Scope — 3

by pieterseuren

Let’s recapitulate. I began by stating that the notion of scope, though universally acknowledged and used, has so far remained undefined, whether in logic, where it originated, or in semantics. All one finds is the statement, or implication, that scope is a property of logical operators (constants). I then decided to follow custom and rely on some intuitive understanding of the notion based on examples. I showed that scope phenomena extend way beyond the logical elements in sentences and that nonlogical lexical elements, such as high-peripheral adverbials or complement-taking verbs such as cause or allow, interact with each other and with logical operators in exactly the way logical operators do among themselves and I concluded that these nonlogical elements have scope the same way as the logical operators do. I argued that at least logical scope differences are best represented as hierarchical differences in the tree structures of some logico-semantic language that may be taken to represent meanings, the language of semantic analysis (SA) and I concluded that the nonlogical scope-bearing elements in sentences should be represented the same way, thus blurring the distinction between purely logical representations and semantic representations in a more general sense. This gave rise to the idea, based on McCawley’s work, that it pays to treat all lexical items with semantic content as predicates at SA-level, letting the grammar define their surface category. We have seen that in many languages what are causative predicates at SA-level regularly manifest themselves as morphological causal affixes in surface verbs (in English often as zero morphemes, as in verbs like drop, close, burn, explode), but that nevertheless some scope-bearing adverbials can take the complement of SA causative predicates as their scope, even when the surface causative verb has been fully lexicalized, provided the scope-bearing adverbial is in right-peripheral (clause-final) position. All this has given rise to a whole lot of questions, some of which I will discuss now.

I want to focus first on the question of how to define the notion of scope, because I think we are a bit closer to an answer now. Given the assumption that all semantically significant surface categories are predicates at SA-level, we make a distinction, at SA-level, between abstract predicates and matrix predicates (what I call “matrix” here was called “nucleus” in my 1969 PhD-thesis Operators and Nucleus). Matrix predicates occur as the main predicate (verb or adjective) in surface sentences or clauses. Abstract predicates are incorporated by the grammar into the main clause or sentence in a bottom-up cyclic way (this is what I usually call “matrix greed”). They end up in surface structure as supplementary elements under some surface categorial label, often of an adverbial kind. Logical operators are SA-level abstract predicates. Most surface sentences have incorporated one or more abstract predicates, which are organized at SA-level in a hierarchical scope structure. Thus, in a sentence like Last year, most tourists preferred Italy, the surface elements last year, most, and the past tense suffix -ed represent abstract SA-predicates, whereas prefer is the matrix predicate of the tenseless matrix structure prefer(x, Italy). At SA-level, every abstract predicate in an SA tree structure commands at least one matrix line ending in the bare matrix-S. Matrix predicates that take a complement clause in subject or object position start a new matrix line for the embedded clause. In Figure 3, the matrix line of the sentence All protestors studied law in Paris is indicated by lines in bold print.


It is interesting to see in broad outline how the grammar of English transforms the SA-structure in Figure 3 into the surface structure All protesters sudied law in Paris shown in Figure 4 in a fully automatic, algorithmic way. This is a bit technical, but if you take the trouble to follow me you will see that the machinery works beautifully (technical details and definitions are given in my Semantic Syntax, Blackwell, Oxford, 1996). My former assistant Henk Schotel turned the apparatus as applied to English into a computer program.

The grammar starts at the lowest S or NP/S structure, in this case the matrix structure NP/S4, and subsequently goes through all higher S or NP/S structures in a cyclic way. This has been known since the mid-1960s as the transformational cycle. The predicate of each S or NP/S structure is lexically defined for the rule or rules it induces. It assumes its surface category as the cyclic mechanism passes through its S or NP/S. On the NP/S4-cycle, no rule is activated because the predicate study induces no cyclic rule. Yet since the cyclic mechanism passes through its NP/S, study is relabelled Verb (V). The predicate past of the NP/Ss-cycle induces two rules, first Subject Raising (SR), then Lowering (L). SR raises the subject term of the lower S or NP/S, in this case NP[x] in NP/S4, to the subject position of the higher S or NP/S. As a result, what remains of the matrix structure is automatically relabelled Verb Phrase (VP) and moved to the right. VPs are thus S-structures bereft of their subject term. Next, still on the NP/S3-cycle, Pred[past] is incorporated into the matrix-VP, where it ‘lands’ on Verb[study], as Affix[past], forming one cluster with Verb[study]. The node NP/S3 now dominates NP[x] followed by VP[Verb[Affix[past] – Verb[study]] – NP[law]]. Then, Pred[in Paris] of NP/S1 is lowered to the peripheral position at the far right of NP/S3 and relabelled Preposition Phrase (PP), giving the matrix structure

