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Unformatted text preview: Hayes Introductory Linguistics p. 263 OPERATORS AND SCOPE 12. Operators and scope in formal logic The idea of operators and scope was incorporated into linguistics from the field of formal logic, a branch of philosophy.94 Logicians express (certain aspects of) meaning with formulas like the following. For all x P is true of x x(P(x)) The meaning of the formula is, “for all x, P is true of x”. If we were applying this formula to a real-life situation, we might image a universe in which x represents only students in Linguistics 20, and P represents “has the flu”. The formula could then be interpreted as “Every student in Linguistics 20 has the flu.” In the formula, x is an operator, x is a variable, and P is a predicate (just like we saw with predicate-argument structure). To see the concept of scope, let us compare two formulae that are more complex. Here is the first one. I. For all x P is true of x implies that Q is true x(P(x)) Q Pursuing our real-life interpretation, we might suppose that Q means “the professor postpones the exam”. The symbol means “if … then”. The interpretation would then be “If every one of the students in Linguistics 20 has the flu, then the professor will postpone the exam.” Now consider a similar formula, with a different location for the right parenthesis. II. For all x P is true of x implies that Q is true x(P(x) Q)
94 You can study the basics of this field in UCLA’s Philosophy 31. Hayes Introductory Linguistics p. 264 With the parenthesis relocated, “for all” now covers the entire rest of the formula, rather than just P(x). Thus, in the real-life interpretation of the formula, this would be “For every student, if that student has the flu, then the professor will postpone the exam.” — in other words, the professor will postpone the exam even if there is just one case of the flu in the class. One can speak here of an operator having scope. In the first formula, the scope of the operator x is just P(x) (informally, “x has the flu”) whereas in the second formula the scope of the operator x is P(x) Q (informally, “if x has the flu, the professor will postpone the exam”). The operator x is of a particular kind, called a quantifier. It means “all” (symbol: inverted A). The other quantifier most often used in elementary logic is x, which means “at least one x” (inverted E, “exists”). In logic, these concepts are employed in the study of the principles of valid reasoning. For example, the formula ~x(P(x)) y(~P(y)) (which means “If it’s not so that P is true of all x, then there must be a y of which P is not true”) represents a case of valid reasoning. It is true irrespective of how we interpret the elements it contains. Over the centuries, logicians have provided mathematical proofs for a vast number of such formulae, thus providing a more solid basis for reasoning. 13. Operators and scope in language: some examples In linguistics, the focus is less on proofs of validity, and more on using logic to provide a precise and interpretable characterization of meaning. In fact, linguistic meaning is much richer than what can be expressed with the logic taught in beginning courses, and finding a rich enough formal system to characterize human language continues to be a research challenge for logicians and linguists alike. We can start by seeing that the logical notions of quantifier, scope, and variable are expressed fairly directly in English (or indeed in any other language). Here is an example: Every boy sang. Here, we have the following: Every boy x sang a kind of (restricted) universal quantifier (x, x a boy) a predicate Putting these together, we get something like: (x, x a boy) (x sang) Generally, linguists, just like logicians, put operators at the left of the domain over which they have scope; this is a matter of convenience and convention. So, for instance, a sentence like: Hayes Introductory Linguistics p. 265 Jane taught every student. would be expressed as: (x, x a student) (Jane taught x) In principle, we could integrate such expressions with the predicate-argument structure developed above, so that the meaning would appear like this: (x, x a student) (TEACH ( (Teacher Jane ) (Teachee x) ) For brevity (and to avoid unwanted complications), in what follows I will skip this step and simply place the quantifiers and variables into ordinary syntactic structure. 