that the sentence John laughed is a constituent of the NP the claim that John ... morphemes are semantically interpreted as semantic invariants (a notion we de-.
Syntactic Invariants Edward L. Keenan Edward P. Stabler
November 1997 Generative grammar is primarily concerned with modeling the structure of languages and their expressions. A proposed grammar G for English would be judged structurally complete if each expression of English had the same syntactic structure as an expression generated by G.1 For example, once G had been designed to generate say 100 appropriately chosen proper nouns there would be no point in adding in the additional parts of the telephone directories of New York, London, Sydney, : : : as the additions would tell us nothing structurally new about English. But what precisely is meant by saying that two expressions have the same (syntactic) structure? In the rst part of this paper we provide a rigorous, and grammar independent, answer to this question. This enables us to characterize the syntactic invariants of a language determined by a grammar for that language. We use the formal notion of (syntactic) invariant to answer such pretheoretical questions as the following: In a grammar of English, is the property of being an NP (an S , a VP , : : : ) a \structural" one? That is, in our terms, is it syntactically invariant? One expects that the answer is: Yes. But is it true in any grammar that the property of having a given category C is a structural property? The question assumes that we have a de nition of \structural property", and our intuitions as linguists are not pretheoretically decisive here. For example, one of the intuitions behind X 0 theory is that expressions of di erent categories like NP; DP; VP; IP (= S ); PP , have the same internal structure. This suggests the possibility that in some particular X 0 grammar the S 's and the NPs might be isomorphic { that is, they could not be distinguished structurally, whence the property of being an S would not be invariant. If the property of being of category C is not universally structural (that is, invariant in all grammars for natural languages) do we want to impose it as an axiom on the class of possible human grammars? Certain syntactic relations between expressions are more generally felt to be inherently structural. An example is the relation is a constituent of. The fact that the sentence John laughed is a constituent of the NP the claim that John laughed and is not a constituent of the NP the man who saw John laughed is a structural fact about English. And this relation (de ned later), is provably 1 This statement enables us to simplify the goals of generative grammar somewhat. Normally we require that a proposed grammar G for an empirically given language L be complete in that it generates all the expressions judged grammatical by native speakers. We now only require that G generates representatives of all the structurally distinct expressions.
1
a universal invariant. So is the more technical relation C-commands. But a pretheoretically less clear case is the Anaphor-Antecedent (AA) relation, the three place relation given by \X is a possible antecedent of an anaphor Y ". The relation as given is a semantic one; in an expression Z i whether it can be syntactically characterized in a given language (much less all languages) is not obvious. (Linguists usually assume an armative answer, but many dicult cases remain unanalyzed.) And in general our notion of syntactic invariant enables us to ask non-circularly whether semantically de ned relations in a given language are structurally characterizable. A grammar independent de nition of syntactic invariant enables us to raise a variety of basic, even foundational questions concerning the nature of grammars. For example, given our fully general de nition of a grammar one shows easily that isomorphic grammars which generate the same expressions determine the same invariants.2 But the converse fails: we can nd (easily) non-isomorphic grammars generating the same expressions which do determine the same invariants. It may then be interesting to consider grammars for the same set of expressions to be be structurally equivalent if they yield the same invariants. For example we might have one grammar for a given L given in terms of standard trees and another given in apparently incomparable terms of feature-value matrices, yet they could be shown to be structurally equivalent (even though they were not \notational variants"). The questions we pursue in this paper are less abstract. We exhibit some small \model" languages designed to represent certain observed properties of natural languages and we study their invariants with the end in mind of providing formal answers to several linguistic hypotheses extant in the literature. The rst concerns the relation between \morphology" (bound or free) and \syntax." Our notion of invariant covers much more than properties of expressions and relations between them; in particular, it covers expressions themselves. That is, it makes sense on our de nition for a single expression to be a grammatical invariant. This, we claim, corresponds to the traditional distinction between \grammatical morpheme" or \function word" on the one hand and \content" word on the other. For example in English, plausibly in nitival to, as in John wants to play chess, and gerundive -ing, as in playing chess is fun are grammatical morphemes, whereas John and play are not. Current linguistic theories cannot show that identity of morphemes can be structural, independent of classical structural relations like is a constituent of. \Functional" categories are represented simply as part of the hierarchical (that is, constituency de ned) structure of expressions. Equally current theories are ill equipped to support classical observations like the covariation of word order freedom and quantity of case marking on NPs. In this paper we provide a model grammar for a case marking language and show that identity of case markers 2 The grammars are \general" in the sense that they can represent any set of categorized expressions.
2
is structural property independent of constituency relations. And consequently, the Anaphor-Antecedent relation in the language we present is provably structural when given in terms of particular case markers, and is not naturally given in terms of hierarchical relations. Speci cally an anaphor need not be a constituent of the expression its antecedent combines with. The possibility of correctly interpreting anaphors \higher up the syntactic tree" than their antecedents depends on interpreting the nominative case marker in a certain way. Curiously this seems to run counter the spirit of other claims in the linguistic literature to the e ect that case is uninterpretable.3 The latter part of our paper is devoted to the study of the semantic interpretation of syntactic invariants. We argue in particular that syntactically invariant morphemes are semantically interpreted as semantic invariants (a notion we de ne). This is true both for case marking and for agreement morphology which we introduce with another model grammar.
1 A theory independent de nition of Language and Syntactic Invariance We think of a language as a set of expressions built from a given, ad hoc set called a lexicon by applying various structure building functions (called rules). For example, a grammar of English might contain a function F that combines an expression s of category NP with an expression t of category VP to yield a compound expression s_ t of category S . F might also combine NPs with transitive verbs to make VP s. But no function in English combines Prepositions with VPs to make S s. So the structure building functions make essential use of categories (NP; VP; S; TV; : : : ) to constrain what they apply to, and the category assigned to an expression constrains (and sometimes determines) its distribution, that is, what functions may apply to it to form other expressions. Linguists often distinguish the internal structure of an expression (how it is built up) and its distribution. Both these aspects of structure are determined by the structure building functions. We shall treat the category of an expression as an essential part of the expression. This enables us to distinguish, in particular at the level of lexical items, expressions of distinct categories with the same pronunciation. For example in English respect is both a noun and a verb. The lexicon of English might then contain the distinct expressions hrespect,Ni and hrespect,Vi. And since the two expressions are distinct we can semantically interpret them differently. Formally then we represent expressions (lexical or derived) as pairs consisting of a phonological representation and a category symbol. 3 For example, in HPSG, case is not a \content attribute" (Pollard and Sag 1994, p80). In Chomsky's recent work, case is treated as an uninterpretable feature (Chomsky 1995, pp278279).