NP/S[NP[x] – VP[V[V[study]+Affix[past]] – NP[law]] – PP[Prep[in] N[Paris]]]

which stands semantically for ‘the set of x such that x studied law in Paris, or, more simply, ‘the set of those who studied law in Paris’. Node NP/S1 has become vacuous and is deleted. Then, Pred[allx] of S0 incorporates the object term NP/S[protestor x] according to the common cyclic rule of OBJECT INCORPORATION and thus gives the complex predicate [Pred[all]+NP/S[protestor+x]]. This complex predicate is then lowered onto the variable x in the matrix structure, giving the following string in terms of labelled bracketing, corresponding to the tree structure in Figure 4:

S[NP[Det[all] N[protestor]] – VP[V[V[study]+Affix[past]] – NP[law]] – PP[Prep[in] N[Paris]]]


After this cyclic machinery, each language has a set of postcyclic rules to prepare the structure for the morphology, if necessary, and for possible further cosmetics. Again, for technical details and ample illustration taken from a variety of languages, see my Semantic Syntax (Blackwell, Oxford, 1996). I think that if a metric were to be applied for the ratio between data coverage and generalizations, this system would not score too badly.

But back to the notion of scope. Some abstract predicates, in particular the binary logical connectives and and or, take two parallel matrix lines, which means that such binary connectives have a twin scope. Most languages lower such twin scope predicates to the position between the two matrix constituents. Twin scope also occurs with those matrix predicates that can take an S-structure in both subject and object position. Examples are suggest, mean, prove, entail, etc., as in That the butler had blood on his hands proved that he was the murderer. Here, the lexical matrix verb prove occupies the normal position of the finite verb in English sentences. Curiously, all lexical matrix verbs that take both a sentential subject and a sentential object term are subject to the rule that the sentential subject term is factive in the sense that the whole sentence presupposes that what is said in the subject-S is, in fact, true. Why this should be so is unknown.

Given all this, we can now tentatively define the notion of scope in a simple and general way as follows:

– The scope of an SA-predicate P is the matrix-line sentential argument(s) of P.

Whether this definition will stand up to future scrutiny is a question I cannot answer now, but, as far as I can see, it does stand up to all past scrutiny. An important corollary is that the logical operators are treated as abstract predicates in the semantics of natural language, just as was proposed by McCawley during the late 1960s. The logical operators are thus lexical items like the others and belong in the lexicon of a language. They distinguish themselves from nonlogical predicates in that (a) they are, probably in all languages, abstract predicates and thus do not occur as matrix verbs, and (b) their (core) meanings are definable in the mathematical terms of the human cognitive version of set theory. This is, in fact, the perspective in which I have developed my theory of the natural logic of language and cognition over the past 15 years.

Another corollary is that definite NP terms do not have scope. This has to be said because, in the wake of Russell’s (1905) analysis of definite descriptions as existentially quantified terms, it has become common to treat definite determiners such as English the, that, etc. as quantifiers. I will not expand too much on this here, but I consider that to be a fatal distortion of the reality of language. For one thing, definite determiners do not participate in the game of scope differences: there is no semantic difference between, for example, English I don’t know that man and German Ich kenne diesen Mann nicht, despite the fact that in English the negation precedes but in German it follows the definite object term. By contrast, English I don’t know all men corresponds to German Ich kenne nicht alle Männer, with the negation not in final position but preceding the universally quantified object term, as in English. But German also has Ich kenne alle Männer nicht, meaning ‘for all men x it is not so that I know x’, which is impossible in English: *I know all men not is ungrammatical in modern English, which, but for a few well-defined exceptions, wants the negation to be constructed with the finite verb form. Crucially, those who want to treat definite determiners as quantifiers have not provided a semantic definition for them in the terms of a binary higher order predicate, as we have done for the standard quantifiers. … to be continued