“All doors will not open.” — operator, Amtrak Northeast Regional train, Charlottesville
In language, scope ambiguities are one of the trickiest parts of semantic theory. Semantic meaning is famously said to be determined compositionally: the meaning of a larger sentence is determined by the meanings of its smaller parts, as well as by the way these smaller parts are assembled into a whole. Even so, there can be interactions between these parts, in the sense that certain words can exert control over other words. When one word influences how another is interpreted, it is said to hold that word in its scope.
Every king admires himself. 
In this situation, the reflexive pronoun himself is given meaning by the separate noun king, which holds himself in its scope.
Let’s consider another example:
Puck didn’t solve one problem. 
What does this mean? It actually depends on how scope is assigned. It’s ambiguous. Indeed, this sentence, due to Fromkin, et. al.  (with paraphrases partially my own), carries two distinct meanings:
- Puck didn’t solve one problem.
- It is not the case that Puck solved one problem. He solved none, or perhaps two or three.
- There is one particular problem that Puck didn’t solve. He may have solved other problems, though, perhaps even exactly one other problem.
The key difference here, as we’ll see, is whether the scope of the negation contains, or does not contain, one problem.
In this article, following the pattern of the last, I’ll begin by exploring some examples of delightful scope ambiguities I’ve come across in the literature. I’ll also attempt to explain some of the delicate theory surrounding these constructions. In the second part, we’ll play some games.
I’ll begin with an important technical point. The possible scope readings permissible to a sentence can be constrained by so-called c-command relations. Roughly speaking, one word is said to c-command another when the syntax subtree formed by taking the first word’s parent node as root contains the second word. Words higher in the syntax tree tend to c-command words lower in the tree. For example, in
Puck didn’t solve one problem,
the negation not c-commands the noun phrase one problem.
In a scope ambiguity involving two words with the property that one c-commands the other, one of the two scope readings is held to be more natural than the other. Indeed, the reading in which the c-commanding word outscopes the other is called the linear, or surface scope reading; the opposite reading is called the inverse scope reading. We’ll soon see that an inverse scope reading is not always permissible.
The precise meaning of the word outscope – that is, what scope means at all – is best understood when one writes a sentence’s various meanings in first-order logic. First, one should rather try to grasp this meaning intuitively. Later, we’ll see precise logic at work.
Negation and determiner phrase
Many scope ambiguities, like Puck’s above, involve interchanging the scope of a negation with that of a determiner phrase. Consider the following two examples (and paraphrases), again by Fromkin :
- “Richard III didn’t murder fewer than three squires.”
- “It is not the case that Richard III murdered fewer than three squires; he murdered at least three squires.”
- *”There are fewer than three squires whom R. III didn’t murder.”
- “Fewer than three squires didn’t obey Richard III.”
- *”It is not the case that fewer than three squires obeyed R. III; at least three squires obeyed him.”
- “There are fewer than three squires who didn’t obey R. III.”
In sentences 2 and 3, the negation alternates in scope with the determiner phrase fewer than three squires.
In sentence 2, the surface scope reading is that in which not outscopes fewer than three; in sentence 3, the surface scope reading is that in which fewer than three outscopes not. (This can be seen by observing the sentences’ syntax trees.) Fromkin, et. al. place stars next to paraphrases 2.B. and 3.A., indicating that the meanings suggested by these paraphrases — the inverse scope readings — aren’t actually furnished by the English language. Indeed, some scope ambiguities aren’t really ambiguous in the sense that the alternate (in this case the inverse) among two meanings is so ridiculous that no English speaker would acknowledge it as legitimate.
Fromkin, et. al  provide a sophisticated account of the circumstances under which an inverse scope reading is permissible. (In this case, it has to do with the so-called decreasingness of the determiner fewer than three .) The literature only takes this further (I can only offer the excellent paper ). While this question is rich and fascinating, I won’t take it up. My interest lies not so much in which readings are acceptable to English speakers as in how many readings we can create, acceptable or not. We’ll see more of this in the second part.
In the sentence “All doors will not open”, above, the hapless train operator utters a sentence in which the inverse scope reading is the intended one. (We hope.)
Verb and determiner phrase
A second class of scope ambiguities I’ve found seems to involve, roughly, the interchanging of the scopes of a determiner and a verb. I lack the sophistication required to describe this class adequately; I only refer the reader to Park . Consider the following sentence, adapted from a sentence originally due to Hendriks (1993) and considered by Park :
Fred believes that a mathematician wrote “Through the looking glass.”
