Scope — 4
In the previous post I defined scope as the matrix-line sentential argument(s) of a predicate at SA-level. This is a simple and straightforward definition, but it requires a Semantic Syntax ( SeSyn), i.e. erstwhile Generative Semantics, theory of grammar. This definition is not only formal, in the sense that it is cast in terms of SA-structures, but also semantic, in that it defines the semantic function of scope in a general way. The semantics of scope is defined as follows:
If a predicate in general denotes a property C of any kind assignable to the referent(s) of the n-tuple of its argument terms in a proposition, a scope-bearing predicate, when used in a proposition, assigns C to whatever is denoted by (the n-tuple of) its matrix-line sentential argument term(s).
In SeSyn, the classical quantifiers are defined for two terms, the matrix term (i.e. their scope) in subject position and the restrictor term in object position, as shown in Figure 1-b of post Scope—2. Both terms denote sets (‘the set of x such that…’) and the property assigned by the universal quantifier to this pair of terms is, in general terms: “the set denoted by the restrictor term is included in the set denoted by the matrix term”. Likewise for the existential quantifier: “the set denoted by the restrictor term and the set denoted by the matrix term have a nonnull intersection”. One can, of course, modify these semantic definitions, for example, by adding to the universal quantifier the condition that the R-set be nonnull, or by adding to the existential quantifier the condition that the intersection is partial for both sets. Such modifications may deliver, under conditions of consistency, logical systems that differ from standard modern predicate logic and may be closer to natural intuitions. This, however, is not immediately relevant at this point (though it has been the main focus of my research for the past fifteen years).
If the matrix term (M-term) under a scope-bearing operator O contains a further scope-bearing operator Q, then Q is in the scope of O. But if the restrictor term (R-term) under O contains a further operator Q, as in All students who have published some poetry will be considered for the prize, then Q is not in the scope of O. This is shown by the fact that the Conversions are not valid when the internal negation stands over the R-term. The Conversions say that All students did not read this book is equivalent with No student(s) read this book, and Some student(s) did not read this book is equivalent with Not all students read this book. That is, ‘all not’ is equivalent with ‘not some’ (= ‘no’), and ‘some not’ with ‘not all’. But these equivalences do not hold when the internal negation stands over the R-term: All of those who were not students failed is not equivalent with None of those who were students failed, and Some of those who were not students failed is not equivalent with not all of those who were students failed. The reason is that for the Conversions to hold the internal negation must be in the scope of the higher quantifier, which, according to my definition, it isn’t when the internal negation stands over the R-term.
With this, I provisionally consider the notion of scope well-defined, ‘provisionally’ because, of course, many questions remain. Let us now turn to question (b) of post Scope—2: how to account for the fact that some scope readings are legitimate for some surface structures and others are not. This question has dominated the entire scope literature since its inception during the late 1960s. Yet, despite the considerable amount of research devoted to this question, no convincing answer has so far been provided. Perhaps this is not so surprising, not only because the notion of scope has never been properly understood, but also because, of the two poles of the relation, the logico-semantic structure has at best been specified only in the approximate terms of what is called ‘the language of modern logic’, which leaves it open which variety of that ‘language’ suits our purpose best. Also, opinions differ vastly regarding the nature of the grammatical machinery into which the relation between logico-semantic scope and its surface manifestations is to be integrated.
Even so, a variety of answers have been tried out. One answer, current among cognitivists to the extent that they are concerned with scope questions at all, is that interpretive scope assignments are a question of ‘information structure’, that is, the packaging of information in a wider context into a sentence with some grammatical form—a process taken to be restricted to surface structure operations (see, for example, Adele E. Goldberg, Constructions: A Construction Grammar Approach to Argument Structure, Chicago University Press, Chicago, 2006, p. 161). This answer is surprising in that it implies that differences in information structure correspond with truth-conditional scope differences and vice versa, which strongly contrasts with their explicit or implicit assumption, which they share with the pragmaticists, that information structure is merely a matter of ‘packaging’ given truth-conditional content for the purpose of context-bound interpretation and thus has no truth-conditional implications (I hope to show in a later blog that this is false). In general, cognitivists seem to be blissfully unaware of the importance of truth conditions in semantics and of ways of representing scope. Moreover, they fail to specify in any precise way how this machinery is supposed to work. All we get is vague allusions. I will, therefore, pay no further attention to cognitivism in this discussion.
Chomskyans want scope differences to be reducible to structural conditions and variations in surface structure. They have in principle two strategies to reach a solution, the Quantifier Raising (QR) and the surface C-command strategy, possibly combined into one umbrella strategy. The former, Quantifier Raising, is a bottom-up machinery that isolates quantified terms and negation from surface structures and gives them a place in a logical scope hierarchy. It goes back to Robert May’s 1977 MIT PhD-thesis The Grammar of Quantification (which provoked some hilarity as it proposed the converse of what had been current theory in Generative Semantics, namely top-down Quantifier Lowering, whereby quantifiers are lowered from SA-structures into surface structures, without properly analysing the differences between the two methods). Globally speaking, the surface C-command strategy implies that scope is subject to C-command in surface structure. For those who are not familiar with the term C-command: in a tree structure, a node A C-commands a node B iff the first node up from A dominates B. Thus, if A and B are sister nodes, they C-command each other. In these terms it is said that an operator Q takes scope over an operator R iff Q C-commands R in surface structure. If Q and R C-command each other, there is scope ambiguity.
