunately' is a qualifier applied to the sentence 'John saw Mary', animating that the circumstance adverted to by the sentence is a fortunate one. If they did they had a venerable authority on their side, because this is exactly what it says in the OED⁽⁴⁾. A sentence occurring within another sentence should surely qualify as a constituent of the latter.

One wonders what answer these students received from their tutors. They were probably told that it is bad manners to appeal to considerations of meaning in the study of syntax. Maybe the example was chosen deliberately to nip the reader's meaning-based intuitions in the bud.

But then there are purely syntactic considerations militating very strongly against (1.4). For one thing, it used to be good orthography to write 'Fortunately, John saw Mary',⁽⁵⁾ thus highlighting the fact that what follows the comma is a self-contained part of the whole sentence. Although in modern usage the comma tends to drop out it is still de rigueur in 'John saw Mary, fortunately'. Surely the status of 'John saw Mary' as a constituent does not depend on whether the adverb 'fortunately' appears at the beginning or at the end of the sentence. Another point is that 'John saw Mary' can be made the focus of a question by asking (if not very elegantly) 'Fortunately what ?' Clearly nothing of the sort can be done with something that is not a constituent.

But more sophisticated arguments can be given. Chomsky himself has argued that the possibility of forming coordinate structures like

X and Y

is a sure sign that both X and Y are constituents:

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If we have two sentences Z+X+W and Z+Y+W ... we can generally form a new sentence Z+X+and+Y+W. If X and Y are, however, not constituents, we generally cannot do this.⁽⁶⁾

Now 'Fortunately Mary saw John' is as good and English sentence as is (1.2). Furthermore, the two can be combined into 'Fortunately John saw Mary and Mary saw John'. Hence by Chomsky's own test (taking 'fortunately' as Z and an empty string as W), 'John Saw Mary' is a constituent of (1.2), contrary to (1.4).

Just in case the reader thinks that this is making too much of what may be a mere slip, or perhaps a deliberate simplification, in an introductory text,⁽⁷⁾ let us consider the sentence.

(1.5) The man came,

to which Chomsky assigns a phrase-marker containing this:

(1.6) [_S[_NP[_DETthe][_Nman]][_INFLpast][_VPcome]]].

It seems natural to assume that the past tense particle modifies the infinitive root of the verb 'come' into its past-tense form. After all, the combination of the two reaches the surface in the form of a single word, 'came'. So the untutored mind could be excused for expecting that the combination of [_INFLpast] and [_VPcome], will be parsed as a self-contained phrase. But if Chomsky's formula is right, the untutored mind is wrong. For a part of an expression to be a constituent, it must be possible to 'trace' it up through the phrase-marker 'to a node that exclusively dominates it' (see Jacobsen II, p. 52). But there is no node in (1.6) which dominates just 'past' and 'come': any mode dominating these two words also dominates 'the man'. Thus, according to (1.6), the word 'came' (ie the surface form on the combination of 'past' and 'come') does not qualify as a grammatical constituent of the sentence.

Yet 'The man will come again' is undoubtedly as good an English sentence as is (1.5). Moreover, the two can be combined into 'The man came and will come again'. Hence by Chomsky's own coordination test (taking 'the man' as Z and an empty string as W), 'came' is a constituent of (1.5), contrary to (1.6). If so, Chomsky' phrase-structure rule

S ® NP INFL VP,

which compels us to analyze (1.5) in the form (1.6), cannot be correct.

Anaphora

Although autonomous syntax is supposed to analyze sentences 'without reference to interpretation', a great deal of the work emanating from Chomsky and his followers under that heading comes in fact with a heavy admixture of considerations which are unmistakably semantic in nature. In discussing pronouns and anaphora, for example, the theorists do not hesitate to enlist our intuitions concerning coreference, a notion which by no stretch of the imagination can be called syntactic. But the fact that

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the discussion is conducted under the heading of autonomous syntax provides an excuse for not taking coreference quite seriously. What results are analyses which postulate anaphoric relations where, from a logical point of view, none can be present.

Sentences containing reflexives are a typical case at issue. The autonomous syntacticians never question the dogma that in a sentence like

(1.7) Fred washed himself

the reflexive pronoun 'himself' is anaphoric of, and hence coreferential with, 'Fred'. To express this typographically they add coreferentiality subscripts thus:

(1.7') Fred₁ washed himself₁.

It is thus taken for granted that in (1.7), Fred is referred to twice: once in the subject and again within the predicate 'washed himself'. One may well ask how a thesis of this sort can find its place in a theory which calls itself 'autonomous syntax'. But, setting this aside, its the thesis defensible?

Not if 'reference' means reference. To see this, consider

(1.8) Fred washed himself and so did Mary.

In this sentence the right-hand conjunct ascribes to Mary the very same property that the first conjunct ascribers to Fred. Thus 'so did' is undeniably anaphoric to 'washed himself'. But if the predicate 'washed himself' referred to the property of having washed Fred - as it would if 'himself' was anaphoric to the clausal subject 'Fred' - then the right-hand conjunct would ascribe that property to Mary, which it clearly does not.

What the right-hand conjunct in fact ascribes to Mary is having washed oneself: a property which is nothing to do with Mary, Fred, or any other individual specifically. The property is instantiated by countless individuals and in each case it is the very same property that is instantiated. It is this property that the left-hand conjunct of (1.8) ascribes to Fred. Just as the sentence 'Fred sang and so did Mary', (1.8) refers to Fred only once. The reason that the pronoun in the left-hand conjunct takes its masculine form is not because it refers to a male individual but because it is part of a predicate which must gender-agree with its subject. Seeing anaphors where there cannot be any is a pervasive feature of Chomsky's method. Even in a simple sentence like

(1.9) John was chosen,

John is, according to Chomsky, mentioned twice: His analysis of (1.9) is

(1.10) John₁ was [PRO chosen t₁],

where t₁ is an anaphoric term whose antecedent is John. One cannot actually see the term in the sentence (1.9) itself, but Chomsky is right, the term is nevertheless there and refers to John.

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Note that by offering (1.10) as an analysis of (1.9), Chomsky is telling us that aside from 'John', (1.9) actually contains two invisible referring terms: PRO and t₁¹. How does he come to make this astonishing claim? It is simply the result of an attempt to substitute syntax for semantics.

It is an undeniable fact that in order for John to have been be chosen, there must have been be someone or other who chose John. Similarly, one may observe that in order for John to have received a letter, there must have been be somebody who sent him one. Each of the two observations concerns the meaning of an English verb phrase: the former concerns the meaning of 'was chosen' and the latter of 'received a letter'.

Chomsky, we have seen, takes himself to be doing syntax, which he considers to have nothing to do with meaning. Yet the insight concerning 'was chosen' is too good to pass up. So he artificially turns what is patently a matter of meaning into a matter of syntax. The results is the incredible claim that the predicate of (1.9) actually contains two invisible terms, one (PRO) referring to an unspecified chooser and one (t₁) to John, the already mentioned target of the choosing action. One might as well argue that the predicate of the sentence 'John received a letter' contains two invisible terms, one referring to the unspecified sender and one making a repeated reference to John.

To make matters even more mysterious, Chomsky further posits a whole invisible sentence,

(1.10') e was [PRO Chosen John],

the so-called 'd-structure' of (1.10). Although nobody ever utters (1.10'), Chomsky deems in the origin, or derivational source, of (1.10). He tells us that (1.10) arises from (1.10') as a result of the term 'John' moving from its original position to the 'landing side' marked e. The invisible anaphoric term t₁ in (1.10) is a 'trace' the term leaves behind on take-off.

Chomsky never makes it clear where the alleged flight of the word 'John' is supposed to take place. His repeated characterisations of autonomous syntax as a branch of psychology seem to suggest that he thinks of the movement as a psychological event, that the journey from the 'd-structure' (1.10) to the 's-structure' (1.10'), is one that the speaker/hearer makes in his mind.

Something of this sort was arguable in the context of Chomsky's original syntactic theory, as set out in his Syntactic Structures. There a passive like (1.9) was obtained through the so-called Passive Transformation from the corresponding well-formed active-voice sentence ('Someone chose John'). The transformation was reasonably well motivated. For one thing, a passive-voice sentence normally reports the same state of affairs as its active-voice counterpart. Secondly, postulating the transformation resulted in significant theoretical economy. This is because the formation of the active sentence is subject to various restrictions which an explicit grammar must spell out, and

[i]f we try to include passives directly in the grammar [Chomsky wrote] we shall have to restate all these restrictions in the opposite order... This inelegant duplication, as well as the

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special restrictions involving the element be+en can be avoided only if we deliberately exclude passives from the grammar of phrase structure, and reintroduce them by [means of a transformation rule].⁽⁸⁾

But most importantly, it is not implausible to assume that the transformation rule has some sort of psychological reality. It is arguable, if by no means obvious, that someone who ends up uttering the passive sentence (1.9) starts (perhaps subconsciously) with its active counterpart and (also subconsciously) reshuffles the terms in conformity with the alleged Passive Transformation.

But none of these arguments have any validity in the context of Chomsky's more recent theory, whereby the source of the passive sentence (1.9) is no longer its active counterpart but rather (1.10'). To an untutored speaker of English, (1.10') is a place of unintelligible word salad. Chomsky himself advocates a theory, called Case Theory, which offers an explanation why (1.10') is ungrammatical. According to that theory, every noun phrase must appear in the context of a verb which assigns it a definite case. As a transitive verb, 'choose' assigns the accusative case to the noun phrase which follows it (in 'Somebody chose John', 'John' has the accusative case). Now the participial suffix '-en', according to the Case Theory, robs the verb of its ability to assign case to a noun that follows it, hence any noun-phrase which follows 'chosen' in a sentence is a grammatical dangler. This is what makes (1.10') unintelligible.