13.1 Pronouns as variables So far, we have treated pronouns as NP’s that refer to things. When a pronoun is coindexed with another NP, (Billi thinks hei is tall.) it is meant to refer to the some real-world thing as that NP. When a pronoun has its own distinct index (Well, hei’s gone), it is meant to refer to some real-world thing assumed to be identifiable by the real-world context. However, not all pronouns refer to things. The other use of pronouns is as the linguistic manifestation of logical variables. This can happens when there is a logical operator, such a quantified, elsewhere in the sentence. Consider the following sentence. Every boy thinks that he is smarter than average. The interesting reading here is the one where Fred thinks Fred is smarter than average, Bill thinks Bill is smarter than average, and so on. In this reading, the pronoun he does not refer to anyone. Rather, we can give it a sensible interpretation provided we set up a structure that has two variables, as follows. Every boy is the quantifier (x, x a boy) he is a bound variable (x) x thinks x is smarter than average is a complex predicate, with two variables Putting these together, we get the following structure: (x, x a boy) (x thinks x is smarter than average) Thus, the pronoun is not referential but rather is the linguistic means for expressing the second instances of the variable. (The first variable simply occurs in the syntactic location of the quantifier phrase; see rules below for how this can be derived). Hayes Introductory Linguistics p. 266 Observe now that the sentence under discussion, Every boy thinks that he is smarter than average, is ambiguous, because the pronoun he does not have to act as a bound variable. It can also be an ordinary pronoun, which can refer to some male person who happens to be under discussion. Thus, the other reading is as follows: (x, x a boy) (x thinks hei is smarter than average) where hei is an ordinary pronoun referring to someone in the ordinary way. Some terminology: we say that in the first reading, he acts as a variable that is bound by the quantifier. In sum, the pronouns of a language play two roles: they either simply refer to other entities, or they act as bound variables. What is the mechanism whereby pronouns get interpreted as bound variables? As a rough approximation, we can make use of the discussion of pronoun reference from earlier in this chapter. There, we studied rules that assign indices to pronouns and their antecedents, to express ordinary coreference and non-coreference. The extension of this idea in the present context is this: if a pronoun gets coindexed with a quantified NP, then the relationship is then semantically interpreted not as coreference, but as an operator-variable relationship. Thus, for instance, the rules of anaphoric interpretation permits the following coindexation for the NP’s in the sentence we are working with (he is not the clausemate of every boy, so Regular Pronoun Interpretation is satisfied; and he does not c-command every boy, so Full Noun Phrase Interpretation is satisfied). S VP S ¯ S VP AP PP NPi NP | | Art N V Comp Pro V A P N | | | | | | | | | Every boy thinks that he is smarter than average.95 NPi This tree is not compliant with our phrase structure rules. The additional rule needed is pretty straightforward: AP A (PP). All comparative adjectives (“X-er”) can take a PP with than. 95 Hayes Introductory Linguistics p. 267 Because every boy is a quantified NP, this must be further translated to (x, x a boy) (x thinks x is smarter than average) More specifically, when a quantified NP is logically interpreted as an operator-variable combination, any pronouns coindexed with it must be assigned the same variable. The following match-up illustrates this: [ Every boy ]i thinks that [ he ]i is smarter than average. (x, x a boy) (x thinks x is smarter than average) 14. Logical form It is time now to integrate the discussion into a general approach to semantics. Note that the following is just one (well represented) viewpoint among many. The code idea is that the rules of the semantics create from syntactic representation96 a separate representation of the sentence’s meaning (or, in cases of ambiguity, more than one representation). Such a semantic representation is often called the logical form of a sentence. Logical form is meant to be specifically linguistic in character; it only represents the contribution of language to meaning and is certainly not the “language of thought”, if such a thing exists—our thoughts involve all sorts of non-linguistic inferences and associations, in addition to language. Here are some of the steps that would be needed to construct a logical form from a syntactic structure. As some (probably early) stage we would establish the possible references of pronouns and reflexives through the assignment of indices, using the rules of Reflexive Interpretation, Regular Pronoun Interpretation, and Full Noun Phrase Interpretation, given earlier in this chapter. Another step would be to convert quantified NP’s into operator-variable pairs, to indicate scope, as described in the previous section. Yet another step would be to establish precisely “who is doing what to whom” by replacing the syntactic tree with an appropriate predicate-argument structure. . Here are a couple of examples of how all this might work. In Every boy thinks he is smart the rule of Regular Pronoun Interpretation would (as one of its options) coindex every boy and he, thus:
Most likely, from surface structure. The traces left by movement rules generally make it possible to cover the effects of deep structure on meaning; they serve as a “memory” for the location of phrases at the deep structure level.