3
So to de ne a grammar G we must state four things: (1) what the basic vocabulary items VG are; (2) what the set of categories CatG for that grammar is; (3) what subset of V Cat, the set of possible expressions, LexG is; and (4) what the set Rule of structure building functions is. L(G) then is de ned to be the the closure of Lex under Rule. That is, it is the set of expressions we construct beginning with Lex by applying the structure building functions nitely many times. Formally,
De nition 1 A grammar G is a 4-tuple hVG ; CatG; LexG; RuleGi, where
(omitting subscripts): a. Lex V Cat b. Each F 2 Rule is a partial function from (V Cat) to V Cat. Elements of V are called vocabulary items, those of Cat categories, those of V Cat possible expressions, those of Lex lexical items, and those of Rule structure building functions. L(G) or the language generated by G =df the closure of Lex with respect to the structure building functions Rule. 2
So L(G) is the set of expressions that can be built starting with Lex and applying the generating functions nitely many times. A possible expression s is in LG i either s is in Lex or there is a tuple of expressions in L(G) and a structure building function F such that s = F ().4 Where s is a possible expression (an element of V Cat) we write Cat(s) for its category, that is, its second coordinate, and we write string(s) for its rst coordinate. For each C in Cat, PH (C ), the set of phrases of category C is, fs 2 LGj Cat(s) = C g. String(C ), the set of strings of category C , fs 2 V j hs; C i 2 LGg. This formalism is completely general, in the sense that for any sets V; CAT and any subsets set L of V CAT , there is a grammar G such that L(G) = L. In an account of human languages, or of any particular language, any special constraints must be stated explicitly. We turn now to our crucial de nition, that of syntactic invariant. We de ne this notion in terms of the more primitive relation \s is grammatically the same as t". Here rst is an outline of the steps in our de nition. First, for an arbitrary grammar G we de ne a set of functions, called structure (preserving) maps, from L(G) to L(G). The idea here is that for any expression s and any structure map h; h(s) has the same structure as s. Crucial here then is the statement of the precise conditions a function must meet to be a structure preserving map.5 Once the structure maps are de ned then, informally, we consider the \structure" of 4 Given a grammar G and F 2 Rule, it will be convenient to write simply F for F L(G), where F L(G) =df fh; ij 2 Dom(F ) \ L(G) & = F ()g: Clearly, this restriction of F leaves the language L(G) unchanged. 5 These maps would more usually be called automorphisms in the algebraic tradition
4
an expression to be what it has in common with the expressions that have the same structure as it (those it can be mapped to by a structure preserving map). \Structure" is what doesn't change under the action of the structure maps. Let us see abstractly rst just what this means. Suppose we are given a function h from a set A to itself. It may be that there are elements x of A such that h maps x to itself. That is, h(x) = x. Such an x is said to be xed, or invariant, under the action of h, and is called a xed point of h. As long as A has at least two elements there will be functions from A to A which have no xed points. But if we limit the functions h to ones that satisfy certain conditions, as we will when A is taken to be some LG and the h's meet the stringent conditions for being structure preserving maps, then it may be some x's in A are xed by all the h's. These items will be \grammatical constants", expressions that must be mapped to themselves under changes that preserve structure. An important case that arises here is when we consider the action of h on subsets of A. Given h from A to A, we standardly think of h as derivatively mapping subsets of A to subsets of A, as follows:
De nition 2 Given a function h from A to A, then for each subset K of A, h[K ] =df fh(x)j x 2 K g And of course K is said to be invariant under h i h[K] = K.
2
It is now meaningful to ask which subsets K of A are invariant under a given h (or even all of the h's). And when A is taken to be some L(G) the subsets of the language are determined by the properties of expressions. For example, LexG is a subset of L(G) and we can ask whether LexG is invariant under the action
of all the structure maps. If it is then the property of being a lexical item is a structural property { one that cannot be changed without changing structure. Similarly we can meaningfully ask whether the property of being an NP in English is a structural property or not. That is, does h[PH (NP)] = PH (NP) for all structure maps h for a given grammar of English? In general if P is a structural (invariant) property of a grammar then an expression has P i each expression it is isomorphic to has P . In fact the structural properties are those which cannot distinguish between expressions having the same structure. Pursuing this line one step further, any function h from A to A also carries sequences of expressions to sequences of expressions. For example if x and y are elements of A then h maps the sequence hx; y; xi to the sequence hh(x); h(y); h(x)i. And more generally,
De nition 3 Given a function h from A to A, then for each sequence s = hs1 ; : : : ; sn i of elements of A, h[hs1 ; : : : ; sn i] =df hh(s1 ); : : : ; h(sn )i. 2 Combining the e ect of the previous 2 de nitions, we can apply such h's to relations R on A and ask which are invariant under the action of h. An 5
n-ary relation R on A is given by a set of n-ary sequences of elements of A. So h[R] = fhh(s1 ); : : : ; h(sn )ij hs1 ; :::; sn i 2 Rg. And, as usual, to say that R is invariant under some h is just to say that h[R] = R. For example, when A is some L(G) and h is a structure map for G it turns out that h(CONG ) = CONG , where CONG is the binary relation is a constituent of on expressions of L(G).
We turn now to the case of interest: structure maps for grammars. Given a grammar G, just what conditions should a function h from L(G) to L(G) meet in order to justify that an expression s and its image h(s) are \syntactically identical"? We begin by adopting two standard conditions for structure preserving maps. First, each such h should be one to one (injective). Otherwise it might map reasonable expressions like (1a) to ones like (1b), whose grammaticality is doubtful at best. (1)
a. Every student and every teacher came to the party b. Every student and every student came to the party
Secondly, structure maps should be onto (every expression in L(G) has something mapped to it). The motivation here is one that concerns the structure of the entire language, not just individual expressions. If a structure map h is not onto then h[L(G)] would be a proper subset of L(G) and might then lack many structurally signi cant expressions as pretheoretically judged. For example a \language" like English except that it lacked all re exive pronouns (himself, herself, : : : ) would arguably be structurally di erent from English, as the re exives have several grammatically distinctive properties. For example they do not occur as subjects *Himself laughed or as possessors *He lost himself's keys. But they do occur as objects He criticized himself and they coordinate with NPs He criticized both himself and the teacher, so they share some but not all distributional properties with de nite NPs. A function from English to English which mapped nothing to any re exives would omit some structurally distinctive expressions, whence it would be unreasonable to think that h[L(G)] had the same structure as L(G). Third, and most important, a structure map h cannot change a structure building function F . That is, we require that h(F ) = F . It is the structure building functions that determine the internal structure of an expression (how it is built up) and its distribution since whether certain expressions may combine to form another depends on whether they are in the domains of the generating functions. Recall just what it means to say that h(F ) = F . Think of F as a relation on L(G), so h(F ) is just the fhh(x); h(y)ij hx; yiinF g. To say that hx; yi in F just says F (x) = y, where x of course may be a sequence of expressions. Thus to say that h(F ) = F is to say that F (x) = y i F (h(x)) = h(y), which in turn just says that h commutes with F : F (h(x)) = h(F (x)). 6
Formally then we de ne:
De nition 4 Given a grammar G = hV; Cat; Lex; Rulei a structure map (rel-
ative to G) is a bijection h from L(G) to L(G) which xes each F in Rule. 2
It follows immediately that the identity map is always a structure map for any grammar G. And if h is a structure map, is so is its inverse. And if h and k are structure maps so is their composition.
De nition 5 For all expressions s; t of L(G), s = t (s has the same structure
as t) i there is a structure map h such that h(s) = h(t).
2
We note without argument that = as de ned here is an equivalence relation. Finally,
De nition 6 The syntactic invariants of G are those objects in the set hierarchy built from L(G) which are xed by all the structure maps.
2
The objects in the set hierarchy include, at the lowest level, elements (i.e. expressions) of L(G), then sets of expressions, then relations between expressions, and so on. Note that, with this de nition of invariants, expressions, sets of expressions, relations between them, and so on, are all invariant in the same sense: namely, they are xed by the structure maps. We shall present shortly a variety of theorems concerning \universal invariants" such as relations like is a constituent of and C-commands whose instantiation in any grammar G is always invariant under the structure maps for G. But rst we turn to a model of a case marking language which makes provably structural use of bound morphology.
2 Morphology is structural We present a grammar LK \Little Korean" which models a common language type: verb nal, nominal case marking with free order of arguments preverbally, and crucially, case marking morphology is \structural" (invariant). In terms of it we can prove that the Anaphor-Antecedent (AA) relation in LK is structural, even though anaphors may be strictly higher than their antecedents in their derivation trees, as in (18) below. So our conception of grammar enables us to state non-trivial generalizations relating morphological form and classical syntactic form.