It is conceivable that aversion to case-theoretic danglers is an inborn feature of the human mind, or, as Chomsky would say, that the principle whereby every noun phrase must be assigned Case is part of Universal Grammar. But, assuming that this principle is written into our genetic make-up, it is difficult to understand why the 'linguistic organ' should tolerate (1.10') and use it as a starting point of a mental process eventuating in the production (or comprehension) of (1.9). It is also difficult to understand why the trace t₁, which, according to Chomsky, is a tacit anaphor linked to the noun 'John', should be exempt from the principle and require no case not only in the d-structure (1.10) but even in the s-structure (1.10'). One can thus readily sympathise with A. Radford who rejects any interpretation of the transformations postulated by Chomskian syntactic theory as descriptions of 'what a native speaker actually does in his mind when he produces a sentence', and brands any suggestion to this effect as 'patently absurd', 'perverse', 'pointless', and 'utterly misguided'.⁽⁹⁾

So perhaps the alleged movement of 'John' from the object position to the subject position is not meant to correspond to anything psychologically real. Maybe it is just a convenient technical device facilitating the definition of the class of well-formed s-structures. If this is the case, however, the positing of the movement has no explanatory value. If (1.10) is postulated solely in order to put (1.10') in the class of surface structures, then any other definition which eschews postulating (1.10) and still successfully defines the same class should be equally satisfactory. It seems that a definition of this sort should not be difficult to come by. After all, a sentence like

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(1.11) John was present

is presumably also in the class of well-formed surface structures, hence any definition of the class will have to assign it to the class somehow. Chomsky is unlikely to propose 'e was [PRO present John]' as the d-structure source of (1.11). His wiew would be, probably, that in the case of (1.11) the deep and surface structures roughly coincide. But no matter how the well-formedness of (1.11) is accounted for, surely the well-formendness of (1.9) can be accounted for in the same manner, because the two sentences are word-by-word isomorphic. Indeed the fact that 'present' and 'chosen' can be combined into coordination structures like 'John was present and chosen to chair the meeting' seems to indicate that (1.9) and (1.11) should have parallel analyses.

Yet it is far from clear that Chomsky's deep structures can be sensibly construed as a technical aids to circumscribing the class of surface structures. For some of Chomsky's statements strongly indicate that, on his view, the notion of surface structure comes logically before that of deep structure. Chomsky repeatedly says⁽¹⁰⁾ that an expression counts as a deep structure only if it can be transformed into a well-formed surface structure, which seems to indicate that one has to know what the well-formed surface structures are before one can judge the well-formedness of deep structures. Yet not even that can be said with any certainty because elsewhere Chomsky assigns deep structures, perplexingly, also to ill-formed surface structures. One can see what Thomas Wasow is referring to when he says, euphemistically, that in Chomsky's post-1965 output the 'standards of explicitness and rigour were relaxed'.

But whatever the purpose of postulating d-structures in general, postulating (1.10) in the case of (1.9) seems definitely idle. Even if one should agree with Chomsky that the passive nature of 'is chosen' in (1.9) requires a syntactic account, and that such an account is best given by positing an invisible term anaphoric to 'John', the purpose of (1.10') is still unclear. In the surface structure (1.10) 'is chosen' is followed by the 'empty category' t₁ which is co-indexed with 'John'. Is this not a sufficient syntactic indication of the fact that the choosing action has 'John' for an object rather than subject? It is entirely unclear what more is achieved by postulating the completely unfamiliar sentence (1.10') and spinning the movement story. It is not as if (1.10) could not be generated other than via (1.10'). As we have seen, deep structures seem to be defined in Chomsky in terms of surface structures. Besides, Chomsky expressly subscribes to Edmond's Preservation of Structure Principle,⁽¹¹⁾ which says, essentially, that the result of carrying out a transformation on a base-generated deep structure must also be base-generable.

But the main point is that s-structure (1.10), however generated, is completely indefensible an as analysis of (1.9). It is vulnerable to the very same objection that was levelled above against the construal of 'washed himself' in (1.7) as containing an anaphoric reference to Fred. Having been chosen is a single general condition which can be satisfied by any number of individuals. This is why

(1.12) John was chosen and Bill was chosen

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can be condensed to

(1.12') John was chosen and so was Bill.

This would be hardly possible if the surface structure of (1.11) was

(1.12'') [John₁ was [PRO chosen t₁]] and [Bill₂ was [PRO chosen t₂]],

where t₁ (the trace of 'John') referred to John and t₂ (the trace of 'Bill') to Bill. For then the second occurrence of 'was chosen' in (1.12) would not be synonymous with the first and could not therefore be anaphorised.

But an autonomous syntactician is unlikely to be disturbed by arguments of this sort. After all, he subscribes to the thesis that 'syntax can be essentially defined without reference to interpretation.' This enables him to help himself to a semantic concept like coreference whenever convenient without feeling committed to the logical consequences of what he says about it.

Chomsky envisages, to be sure, another 'module' of grammar where the strings of letters generated by the syntactic 'module' will be assigned 'meanings'. But when one looks closely at the few heuristic hints that Chomsky has given on this matter (and at the few attempts on the part of his followers to implement those hints) it turns out that what he plans to assign to the surface strings are not so much meanings as other strings of letters. The semantic module is to translate the strings into their 'logical forms', which are expressions in an artificial language.

But to translate a sentence into another language hardly amounts to revealing what its meaning is; it only amounts to shifting the task to the other language. If someone wants to know how his watch works, his curiosity will hardly be satisfied to being told that it works on the same principle as the clock on the wall. By the same token, if someone wants to know how a sentence works semantically, his curiosity will hardly be satisfied by being told that it works in the same way as a certain sentence of another language. One can know that two sentences are intertranslatable without knowing what either of them means.

1.2 DETERMINERS

Chomsky's programme is undoubtedly in harmony with the nominalist ethos: much of twentieth-century philosophy agrees with W.V.O. Quine that there are no such things as meanings 'over and against their verbal embodiments'.⁽¹²⁾

Linguists are by and large less hardnosed than philosophical nominalists an indulge in a kind of double-talk. They concede that linguistic expressions have meanings. But they deny that this concession legitimizes the question What meaning? It would be in vain to look in modern linguistic literature for attempts explain what meaning is. What one finds instead are numerous arguments to she that any effort in this direction would be misguided. Jerrold Katz, for example, is of the opinion that

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the question 'What is meaning' does not admit of a direct 'this and that' answer; its answer in instead a whole theory. ... [I]t is a theoretical question, like 'What is matter?', 'What is electricity?', 'What is light?'⁽¹³⁾

The tacit premise seems to be that by producing a theory one is somehow exempt from the task of giving 'this and that' answers. Yet the accepted theory of electric would be hardly illuminating if it did not yield the 'this and that' definition electricity as a stream of electrons, and the accepted theory of light would be hard illuminating if it did not yield the 'this and that' definition of light as electromagnetic waves in a certain frequency band.

Bent Jacobsen suggests that we get rid of the obnoxious question by 'breaking it down':

[Transformational-generative] semanticists have argued from the beginning that the question 'what is meaning?' asked at one swoop is not likely to result in an adequate semantic theory. Rather, this question must be broken down into smaller, more modest ones of the following kind: 'What is the sameness of meaning (i.e., synonymy)?"' - 'What is semantic ambiguity?' - 'What is contradictoriness?'⁽¹⁴⁾

One may well wonder what makes the said semanticists a priori certain that an answer to the question 'What is meaning?' would yield an inadequate semantic theory. It would seem that once that question were satisfactorily answered, all the 'more modest' questions would be satisfactorily answered as well. Once we knew what meanings were, synonymy, ambiguity, etc. would present no further problems. We could simply say that expressions are synonymous if they have the same meaning and that an expression is ambiguous if it is fraught with at least two different meanings. Nor is the conceptual situation symmetrical: merely knowing that two expressions are synonymous does not by itself tell one what either of them means, and knowing that an expression is ambiguous does not tell one what it means on either of its readings. A mere theory of synonymy and ambiguity thus hardly amounts to a theory of meaning.

Indeed it is not easy to see how one could even begin to answer the 'modest' questions without having first answered the 'immodest' one. Imagine that someone argued that the question 'What is a book?', asked at one swoop, is not likely to result in an adequate theory of book ownership, and that we should rather ask more modest question like 'What is it for John and Bill to own the same book?' and 'What is it for John to own two different books?' But how can one even begin to answer these question if it is unclear what a book is? For suppose John has a copy of War and Peace and so does Bill. Do they own the same book? Or suppose John has two copies of War and Peace. Does he therefore own to different books? It is clearly futile to try to puzzle over such questions until it is decided what is to count as a book. By the same token, it is futile to puzzle over sameness or multiciplity of meaning until it is decided what meaning is.

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Propositions, Properties, and other Determiners

Given that sentences serve to convey information, a simple answer to the immodest question seems to suggest itself: the meaning of a sentence it its informational content, that is, the proposition it expresses. But what is a proposition?

Traditionally a proposition was defined as something for which the question of truth-values arises. Some proposition are true, others are false. We can thus say that a proposition is a truth-value determiner: depending on the facts, in determines one or the other of the two truth-values.

The proposition and the truth-value it determines have to be kept apart. The proposition that Bill Clinton is white determines the very same truth-value as the proposition that Jesse Jackson is black. Yet the two propositions are distinct, conveying as they do completely different information. One can be acquainted with a proposition without knowing what its truth-value is. All this would be impossible if a proposition and its truth-value were one and the same item.

The question 'What is meaning?' is, of course, not fully answered by asserting that the meaning of a sentence is a proposition. Sentences have parts which, although not sentences themselves, are nevertheless meaningful: common nouns, descriptions, adjectives, verb phrases, adverbs, and so on. But the fact that sentences are not the only meaningful expressions is matched by the fact that propositions are not the only determiners.

Properties are another typical kind of determiner. A property is an item which determines a class of objects, namely those objects which instantiate that property. It is readily seen that a property must be distinguished from the class it determines, or, as it is also called, its extension. The property of having a heart, for example, happens to determine the same class of objects as does the property of having kidneys; yet the two properties are undeniably distinct. One can be acquainted with a property without knowing which particular class is its extension. This is because having a definite class for an extension is not intrinsic to the property itself, but something determined by the property and matters of contingent fact. The circumstance that the house at 1600 Pennsylvania Ave., Washington, D.C. belongs to the extension of the property white, cannot be excogitated from the nature of the colour itself because it is partly due to the contingent fact that the house has been given a coat of a certain kind of paint.

While white is something that determines a class, and can thus be called a class-determiner, the U.S. president determines an individual, and can thus be called an individual-determiner. This individual determiner is not to be confused with the individual, Bill Clinton, whom it happens to determine. One can be acquainted with the determiner without which individual is determined by it. A person may regret, as Jerry Brown undoubtedly does, that he is not the U.S. president, without regretting that he is not Bill Clinton. The U.S. president, on the one hand, and the husband of Hillary Clinton, on another, are clearly two different determiners despite the fact they happen to determine one and the same man. Thus most of what has been said about white applies to the U.S. president. They differ in that the former determines a class and the latter an individual; but otherwise they are very much alike.