96 Hayes Introductory Linguistics p. 268 [ Every boy ]i thinks [ he ]i is smart Next, the quantified NP every boy would be converted to an operator-variable combination. Since he is coindexed with every boy, it is a assigned the same variable x: (x, x a boy) ( x thinks x is smart ) Then the whole expression could be converted to a predicate-argument structure, yielding a logical form: (x, x a boy)((THINK((Thinker x), (Proposition SMART((Assessee x))))) For the sentence Mary seems to like every boy, the same processes would yield: SEEM (Proposition (x, x a boy)( LIKE ( (Liker Mary ) (Lik-ee x ) ) ) ) ) This is, of course, only an outline scheme. In the pages below, I’ll discuss briefly the rules for converting quantified NP’s into operator-variable pairs, which will flesh out the scheme a bit. However, we will henceforth skip the step of creating predicate-argument structure from syntax. 15. Sentences with two operators Many sentences contain two operators. When this happens, the two operators often interact with each other. For example, speaking of an archery tournament, we could say: At least two arrows hit every target. This sentence is ambiguous, in the following way. Suppose that the archers present are so impoverished that between them they could bring a total of only five arrows to the tournament. Thus, each arrow has to be used repeatedly. Suppose further that the archers used a total of five targets. Here is one reading: at least two of the arrows (perhaps the straightest ones) were used so successfully that during the course of the tournament they penetrated every one of the five targets. Hayes Introductory Linguistics p. 269 In the other reading, we would find that inspecting the targets at the end of the tournament, each has at least two holes in it. The two readings of At least two arrows hit every target can be summarized as follows: Hayes Introductory Linguistics p. 270 ‘There were at least two arrows such that they hit every target.’ True only of the first diagram. ‘For every target, it is the case that at least two arrows hit it.’ True of both diagrams. The ambiguity we have just seen is within the capacity of the system we are developing. To handle it, we use two operators. The word every is a real-language version of the universal quantifier x. The phrase at least two is not an operator that is normally taught in introductory logic, but I think it is intuitively clear that it is an operator of some kind. Thus, by putting the operators in the right structural locations, we can characterize the ambiguity. a. ( For at least two x, x an arrow ) ( ( for every y, y a target ) ( x hit y ) ) ) ‘There were at least two arrows such that they hit every target.’ b. ( For every y, y a target ) ( ( for at least two x, x an arrow ) ( x hit y ) ) ) ‘For every target, it is the case that at least two arrows hit it.’ This is an example of a scope ambiguity. In (a), the scope of the operator ( For at least two x, x an arrow ) is ( ( for every y, y a target ) ( x hit y ) ). In (b), the scope of the operator ( For every y, y a target ) is ( ( for at least two x, x an arrow ) ( x hit y ) ). Another way of saying that same thing is that in (a), ( For at least two x, x an arrow ) takes scope over ( for every y, y a target ), because ( for every y, y a target ) is inside the scope of ( For at least two x, x an arrow ). In (b), ( for every y, y a target ) takes scope over ( For at least two x, x an arrow ). 15.1 The local “universe of discourse” As you can see above, in language operators often consist of two parts, one the quantifying expression itself (at least two, every), and the other an expression of the set of entities (arrows, targets) being quantified over. The latter set is grounded in the local “universe of discourse”— when I say every target, I mean, “every target in the set of targets relevant to the conversation we are having”; hence, in the present context, every target that was present at the archery tournament. Clearly, speakers interpret quantifiers making use of their real-world knowledge, which permits them to infer the set of relevant targets (or whatever) from the context. 16. Operator scope in multiclause sentences Operators can have scope not just over other operators, but over particular clauses in a sentence that has more than one clause. These cases are of special interest for us because thy can be used to show the close relationship of operator scope with syntactic structure. Here is an example. The sentence at hand is: Sue shouted [ for us to give water to each runner ] Hayes Introductory Linguistics p. 271 We need briefly to cover the syntax here. In one commonly-used analysis, for us to give water to each runner is an S, for is a Comp, us is the NP subject of for us to give water to each runner ¯ and to is a particular sort of Aux used only in verbal infinitives. Shout is a verb that subcategorizes for this particular kind of S (often called an “infinitival clause”, since to give is ¯ the infinitive form of give) Here is the proposed parse: Now, let us consider the meanings at hand. The easy reading here is the one where Sue shouts just once, at the beginning of a marathon, “Hey! Give water to each runner!”