2.1 Some properties of Korean
First some observable properties of Korean which motivate our grammar. In (2a,b) we see that the relative preverbal order of arguments of a transitive verb 7
is free. These arguments carry postpositions glossed -nom and -acc. -nom has the shape -i on consonant nal NP s and -ka otherwise. Similarly -acc is -ul consonant nally and -lul vowel nally. In our formal model, LK, we ignore this phonologically based variation. (The Topic marker -nun/-un is similarly conditioned). (2)
a. John-i Mary-lul pinanhayssta John-nom Mary-acc criticized `John criticized Mary' b. Mary-lul John-i pinanhayssta Mary-acc John-nom criticized `John criticized Mary'
Our major claim about this word order variation is that (2a,b) are free variants of each other, neither being syntactically more complex than the other and neither being understood as derived or interpreted as a function of the other. In our grammar LK the derivation trees (but not the expressions themselves) for these two expressions are isomorphic. We show rst that the word order variation in Korean di ers from English Ss which present the Patient before the Agent, as Passive in (3b) and Topicalization in (3c). (3) a. John criticized Mary b. Mary was criticized by John c. Mary(,) John criticized Topicalized S s like (3c) are perhaps more familiar from examples like Beans I like (not tofu). In Korean S s like (2a,b) the -(l)ul marked NP , the Patient, can be a (complex) re exive: (4)
a. John-i caki-casin-ul pinanhayssta John-nom self-emph-acc criticized `John criticized himself' b. Caki-casin-ul John-i pinanhayssta self-emph-acc John-nom criticized `John criticized himself'
By contrast in English (3b) the Patient NP cannot be re exively bound to the Agent, (5a): (5) * Himselfi was criticized by Johni b. Himselfi Johni likes (no one else) a. 8
Thus the Korean variation is not like the active/passive variation in English (and to our knowledge, no one has ever suggested that it was). But (5) allows that the Korean variation might be like Topicalization in English. The data in (6)-(13) argue against this. First, the antecedent of the re exive may be quanti ed, (6,7), or interrogative, (8), and the relative order of anaphor an antecedent is still free: (6)
(7)
(8)
a. Nwukwunka(-ka) caki-casin-ul pinanhayssta someone-nom self-emph-acc criticized `Someone criticized himself' b. Caki-casin-ul nwukwunka(-ka) pinanhayssta self-emph-acc someone-nom criticized `Someone criticized himself' a. (Motun) haksayng-tul-i caki-casin-ul pinanhayssta (all) student-pl-nom self-emph-acc criticized `(All) the students criticized themselves' b. Caki-casin-ul (motun) haksayng-tul-i pinanhayssta self-emph-acc (all) student-pl-nom criticized `(All) the students criticized themselves' a. Nwuka caki-casin-ul pinanhayss-ni who self-emph-acc criticized? `Who criticized himself?' b. Caki-casin-ul nwuka pinanhayss-ni self-emph-acc who criticized? `Who criticized himself?'
But Topicalization of re exives in these contexts in English varies from marginal to bad: (9) a. *? Himselfi someonei criticized b. * Himselfi whoi criticized? Equally the re exive- rst order is natural in subordinate clauses in Korean, whereas Topicalization in English is largely a root clause phenomenon: (10) a. Himselfi Johni criticized at the meeting b. * the meeting at which himselfi Johni criticized (11) a. Caki-casin-ul John-i hoyuy-eyse pinanhayssta self-emph-acc John-nom meeting-loc criticized `John criticized himself at the meeting' 9
b. Caki-casin-ul John-i pinanhayssta hoyuy-ka ecey self-emph-acc John-nom criticized meeting-nom yesterday iss-ess-ta exist-past-decl `there was a meeting yesterday at which John criticized himself' And of course, the Topicalized (3c) di ers semantically/pragmatically in a striking way from (3a). It incorporates all the meaning of (3a), indeed it logically entails it. But in addition it contrasts the Patient with other plausible candidates in context, which (3a) does not do. In Korean, merely placing an NP in clause initial position does not force contrast or emphasis. Rather to contrast an NP denotation we use the morphological resources of the language, marking the focused NP with the \topic" marker -(n)un. And the immediate preverbal position for the topic marked NP is at least as natural as clause initial position. (12) John-i Mary-nun pinanhayssta John-nom Mary-top criticized `John criticized MARY (not someone else)' Lastly, we see that while the relative order of anaphor and antecedent may vary, as in (4a,b), their relative case marking does not so vary. Speci cally, while a -nom marked NP (-i/-ka) or a topic marked one (-nun/-un) may be interpreted as the antecedent of an -acc (-lul/-ul) one, we do not in general nd -acc marked NP s interpreted as antecedents of -nom marked re exives (even if they precede them). Thus the expressions in (13) are generally bad, but reversing the -nom and -acc marking produces fully grammatical ones. (13) a. * Haksayng-tul-ul caki-casin-i pinanhayssta student-pl-acc self-emph-nom criticized `The students criticized themselves b. * Nwukwu-lul caki-casin-i pinanhayss-ni who-acc self-emph-nom criticized? ` Who criticized himself? c. * Caki-casin-i pinanha-n haksayng-ul manna-ss-ta self-emph-nom criticize-Adnom student-acc meet-Past-Decl `I met the student who criticized himself' d. ?? John-ul caki-casin-i pinanhayssta John-acc self-emph-nom criticized `John criticized himself' (14)
The Local Anaphor-Antecedent Relation in Korean
In a transitive S, x is a possible antecedent of y i x and y are coarguments and y is in the coordinate closure of (l)ul suxed NPs. 10
(14) of course does not pretend to be a complete characterization of the AA relation in Korean. We have not considered re exives in ditransitive S s or in adjuncts, or as possessors, not to mention the long distance anaphoric possibilities of bare caki. But (14) is sucient as a challenge to current theories, and more extensive treatments must be extensionally equivalent to it for the restricted set of data (transitive S s) it covers. We note that the Korean pattern in (4)-(13) extends to Bengali and Hindi, modulo the absence of topic marking axes and some di erences in the case marking patterns. We now model these properties with our grammar (syntax + semantics) of Little Korean and we show that (14) is, as given, a properly structural statement of the AA relation in Little Korean, even though it mentions a particular morpheme. To back this up of course we must semantically interpret S s with re exives and show that binding does indeed obtain in S s like (4a,b). (See Keenan and Stabler 1997 for a more thorough discussion of the empirical motivations for various aspects of this grammar.)
2.2 Syntax
LK is the grammar hV; Cat; Lex; Rulei, where: V is the set whose members are: john, bill, himself, laughed, cried, praised, criticized, -nom, -acc, both, and, either, or, neither, nor Cat is the set whose elements are: K, NP, KPa, KPn, P2, P1a, P1n, S, CONJ Lex is the following subset of V Cat: NP: john, bill, himself K: -acc, -nom CONJ: and, or, nor P1n: laughed, cried P2: praised, criticized (The rst line here indicates hjohn; NP i 2 Lex; hbill; NP i 2 Lex, etc.) Finally, Rule consists of three functions: Case Marking (CM ), PredicateArgument (PA), and Coordination (COORD) de ned as follows: (15) Case Marking: Domain
CM
-nom t
Value
Conditions
t-nom K NP ??! KPn t 6= himself -acc t ??! t-acc (none) K NP KPa 11
We read (15) as follows: The domain of the partial function CM is all the pairs listed under \Domain" which satisfy the conditions on the right. So CM maps two possible expressions, the rst h-nom; K i and the second ht; NP i to ht-nom; KPni provided t is any string over V except himself. The second clause is understood analogously. (16) Predicate-Argument: Domain
(17)
PA Value Conditions
s t s_ t KPx P 1x ??! S x 2 fn; ag s_ t s t KPx P 2 ??! P 1y x 6= y 2 fn; ag Coordination:6 For all C 2 fS; P 1n; P 1a; KPa; KPn; P 2g, Domain
and CONJ or CONJ nor CONJ
t C t C t C
COORD
u C u C u C
??! ??! ??!
Value
Conditions
both_ t_ and_ u t 6= u C _ either t_ or_ u t = 6 u C neither_ t_ nor_ u t 6= u C
Little Korean derivations can be depicted with trees like the following: (18) a. S KPn
P1n
NP
K
john
-nom
KPa
P2
NP
K
himself
-acc
criticized
6 We write both x and y for readability in English. Normally if a verb nal language has as discontinuous coordinator it is postpositional, as though we said x and y both, though usually the two parts are identical, as in the French et Jean et Marie. Korean is no exception: Yongho wa Mica wa ka kyelhonhayssta Yongho and Mica and nom married `Yongho and Mica are married'
12
b.
S KPa
P1a
NP
K
himself
-acc
KPn
P2
NP
K
john
-nom
criticized
2.3 Syntactic invariants of Little Korean
Theorem 7 a. h-nom; K i is invariant; so is h-acc; K i. In other words, every structure map for LK maps h-nom; K i to itself and also maps h-acc; K i to itself. So these lexical items are structural. In distinction hJohn; NP i is not invariant.
b. For all C in Cat(LK ), PH (C ), the set of expressions of category C , is invariant. So for example the property of being a KPa, an accusative Kase Phrase, is invariant. c. Now let's de ne the co-argument relation, a relation among triples s; t; u of expressions in the language, as follows: s is a co-argument of t in u i for some v of category P 2, either PA(s; PA(t; v)) or PA(t; PA(s; v)) are constituents of u. We can then prove: the co-argument relation in LK is invariant. d. Finally we characterize the relation is a possible antecedent of, noted PAA, as follows: x is a possible antecedent of y in z i x is a co-argument of y in z and Cat(y) = KPa. Then PAA is invariant.