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Similar comments apply to number-determiners, or magnitudes. The temperature (in Centigrade) at Piccadilly Circus on 1/1/1982, at 12 noon, for example, is not a number, but something for a number to be. It determines a unique number, but is distinct from that number. For otherwise to assert that the temperature (in Centigrade) was, say, 10, would either amount to asserting the truism that 10 is identical with 10 (if the temperature at the palace was 10 degrees), or to asserting the absurdity that some number other than 10 is identical with 10 (otherwise).

Intentionality

In modern philosophy the notion of determiner emerged in reaction to epistemological psychologism, the view that it is our ideas, ie events or states taking place in our own minds, that constitute the objects of our knowledge. There are two major objection to psychologism, or, as it is also known, idealism. Firstly, it seems to isolate the knower from the world. If the subject matter of cognition is our own cognitive states, then the world out there is left out of our purview. Secondly, psychologism seems to isolate the knower from his fellow knowers. If my subject matter is my ideas then your subject matter is your ideas and there is no thematic overlap. There is nothing on which we can genuinely disagree, and any dispute would be just so much talk at cross purposes. Psychologism leads directly to solipsism.

The reason that not all practitioners of psychologism ended up as solipsists was because many imagined that somehow or other their ideas were correlated with the world at large and also with the ideas of their fellow human beings. But they were unable to give any plausible account of either correlation. Some have suggested that our ideas are connected with things by virtue of resembling them. But if an idea was an idea of something by virtue of resembling it, then the idea of motion, for example, would have to move and the idea of rot would have to be rotten. Also, the idea of a square would have to be square and the idea of a circle circular; consequently, there could be no idea of a square circle for the very same reason that there can be no square circle.

The doctrine of intentionality developed by Franz Brentano and his realist school opened a new avenue to solving the epistemological puzzle. According to this doctrine it is of the essence of mental events to be directed to objects; this is what distinguishes psychological from non-psychological phenomena. A mental event does not refer to itself, as the idealist would have it, but to a target beyond itself.

The intentionality thesis removes automatically the interpersonality problem. Although every mental act is private to the person in whose mind it occurs, mental acts occurring in different persons' mind can be directed to one and the same object. This is why individuals who have no access to each other's minds can nevertheless engage in collective inquiry and rational debate.

The objects which serve as targets of intentional reference, however, cannot be simply identified with the concrete particulars - trees, mountains, animals, persons - which populate the world around us.

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For mental reference is often to items which do not exist. One can consider and reflect upon purely imaginary mountains, mountains which are not to be found anywhere in the world.

A non-existent mountain can become a legitimate object of thought as soon as it is clear what it would be like if it did exist. We can think, for example, of the golden mountain. When we do, the focus of our cogitation is a collection of traits (mountainhood and goldenness) which would pick out a mountain if there were a unique one having those traits. What the mental act is directed to is thus not a mountain but a determiner of a mountain. The circumstance that is fails to single any mountain out does not disqualify the determiner as an object of mental reference. This is particularly obvious in the case of negative existence judgments. A judgment which denies that the golden mountains exists is clearly a judgment about a mountain-determiner: it asserts that the determiner is an improper one inthe sense of failing to single out any concrete particular.

When a determiner fails to pick out a concrete particular, it is a factual matter. As such it is external to mental acts and cannot affect their nature. If the determiner which in fact fails to pick out a mountain succeeded in doing so, the very same mental act which as a matter of fact refers to a non-existent mountain would refer to an existing one. It follows therefore that even in the case of an existing mountain - say the highest mountain on Earth, or the mountain depicted in a certain Cézanne painting - the immediate focus of our mental act is not something made of lava or sandstone, but rather a determiner which picks out such a thing. A dispute over the existence or otherwise of a mountain which in fact does exist cannot be a dispute about a definite lump of lava, for otherwise those who denied its existence would have to be either confused as to what the dispute is about or simply insane. The topic of such a dispute must be a determiner of a mountain and the question at issue must be as to whether any geological formation is in fact determined by it. The actual mountain which the determiner happens to pick out (should there be one) is of no direct concern, for the dispute might conceivably be settled even without determining which particular mountain it is.

Thus when a mental act refers to an existent object like the highest mountain on Earth, two items come into play: the determiner and, over and above, Mt Everest, the mountain the determiner happens to single out. But, as we have seen, it is the former which serves as a focus to which the mental act is directed.

Failure to see this point was the Achilles' heel of realism. Brentano, Meinong, and Twardowski - driven by their anti-idealist zeal and eagerness to show that our cogitations engage objects in the real world - all took it for axiomatic that to contemplate the highest mountain on Earth is to perform a mental act whose object is the mountain itself. Twardowski, the most incisive among the realists, did entertain an intermediary between the mental act and the lump of matter: he called it the content of the mental act. But the content, according to Twardowski, was not the target of the mental act but only something through which the mental act refers to the lump of matter itself.

But on Twardowski's theory it seems to follow that the act of contemplating the golden mountain has to be objectless. Certainly no mountain will qualify as its object, for it is a well-established fact that no mountain happens to be golden.

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The realists' way out of this difficulty was to defy that fact. They bit the bullet and denied that no mountain is golden. On their view, mountains came in two kinds: existing ones and those which fail to exist. They agreed that no mountain of the former kinds is golden. Bud they insisted that, as far as objects of knowledge go, mountains of the latter sort are as good as those of the former.

Here in a nutshell is what may be called the soft theory of non-existence. On this theory, to say that an object of a certain sort does not exist does not amount to saying that there is no such object. Failure to exist does not make an object a mere nothing. On the contrary, is hat has no effect on the object, qua object, whatsoever. In particular, it is no obstacle to the object's instantiating properties or bearing relations to other objects. It is thus no obstacle to the object's serving as the subject-matter of a body of knowledge. Meinong famously complained about an ingrained 'prejudice in favour of the actual'⁽¹⁵⁾ which makes people take one-sided interest in existent mountains as if non-existent ones were less interesting or less worthy of scientific investigation.

Meinong is often accused of ontological extravagance, of recklessly espousing all kinds of weird entities, from the golden mountain to the 27-eared pink elephant. But this accusation is unfair if Meinong's universe of 'pure objects' is looked upon as a universe of determiners, of things a concrete particular could possibly be. Neither the golden mountain nor the 27-eared pink elephant should be missing from anybody's list of things a particular could conceivably turn out to be. On this interpretation Meinong seems to have sinned not on the side of ontological largesse, but on the side of ontological parsimony. For he did not allow, over and above the determiners, for concrete individuals capable of being determined by them. Existence, according to Meinong, did not consist in a pure object's being embodied, ie having an individual corresponding to it. Rather, it was a non-relational feature of the pure object itself. A pure object consisted of a Sosein, a bunch of attributes definitive of what object it was. Some pure objects were, over and above, endowed with Sein, or existence. But the attributes in the Sosein as well as the attribute of existence were instantiated by the pure object itself. They were of the same logical type. Hence nothing but a completely ad hoc stipulation could preclude the formation of a Sosein including existence. One could, for example, consider the existing golden mountain. Now just as the golden mountain must be, according to Meinong, both golden and mountainous, so the existing golden mountain must be golden, mountainous, and existent. It seems to follow that not only are there golden mountains, but some of them exist. This, of course, is Russells' fatal objection to Meinong's Objektlehre.⁽¹⁶⁾

The root of the difficulty is Meinong's failure to distinguish between determiners and the concrete particulars determined by (some of) them. Once this distinction is made another one suggests itself. A determiner picks out its determinee (if any) through a bunch of properties. These may be called the determiner's requisites, because they are properties which an individual is required to have if the

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determiner is to pick it out. Mountainhood, for example, is a requisite of the determiner the golden mountain. But note that it is the individual (if any) which the determiner picks out that is required to instantiate the determiner's requisites, not the determiner itself. A determiner of a mountain is not itself a mountain, just as the property of being white is not itself white.

Like any other item, the determiner itself will, of course, also instantiate a variety of properties, but of properties of a higher logical type - not its own requisites. Determining something, and determining nothing, are typical properties instantiable by determiners. When it is asserted that the highest mountain on Earth exists, the former property is ascribed to the determiner the highest mountain on Earth. Nothing is said in this case of Mount Everest. To be sure, the mountain does come into play in an indirect way. It is because Mount Everest, that lump of rock, has what it takes to be the highest mountain on Earth that it is true to say that the highest mountain on Earth exists. But the statement would be equally true if any other lump of rock turned out to be the highest mountain on Earth. So the statement says nothing specifically about any particular mountain.

It is unclear anyway what an ascription of existence to Mount Everest itself could possibly amount to. It could hardly be a statement of contingent fact, a report of something established as a result of empirical examination of the mountain. It is absurd to imagine that the net result of examining a mountain might be the conclusion that the mountain does not exist. A non-existence claim can only be the result of an investigation whose starting point is something capable of determining a mountain: a description (pictorial, verbal, or cartographic) of a mountain. When empirical investigation reveals that the description does not happen to describe any mountain, it is a finding about the determiner specified by the description, not about any non-existent geological formation.

An Explication of Determiners

We thus see that determiners of various sorts - truth-value determiners, individual-determiners, class-determiners, number-determiners - are among the items that many assertions we make are about. What sort of objects are determiners in general?

The general notion of determination has found its rigorous explication in the mathematical concept of mapping. Given an argument, ie an object at which it is defined, a mapping assigns to it, and in this sense determines, the corresponding value. The mapping thus represents the dependence of its values on its arguments.

Determiners, we have seen, depend for what they determine on factual circumstances. The American president, for example, is one person in one set of factual circumstances and another person in another. It is thus natural to explicate determiners as mappings defined on possible sets of factual circumstances. But if such sets are to serve as arguments of mathematical mappings they must be given a rigorous mathematical explication in their turn.

In order to see how this can be done, let us imagine that we wish to undertake an empirical inquiry. To set the parameters of the inquiry, we have to make two decisions. Firstly, we have to decide what

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range of objects we wish to examine. Secondly, we have to agree what traits of those objects we are interested in. We have to agree, in other words, what properties of those objects, what relations between them, etc. are to count as relevant. The range of objects will be spoken of as the universe of discourse and the collection of relevant traits as the intensional base of the inquiry in question. Together the universe of discourse and the intensional base form what may be called an epistemic framework, a framework that is which sets the scope and limits of the empirical investigation in question.