I won’t discuss in detail the fairly large literature on bottom-up scope determination, with surface structures as input, as every proposal made so far either overgenerates in that it assigns scope readings that are in fact impossible, or undergenerates in that it fails to allow for scope readings that native speakers assent to, or both. Moreover, certain surface phrases are identified as quantifier phrases whereas they may well not be. I will give some examples in a moment.
Generative Semantics (Semantic Syntax) tends to take the obvious correspondence between operator scope and left-to-right order in surface sentences as the default point of departure, proposing a system of Operator Lowering from SAs to surface structures subject to the Scope Ordering Constraint (SOC), which says that a higher operator, when lowered, has to land in a position to the left of the operator lowered earlier. I myself have proposed and defended a system of that nature in many publications over the years. However, such a system still needs so many hedges and extra provisions that, for the moment at least, I can’t see my way through to a rule system that is both empirically adequate and sufficiently compact and transparent to carry conviction. Yet it must be observed that prima facie a strategy of operator lowering is the only option in any overall theory, such as SeSyn, that sees a grammar as a topdown machinery converting meaning representations into surface structures, reflecting the intuition that sentences are type-level schemata for the expression of cognitive propositional content. In such a theory it would be anomalous to treat the clearly truth-conditional phenomena of operator scope in a bottom-up fashion.
SeSyn is the only theory that advocates a top-down rule system. All other approaches, whether emanating from the Chomskyan camp or from any of the logic-oriented approaches to grammar and semantics, such as Montague Grammar (MG) or Categorial Grammar (CG), attempt to specify scope in a bottom-up fashion, from surface to SA. Logic-oriented approaches, despite the impressive display of formal and mathematical prowess, allow in principle for any scope assignment, which is empirically untenable. Like all other theories, they moreover fail to distinguish between dominant and recessive scope readings. What I mean by the latter is this. A sentence like Nobody here speaks two languages is normally (dominantly) interpreted as saying that nobody here is bilingual: ‘not – there is a person x here such that – there are two languages y such that – x speaks y. In this interpretation SOC is fully observed. However, it is also possible, with rising intonation on two languages, to assign a recessive reading to the sentence in which it says that there are two languages that nobody here speaks, assigning wide scope to two languages, thereby violating SOC. Often, however, as in some of the examples given below, there simply is no recessive reading, which makes the logic-oriented theories empirically inadequate.
I can only give a very partial illustration of the problems encountered in the various approaches. Consider a sentence like All books of some students were stolen. This has a dominant reading with some students as the highest operator: ‘there are some students all of whose books were stolen’. The recessive reading ‘all books belonging to some student or other were stolen’ is hard to get. Yet all linguistically oriented bottom-up theories exclude the dominant reading on the grounds that it violates an island constraint. Since it also violates SOC, SeSyn is equally at a loss. Another, hackneyed, example is taken from Ed Klima’s 1964 study on negation: Many smokers don’t chew gum is unambiguous and differs by scope inversion from the equally unambiguous Not many smokers chew gum. Logic-oriented and C-command theories fail in that they generate both readings at least for the second sentence. Only SOC provides an answer here. Likewise for the German Ich habe kein Wort verstanden (‘I didn’t understand a word’) versus Ich habe ein Wort nicht verstanden (‘there was one word I didn’t understand’), both entirely unambiguous, or the unambiguous English sentence For many reasons I did not go out versus the ambiguous I didn’t go out(,) for many reasons (the reading without the comma stands out when a phrase like but for one reason only is added). Sometimes a reading can indeed be attributed to something like ‘information structure’, especially when it takes the form of topic-comment structure (TCS). But in many such cases it is doubtful that the comment is indeed a quantifier. Take well-known examples like A rose adorned every table or A ballerina escorted every officer. These look as if they assign higher scope to every and lower scope to the indefinite article, taken to represent the existential quantifier. But does the indefinite article represent the existential quantifier in such cases? Couldn’t it represent so-called ‘isa’ predication of the type John is a soldier? The underlying structure would then be something like ‘the x such that x adorned every table was a rose’. The predicate a rose is then lowered to the position of the variable in the matrix clause, giving A rose adorned every table, with predicate accent on a rose. This would then not be Operator Lowering but Comment Lowering. Montague treated a soldier as in John is a soldier as an existential quantifier, but he was immediately corrected by Barbara Partee, who considered this to be nonsensical. I think Partee is right, but then, why does the indefinite article systematically occur with the ‘isa’ sort of predication in so many different languages? We don’t have an answer. Given all this, I fear that, at least for now, the upshot is that we have not, so far, been able to resolve the question of what determines scope readings in surface structures in any satisfactory way. I have the feeling that we still need a whole lot of innovating research before we will be in a position to tackle this problem.