Proofs of the above theorems rely on certain information speci c to the grammar LK. But they also draw heavily on the following theorems regarding universal invariants:
Theorem 8 Universal Invariants. Let G = hV; Cat; Lex; Rulei be an arbitrary grammar. Then,
a. ; and L(G) are invariant subsets of L(G). More generally, for each nite n, L(G)n is an invariant set of n-ary sequences of expressions of L(G). The rst of these claims tells us that the property of being grammatical in G is a structurally invariant property of expressions. If this were not so we would just have the wrong de nition of \structural invariant".
13
b. Each generating function F in Rule(G) is invariant (trivially); the domain and range of each generating function F is invariant, and more generally, the set of ith coordinates of F is an invariant subset of L(G). c. The collection of invariant subsets of L(G)n is closed under arbitrary intersections, unions, products and relative complements. So if two properties P and Q are invariant so are the properties expressed by their conjunction, disjunction, and negation. For example, it is plausible that the property of being a masculine singular proper noun is a structural property of English. Plausibly (this is not an argument, just a thought experiment to help build our intuitive understanding of the notion \structural property") a function that mapped John to Bill, Bill to Frank and Frank to John, and made no other changes except those induced by these (i.e. the function must map the complex expression John laughed to Bill laughed, etc.) is a structure map for English. This would mean for example that no structure map for English could map John to Mary, or sings, or All cats are black, which is reasonable. Rather h(John), the image of the proper noun John under an arbitrary structure map h, must always be a masculine singular proper noun. d. For each d 2 L(G), the set [d] of expressions isomorphic to d is an invariant subset of L(G). It is always non-empty (since d is in it) and no non-empty proper subset of [d] is invariant. That is, [d] is an atom in the complete atomic boolean lattice L(G). Similarly L(G)n is a complete atomic boolean lattice. e. The value of an invariant function at an invariant argument is itself invariant. f. The binary relation is a constituent of, CON , is invariant, where this is de ned as follows. For all expressions s; t, sCONG t i either s = t or there is a structure building function F and a sequence d1 ; : : : ; dk of expressions such that t = F (d1 ; : : : ; dk ) and for some i, sCONG di . (Notice that this relation is re exive.) g. The 3-place relation is a sister of is invariant, where we say expressions s and t are sisters in an expression v i for some F 2 Rule and some tuple u 2 DomF , F (u) is a constituent of v where s and t are distinct elements of the tuple u. (Notice that this relation is irre exive.) h. The 3-place relation C-commands is invariant, where an expression s Ccommands t in v i for some u, s and u are sisters in v, and t is a constituent of u. i. Notice that in the grammar LK , the rst argument to the 2-place structure building function PA is always KPn or KPa, and these are always produced by an application of the 2-place function CM . We can compose CM and the rst argument of PA to get a new 3-place function, which we might call
14
CMPA, whose domain is the triples hs; t; ui such that hs; ti 2 DomCM and hCM (s; t); ui 2 DomPA. This function is invariant.
In general, any composition of generating functions (at any positions) is invariant, as are compositions of compositions, etc. That is, the set of invariants is closed under all generalized compositions of generating functions. This means that closing Rule under generalized composition yields a grammar with the same structure as the original, even though every non-lexical expression has a one-step derivation from lexical items. This provides another perspective on a point mentioned earlier: the structure of an expression is not given by any one of its derivation trees.
See Keenan 1996 for much more systematic discussion of semantic invariants. Cf. also van Benthem 1989. We note that some sets which one might have thought would be universally invariant are not. For example the property of being a lexical item may fail to be structural. Similarly in some grammars the property of being an expression of a given category C is not invariant. That is, from some C 2 Cat, the set of phrases of category C , PH (C ) =df fs 2 L(G)j Cat(s) = C g is not universally invariant. For example, in the language Little Spanish described later, the masculine nouns, of category Nm can be interchanged with the feminine nouns, of category Nf, preserving structure (though this property is \unstable" in that it requires that many other highly speci c conditions be met, speci cally that the number of masculine and feminine nouns in the Lexicon be the same. So the property is lost if we add just a single new lexical item to just one of those categories). Closer to hand, one might have thought that in LK an automorphism could interchange h-nom; K i and h-acc; K i. They have the same category and seem to play comparable syntactic roles in LK. But in fact we cannot. One problem arises for example with S s built from intransitive verbs, like john-nom laughed. If we traded in -nom for -acc and so john-nom for some KPa the resulting KPa could not combine with a P 1n like laughed. The only option would be to also trade in laughed for some P 1a, but all of them are syntactically complex and it is easy to show that no automorphism maps a simplex expression in LK to a complex one. Equally in LK, S s like (4a) and (4b) are not syntactically equivalent: no syntactic automorphism of LK maps one of these S s to the other. The immediate reason is that we provably cannot interchange P 1n's with P 1a's, and ultimately that derives from the fact hhimself; NP i has a restricted distribution. 15
3 Semantically interpreting anaphors in Little Korean We have so far been a little cavalier in claiming that we have structurally characterized the Anaphor-Antecedent relation in LK. We have to be sure well de ned a ternary relation x is a possible antecedent of y in z and claimed (correctly) that that relation is an invariant of LK. But we have not shown that S s like (18a,b) which contain expressions like hhimself -acc; KPai are interpreted correctly and as a function of their syntactic structure (compositionally). That is, we must be able to show that (18a,b) are both true in a situation i the object denoted John stands in the criticize relation to itself. So we now show this. The reader will see that just as the case markers are crucial in de ning the expressions of LK { for example in ruling out simple S s built from two KPn's or two KPa's { so also getting the correct semantic interpretation of S s like (18b) in which the anaphor C-commands its antecedent. Here we present the essentials of our interpretation of LK. A more comprehensive formal statement, which must cover several issues not speci c to case marking or anaphors, like the interpretation of coordinate expressions, is given in the Appendix. Standardly we interpret a language relative to a model (situation) M = hE; 2; mi, where E is the universe of objects under consideration, 2 is the two element set fT; F g of truth values, usually noted f1; 0g, in which sentences take their denotations, and m is a function which gives the denotations of the lexical items. Then an interpretation IM of L relative to M is a function assigning denotations to all expressions of L in such a way as to agree with m on the lexical items; the value of IM at a complex expression is given as a function of its values at the expressions d is built from. So, crucially, our de nition of semantic interpretation is compositional. Expressions of category P 2 denote binary relations on E , and expressions of category P 1n denote subsets (unary relations) of E . So in particular in any model M for LK, m(laughed; P 1n) is a subset of E , and m(criticized; P 2) is a subset of E E . (Note that we have not yet said what sorts of things P 1a's denote. Recall that they are are the kind of P 1 which combines with accusative Kase Phrases to form a sentence. All P 1a's are derived, there are no lexical items of this category). NP s denote generalized quanti ers (GQs) over E , that is, functions mapping n +1-ary relations on E to n-ary ones. Speci cally they map subsets of E (unary relations) to truth values (zero-ary relations), and they map binary relations to unary ones. Special restrictions apply to NP s which are lexical items (which happens to be all of the NP s in L(LK )). First, the interpreting function m of a model is required to interpret the proper nouns, hjohn; NP i and hbill; NP i as those generalized quanti ers we call individuals, de ned as follows. 16
De nition 9 For each b in E , Ib is that generalized quanti er given by: a. for each subset P E , Ib (P ) = T i b 2 P b. for each binary relation R on E , Ib (R) = fa 2 E j aRbg 2 Second, m is required to interpret hhimself; NP i as any generalized quanti-