Once an epistemic framework is fixed, a number of possibilities arise as to how the traits in the intensional base are distributed through the object in the universe of discourse. For the sake of illustration, let us consider a miniature example. Suppose that we are interested in two individuals only, say Bill Clinton and Jesse Jackson, and that the only traits we are interested in are the colours black and white. From a purely combinatorial point of view, there are altogether sixteen distributions of the two colour attributes through the two-element universe, starting with both Clinton and Jackson having both the colours, down to neither of the individuals having either colour. In view of the incompatibility of black and white, not all of these distributions are realisable or possible. If the impossible ones are weeded out, we are left with nine possible distributions, starting with Clinton black, Jackson black, followed by Clinton black, Jackson white, and so on down to neither of them black or white.

The nine possible distributions are known as the possible worlds with respect to the epistemic framework at issue. A possible world is thus one conceivable set of circumstances among the members of the universe of discourse as regards the traits in the intensional base. The class of possible worlds is known as the logical space generated by the epistemic framework.

Only one of the possible worlds in the logical space is actualised, namely the one where the traits from the intensional base are distributed through the individuals in the universe of discourse as they are in fact. In our miniature example it happens to be the world in which Clinton is white and Jackson is black. It is, however, no part of the definition of the framework to specify which possible world is the actual one. To establish which one it is rather the ultimate aim of the empirical inquiry conducted within the framework. The framework sets the question, but reality provides the answers.

Although the notion of possible world has recently been one of the most widely used tools of philosophical analysis, there is little agreement among philosophers on the nature of possible worlds. But it is worth noting that thinking of possible worlds as just suggested - that is, as distributions of traits through a universe of discourse - has the advantage of resolving the interminable agonizing as to whether there really are such entities as non-actualised possible worlds or whether they are just figments of the imagination. Once it is agreed that the actual world with respect to the Clinton-Jackson-white-black framework is the combination associating Clinton with white and Jackson with black, it would be defying a simple mathematical fact to deny that this world is one of

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several. The other combinations - such as the one which associates Clinton with black and Jackson with white - lack a certain contingent property, that of being actualised. But this is not to say that there are no such combinations, that the actual combination is the only combination there is. This would be analogous to claiming that since the number of planets is in fact nine, there are really no other numbers. (There are philosophers, of course, who disown numbers, combinations, and any other entities studied in mathematics wholesale. These philosophers, however, should disown the actual world among the non-actualised ones. Reality, if there be such, clearly consists in definite objects' exemplifying, and thus being combined with, definite traits. Hence some one who disowns combinations in general is thereby disowning reality in particular.)

Once a logical space is generated, the notion of determiner can be rigorously explicated as a mapping from possible worlds to set-theoretical objects of some given type. Properties can be explicated as mappings from possible worlds to subclasses of the universe of discourse. The property of being black, for example, takes every world to the class of individuals who are black in that world. At the actual world (with respect to our miniature framework) the value of the mapping is the class whose only member is Jackson. But, as noted above, to be acquainted with the colour it is neither necessary nor sufficient to know which objects are actually black and which are not. To be acquainted with the colour one must be able to tell which objects would be black under any specified circumstances, real or imaginary. This ability is adequately represented by the mapping which takes each of those circumstances to the appropriate class.

The logical space enables us to explicate not only the properties which are in the intensional base, but also a host of secondary, or derivative, properties. Indeed any mapping from worlds to classes of individuals represents a property of individuals. For example, the mapping which takes every world to the class of objects which are white or black in that world, represents the (derivative) property being white or black.

Similarly for determiners of other types. Take for example the mapping which takes a world to the individual who is black in that world, if exactly one individual is black in that world, and which is undefined otherwise. This mapping explicates the individual-determiner the black individual. But again, any mapping from possible worlds to individuals will count as an individual-determiner. For another example, consider the mapping which takes a world to the truth-value truth if Bill Clinton is black in that world and to falsehood otherwise. This mapping explicates the proposition that Bill Clinton is black. Any other mapping from worlds to truth-values also counts as a proposition.

The tiny epistemic framework invoked above as an example is, of course, too small to be of practical use. What makes it unrealistic, however, is not just the paucity of individuals and traits. It is also the circumstance that is does not allow for change, for the familiar fact that what is white at one time may become black later. To allow for change, the notion of possible world must be broadened. Instead of thinking of a possible world as an instantaneous state of the world (relative to a

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framework), we must think of it as a world history, a whole course of events unfolding in time. In other words, a world must be conceived of not as a single distribution of the traits from the intensional base through the universe of discourse, but as a series of such distributions, one for each moment of time.

A determiner picks out its determinee relative not just to a world, but relative to a world and a particular moment of time. What is picked out by the determiner the American president in the actual world, for example, is not a single person but a succession of persons. This succession may be called the chronology of the determiner in the world in question, and is best identified with a mapping taking moments of time to individuals. The determiner itself can the be represented by the mapping which takes every possible world (history) to its chronology in that world.

We thus see that the objects representing various entities relative to an epistemic framework are built up from four basic collections: the collection consisting of Truth and Falsehood, the universe of discourse, the logical space, and the time scale. Call these classes o, i, w, and t, respectively. The members of o (Truth and Falsehood) are called truth-values, the members of i individuals, the members of w possible worlds, and the members of t moments of time. (In view of the perfect isomorphism between moments of time and real numbers, the members of t can double as real numbers.)

Objects of types other than o, i, w, and t will be represented as mappings defined over these basic sets. Let x₁,x₂, ... x_n be an ordered n-tuple of some collections. An ordered n-tuple X₁,X₂, ... X_n is said to be drawn from x₁,x₂, ... x_n if X₁ is a member of x₁, X₂ is a member of x₂, ... and X_n is a member of x_n. A mapping from classes x₁,x₂, ... x_n into class is an assignment of unique members of to ordered n-tuples drawn from x₁,x₂, ... x_n. The member of assigned to a given n-tuple X₁,X₂, ... X_n is called the value of the mapping at the arguments X₁,X₂, ... X_n. A mapping from x₁,x₂, ... x_n which assigns a value to every ordered n-tuple drawn from x₁,x₂, ... x_n is called total. It if fails to assign a value to some of then, it is called a partial mapping from x₁,x₂, ... x_n into .

A mapping is often specified by means of a rule or method whereby its value can be calculated from its arguments. But the mapping is not be identified with the rule. The mapping is rather the correspondence between arguments and values induced by the rule. Two different rules can induce one and the same mapping. Mappings are thus individuated extensionally in the sense that if mappings F and G from x₁,x₂, ... x_n into take the same arguments to the same values then they are not two mappings but one and the same mapping.

The following is an inductive definition of types over (o, i, w, t):

1. o, i, w and t are types over {o, i, w, t},

2. If x₁,x₂, ... x_n and are types over {o, i, w, t}, then the collection - call it (x₁x₂...x_n) - of all (total and partial) mappings from x₁,x₂, ... x_n into is also as a type over {o, i, w, t}.

3. Nothing is a type over {o, i, w, t} unless it so follows from 1 and 2 above.

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Where x is any type over {o, i, w, t}, members of x are called objects of type x, of briefly x-objects, over {o, i, w, t}. (In what follows the qualifier 'over {o, i, w, t)' will often be omitted.)

Some types of objects feature prominently in logic, mathematics and philosophy, and have well established names. Objects of type (ox), ie mappings from x-objects to truth-values, are known as classes x-objects, or briefly as x-classes. If C is a x-class, those x-objects at which the value of C is Truth are called members or elements of C and those at which the value of C is Falsehood are called counter-members or counter-elements of C. (Note that if C is not total them some x-objects will be neither members nor counter-members of C.) Objects of type (ox₁x₂), ie. binary mappings from x₁,x₂ to truth-values, are known as (two-places) linkages (or relations-in-extension) between x₁ and x₂, or briefly as x₁,x₂-linkages. Where X₁ is a x₁-object and X₂ a x₂-object, x₁,x₂-linkage L is said to link or counter-link X₁ with X₂ just in case L takes the couple X₁,X₂ to Truth or Falsehood. Similarly for three-place, four-place, etc. linkages.

Mappings of type (xt) are called x-chronologies and those of type ((xt)w) x-determiners. For brevity, we shall speak of the type (xt) also as x_t and of the type ((xt)w) as x_tw. Where D is a x-determiner and W a world, the value of D at W is the chronology of D in W. Let C be the chronology of D in W and T a moment of time; then the value of C at T is D's determinee in W at T.

o-determiners, that is, objects of type o_tw, are also known as propositions. Propositions P is said to be true or false in W at T according as P's determinee in W at T is Truth or Falsehood.

Determiners of mappings will be called operations; more particularly, a x₁x₂...x_n-determiner will be called an operation from x₁,x₂, ... x_n into . Operation O is said to take X₁,X₂, ... X_n to Y in W at T if the mapping by O in W at T takes X₁,X₂, ... X_n to Y.

Operations from x to o (ie objects of type (ox)_tw) are also known as properties of x-objects, or briefly, x-properties. x-object X is said to instantiate or counter-instantiate x-property Q in W at T according as Q takes X to Truth or Falsehood in W at T. Operations from x₁,x₂ to o (ie. objects of the type (ox₁x₂)_tw) are also known as (two-place) relations between x₁-objects and x₂-objects, or briefly as x₁, x₂-relations, x₁-object X₁ is said to bear or counter-bear x₁,x₂-relation R to x₂-object X₂ in W at T according as R takes x₁,x₂ to Truth or Falsehood in W at T. Similarly for three-place relations, four-place relations, etc.

1.3 MONTAGUE'S INTENSIONAL LOGIC

Not all linguists subscribe to Chomsky-style autonomous syntax. So-called truth-conditional or possible-world semantic, initiated by philosophical logicians but increasingly popular with linguists as well, is based on the assumption that syntax and semantic are inseparable and must go hand in hand.

The adherents of this program see the aim of linguistic analysis as that of pairing expressions with meanings and they do not mean by that (as the autonomous syntacticians do) pairing them with other

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expressions. In their view, meanings are determiners. In particular, the meaning of a sentence is a proposition, the meaning of a predicative expressions is a property or relation, etc. These are not linguistic expressions but abstract objects in the set-theoretic hierarchy over the base {o, i, w, t}, as defined in the foregoing section. To interpret a linguistic expression means, according to the possible-world semantic, to associate with it a non-linguistic object of this sort.