I’ll offer paraphrases. For the reader’s benefit, I’ll also attempt to write each scope reading in first-order logic.
- Fred believes that a mathematician wrote “Through the looking glass.”
- believe(F, ∃u[math(u) ∧ write(u, L)])
According to Fred’s belief, there exists some mathematician such that that this mathematician wrote “Through the looking glass.”
- ∃u[math(u) ∧ believe(F, write(u, L))]
There exists some person, a mathematician, such that Fred believes that this person wrote “Through the looking glass.”
- believe(F, ∃u[math(u) ∧ write(u, L)])
In the second scope reading, the determiner phrase a mathematician attains inverse scope over the verb believes.
These readings don’t in fact amount to the same thing. Reading 4.A. could be true while 4.B. remained false, if, say, the mathematician featured in Fred’s belief did not really exist. On the other hand, if, say, Fred believed that a certain person wrote “Through the looking glass” while failing to realize that this person was a mathematician, then reading 4.B. would be true while reading 4.A. remained false.
I’ll offer another, and slightly subtler, example. Consider the following sentence, again adapted from Park :
Fred claims that every schoolboy is intelligent.
I’ll offer paraphrases and first-order logic expressions:
- Fred claims that every schoolboy is intelligent.
- claim(F, ∀v[boy(v) → intelligent(v)])
According to Fred’s claim, each schoolboy is intelligent.
- ∀v[boy(v) → claim(F, intelligent(v))]
For each schoolboy which exists, Fred claims that this schoolboy is intelligent.
- claim(F, ∀v[boy(v) → intelligent(v)])
In the second reading, the determiner phrase every schoolboy attains inverse scope over the verb claims. These readings are also independent from each other, a fact whose details I’ll leave to the reader.
Two or more determiner phrases
A large class of scope ambiguities arises from interchanging the scopes of two determiner phrases in some sentence. Consider the following example and paraphrases, from Fromkin :
- “Two fairies have talked with every Athenian.”
- “There are two fairies such that each has talked with every Athenian.”
- “For every Athenian, there are two possibly different fairies who have talked with him/her.”
Here, the determiner phrase two fairies c-commands the determiner phrase every Athenian, and yet the two phrases freely alternate in scope. Sentence 6.B. paraphrases the inverse scope reading. These sentences don’t amount to the same thing. Sentence 6.B. could be true while sentence 6.B. remained false, if, say, each Athenian were visited by both members of precisely one among a collection of pairwise disjoint pairs of fairies. In that case, (using the surface scope reading) no fairies would have talked with every Athenian.
Sentences with two statements of quantity can behave in analogous ways:
Three Frenchmen visited five Russians. 
Barring the so-called cumulative reading — under which this sentence would be true of an encounter which took place between two groups (see the comments to this piece) — this sentence has two standard, ambiguous scope readings. This readings are also semantically independent of each other, in the above sense that one can construct situations for which any one holds true while the other does not. For example, consider the following two situations:
- Three Frenchmen visited five Russians.
- There exist three Frenchmen such that each of these three Frenchmen visited each member of precisely one among three pairwise disjoint collections of five Russians.
- Each member of a collection of five Russians was visited by each member of precisely one among five pairwise disjoint collections of three Frenchmen.
In situation 7.A., the surface scope reading holds while the inverse reading does not; in situation 7.B., the inverse scope reading holds while the surface scope reading does not.
Park studies still wilder examples. In the following sentences, collected by Park, three distinct determiner phrases compete for scope. Park, and those he cites, claim that not all possible readings are available in English (see the comments above). Nonetheless, the number becomes quite high.
- Every representative of a company saw most samples.
- Some student will investigate two dialects of every language. 
Very generally, it appears that a sentence containing multiple determiner phrases must assemble these phrases into one among the many possible partial orders admissible on them. This is to say that in any logical expression representing such a sentence, the relation outscopes defines a reflexive, transitive, and anti-symmetric relation on the set of all determiner phrases. There are a number of such orderings on any nontrivial set.
In common language, this means that in any scope reading of a sentence containing multiple determiner phrases, while certain pairs of determiner phrases might be such that one outscopes the other, there may be other pairs whose elements are parallel, or between which no relation exists. (We can call such determiner phrases incomparable.) We will see examples of this below.
I’ve been unable to find, in the literature, corroboration for the possibility that a sentence’s scope ordering may represent a partial, as opposed to a total, order. Perhaps the examples I present below get something wrong.