er whose value at a binary relation R is given by the function self below: (19) SELF (R) =df fa 2 E j aRag Now, if we were doing \Little English" rather than Little Korean we could now say directly how the combination of an NP and a P 2 or a P 1n would be interpreted, as NP s denote functions taking P 2 and P 1n denotations as arguments. So for example the interpretation of hJohn laughed; S i would just be the truth value that the interpretation of hJohn; NP i assigned to that of hlaughed; P 1ni. (In general, here and later, when an expression c is built from two expressions, one of them denotes a function whose domain includes the denotation of the other, and c itself is interpreted as the value of that function at that argument). However, in LK NP s do not combine directly with predicates, they only combine with Case Markers h-nom; K i and h-acc; K i to form Kase Phrases. By compositionality the interpretation of these derived KP s is given as a function of the interpretation of the NP s and Case Markers they are derived from. So let us say how h-nom; K i and h-acc; K i are interpreted. (20) m(-acc; K ) is that function acc mapping generalized quanti ers to generalized quanti ers as follows: acc(F ) = F . So the KPa John-acc can be interpreted in the P 1a John-acc criticized as follows: [[John-acc criticized; P 1n]] = Ij (criticize) = fa 2 E j acriticizej g [[John-acc; KPa]] = acc(Ij ) = Ij
[[criticized; P 2]] = criticize
[[john; NP ]] = Ij [[-acc; K ]] = acc As shown here, the P 1a John-acc criticized is interpreted as denoting the set of objects in the model that stand in the criticize relation to j . Similarly, replacing John with himself above the reader may compute that hhimself -acc criticized; P 1ni is interpreted by self(criticize) = fa 2 E j acriticizeag, the set of those objects in the universe that stand in the criticize relation to themselves. Note that the interpretation of h-acc; K i is rather trivial, the identity map on generalized quanti ers. We could have simpli ed our syntax of LK by eliminating it entirely. We included it because (1) real Korean has it, and (2) as we enrich LK to include predicates of higher arity (ditransitive verbs) h-acc; K i plays more signi cant of a role. To discriminate the interpretations of n + 1 17
argument denoting expressions only n of them need be overtly marked. So we need only one Case Marker with two arguments, but two with three arguments. Still, for the limited fragment of Korean represented abstractly by LK, interpreting the accusative Case Marker h-acc; K i as the identity map on GQs seems, and is, a trivial way to satisfy our claim that case marking is semantically interpreted. This claim would not be substantive if morphology were always interpreted as pertinent identity functions. That would just say that there is no di erence in the interpretation of an X with and without its attached morphology. In our case however we see that the interpretation of nominative morphology, is not trivial. Indeed our ability to interpret arguments that form a constituent with a P 2 as antecedents of sisters to the derived P 1 relies crucially on the interpretation of h-nom; K i, which is given by: (21) In each model hE; 2; mi; m(-nom; K ) is that function nom whose domain is the set of generalized quanti ers over E and whose values are given by: a. For each subset P E; nom(F )(P ) = F (P ) b. For each binary relation R on E; nom(F )(R) is that function which maps each generalized quanti er G to a truth value, as follows: nom(F )(R)(G) = F (G(R)) Observe now the (compositional) interpretation of (18b), the sentence himselfacc john-nom criticized: [[himself -acc John-nom criticized; S]] = nom(Ij )(criticize)(self) [[himself -acc; KPa]] = acc(self) = self [[-acc; K ]] = acc
[[John-nom criticized; P 1aD]] = nom(Ij )(criticize)
[[himself; NP ]] = self
[[John-nom; KPn]] = nom(Ij ) [[criticized; P 2]] = criticize [[-nom; K ]] [[john; NP ]] = nom = Ij We can calculate, according to our de nitions: nom(Ij )(criticize)(self) = Ij (self(criticize)) = Ij (fa 2 E j acriticizeag) = T i j 2 fa 2 E j acriticizeag = T i j criticizej Clearly this is the correct anaphoric interpretation of the S . It is true just in case John stands in the criticize relation to himself. Similarly one computes that the interpretation of hjohn-nom himself-acc criticized,Si in (18a) is identical to that of (18b) above. So the two S s receive 18
the same interpretation, despite the fact that the antecedent only c-commands its anaphor in one of the cases. This completes our basic treatment of the interpretation of LK. Wemight only note that our interpretation of h-nom; K i above implies that expressions of category P 1a are interpreted as functions mapping generalized quanti ers to truth values. The remainder of the de nition of interpretation concerns how to interpret coordinations of expressions in the various categories, not a point central to our interests here, though do note that we interpret complex anaphors like hboth john-acc and himself -acc; KPai.
4 Interpreting grammatical morphemes The interpretations of the grammatical morphemes of LK, namely h-nom; K i, h-acc; K i and hhimself; NP i have a property in common. Namely, (22)
given the universe E of a model, they are constant in their denotation.
Furthermore, in a sense we make explicit shortly, (23)
the functions these morphemes denote are themselves \semantic invariants". That is, they are invariant under the action of \semantic isomorphisms".
These observations give rise to two questions. First, concerning the relation between syntax and semantic interpretation, are properties (22) and (23) independent? We show that the answer is: No. Relative to one very generally obeyed condition on the closure under isomorphism of the models for a language, we show that (22) entails (23), though the converse fails. And second, is it simply an accident of our grammar and semantics for LK that the grammatical morphemes share the semantic condition given in (22) and (23) or is this correlation in some way principled? We feel that it is principled, but can at the moment do no better than formulate the principle clearly. Further work is needed to nd a more general principle from which this one would follow as a special case. In order to give ourselves some con dence that the semantic properties associated with the grammatical morphemes of LK are not simply accidental we illustrate, brie y, a formal model of gender agreement and we provide two plausible ways of interpreting the agreement morphology according as it indicates \natural gender" (Pollard and Sag 1994) or is purely \syntactic". In each case its denotation is semantically invariant. We call our language \Little Spanish". Once the syntax and semantics for LS is given we conclude with a properly formal statement of what semantic invariance means and then provide the principle governing the interpretation of grammatical morphemes together with a few other related principles relating form and meaning. 19
5 Interpreting Agreement: Little Spanish We consider here a little language with a simple gender agreement relation between nouns, adjectives and determiners. The syntax is very simple, but, surprisingly perhaps, the agreement morphology in our example language is not quite structural (invariant). Speci cally in our example it is possible to systematically interchange the masculine and feminine suxes preserving structure (though several other changes must be made as well). Our feeling is that this structural possibility that e.g. the masculine and feminine endings in a language be structurally comparable is allowed by a natural language. It is worth noting however that many other conditions would have to be met in order to interchange them preserving structure. Speci cally in our example the number of masculine nouns in the lexicon must equal the number of feminine ones. So the lack of invariance of the agreement endings is not preserved under what we would otherwise consider structurally trivial changes { namely adding more lexical items to categories already instantiated by more than one lexical item. However, regardless of whether the agreement morphemes are grammatical morphemes or not, their denotations on both ways of interpreting them are still \semantic invariants".
5.1 Little Spanish syntax Consider the generic agreement language LS = hV; Cat; Lex; Rulei where
V = fman ; woman ; obstetrician ; doctor ; -a ; -o ; gentle ; intelligent ; every; some; very; moderatelyg Cat = fNm ; Nf ; A; Am ; Af ; Amod ; NPm ; NPf ; Agr ; D ; Dm ; Df g Lex is the following subset of V Cat: Nm: Nf: A: D: Agr: Amod: NP: K: CONJ: P1n: P2:
man, doctor woman, obstetrician gentle, intelligent every, some -a, -o very, moderately john, bill, himself -acc, -nom and, or, nor laughed, cried praised, criticized Rule = fGM; PAg as follows:
20
(masculine nouns) (feminine nouns)
Domain GM Value Conditions
s A s A s D s D
-o Agr -a Agr -o Agr -a Agr
Domain
_ ??! sAm-o
_ ??! sAf-a _ ??! sDm-o
_ ??! sDf-a
PA Value Conditions
s t Ax Nx ??! s t Dx Nx ??! s t Amod Ax ??!
s_ t x 2 fm; f g N s_ t NPx x 2 fm; f g s_ t x 2 fm; f g Ax
This grammar is similar to previous ones, except that the nouns are subcategorized according to \gender," and modifying adjectives and determiners are correspondingly marked. No rules are provided for coordination. The following tree depicts a sample derivation: NPf Df D every
NPf Agr -a
Af
NPf
A
Agr
gentle
-a
Af Amod very
Nf Af
obstetrician
A
Agr
intelligent
-a
It is interesting to note that there is a structure map h for Little Spanish which maps h-o; Agri to h-a; Agri and h-a; Agri to h-o; Agri. This function h must also interchange the masculine nouns in the lexicon with the feminine ones, and otherwise it can be the identity map on the lexicon. 21
It can also be shown that for any structure map h for Little Spanish, the value of h at an agreement morpheme is an agreement morpheme, so all it can do is map the agreement morphemes to themselves or it can interchange them. So while h-o; Agri and h-a; Agri are not grammatical constants, they have only one degree of freedom.