What has just been said is easily obscured by a methodological precept used by most truth-conditional semanticists. Following Richard Montague, they don't assign entities as meaning to English expressions directly, that is, by referring to those entities in their metalanguage, but rather by way of another object language. The procedure is to define a formal language of intensional logic and then to interpret English expressions by translating them into that language.

This would seem an unnecessary complication, because the artificial language itself has to be semantically interpreted before in can play its role as an aid in interpreting English. It is true that a streamlined artificial language is easier to interpret than a natural one. But it would seem that any saving of labour which accrues from the decision to interpret the artificial notation rather than directly English itself, is bound to be offset by complications arising from the additional task of translating every English expression into the artificial notation.

The Referential Fallacy

This methodological quibble would have little force, however, it the translations offered by the truth-conditional semanticists were acceptable. Unfortunately, they are deeply flawed. All the artificial language, including the language IL (Intensional Logic) proposed by Montague, implement a thesis originally propounded by Frege and repeated by theorist after theorist ever since, despite the fact that Russell exposed it as a fallacy almost ninety years ago. It is the thesis that an expression whose meaning can be represented by a determiner is not a name of the determiner itself but rather of the object, if any, determined by it. The thesis is enshrined in Frege's own cleavage between Sinn and Bedeutung, in Carnap's distinction between intension and extension, and, more recently in Kripke's notion of non-rigid designator. I shall refer to it as Frege's Thesis despite the fact that Frege himself, of course, did not explicate his determiners (senses) in the possible-worlds style.

We have seen that the meaning of a predicate like 'white' can be represented as a class-determiner, a mapping from world-times to classes. But according to Frege's Thesis it is not the determiner itself that receives reference in sentences where the predicate occurs, but rather the class it happens to determine. This makes the subject matter of the sentence

(1.13) Bill Clinton is white

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not an individual and a property, but an individual and a class. The whole sentence in thus interpreted as saying that the individual is a member of the class.

Let us call the class of things which happen to be currently white C. It is readily seen that in order for Clinton to be a member of C he need not be white. Should he turn black tomorrow he would still be a member of C. A class is not a club which one can join and later resign from, because a class, unlike a club, is individuated by its membership; any class which does not have Clinton for a member is numerically distinct from C. This means that in a possible world W in which Clinton is black now, he is, as of now, a member of C. Thus if (1.13) said that Clinton is a member of C, then the sentence would be currently true in W, which it obviously is not. A sentence clearly cannot fail to be true at a time or in a world where what is says is the case. Hence the message carried by (1) cannot be to the effect that the class C has Clinton for a member.

It is not easy to think of anything else that (1.13) could be imagined, with a modicum of plausibility, as saying about C. But once it is conceded that (1.13) does not really say anything about the class C specifically, the question arises as to why a sentence which has nothing to say about an object should be construed as containing a name of that object.

Parallel comments apply to a definitely descriptive term like 'the U.S. president'. The meaning of that term, we have seen, is arguably represented by an individual-determiner, a (partial) mapping from world-times to individuals. But according go Frege's thesis, it is not the determiner itself that receives reference in a sentence where the term occurs, but rather its determinee, Bill Clinton.

It is a sobering thought that, nine decades after 'On Denoting', it is still necessary to argue against this view. So many counterarguments suggest themselves that it is hard to know where to start. If 'the U.S. president' designated Bill Clinton then each of the sentences

(1.14) Brown wanted to be the U.S. president

(1.15) Fred thinks that the U.S. president is white

(1.16) Brown is the U.S. president

(1.17) The U.S. president exists

would say something about Clinton. What could that possibly be? (1.14) certainly does not say that Brown wanted to be Clinton. (1.15) does not imply that Fred has any view concerning Clinton or that he is even acquainted with, or aware of, the man. (1.16) does not allege that Brown is identical with Clinton and (1.17) does not ascribe existence to Clinton. Indeed one can understand (1.14)-(1.17) perfectly well without having any idea that the term 'the U.S. president' is anything to do with Clinton. But how can one understand a sentence without knowing what it is about?

Note, on the other hand, that each of the sentences (1.14)-(1.17) tells us something about the U.S. presidential office: (1.14) tells ut that Brown wanted to hold the office, (1.15) tells us that Fred thinks that the office is held by a white person, (1.16) tells us (erroneously) that it is occupied by

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Brown, and (1.17) simply that it is occupied. Someone who fails to extract this information from these sentences does not quite understand them. But if the term "the U.S. president' refers to Mr Clinton, there is nothing left in the sentence to refer to the presidential office. How it is possible that one can acquire information about an item from sentences in which that item is never mentioned?

Adherents of Frege's Thesis have invested a great deal of intellectual energy in fending off such obvious counter-examples. In (1.14)-(1.17), they tell us, the descriptive term does not refer to what it describes, because it appears in a special kind of context, called variously 'oblique', 'opaque', 'non-referential' or 'intensional'. This sounds interesting and makes one anxious to learn more about opaqueness. But there is in fact nothing to learn. A context is by definition oblique (or opaque etc.) if it fails to obey Frege's Thesis. The thesis is thus made invulnerable to counter-examples by being watered down to the following tautology: a descriptive term refers to its descriptum except where it does not. It is rather like someone claiming that all cows are black, securing his theory from counter-examples by watering it down as follows: all cows are black except for those which are special in being some other colour.

So attenuated, the thesis is, of course, perfectly compatible with the view that a descriptive term never refers to what it describes. What saves the opacity doctrine as a whole from the abyss of triviality is the fact that it comes with a sample of contexts which are allegedly non-special or 'transparent'. There is a universal consensus, for example, that a sentence like

(1.18) The U.S. president is white.

represents such a context. In such a sentence, it is claimed, the term 'the U.S. president' definitely refers to Bill Clinton.

It would be absurd for anyone who takes this view to maintain that the sentence nevertheless asserts nothing about Clinton. Surely there is little point in referring to a man if one has nothing to say about him.

Let us then ponder what it may possibly be that the sentence says about the man. The only conceivable answer seems to be that it ascribes to him the property of being white. But if this was so then the assertive content of (1.18) would be not different from that of

(1.19) Bill Clinton is white.

Both would mention the same man and ascribe the same property to him. How can this be squared with the obvious fact that the two sentences express two logically independent facts? For nobody would deny that it is perfectly possible for Bill Clinton to be white without the American president's being white, and equally possible for the American president to be white without Bill Clinton's being white. How can it be squared with the fact that (1.18) conveys factual information about the American presidential office - implying as it does that the office has a white incumbent -while (1.19) does no such thing?

It is natural to think that it is an a priori matter, as regards a factually non-trivial sentence, to determine what particular factual claim it makes, and an a posteriori matter to determine whether

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that claim is true. It is interesting to note that if Frege's Thesis were right them things would be exactly the other way around.

On the one hand, more often then not speakers would not know what they are talking about. A detective, who says 'Mrs Smith's murderer is white', names, according to the thesis, a particular person and ascribes whiteness to that person. Yet, unless he has already cracked the case, he has no idea which particular person it is. The detective thus does not know what particular factual claim he has himself made.

On the other hand, it can be shown that, if the thesis were right, any fact could be knowable a priori. For consider any factual statement, say

(1.20) There is life on Mars,

call it S for short. We can define a property p in the following way:

x is a p iff_df x is 1 and S is true or x is 2 and S is false.

It follows from the definition that

(i) 1's being a p logically entails that S is true

and

(ii) 2's being a p logically entails that S is false.

Clearly one and only one number is a p. This number answers to the description 'the p' and is bound to be a p. Hence no factual inquires are needed to see that the sentence

(1.21) The p is a p

is true: we know (1.21) a priori. Now there are two possibilities: either there is life on Mars or there is not. Let us consider them in this order.

First suppose that there is life on Mars. In this case 1 is the unique bearer of p and, according to Frege's Thesis, the term 'the p' refers to it. Hence (1.21) states that 1 is a p. But by (i) this logically entails that there is life on Mars. Now suppose that there is no life on Mars. In this case 2 is the unique bearer of p and, according to Frege's Thesis, the term 'the p' refers to it. Hence (1.21) states that 2 is a p. But by (ii) this logically entails that there is no life on Mars.

Thus in either case, the correct answer to the question 'is there life on Mars?' is logically entailed by something that we know a priori, namely (1.21). It would seem futile to send expensive space probes to Mars in search of information which is logically entailed by something we already know.

But of course (1.21) is a factually empty tautology and nothing non-trivial follows from it. This is because, contrary to Frege's thesis, it makes no reference to any number. Its subject term 'the p'

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refers to a number-determiner, a mapping which takes every world-time either to 1 (if there is life on Mars in that world at that time) or to 2 (otherwise). The whole sentence says something about that mapping: namely, that it takes the actual world and the current moment of time to a individual number p. But it leaves it open which number it happens to be.

Analogously with (1.18). Its subject term 'the American president' does not name Clinton or any other individual. It refers to an individual-determiner, a mapping which takes a world-time to whoever (if anybody) is the American president in that world at that time. That mapping represents something for an individual to be, a status or office occupiable by an individual. The whole sentence says something about that office, namely, that it happens to be occupied by a white individual; but it leaves it entirely open which individual it happens to be. Otherwise the sentence would not be qualifiable by temporal adverbials like "invariably', 'often', and 'hardly ever' or modal adverbials like 'necessarily', 'possibly', and 'it is unlikely that'. The sentence 'The American president is necessarily white' is not false because Clinton might conceivably be black (although he might), but because American presidency might be occupied by a black person.

According to a routine objection, the subject matter of (1.18) cannot be the presidential office because the office is neither white nor any other colour. The objection, however, is based on the naive principle that the predicate of a statement must invariably be ascribed directly to whatever item the statement speaks of. On this principle one could similarly argue that the statement that every swan is white cannot be about swanhood, the property, because the property is not the sort of thing that can be white or any other colour. It is obvious, however, that the statement does say something about swanhood. It says that swanhood is related to another property, white, in such a way that every object instantiating it also instantiates the other property. But the statement leaves it completely open what those instances are and therefore does not refer to any of them. The statement that the American president is white is analogous in structure. It speaks of the American president, the individual-determiner, and whiteness, the class-determiner. What it says is that the determinee of the former is an instance of the latter. But it leaves it entirely open who the determinee is, and therefore involves no reference to that individual.

