Scope — 1

by pieterseuren

In the next few postings I intend to deal with the notion of scope in language—which will necessarily take me to the notion of scope in logic. I had promised Mark Brenchley to devote one posting to Chomsky’s defective citation practices, having admitted that I was wrong in saying that Chomsky had failed to give Karl Lashley his proper due in the review article against Skinner in Language of 1959: Lashley was referenced in that article in a perfectly adequate way, mea culpa. But I think it has been made sufficiently clear in the avalanche of comments to subsequent postings that it was indeed Chomsky’s practice not to refer to his sources adequately. So let’s leave that topic for what it is and turn to something that is intellectually more exciting and morally more uplifting.

Since the mid-sixties, the notion of scope has played an important role in a large variety of linguistic theories, starting with Transformational Generative Grammar. Yet if you look for a definition, or at least a reasonably precise description, of what is meant by ‘scope’ in the linguistic literature, you will find nothing. Since the notion stems from modern logic, one would expect a precise definition in the logical literature. Yet again, one draws a blank. I have consulted a fair number of advanced logical handbooks, but all I found was a casual use of the term, usually illustrated with a few examples and usually restricted to the quantifiers all and some. Occasionally, there is talk about the scope of the propositional connectives (not, or, and, if) or of the operators of modal logic, but no definition of what scope does in logic generally. There is an implicit convention to speak of scope as a property of logical operators—the so-called ‘constants’ of any logical system, as opposed to the variables that are used.

How about formal semantics, where scope is all-important? In the admirable Introduction to Montague Semantics by David Dowty, Robert Wall and Stanley Peters (Reidel, 1981, pp. 63–6), scope for quantifiers is described, not defined, in terms of the machinery of variable substitution in Montague formulae, but what scope actually is, is not revealed. Nor is scope described or defined, in that book, for other logical operators than the quantifiers. Sometimes one finds scope defined in terms of the notation used: count the number of brackets or full stops or whatever, and you will be able to see what is in the scope of what. But what it means to be in the scope of something is never revealed. I myself am equally guilty: I have written a great deal about scope over the past 45 years, but without any proper definition of that notion other than the implicit assumption that scope is a property of (logical and other) operators. I may be wrong, of course. There may be some publication where scope is properly explained. If so, please let me know. But so far I haven’t found any.

The way out of this curious situation is far from simple. In fact, I will argue that scope is a necessary property of all lexical and logical predicates that take an embedded proposition or propositional function as one or more of their terms, which means that one has to break loose from a number of established basic logical notions if one wishes to provide an adequate and general definition of scope. Part of what my argument will show is that it is very hard (read: impossible) to escape from the conclusion that it makes a lot of sense to treat all logical and nonlogical operators as (abstract) predicates. For the moment, however, I will do what everybody has done so far: I will give you a feel of what scope is simply by giving a few examples. Consider the difference between (1a) and (1b) (I have italicised the surface representatives of the scope-bearing elements):

(1)     a.         One student did not pass the test.

—      b.         Not one student passed the test.

It is universally agreed that the difference in meaning between these two sentences consists in the different scopes of the operators one and not: in (1a) one takes scope over not and in (1b) it is the other way round. The difference is, of course, straightforwardly truth-conditional. Logical paraphrases make the difference explicit. For (1a): ‘there was one student s such that – it is not the case that – x passed the test’ and for (1b): ‘it is not the case that–  there was one student x such that – x passed the test’. It seems obvious that a proper theory of meaning will have to specify what scope is, and so far no theory of meaning has achieved that.

I have chosen the examples (1a) and (1b) because they are both unambiguous: there is hardly a way (1a) can be taken to have the meaning of (1b) or vice versa, no matter how one plays around with accent, intonation or context. Often, however, there is a ‘dominant’ and a ‘recessive’ reading. It is a standard observation that in a sentence pair like (2a) and (2b), the one can, though with some difficulty, have the meaning of the other under certain intonation patterns and/or in certain contexts, though the dominant meaning of (2a) is ‘there is nobody x here such that there are two languages l such that x knows l’ and of (2b): ‘there are two languages l such that there is nobody x such that x knows l’:

(2)     a.         Nobody here knows two languages.