5.2 Little Spanish semantics 1
A simple model theory for Little Spanish has the structure M = hE; 2; mi used for previous languages, with E is a non-empty universe, 2 the set fT; F g of truth values, and m a function mapping elements hv; C i of Lex into DenE (C ), the set of possible denotations of expressions of category C in M. The possible denotation sets for Little Spanish expressions can be de ned as follows. A category C 2 Cat will denote in DenE C as follows: DenE N = DenE Nf = DenE Nm = P E (the power set of E ) DenE A = DenE Af = DenE Am = f 2 [P E ! P E ]j f (p) pg (restricting functions on P E ) DenE Amod = ff 2 [DenE A ! DenE A]j f (p) pg (restricting functions on DenE A) DenE P 2 = R2 = P (E E ) (the binary relations) DenE NPn = DenE NPf = [P E ! 2] (generalized quanti ers) DenE Agr = fidA _ idD g (identity) That is, m(-o; Agr) = m(-a; Agr), each denoting the identity function whose domain is the union of the set of possible adjective denotations with the possible determiner denotations. A model for LS M = hE; mi then satis es the following conditions:
Lexical Conditions: a. for all hs; C i 2 L(G), m(s; C ) 2 DenE C Conditions on Derived Expressions: b. for F 2 fGM; PAg, and any ; 2 DomF ,
m(F (; )) = m()(m( )) In these models, agreement is semantically invariant and semantically trivial. Still it is the case that the agreement morphology is semantically interpreted and the denotations of h-o; AGRi and h-a; AGRi are \semantically invariant" (a notion we still have not de ned), since they are just identity functions. If, speaking Little Spanish with semantics 1, we said \every-a intelligent-a," leaving o the noun, you might be able to make an inference about the subset of things I might have been speaking about { it could only be some of those things which happen to be denoted by constituents of category Nf { but this inference 22
is pragmatic, not something that re ects a basic categorization of elements of the universe. The rather arbitrary gender systems of the Romance languages are sometimes given such a \syntactic" treatment.
5.3 Little Spanish semantics 2
A di erent semantic theory makes agreement \semantic," with models that distinguish (possibly overlapping) subsets of the universe which can be named by masculine and feminine nouns. The adjectives then have their domains appropriately restricted by the agreement markers. Gender agreement in languages like English is often held to be \semantic" in something like this sense. Speci cally the choice of he versus she is not determined by the gender of the head noun used to refer to their referents but rather by the the sex of the referent, with it or that being used for asexual referents. (We are deliberately ignoring several messy details.) But again, on a natural approach, the denotations of the agreement markers is semantically invariant. A model is a ve-tuple M = hE; Ef; Em; 2; mi in which Ef E; Em E . Intuitively here Ef is the set of \feminine objects" in the model, Em the set of masculine ones. Feminine nouns will be required to denote subsets of Ef , masculine ones subsets of Em. So the set of masculine and feminine things are part of the model structure, and must be xed by the semantic analogue of \structure maps", called model automorphisms. So automorphisms of such Ms must x E; Ef; and Em and 2. Denotation sets are de ned as follows:
DenM (Nm) = Em; DenM(Nf ) = Ef DenM (ADJ ) = the restricting maps from P (E ) to P (E ) as before DenM (DET ) = the conservative maps from P (E ) into [P (E ) ! 2] DenM (AGR) = [DenM (ADJ ) ! DenM (ADJ )][ [DenM (DET ) ! DenM (Det)], where m(o; AGR)(f ) = f P (Em) and m(a; AGR)(f ) = f P (Ef ): For x 2 fm; f g,
DenM (ADJx) = the restricting maps from P (Ex) to P (Ex); DenM (DETx) = the conservative maps from P (Ex) into [P (E ) ! 2] DenM (NPx) = the maps in [P (E ) ! 2] that live on Ex Of course automorphisms of M must now not only x E and be boolean automorphisms of 2 they must also x Em and Ef .
Fact 10 In models of Little Spanish with Semantics 2, h-o; AGRi and h-a; AGRi
may be interpreted di erently, and they are typically not interpreted as identity maps, but they are always interpreted as AI objects and are in fact ISOM.
23
Fact 11 Expressions of the form hd; NPmi live on Em; that is, for all models M = hE; Em; Ef; 2; mi; m(d; NPm)(p) = m(d; NPm)(p \ Em). Similarly hd; NPf i lives on Ef . If we add say John and he to Lex of category NPm and Mary and she in category NPf then we would require that m(john) and m(he) are interpreted as individuals I mapping Em to 1; similarly m(mary) and m(she) are individuals mapping Ef to 1.
6 Semantic Invariants We begin rst with a simple example. Consider an arbitrary mathematical structure, say hA; +; Ri, where A is a set (the domain of the the structure), + is a two place function on A, and R is a binary relation on A. + and R represent a certain structure de ned on A. Now consider the analogue of a structure map for hA; +; Ri. It would be a permutation of A, that is a one to one and onto function h from A to A which does not change the structure. That is h(+) = + and h(R) = R. We have already said what these equations mean. For example, h(R) is just fhh(x); h(y)ij xRyg. So to say that h(R) = R is to say that for all x; y in A, xRy i h(x)Rh(y). So h might trade in a for b, b for c, and c for a, but it does not change their relatedness under R: if x stands in the relation R to y then the object h maps x to, h(x), stands in the relation R to h(y), and conversely. Similarly to say that h(+) = + is just to say that x + y = z i h(x) + h(y) = h(z ). So h preserves the \addition" relation among elements. Such structure preserving maps are, standardly, theautomorphisms of hA; +; Ri. And in general, where A is a set with relations R1 ; R2 ; : : : de ned on it (some of the relations may be functions) then an automorphism of the structure is a permutation of A which xes all the relations. Applying this idea to our semantic analysis, consider again the standard notion of extensional model that we have used for Little Korean and Little Spanish. A model M consists of two semantic primitives, E and 2, henceforth called the ontology of the model, and a function m that interprets lexical expressions as objects in various of the sets in the type hierarchy built from E and 2. By de nition this hierarchy includes as members E and 2 and all the sets constructible from these by forming power sets, function sets, and nite cross products. That is, whenever a set A is in the hierarchy then so is P (A); whenever A and B are in the hierarchy then so are A B and [A ! B ], the set of functions from A into B . Now consider the initial members of the type hierarchy, E and 2 (and assume for simplicity that they are required to be disjoint). 2 is a small set but it has much boolean structure (which is implied when we say that 2 = fT; F g where T is true and F is false). For example it possesses a binary function ^ \meet", used in the de nition of interpretation of a conjunction of S s built with \and". 24
Meet maps the pair hT; T i to T and all other pairs to F . That is a conjunction of S s is true i each conjunct is, otherwise it is false. Similarly it has a binary relation R which corresponds to material implication. That is, we say that a truth value x bears R to a truth value y i an arbitrary sentence of the form If P then Q is true when P has truth value x and Q truth value y. Extensionally this comes down to saying that T bears R to T and F bears R to both T and F , but T does not bear R to F . An automorphism of 2 is a permutation of 2 which xes the \implies" relation (from which it follows that it xes the meet operation). Provably the only automorphism of 2 is the identity function on 2. (The only other bijection of 2 is the denotation of \not"). E on the other has no structure imposed on it, so an automorphism of E is simply a permutation of E . Now, a semantic automorphism of an ontology hE; 2i is a function h which is a permutation of E and the identity map on 2. And h extends in familiar ways to all sets in the type hierarchy built from E and 2. E.g. for K a subset of E , h[K ] =df fh(x)j x 2 K g. And we may de ne the semantic invariants of an ontology as those members of sets in the type hierarchy that are xed by all the semantic automorphisms. For example True and False are xed by allsemantic automorphisms. Among the subsets of E , ; and E itself are xed by all semantic automorphisms. Among the binary relations just ;; E E , the identity relation on E (fhx; xij x 2 E g and the complement of the identity relation fhx; yij x 6= y 2 E g. As we move into more complex elements of the type hierarchy we the number of automorphism invariant (AI) elements grows. For example in a nite universe of cardinality n, there are 2n+1 AI functions in [P (E ) ! 2], that is, 2n+1 possible subject NP denotations. See Keenan 1996 and references cited there for extensive discussion. We can now state our earlier claims regarding Little Korean and Little Spanish:
Theorem 12 a. In all models hE; 2; mi for LK, the denotations of h-nom; K i, h-acc; K i and hhimself; NP i are semantic invariants. b. In all models hE; 2; mi for Little Spanish, on either Semantics 1 or Semantics 2, the denotations of h-o; Agri and h-a; Agri are semantic invariants. We conclude now with four general constraints relating syntactic form and semantic interpretation.