. In this reading, the scope of each is the embedded clause that reports what Sue shouted. Here is a possible logical structure for this reading: Sue shouted ( ( for each x, x a runner ) ( for us to give water to x ) ) In the other reading, which is a bit harder to get, Sue is a more hands-on manager, who drives around monitoring the individual water stations. Here is a context: We were covering the water station at Mile 23. By this point, the runners were fairly far apart from one another. Sue, watching continuously, shouted for us to give water to each runner, and every time we heard this shout, we complied.. I believe this could fairly be given the reading: ( For each x, x a runner ) (Sue shouted ( for us to give water to x ) ) In other words, for each passing runner, there was a “shouting event”, in which Sue directed the workers to give that runner some water. Hayes Introductory Linguistics p. 272 Some terminology: we can say that in the first reading, that is Sue shouted ( ( for each x, x a runner ) ( for us to give water to x ) ) the operator each has narrow scope, namely the embedded clause; whereas in the second reading ( For each x, x a runner ) (Sue shouted ( for us to give water to x ) ) the operator each has wide scope—here, scope over the whole sentence. The general point that here we have a sentence that has just one variable, but it is ambiguous. This is because the sentence has two clauses, and thus two locations for the operator to go. 17. Creating operator-variable pairs from quantifiers in logical form We can now consider what is needed to derive the logical form of quantified sentences. We know, up front, that the rules need to have some flexibility, because of sentences like Sue shouted for us to give water to each runner, where a single syntactic structure yields two different interpretations for quantification. We first need a rule that translates quantified NP’s into operators. Quantifier Translation Replace [ every N ]NP [ some N ]NP … with with [ for every x, x an N]NP [ for some x, x an N]NP and similarly for other quantified expressions. If the variable x is already in use, use y instead; etc. The other rule we need is more dramatic: it lets us pick the clause over which the operator will have scope, moves it there, and creates a variable in the location that the moved NP left behind. Quantifier Raising Left-adjoin a quantified NP to S, leaving behind a variable in its original location. This rule has an undefined term in it, adjunction¸ which is defined as follows: Hayes Introductory Linguistics p. 273 Left-Adjunction Given a constituent A, containing a B, and (optionally) C, the mother of A: Form a new constituent, which is: has the same node label as A has as its daughter nodes a copy of B, followed by A if A was the daughter of C, the new constituent becomes the daughter of C Here are two simple cases of left adjunction. left adjoin B to A (A is not the daughter of any node) left adjoin B to A (A is daughter of C) The purpose of left adjunction here is simply to provide a slot in which the logical operator can reside. 18. Deriving distinct meanings with Quantifier Raising Let us return to Sue shouted for us to give water to each runner, whose surface structure is repeated below. Hayes Introductory Linguistics p. 274 First applying Quantifier Translation to each runner, we get the following. A triangle is used to avoid worrying about the inner details of the quantifiers. Next, we note that the clue to the multiple meanings is that the sentence has two clauses, hence two S nodes that the Quantifier Raising can adjoin each runner to. If we pick the lower S, adjunction will look like this: Hayes Introductory Linguistics p. 275 Adjoin here Expression to be moved Inserting the new S node, and rearranging the tree in the way required, we get the following: New copy (adjunction) Old copy moved quantifier variable inserted by Quantifier Raising Note the variable: it is the logical place marker formerly occupied by each runner, and it is bound (shown by the shared index x) by the raised operator each runner. This yields a logical form for one of the meanings, that is, a single act of shouting, telling us to attend to all of the runners. This is the reading of (11)-(12) above. If we pick the upper clause, adjunction will look like this: Hayes Introductory Linguistics p. 276 Adjoin to this node Quantifier to be adjoined Study Exercise #33 Show the final output of the derivation. Hayes Introductory Linguistics p. 277 Answer to Study Exercise #33 ———————————————————————————————————— 19. Logical form in sentences with two quantifiers Let us now return to the topic of section 15, namely sentences that include two quantifiers. Suppose we start with a simplified version of our arrow-target sentence: Many arrows hit every target. This sentence is ambiguous, and could mean either “Many were the arrows that hit every target”; or “For every target, many arrows hit it.” The syntactic