—      b.         Two languages are known by nobody here.

That is, if in (2a) the NP two languages is given high pitch accent, it may have the meaning of (2b), and a similar scope-inverting effect can be observed when (2b) is read with low accent on two languages and high accent on nobody here.

One regularity strikes the observer immediately: there is a predominant trend to assign the highest scope to the first occurring operator in the surface sentence, the second-highest scope to the second occurring operator, etc. This is nicely illustrated in a sentence like:

(3)     John may not have been able to buy two books.

Here we identify four scope-bearing operators: the epistemic possibility operator may, the negation operator not, the agentive possibility operator be able, and the existential operator two. And indeed, the natural interpretation of (3) is: ‘it is possible that – it is not the case that – John was in a position to  bring about that – there were two books b such that – John bought b’. It is possible, under special intonation, to assign highest scope to two books: ‘there are two books such that John may not have been able to buy them’, but it is not possible to conjure up a reading where two books has some intermediate scope in between two other operators: it has either the highest or the lowest scope.

As I alluded to above, scope turns out not to be exclusively reserved for logical operators: scope differences are also observed with nonlogical expressions, as is clear from the following:

(4)    a.         Because of the rain I did not go out. (unambiguous)

—     b.         I did not go out(,) because of the rain. (ambiguous)

—     c.         Not because of the rain did I go out. (unambiguous)

Phrases like because of the rain are not dealt with in logic, but they occur in language and turn out to have scope there. So if we are looking for a general definition of scope, we will have to make sure that the definition covers nonlogical operators as well. (I’ll keep calling all scope-bearing elements operators, whether logical or nonlogical.) I will also call prepositional phrases (PPs) that have the structural status of because of the rain HIGH-PERIPHERAL PPs (HPPP), to distinguish them from PPs in other structural positions, such as in London in Harry lives in London, or about wildlife in Harry writes about wildlife, or on the other hand, as in On the other hand, Harry never writes about wildlife, or in addition, as in In addition, Harry writes about wildlife, etc. (In my Semantic Syntax, Blackwell, 1996, pp. 116–28, I distinguish at least six levels at which English adverbs can function at sentence level. PPs differ from adverbs in certain ways, but they too can  occupy positions of different status  in English sentences.) HPPPs are typically scope-bearing operators.

The examples (4a–c) again illustrate the tendency for operator scope to be reflected in the left-to-right order of the operators (or their representatives) in surface structure: in (4a), because of the rain precedes not, but in (4c) the latter precedes the former. I will henceforth call that tendency the Scope Ordering Constraint (SOC). SOC is not an absolute constraint, as has been observed by a great many authors, but I defend the thesis that it is a, probably universal, default constraint on the ordering of operators in surface structures. This default constraint can be overruled under certain conditions, which will have to be carefully charted in the hope that the violations of SOC will be explained by the interference of other systematic factors. One case where SOC is overruled is (4b), which is clearly ambiguous—an ambiguity that is naturally resolved by means of different intonation patterns. In one reading, (4b) means what (4a) means; in the other it has the meaning of (4c). In the meaning of (4a), because of the rain has the status of HPPP and  a comma is preferred between out and preferred. In the other reading, not takes scope over because of the rain, which is, therefore, not a HPPP. The difference between the two readings is, of course, truth-conditional and not a matter of pragmatics in whatever sense.

The difference is systematic. It is, for example, also found in pairs like:

(5)   a.         Every morning I read two poems. (unambiguous)

—    b.         I read two poems every morning. (ambiguous)

(6)   a.         For two years the sheriff jailed Robin Hood.  (unambiguous)

—    b.         The sheriff jailed Robin Hood for two years.  (ambiguous)

In (5a) and (6a), the PPs have the status of HPPP (I reckon every morning to be a PP). In (5b) and (6b), they either have HPPP status or they have lower scope: under two poems for (5b), and under … oh dear, under what for (6b)? … to be continued