7 Constraints relating form and meaning
Let G = hVG ; CatG ; LexG ; RuleGi be arbitrary and held constant in what follows. We write MOD(G) for the set of models of L(G). that is, the set of triples hE; 2; mi, where m interprets lexical items of L(G) in accordance with stipulated constraints particular to L(G). 25
7.1 Global Compositionality
Following Keenan and Stabler 1996 and Fulop and Keenan 1997, the following is required of MOD(G): The set fmj 9O; hO; mi is a model of L(G)g satisfying, for each F 2 RuleG, fhm(d); m(F (d))ij d 2 Dom(F ) & m 2 M g is a function. Theorem 13 a. Global compositionality implies Globality: that is, for all d; d0 2 Dom(F ), all m; m0 2 M , if m(d) = m0 (d0 ) then m(F (d)) = m0 (F (d0 )): b. Global compositionality implies standard Compositionality: that is, for all F 2 RuleG, for all models (O; m) of L(G), fhm(d); m(F (d))ij d 2 Dom(F )g is a function.
7.2 Model Closure under isomorphism (MC)
The set of models MOD(G) also has the \model closure" property, where this is de ned as follows: For each model hO; mi of L(G) and each isomorphism of O, hO; mi is a model of L(G). Theorem 14 (Keenan and Stavi 1986, Keenan 1996) If d is semantically constant in the sense of having the same denotation in all models with the same ontology then d denotes a semantically AI object. That is, m(d) is AI, all models hO; mi. Proof. Let d be semantically constant. Show for all M = (O; m), all automorphisms of O, that (m(d)) = m(d). Observe that M = (O; m) is a model, by MC. Whence (m(d)) = m(d) = m(d) Model closure; d is a semantic constant 2
Remark: Contra Etchemendy 1990, x12,
i. denoting AI objects does seem to be a property common to the denotations of \logical constants", and ii. holding interpretations of an expression constant cannot be done freely, i.e. if we hold constant is a senator then in any model M it denotes the empty set or all of E , as they are the only AI subsets of E. 26
7.3 Meaning Recoverability (MR) This condition is actually a purely syntactic one, though it has a semantic motivation: For F 2 RuleG and d; d0 2 Dom(F ), if d ' d0 & d 6= d0 then F (d) 6= F (d0 ) Notice that d ' d0 entails F (d) ' F (d0 ). The intuition is the following. Simple expressions which are distinct but isomorphic may almost always be interpreted di erently: Mary vs Sue; laughed vs cried; Bill laughed vs. John cried. If derivational processes did not preserve distinctness there would be a meaning collapse would be built into the syntax. So e.g. if laugh and cry are isomorphic but distinct we expect (correctly) that their in nitival forms to laugh and to cry are distinct (and isomorphic) and that their gerund forms laughing and crying are distinct (and isomorphic). If e.g. the gerund forms of laugh and cry were both blicking, then the meaning of each would be unrecoverable from the derived form. Similarly the gerundive nominal Bill's laughing is distinct but isomorphic to John's crying; if they were both Sam's talking an inherent loss of meaning would be associated with the nominalization process. Grammatical constants are either lexical items or are built by applying generating functions RuleG to grammatically constant lexical items. That is, writing KGC for the set of grammatical constants in K and CL(K ) for the closure of K under RuleG,
Theorem 15 MR ) L(G)GC = CL(LexGC ) Proof. Show LexGC L(G)GC and L(G)GC closed with respect to RuleG.
i. Let d 2 LexGC LexG . Then d 2 L(G) and d is grammatically constant so d 2 L(G)GC . ii. Let d1 ; : : : ; dn 2 L(G)GC with d = hd1 ; : : : ; dn i 2 DomF . Show F (d) 2 L(G)GC . For any syntactic automorphism, (F (d1 ; : : : ; dn )) = F (d1 ; : : : ; dn ) = F (d1 ; : : : ; dn ), by the IH, = F (d), so F (d) is grammatically constant. Thus CL(LexGC ) L(G)GC .
Set K = fd 2 L(G)j if d 2 L(G)GC then d 2 CL(LexGC )g. i. Lex K . Let d 2 Lex. Suppose d 2 L(G)GC . Then d 2 LexGC CL(LexGC ).
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ii. Let d be a tuple of elements in K with d 2 DomF . Show F (d) 2 K . Suppose F (d) 2 L(G)GC . Show F (d) 2 CL(LexGC ). Suppose, leading to a contradiction, that some di 62 L(G)GC . Let be an automorphism such that (di ) 6= di . Thus (d) 6= d. Since trivially d ' (d) we have by MR that F (d) 6= F ((d)). But F ((d)) = (F (d)) since commutes with generating functions, and (F (d)) = F (d) since F (d) 2 L(G)GC . So F (d) = F ((d)), a contradiction. Thus all di 2 L(G)GC , so by the IH each di 2 CL(LexGC ), so F (d) 2 CL(LexGC ),since this set is closed with respect to F , completing the proof. 2
Notice that this theorem has no semantic analog: Semantic constants may be syntactically complex without being built from (simpler) semantic constants. E.g. Either there exist pink swans or there don't is semantically constant even though pink and swans are not.
7.4 Syntactic Invariants determine Semantic Invariants
Three di erent views: A. if an expression d is syntactically AI then in any model (O; m), m(d) is AI. (Keenan and Stabler 1996) B. Syntactically AI expressions have the same denotation in all models with the same ontology. (These denotations are provably AI, by theorem 14 above.) C. Syntactically AI expressions are isomorphism invariant: models M = (O; m) and M0 = (O0 ; m0 ) and all isomorphisms : M ! M 0; m0 (d) = (m(d)):
7.4.1 First examples
We rst give some easy examples, assuming the relevant expressions to be syntactically AI, and then consider again Case Marking in Little Korean and Agreement in Little Spanish. We assume a standard extensional ontology: the primitive types are just fe; tg. The more novel points here are nding appropriate denotations for \grammatical constants". (See the stimulating but semantically inadequate treatment in Emonds 1985). The re exive himself in (24) interpreted as self, the passive operator in (25) interpreted as pass, and the \agentive" use of by in (26) interpreted as by. (24) Each worker criticized himself in front of his buddies. Here self maps binary relations to properties by: self(R) = faj aRag: 28
(25)
The door was opened =(the door)(pass(open)) pass maps binary relations to properties by: pass(R) = fbj 9a; aRbg = Ran(R):
(26)
The door was opened by Fred =(The door)(pass(open by Fred)) by maps individuals to maps from binary relations to binary relations as follows: by(x)(R) = fha; bij a = x & ha; bi 2 Rg: We leave as an exercise for the reader: compute that (a) and (b) below are logically equivalent: a. John hugged Bill b. Bill was hugged by John = Bill(pass(hug (by John)))
7.4.2 Case marking Lemma 16 For an automorphism of (E; 2), a 2 E and R E E , (aR) = a(R). (We write xR for fyj xRyg). Proof. z 2 (aR) i 9y (y 2 aR & z = y), i 9y (ha; yi 2 R & z = y), i ha; ?1 z i 2 R,i ha; z i 2 R, i z 2 a(R). 2 Theorem 17 The function acc used in Little Korean is semantically AI Proof.