The Perspicuity Problem

The main objection to Frege's Thesis, also due essentially to Russell, is that it portrays languages as lacking perspicuity. In the rest of this section, Russell's often ill-understood objection will be reformulated and illustrated with examples.

Even those who uphold Frege's Thesis, we have seen, agree that there are contexts - such as (1.14)-(1.17) - in which a descriptive term does not refer to what (if anything) it happens to describe. Some semanticists openly declare such contexts imperspicuous (Quine's term is 'opaque')

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and leave the matter at that. They do not try to find something else for the term to refer to in such contexts: they leave 'opaque' occurrences of the term semantically unaccounted for.

Most possible-worlds semanticist, however, do grant definite descriptions the status of referring terms even where they appear in 'special' context. They follow Frege in having them refer to the corresponding determiners. A term like 'the American president' or 'white' thus stands accused of ambiguity. Its reference (and hence, presumably, its meaning as well) is claimed to fluctuate from context to context.

This is far from intuitively obvious, but not outright unacceptable as a claim about natural language. Natural language is certainly guilty of all manner of logical misdemeanours, not least of ambiguity. This, of course, is the reason why logicians invent artificial languages. An artificial notation is, first and foremost, a means of coping with the logical defects of natural language. In particular, it enables us to account for an ambiguous natural-language locution by giving two different translations thereof into the artificial language. This would be hardly helpful if the artificial notation was itself ambiguous. Univocality is thus the first thing that we require of a formal language.

Now the trouble is that as long as it itself implements Frege's Thesis, the artificial language can be kept unambiguous only at the expense of another basic feature which makes a language perspicuous: functionality. This holds in particular of Montague's Intensional Logic, the logical notation most widely used in linguistic research. Or so it will be argued in what follows.

Any language has a stock of primitive expressions and provides means of combining them into compounds. The reason for having compounds as well as primitives is that otherwise we would have to have too many primitives. To enable us to discuss any natural number, for example, such a language would have to have infinitely many primitives. But a language with infinitely many primitives is not learnable and hence useless as a means of communication. Combining primitives into compounds is thus an economy device: it enables us to refer, in the language, to objects for which the language does not have primitive names.

A compound name refers to an object by way of other object, namely those referred to by its components. This is possible thanks to functional relationships between objects. For suppose mapping F takes value Y at argument X, and that we have a name, say 'F', for the mapping and another name, say 'X', for the argument. Then we can refer to Y even if we have no primitive name for it. We can refer to it, via F and X. Or, if a two-argument mapping G takes X₁ and X₂ to Y and we have names 'G', 'X₁', and 'X₂' for G, X₁ and X₂ respectively, we can refer to Y via the compound 'X₁GX₂', ie as the value of G at X₁ and X₂. Thus it is, for example, that we can form a name for any of the infinitely many natural numbers in terms of only three primitive names. If we have names, say '0', '1' and '+', for nought, one, and the addition mapping, we can speak of the number one also as '0+1', of the number two as '(0+1)+1', of the number three as ((0+1)+1)+1', and so on.

A compound name is said to be perspicuous if the manner in which it refers to its nominatum is clear from its shape. If the compound refers to its nominatum, for example, as the value of G at X₁ and X₂ then the compound must consist of names of those three objects and the way those names are put

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together must make it manifest which of them stands fore the first argument and which for the second. Otherwise the expression is not perspicuous.

Many non-perspicuous expressions are innocuous. This is true, in particular, of abbreviations. Suppose, for example, that we decide to simplify the above arithmetical language by introducing the symbol '2', not as a new primitive name but as an abbreviation, for the sake of convenience, of the expression '1+1'. Then in '2' we have an expression which lacks perspicuity. It is to refer in the same manner as '1+1', that is, through the number one and the addition mapping, yet no part of it refers to the number or the mapping. But because the abbreviating convention links '2' explicitly to the fully perspicuous expression '1+1' there is no cause for complaint. In conjunction with the abbreviating convention, '2' is just as good a guide to its referent as is '1+1'. Other cases of non-perspicuity are pernicious. Imagine that someone wanted to enrich the above arithmetical language by introducing the symbol '*' and stipulating that the result of prefixing the symbol to a term shall be a name of the number nought if the therm contains the symbol '0', and a name of the number one otherwise. According to the stipulation, '*(1+0)' for example, stands for nought and '*1' for one.

Thus enriched, the language is no longer perspicuous. The compounds '*(1+0)' and '*1' for example, do not refer to their nominata as values of a mapping at some arguments. If they did '*' would have to stand for a mapping which takes nought as its value at the nominatum of '1+0' and one as its value at the nominatum of '1'. But since the two terms name the same object (the number one) there is no such mapping.

Not that the proponent of '*' has no mapping in mind. In explaining the new symbol he invokes the mapping, call it f, which takes every term of the language to nought or one according as the term does or does not contain an occurrence of '0'. But if '*' was a name of then '*(1+0)' would be an incoherent expression. What remains when the asterisk is taken away, namely '(1+0)' does not present the number nought as a value of a mapping at an argument. It is an intractable expression which leaves us in the dark as to how its nominatum is to be found.

This lack of perspicuity manifests itself in the fact that expressions containing the asterisk do not conform to the Functionality Principle, according to which the referent of a compound is a function of the referents of its components. The expressions '(1+0)' and '1', for instance, have the same nominata, yet '*(1+0)' is a name of 0 and '*1' is a name of 1. Montague's IL contains expressions which violate the Functionality Principle in the same way as '*' does. One of Montague's connectives is '^', the so-called intensionality operator (or 'hat' symbol). Montague explains the operator as follows:

the expression [^a] is regarded as denoting (or having as its extension) the intension of the expression a.⁽¹⁷⁾

For example, let 'H' and 'K' be translations of the verbs 'has a heart' and 'has kidneys' into IL. Then according to the semantic rules of the system, 'H' denotes the class of individuals with a heart and 'K' the class of individuals with kidneys, which, as is well known, happens to be the same i-class. '^H'

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then denotes the property of having a heart and '^K' the property having kidneys, which are two distinct i-properties.

Montague's '^' is thus exactly analogous to the symbol '*' discussed above. The compounds '^H' and '^K', do not refer to their nominata as values of a function at some arguments. If they did '^' would have to stand for a mapping which takes the property of having a heart as its value at the nominatum of 'H', and the distinct property of having kidneys as its value at the nominatum of 'K'. But since the two terms name the same i-class there is no such mapping.

Note that in proposing the hat symbol Montague had no mapping in mind. The explanation quoted above invokes the mapping, call it y, which takes every term of IL to the intension of that term. But '^' cannot possibly be a name of y. If it was, '^H' would be an incoherent expression. What remains when '^' is taken away, namely 'H', is a name of the class of individuals with a heart; but is not defined at that or any other class, it is defined only at IL terms. Hence '^H' does not present the property of having a heart as a value of a mapping at an argument. It is an intractable expression which leaves us in the dark as to how its nominatum is to be found.

Just as in the case of '*', this lack of perspicuity manifest itself in violations of the Functionality Principle. The expressions 'H' and 'K', for instance, have the same nominata, yet '^H' is a name of one i-property sand '^K' is a name of another.

There is an alternative way of looking at Montague's notation. Maybe '^' is to be understood as a mere syncategorematic symbol whose role is not to name anything but merely to create a special context. As we have seen above, Frege and his followers take the view that what an expressions stands for depends on the context in which it occurs. There are special contexts, according to Frege, which affect the reference of expressions occurring in them. In such a context, an expression which normally names a definite object and expresses a definite sense, ceases to name that object and refers instead to the sense. Maybe Montague's hat symbol simply creates such a special context in IL. When prefixed with '^', an expression which normally refers to a definite object and expresses a definite intension, ceases to refer to that object and refers instead to the intension. In '^H', for instance, the letter 'H' itself refers, on the present construal, to the property a having a heart, and in '^K' the letter 'K' itself refers to the property of having a heart.

If this reading is the intended one then IL is a language which tolerates ambiguity. Indeed, every well-formed IL expression will have at least two meanings, depending on whether or not it is prefixed with '^'. As mentioned above, it is by no means clear that natural language suffers from a global ambiguity of this sort ('have a heart' and 'have kidneys', for example, seem perfectly univocal). But assuming that is does (and that 'have a heart', for example, is capable of referring to the class of cordates as well as to the property of having a heart), Montague's IL (on its present construal) simply duplicates that ambiguity. Instead of offering two univocal translations of the ambiguous term, it offers a single one which, in the artificial notation itself, is just as ambiguous as the corresponding natural-language expression.

Frege himself would certainly have had little time for a formal notation of this sort; on his view, in a logically correct language '[i]t is not permissible to designate different things by the same sign, for the first thing that we must require of our sings is that they should be unambiguous.'

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Ambiguity is, of course, always eliminable at the cost of introducing extra primitive names. For example, we could introduce the symbol 'I' to stand for the property of having a heart, and 'J' for its extension, and similarly 'K' for the property of having kidneys and 'L' for its extension. Such notation, however, would fail to do justice to the close relationship which exists between the referents of 'I' and 'J' and between the referents of 'K' and 'L'. The fact that the referent of the second component of each of the two couples is the extension of the referent of the first would be notationally obliterated.

A systematic ambiguity can often be eliminated without notionality obliterating the connection between the object an expression denotes in one kind of context and the object the same expression denotes in another kind of context. Imagine that a bank has four branches in four different cities. The bank's directors might speak ambiguously of both the cities and the branches as 'a', 'b', 'g' and 'd'. Thus in some context, 'a' refers to a city and in other context it refers to the branch located in that city. This ambiguous usage is easily eliminable. In fact it is eliminable in two, equally satisfactory, ways. The directors may agree to reserve the names for reference to the cities and use the compounds 'the branch located in a', 'the branch located in b', etc, to refer to the corresponding branches. Alternatively, they may agree to reserve the names for reference to the branches and use the compounds 'the location of a', 'the location of b', etc to refer to the corresponding cities. Either way, the ambiguity will be rectified. What is more, the connection between a city and the branch located in the city will be unmistakably reflected in the expressions which serve as their names.