surface structure (as well as deep structure) would be as shown: Hayes Introductory Linguistics p. 278 We first translate the NP with quantifiers into appropriate operators, with the rule of Quantifier Translation. Note that it is crucial to use different variables (here, x and y) for the different noun phrases. Although the order in which we perform the operations turns out not to matter here, we can arbitrarily chose first to left-adjoin many x, x arrows to the sentence, as follows: The result has a new S node, copying the original one, and the moved quantifier is the sister of the original S: In the next step, we need to apply the same rule of Quantifier Raising again, this time to the quantified expression every y, y a target, which likewise is a quantified NP. Assuming (again arbitrarily) that it left-adjoins to the highest available S node, the application would look like this: Hayes Introductory Linguistics p. 279 Here is the result: Note that a second variable, y, now appears in the clause. This is the reading we wanted: “For every target, many were the arrows that hit it”. Study Exercise #34 Derive the other reading. Answer to Study Exercise #34 Syntactic structure: Hayes Introductory Linguistics p. 280 Output of Quantified Translation. I also show an arrow that indicates the application of Quantifier Raising to the quantified expression every y, y a target. The structure that results is given below. Next, we apply Quantifier Raising to many x, x arrows. This is shown with the arrow below: Hayes Introductory Linguistics p. 281 The final structure that results is shown below: ———————————————————————————————————— 20. Wh-phrases are operators Earlier in these readings, I mentioned that Wh- questions can differ in the scope of the Wh- phrase, giving the following example: [ What song ] can Sue imagine that Bill sang ___? Sue can imagine [ what song ] Bill sang ___ We can now express this idea more precisely by giving these sentences logical forms similar to the quantifier sentences above. The idea is that wh- phrases are logical operators, which are requests for the listener to fill in the missing information that the variable stands for. Thus we might have the following two logical forms: Hayes Introductory Linguistics p. 282 You can see that the syntactic transformation of Wh- Movement is a kind of observable, syntactic analogue of Quantifier Raising, and has the function of placing the wh- phrase where it bears its logical scope. In languages where Wh- phrases syntactically remain in situ, things will work differently. Here, Quantifier Raising must apply to wh- phrases, so that their scope will be correctly expressed in logical form. Here is an example from Mandarin Chinese, an in-situ language: [S tʂáŋsán tsái [S lìsɯ̂ Zhangsan guess Lisi 1 1 Zhangsan 1 cai ɕìhwán ʂěi ] like who 34 31 2 Li si xi-huan shui This sentence is ambiguous. It can mean “Who does Zhangsan guess that Lisi likes?” This meaning involves raising the Wh- phrase to adjoin to the highest S in logical form. It can also mean “Zhangsan guessed who Lisi likes.” This meaning involves raising the Wh- phrase only to the lower S in logical form.97 We saw above another area where syntax mimics semantics: the use of pronouns as bound variables, as in section 13.1. Here, interpretive rules must convert these pronouns to variables in logical form. These rules are usually optional, resulting the ambiguity of sentences like Every boy thinks that he is smarter than average. For completeness, here is the interpretive rule that could perform this function. 97 Thanks to UCLA graduate students Kristine Yu and Grace Kuo for constructing this example for me. Hayes Introductory Linguistics p. 283 Pronoun-to-Variable Conversion When a NP is converted to an quantifier by Quantifier Translation (p. 272), convert any pronoun coindexed with the NP to the same variable that the quantifier specifies. 21. Summary of operators, variables, and scope Constructions with operators and variables are perhaps the most intricate of semantic phenomena. A basic analysis of them is possible using the rules of Quantifier Translation and Quantifier Raising. These rules apply during the creation of logical form, a hypothesized linguistic level that explicitly characterize linguistic aspects of meaning. Scope differences can be of various kinds: a single operator can be raised to different levels (as in Sue shouted for us to give water to each runner), or there can be two operators that vary in their scope relative to each other (as in At least two arrows hit every target.). Pronouns coindexed with quantifier NP’s often turn into additional variables in logical form (as in Every boy thinks that he is smarter than average) The constructions created in logical form by Quantifier Raising are abstract and not directly observable. Yet they are mimicked by observable constructions in language: Wh- phrases are a sort of quantifier, which in languages like English really does move to the appropriate scope location in surface structure. ...
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