a 2 (acc)(F )(R) i a 2 (accF )(R) i a 2 (accF (R)) i a 2 accF (R) i F (aR) = 1 i (F (aR)) = 1 = 1 i F ((aR)) = 1 i F (a(R)) = 1 i a 2 acc(F )(R)
def ext of to functions
x 2 Y i x 2 Y; all aut. def acc
an auto. of 2 (boolean) lemma
2
Claim 18 There are semantic constants of English which are not grammatical
constants Proof. sixty-seven and sixty-eight are semantic constants but they can replace each other in all syntactic contexts and so are not syntactic constants. 2
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7.4.3 SynInvar ) SemInvar constrains grammars
We could not enrich an adequate grammar of English with the apparently trivial and redundant rule F whose domain was fhJohn; NP ig and whose value at its only argument is hJohn; NP i (or for that matter, anything at all). The reason is that syntactic automorphisms x the domains of generating functions, so English+F requires that all syntactic automorphisms map hJohn; NP i to itself, so it is a grammatical constant, whence by the principles A, B or C of x7.4 above, it is a semantic constant. But in fact it is not a semantic constant; in di erent models (with the same ontology) John may denote di erent individuals.
7.4.4 SynInvar ) SemInvar constrains models
In simple S s the present/past/future tense marking in Malagasy seems interchangeable: (27) m-/n-/hianatra teny angilisy Rabe pres/past/fut study language English Rabe ` Rabe is studying / studied / will study English' But more detailed study reveals that they cannot be interchanged. For example, many verbs select complements in the future tense: (28) n+ibaiko an-dRabe *m/*n/h+anolo ny kodiarana Rasoa past+order acc-Rabe pres/past/fut+replace the tire Rasoa `Rasoa ordered Rabe to change the tire' So future tense marking appears to be a xed point of syntactic automorphisms, whence by SynInvar ) SemInvar it must denote an AI object. This will be so for example in a model in which in addition to E and 2 we have a set T of \times" (or intervals) equipped with a temporal ordering relation >, a distinguished point now with future de ned as > now. (Semantic automorphisms x the temporal order).
7.4.5 Agreement
In both sorts of models for Little Spanish, agreement morphemes have semantically AI denotations. In the rst models, they have a purely syntactic function being interpreted semantically as the identity map. In the second models, the agreement morphemes have nontrivial denotations, again AI.
Fact 19 h-a; AGRi is not a grammatical constant in Little Spanish. Proof. Let be a function from Lex to Lex interchanging h-a; AGRi and h-o; AGRi, hdoctor; Nmi and hobstetrician; Nf i, hman; Nmi and hwoman; Nf i,
and xing all other elements of Lex. Clearly is a permutation of Lex 30
which preserves the domains of all generating functions (Only GM is relevant). Moreover extends to a an automorphism of L(G1) by setting: (F (s; t)) = F ((s); (t)), all F 2 fGM; FAg, all hs; ti 2 Dom(F L(G1)). Given that in PERM (Lex), is well de ned since Lex is disjoint from the ranges of GM and FA, each of GM and FA is 1-1, and their ranges are disjoint. 2
Fact 20 above interchanges PH(Nm) and PH(Nf), PH(ADJm) and PH(ADJf) and PH(DETm) and PH(DETf).
We see then that it is in principle possible to systematically interchange the expressions of two di erent categories, whence the property of being of category C is not a universal invariant.
References Chomsky, Noam. 1995. The Minimalist Program. MIT Press, Cambridge, Massachusetts. Emonds, Joseph E. 1985. A Uni ed Theory of Syntactic Categories. Foris, Dordrecht. Etchemendy, John. 1990. The Concept of Logical Consequence. Harvard University Press, Cambridge, Massachusetts. Fulop, Sean and Edward L. Keenan. 1997. Compositionality: A global perspective. In U. Moennich and H. Kolb, editors, Mathematics of Sentence Structure. de Gruyter, Berlin. forthcoming. Keenan, Edward L. 1996. Logical objects. UCLA manuscript, forthcoming. Keenan, Edward L. and Edward P. Stabler. 1996. Abstract syntax. In Anne-Marie DiSciullo, editor, Con gurations: Essays on Structure and Interpretation, pages 329{344, Somerville, Massachusetts. Cascadilla Press. Conference version available at http://128.97.8.34/. Keenan, Edward L. and Edward P. Stabler. 1997. Bare Grammar. CSLI Publications, Stanford University. Cambridge University Press, NY. forthcoming. Keenan, Edward L. and Jonathan Stavi. 1986. A semantic characterization of natural language determiners. Linguistics and Philosophy, 9:253{326. Pollard, Carl and Ivan Sag. 1994. Head-driven Phrase Structure Grammar. The University of Chicago Press, Chicago. van Benthem, Johan. 1989. Logical constants across varying types. Notre Dame Journal of Formal Logic, 30:315{342.
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Appendix: Semantics of LK
Notation Given a non-empty universe E, i. we write Rn for P (E n ), regarded as a boolean lattice with the order relation given by subset. The elements of R0 are ; and f;g whichwe write variously as F and T , respectively, or 0 and 1. The order relation is just material implication: F F; F T; and T T but not T F . ii. We write [A ! B ] for the set of functions from A into B . iii. The set GQ of Generalized Quanti ers is the set of functions from n + 1-ary relations to n-ary ones:
GQ =df ff 2 [Rn+1 ! Rn ]j for all n; all r 2 Rn+1 ; f (r) 2 Rn g:
1. A model for LK is a pair M = hE; mi where E is a non-empty universe and m is a function mapping elements hv; C i of Lex into DenE (C ), the set of possible denotations of expressions of category C in M, de ned as follows: DenE (S ) = R0 DenE (P 1n) = R1 DenE (P 2) = R2 DenE (NP ) = GQ DenE (KPa) = GQ DenE (P 1a) = [GQ ! R1] DenE (CONJ ) = ff 2 [(DenE (C ))2 ! DenE (C )]j C booleang DenE (KPn) = fnom(f )j f 2 GQg, where nom is de ned by: for each f 2 GQ, nom(f ) is a function with domain R1 [ R2 such that: a. for all P 2 R1; nom(f )(P ) = f (P ), and b. for all R 2 R2; nom(f )(R) is that element of [R2 ! R1] ! R0] given by nom(f )(R)(h) = f (h(R)): For C = KPn; KPa; orP 1a; DenE (C ) is regarded as a Boolean lattice with the relation given pointwise: F G i for x 2 Dom(F ), F (x) G(x). 2. m is required to satisfy certain conditions on the interpretation of elements of Lex: a. for all s 2 Lex, m(s) 2 DenE (Cat(s)) b. m(-acc), noted acc, is the identity map from GQ to GQ c. m(-nom) = nom de ned above d. m(himself; NP ) is any element of GQ whose value at any R 2 R2 is self(R), that is, fb 2 E j bRbg (So we allow semantically that himself-acc could map P 1n denotations to truth values, but we do not allow it to combine with P 1n's in the syntax). 32
e. for all x; y 2 DenE (C ), C boolean, m(and; CONJ )(x; y) is the greatest lower bound of fx; yg, and m(or; CONJ )(x; y) is the least upper bound of fx; yg, and m(nor; CONJ )(x; y) is the complement of m(or; CONJ )(x; y) (Note that we may de ne the negation of x as m(nor; CONJ )(x; x))
3. m extends to a function on LK, called an interpretation of LK relative to M, by setting
i. m(CM (s; t)) = m(s)(m(t)) ( ii. m(PA(s; t)) = m(t)(m(s)) if Cat(s) = KPa and Cat(t) = P 1a, and m(s)(m(t)) otherwise iii. m(BOOL(s; t; u)) = m(s)(m(t); m(u)) This completes our de nition of interpretation in a model for the expressions of LK. Using these de nitions one computes directly that for all models M = (E; m),
m(john-nom himself -acc criticized; S ) = m(himself -acc john-nom criticized; S ) as per our earlier calculation, and thus that the interpretation of himself as a (bound) anaphor does not depend on it being C-commanded by its antecedent.
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