The circumstance that disambiguation can in this case proceed in either of the two directions, however, is contingent upon the fact that the relation between the cities and the branches is one-to-lone. To see this, let us change the story and imagine that two of the bank's branches are located in one and the same city. Hence there are four branches but only three cities to be referred to. On the ambiguous usage, the city with two branches has two names, say 'g' and 'd'. This time disambiguation can go in one direction only. If we reserved the names 'a', 'b', 'g', and 'd' for reference to the cities we could not refer to the branches as 'the branch located in a', 'the branch located in b' etc. The expression 'the branch located in 'g', and 'the branch located in d', would not refer two different branches because 'g' and 'd' would stand for one and the same city. Indeed, for no functor 'F', could the expressions 'the F of g and 'the F of d' denote two different branches. Hence no matter what 'F' we chose, one of the branches would be nameless. Thus the only way to eliminate the ambiguity is by reserving the names 'a', 'b', 'g', and d' for branches and referring to the cities with compounds such as 'the location of a', 'the location of b' etc.

The source of this asymmetry is obviously the fact that while there is a mapping from the bank branches to the corresponding cities such that every city is the value of the mapping at a branch,

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there is no mapping from the cities to the corresponding branches such that every branch is the value of the mapping at a city.

The lesson to be drawn from these examples is as follows. If a class of systematically ambiguous expressions is to be disambiguated, there must exist a mapping from the objects the expressions stand for in one kind of context to the objects they stand for in the other kind of context, such that each of the objects of the latter kind is the value of the mapping at an object of the former kind. Then, and only then, disambiguation can proceed by reserving the originally ambiguous names for reference to objects of the former kind and by referring to objects of the latter kind by means of compounds containing those names.

This means that if such a mapping exists in one direction and not in the other, it is not up to us to choose which of the two kinds of context affecting the reference of the ambiguous names is the ordinary one and which is the special, reference-shifting one. The only correct classification is the one whereby those contexts in which the ambiguous names stand for the arguments of the mapping count as ordinary and those in which they stand for the corresponding values of the mapping count as special, or reference-shifting, ones.

Let us now apply this lesson to Montague's IL . On the construal we are considering, expressions like 'H' and 'K' (the IL counterparts of 'has a heart' and 'has kidneys') are ambiguous. In the one kind of context they stand for determiners (properties), and in another kind of context they stand for the corresponding determinees (classes). There is a mapping which takes each determiner to its determinee. Trivially, every determinee is the value of that mapping for some argument. On the other hand, there is no mapping from determinees to the corresponding determiners such that every determiner is the value of the mapping for some argument. This is because any one object is determined by more than one determiner: the class of individuals with hearts, for example, is determined by the property of having a heart and also by the distinct property of having kidneys.

This is why any attempt to disambiguate IL by reserving expressions like 'H' and 'K' for reference to determinees and referring to the corresponding determiners by means of compounds, is bound to fail. For once 'H' and 'K' are alternative names of the class of individuals with hearts (ie the class of individuals having kidneys), any expression of the form 'the F of H' will name (if anything at all) the same object as the expression 'the F of K'. In particular, the expression 'the determiner which picks out H' will stand for the same property as 'the determiner which picks out 'K'. Hence if they denote the property of having a heart, the property of having kidneys will be nameless, and if they denote the property of having kidneys, the property of having a heart will be nameless.

The only way to disambiguate is thus by reserving like 'H' and 'K' for reference to determiners (properties in the present case) and referring to the corresponding determinees with compound terms, perhaps with 'the object determined by H', 'the object determined by K', etc. Montague has, in fact, introduced a symbol which means the same as 'the object determined by': 'Ú'. Thus, once 'H' and 'K' were names of the properties of having a heart and of having kidneys, respectively, the class

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of individuals having a heart (ie. the class individuals having kidneys) could be referred to as '9H' or alternatively as '9K'.

To disambiguate in this way, however, would amount to considering contexts where the ambiguous IL names stand for determiners as ordinary, and contexts where they stand for the corresponding determinees as special, reference-shifting ones. But this is the exact reverse of how IL works. The notional arrangements of IL are such that the ambiguous symbols are to stand for determinees when they occur ir ordinary context and only switch their reference to determiners (intensions) in special context where they are marked with the hat symbol '^'.

If we abolished '^' and wrote 'H' instead of Montague's '^H' and '9H' instead of Montague's 'H', we would get a notation which would be univocal and obeyed the Functionally Principle as well. It is doubtful, however, that the inverted hat symbol '9' could play any useful role in translating English locutions into the reformed notation. To translate an English sentence into a formula containing '9' is to imply that in mouthing the sentence the English speaker is referring, inter alia, to what '9' stands for, namely a mapping which takes every determiner to its determinee. But we have seen already that only an omniscient speaker can know which mapping that is. If such a translation was correct, English speakers would suffer of chronic ignorance of what they are talking about.

But '9' is redundant. We have seen already that in sentences featuring an expression like 'has a heart' there is no reference to the class of heart-endowed individuals. To grasp the information encapsulated in 'Bill Clinton has a heart' one does not need to have a full inventory of that class because the sentence has nothing to say about it. The sentence tells us that Clinton instantiates a certain property, not that he is a member of any particular class. The same goes even if the predicate is part of the term 'the class of individuals which have a heart'. For this is a descriptive term and we have seen that descriptive terms are not names of what they describe. It seems obvious that the sentence 'Bill Clinton is a member of the class of individuals which have a heart' carries exactly the same message as the sentence 'Bill Clinton has a heart'. Hence if the latter does not say anything about any particular class, neither does the former.

1.4. 'STRUCTURED MEANINGS'

But even if this point were accepted and the inverted hat symbol banned from appearing in formal translations of ordinary-English expressions, the reformed IL would still be inadequate as a medium for interpreting English. This is because it is based on the idea that the meaning of an expressions is a determiner. The semantics of IL associated with an IL expression a determiner over the base {o, i, w, t}, ie a mapping from w and t to objects of some type. Hence if an English expression is interpreted by translating it into IL, all that is (indirectly) associated with it as its meaning is a mapping of that sort. But it is well known that there are expressions whose meaning cannot be captured by any such object.

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This is true most obviously of mathematical expressions. Consider the sentence

(1.22) Veronica tried to calculate 9-5.

Here Veronica is definitely not being related to the number 4. For all the sentence says her attempt may have been unsuccessful, in which case 4 would not come into play at all. Thus '9-5', as it appears in (1.22), does not serve as a vehicle of reference to that number. But it does not serve as a vehicle of reference to the corresponding determiner either. Since the result of subtracting five from nine is 4 independently of any factual or temporal matters, the determiner associated with '9-5' is the mapping which takes every world-time to 4. But if sentence (1.22) related Veronica to that determiner, it could not possibly differ in truth value from

(1.23) Veronica tried to calculate 2²,

because the determiner associated with '2²' is also the mapping which takes every world-time to 4. Yet (1.22) and (1.23) clearly can have different truth-values. Veronica can try to carry out one mathematical calculation without trying to carry out every other calculation which, when no mistake is made, yields the same result.

It seems natural to say therefore that what (1.22) relates Veronica to is a definite arithmetical calculation, a specific way of getting from two numbers to a third. (1.23) can differ from (1.22) in truth-value because in it Veronica is related to a completely different calculation.

The reason why sentences like (1.22) and (1.23), often spoken of as 'hyperintensional contexts', constitute a problem for truth-conditional semantic is because its practitioners are reluctant to admit that there are such things as mathematical calculation or constructions and refuse to take them on board. Ontological parsimony is one of the main passions agitating modern philosophers. They see themselves as hardnosed removers of clutter. The behaviourists, for example, made a living of pretending there are no introspectable mental acts. The nominalists have tried to pretend that there are no sets of things, and when that failed, they present at least that there are no attributes such as colours. The instrumentalists pretend there are no atoms, electrons, or electromagnetic fields. The linguistic formalists (and their literary colleagues, the deconstructionists) pretend there are no such things as meanings.

Mathematics seems a particularly attractive ground for parsimonious posturing. At the time when number-crunching technology is bringing on a second industrial revolution there are scores of philosophers who tell us in all seriousness that mathematics has no subject matter and that science can do without numbers.

These, however, are the fundamentalists. Others are less extreme and grant mathematics a subject matter. They accept the ontology of set theory and take mathematics to deal with sets, mapping, relations and other objects constructible by set-theoretical means. But this is where they draw the line. They have no place in their ontology for mathematical constructions or calculations. The addition mappings and the numbers 5 and 7, for example, are tolerated, but the construction, or

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calculation, which consists in applying the mapping to the two numbers, is beyond the ontological pale. The results of set-theoretical constructions are embraced, but the constructions themselves are spurned. Unspoken pretence that there are no such things is almost universal.

Yet, what point would there be in knowing the proposition that 7+5 is 12 if there was no such thing as the calculation which consists in adding 5 to 7? The whole point of the proposition is clearly to inform us about the outcome of that particular calculation.

It will be objected that the proposition reports a certain relation as holding between 5, 7 and 12. But what relation is it if not the one which holds between three numbers just in case the last one is the result of adding the first two? Hence, for any two numbers there must be such a thing as the construction of adding them, one would think. It will be objected again that the relation need not be defined in this fashion. It is simply a class of ordered triples, call it R, no matter how is it is defined. It is not easy to see why the proposition in question should be constructed as a statement about R, considering that no name of the relation can be discerned in the sentence '5+7=12' which spells that proposition out. But setting this aside, what is a three-place relation between numbers? It is a mapping which is applicable to ordered triples of numbers and which yields 'Yes' or 'No' every time it is so applied. To know that 5+7 is 12 is then to know the outcome of the calculation which consists in applying R to 5, 7 a, and 12. There would be nothing to know if there was no such calculation or construction.

Indeed mathematics is not so much about classes, mappings, and numbers, as about ways in which such objects can be constructed from others. The mathematician does not spend time describing the number 5, or the number 7, or the addition mapping. Individually, these are boring items and little that is at all interesting can be said about them. What is mathematically interesting is, for example, that by applying the mapping to those two numbers one can construct (calculate, or arrive at) the number 12. This is the sort of information that children are encouraged to absorb in their math classes. Constructions (calculations) are what mathematics is all about.

One reason why philosophers of mathematics are so shy of this obvious truth might be because they think of a calculation as an activity. If calculations are events taking place in our minds then they are objects for psychologists to study, not for mathematicians. This, however, is a fallacy caused by an ambiguity in the word 'calculation'. It is an ambiguity the word shares with most epistemic terms, in particular with the terms 'belief', 'wish', 'thought', 'inference' and many others. The term 'belief' can be used to refer to a mental state. This is how it is used, for example, when someone's beliefs are described as 'firm', 'rigid', or 'frivolous'. In this sense of the term, each of us has his own beliefs, distinct from the beliefs of all other people. But each mental state is intentionally directed towards an object. A belief-state, for example, may be directed towards the proposition that it is raining. This proposition itself is not a mental state. It is not the exclusive property of the believer, because any number of people may believe that it is raining and it will be one and the same proposition they all believe. Yet the proposition is also spoken of as each of the believer's 'belief'. In this case the term is used in an objective. sense. To study beliefs in this sense of the word is the task of logic, not psychology.

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The word 'calculation' shows a parallel ambiguity. When it is said that somebody's calculation are swift and brilliant or sluggish and full of mistakes, the reference is probably to events taking place in that person's mind. In this sense of the word, calculations are private performances which the calculators cannot share with one another. Yet when Veronica adds seven to five and John adds seven to five they can be said to have carried out one and the same calculation. Here the word 'calculation' does not refer to mental acts but rather to the common intentional object of those acts, namely, the application of the addition mapping to 5 and 7. This object does not, consist of private cogitations but of mathematical objects: a mapping and two numbers.

It is tempting to liken a calculation to a computer program. A program is also a multiply realisable single thing: any number of computers can carry out the same program. But, despite the term, a computer does not strictly speaking compute. For example, it never really applies the addition mapping to the numbers 5 and 7. A computer is basically a symbol-shuffling machine, one which transforms strings of symbols into other strings of symbols. One can program a computer is such a way that numerals corresponding to 5 and 7, when fed into the machine, will be transformed into a numeral corresponding to 12. But this does not mean that the computer itself added seven to five. It works on the numeral, paying no heed to the number the numeral stands for.

The psychological processes taking place in Veronica's and John's brains may be to a large extent analogous to the electronic processes taking place in the circuitry of a computer. But they differ from their electronic counterparts in being intentionally directed to numbers, mappings, and constructions involving such. The physical events taking place in a computer lack intentionality. Any connection between those events and the realm of mathematical objects has to be mediated by a human operator.

Some semanticists have tried to construe (1.22) as asserting a relationship between Veronica and the expression '9-5'. This, however, is an obvious non-starter. (1.22) may be true even if Veronica is unfamiliar with the standard arithmetical notation and uses some other one instead or none at all. A sentence like 'Veronica tried to calculate the remainder of five from nine' says exactly the same as (1.22) and obviously ascribes to Veronica no attitude to the triple of symbols '9', '-', and '5'. Theorists who cannot bring themselves to recognise constructions as genuine entities thus face a problem. None of the entities they entertain seems to have what it takes to serve as the object of Veronica's attitude reported in (1.22). Linguistic expressions are too fine-grained and, on the other hand, numbers and the corresponding constant determiners are too coarse-grained to fill the bill.

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'Structured Meanings'

One attempt to solve this dilemma comes from M.J. Cresswell. He has suggested that instead of construing (1.22) as relating Veronica to a triple of symbols, '9', '-', and '5', we should construe it as relating her to a triple of mathematical objects, the subtraction mapping, the number nine and the number five.

Cresswell's triple <-,9,5> has much in common with the construction of taking 5 from 9. Like the construction, it contains the substraction mapping and the numbers 9 and 5. But it is not the same item as the construction. A construction is like an itinerary, a way to arrive at something, starting from something else. There surely is a way of arriving at, or constructing, a number by means of -, 9, and 5; one can arrive at 4, or construct it, by applying the mapping - to the arguments 9 and 5 (in this order). But the triple <-,9,5> is not that construction. When Veronica adds seven to five it is not this triple she is intentionally related to. The triple is at best something that can be used to represent what Veronica is related to. A convention is needed which interprets the triple as a proxy for the construction of applying the first component (that is, the substraction mapping) to the other two as arguments. Note that it must be specifically agreed that the second component of the triple (the number 9) is to play the role of the first argument and the third component (the number 5) the role of the second argument of the application, rather than vice versa. But the very need for these conventions shows that the triple and the construction are two different things. The triple merely enumerates the objects of which the construction is composed; it does not combine those objects into the construction. It is just an ersatz of the construction.

But why talk about an ersatz in preference to talking of the thing itself? The only conceivable motive is ontological parsimony. Unlike the construction, the triple is a set-theoretical object. So by talking about the ersatz, rather than what it is an ersatz of, we can go on pretending that there is nothing between heaven and earth which is not a set-theoretical object.

Cresswell introduced his 'structured meanings' to account not expressly for contexts for constructional attitude exemplified by (1.22), but for contexts of propositional attitude like

(1.24) Veronica believes that 5+7=12.

On his view this sentence relates Veronica to the structure

(1.24a) <=,<+,5,7>,12>,

ie. to the ordered triple whose first constituent is the identity relation, whose third constituent is the number twelve, and whose second constituent is the ordered triple consisting of the addition mapping, the number five and the number seven (in this order).

There is a technical difficulty which precludes direct implementation of this idea in Montague's IL. Montague entertains only mappings which are type-theoretically homogeneous in the each of its argument places accepts objects of no more than one logical type over {o, i, w, t). Now an ordered n-tuple is a mapping defined on the first n positive integers (the value of the mapping at 1 being the first component of the n-tuple, its value at 2 the second etc.). The triple <=,<+,5,7>,12> its thus not

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n object over {o, i, w, t), for it takes 1, 2 and 3, to objects of three different types: a linkage between numbers, a triple (which itself is type-theoretically non-homogeneous), and a number.

Cresswell solves this difficulty by the rather drastic measure of scrapping the type theoretical framework of Montague's theory. He collapses the infinite hierarchy of types over {o, i, w, t) to only two types: D₀, which is the set of all sets of possible worlds, and D₁, which he calls the universe of 'things'.

What is a thing? [asks Cresswell and answers:] Obviously, anything at all is a thing: numbers, sets, properties, events, attitudes of the mind and the like.⁽¹⁸⁾

(Note that, comprehensive as it is, 'anything at all' seems to exclude mathematical calculations, such as the one which consists in adding seven to five and then checking whether the result is identical with twelve. For otherwise it would be hard to see why (1.24) should be construed as relating Veronica to the triple <=,<+,5,7>,12> rather than to that calculation.)

This levelling of type distinctions legitimizes the structure <=,<+5,7>,12>, because its components, and components of its components, now all hail from il from the single type D₁. And it is this structure, says Cresswell, that 'we want [to be] the reference of the clause "that 5+7=12"'.⁽¹⁹⁾ And once the referent of the clause is identified with the structure, it is easy to explain why (1.24) may be true and yet

(1.25) Veronica believes that 12-5=7,

false, despite the fact that the sum of 5 and 7 is 12 at the same world-times at which the remainder of 5 from 12 is 7 (namely at all of them). For, on Cresswell's theory (1.25) reports Veronica as taking an attitude to the completely different structure

(1.25a) <=,<-,12,5>,7>.

Thus in this case at least, the 'hyperintensionality puzzle' disappears. There are other cases, however, where Cresswell's theory leaves the puzzle untouched. If (1.24) is true and (1.25) false, then something is wrong with Veronica's grasp of the substraction mapping. But there are other mathematical concepts that Veronica may have trouble with, for instance, functional abstraction. This may be manifested by the fact that although (1.24) is true,

(1.26) Veronica believes that [lx.x+7](5)=12

is false. To account for this on Cresswell's theory, (1.26) would have to be construed as relating Veronica to a structure different from (1.24a). But it is hard to see what sort of structure could possibly fill the bill.

In fact it seems that no structure will. Unlike functional application, abstraction involves bound variables. Veronica has problems with abstraction partly because she suffers a mental block when she is to deal with bound variables.

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Variables are often explained as the letters 'x', 'y', ... which frequently appear in mathematical formulas. But it cannot be letters we are talking about in the present context. Sentence (1.26) does not report an attitude de expressione any more than (1.24) or (1.25) does. As Cresswell rightly notes, (1.24) does not say anything about the sings '5', '7', '12', '+', or '=' because it can be true even if Veronica is unfamiliar with those symbols (using some other arithmetical notation or none at all). Similarly, (1.26) does not say anything about the sign 'x'. Just as (1.24) relates Veronica to the meaning of the expression 'that 5+7=12', rather than to the expression itself, so does (1.26) relate her to the meaning of the expression 'that [l,x,x+7](5)=12' rather than to the expression itself. And the meanings of such expressions are, according to Cresswell's theory, structures built up from mathematical objects-numbers, mappings, relations, and sequences of such-not from letters. But no object in the hierarchy over [o, i, w, t], or in Cresswell's collapsed version thereof, is a variable in an objectual sense for of the word. Thus no structure built up from those objects can represent the meaning of the clause 'that [l,x,x+7](5)=12'.

To be sure, we could represent the bound variable with one of Cresswell's 'things', such as the letter 'x', and try to represent the meaning of the clause, perhaps, with the structure <<<'x'>,<+,'x',7>>,5> or something of this sort. But this would necessitate expanding somehow the convention concerning what is represented by an ordered triple, for as it stands it leaves triples like <+,'x',7> uninterpreted (since 'x' is not an admissible argument for the addition mapping). A further convention would have to interpreted <'x'>. It is not obvious what sort of conventions would be adequate.

NOTES

1. This article contains contents, foreword and the first chapter of, unfortunately, unfinished book, dealing with analysis of natural language. We hold it as a title.

2. Radford (1981), p.167

3. Akmajian and Henry (1975), p.64

4. Vol. IV, p.479

5. The OED cites 'Fortunately, Lord de la Warmet them the day after they had sailed', and Chomsky (1965) himself writes 'Naturally, John will leave' (p.74).

6. Chomsky (1975), p.36

7. Akmajian and Heny are by no means the only ones advocating (1.3) as a 'base rule'. Rules to the same or similar effect can be found in Chomsky (1965) p.72, Bach (1974) p.106, Jacobsen (1977) p.97, Jacobsen (1986) p.34, etc.

8. Chomsky (1957), p.43

9. Radford (1981), p.90

10. See Chomsky (1971)

11. Emonds (1976)

12. Quine (1960), p.76

13. Katz (1972), p.3

14. Jacobsen (1977), p.134

15. Meinong (18??), p.78

16. Russell (19??), p.??

17. Thomason (1900), p.257

18. Cresswell (????)

19. Creswell (????), p.30

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