In any case, this situation presents a number of difficulties. For one, we must be prepared to ask which of a sentence’s possible readings (that is, partial orders) are available in the English, in the sense discussed by Park. A further difficulty arises. Among the available readings, which are independent, in the sense of freedom from semantic entailment described above? Certain readings will resist the construction of situations for which that reading holds and no others do.
I’ll now see how far I can stretch English’s capacity to bear scope ambiguities. As I’ve mentioned, I will not seek to determine which of a sentence’s scope readings are practically comprehensible as English interpretations (most of them won’t be). I’m interested only in whether they logically represent meanings of the sentence.
Game 1: Construct a family of sentences 1, …, n, … such that the nth sentence has n+1 mutually independent scope readings.
Solution: Consider a collection of people P1, P2, …, Pn, … and the family of sentences:
- A mathematician wrote “Through the looking glass.”
- P1 believes that a mathematician wrote “Through the looking glass.”
- P2 believes that P1 believes that a mathematician wrote “Through the looking glass.”
- Pn believes that … P2 believes that P1 believes that a mathematician wrote “Through the looking glass.”
This set of sentences exploits the verb and determiner phrase class of scope ambiguities, which I’ve found particularly amenable to linear growth. In short, in any given member of this family the determiner phrase “a mathematician” gradually escapes from a nested set of beliefs and into reality (see also Park , pp. 27-28). For the sake of example, I’ll offer first-order logic expressions for each of sentence 3’s three possible scope readings:
- P2 believes that P1 believes that a mathematician wrote “Through the looking glass.”
- believe(P2, believe(P1, ∃u[math(u) ∧ write(u, L)]))
- believe(P2, ∃u[math(u) ∧ believe(P1, write(u, L))])
- ∃u[math(u) ∧ believe(P2, believe(P1, write(u, L)))]
Each of this family’s sentences’ various scope readings are semantically independent. Consider an arbitrary sentence n and order its n+1 scope readings by preservation of c-command relations, as above, and pick an arbitrary scope reading, say the ith one. I’ll prove that this ith reading implies no others.
For this, it will suffice to construct a situation about which this ith scope reading, and none of the others, holds, and this I’ll do now. I’ll use a trick of notation to make the proof’s explication more elegant: I’ll use the nonsensical “P(n+1) believes” to mean “it is true that”. Suppose now that this ith scope reading’s logical expression is true, in the sense that we have a nested chain of beliefs from Pn+1, …, P1 according to which, in particular, Pi believes that the person present in P1‘s belief is a mathematician. Suppose now in addition that, again according to this nested chain of beliefs, the Pj for j < i fail to acknowledge that this believed writer is a mathematician that the Pj for j > i do not believe that this mathematician exists. In this intricate situation, the ith reading holds and none of the others do.
Game 2: Construct a family of sentences 1, …, n, … such that the nth sentence has 2n mutually independent scope readings.
Solution: Consider a collection of disjoint groups of people A1, B1, A2, B2, …, An, Bn, … and the following sentences:
- Two A1s visited two B1s.
- Two A1s visited two B1s and two A2s visited two B2s.
- Two A1s visited two B1s and two A2s visited two B2s and … two Ans visited two Bns.
This family operates through a somewhat trivial n-wise conjunction of clauses which each, individually, feature a two-way scope ambiguity. Each among sentence n‘s n clauses may, independently, take either surface or inverted scope. There are as many scope readings as there are binary sequences of n digits.
This family also boasts the property that each of its sentences’ scope readings are independent. Indeed, consider a sentence n and an arbitrary scope reading d represented, say, by the n binary digits d1, … , dn. To construct a situation in which this and no other scope reading holds, we must simply suppose that something like 7.A. applies to all those groups Ai, Bi such that di = 0 and something like 7.B. applies to all those groups Ai, Bi such that di = 1 (for all i = 1, …, n). In such a situation, clearly d holds; conversely, any different reading d’ = d’1, … , d’n where di ≠ d’i, say, fails to hold because its failure on the individual pair Ai, Bi takes down the whole conjunction.
There is no scope relation between the determiner phrases two Ais and two Ajs for i ≠ j. These phrases, in other words, are incomparable. Though this behavior is interesting, it’s achieved in a somewhat trivial way. To distinguish this phenomenon from a more interesting sort, we’ll say that two determiner phrases in a given scope reading are non-trivially incomparable if, between them, no scope relation exists and yet a c-command relation does exist.
Game 3: Construct a family of sentences 1, …, n, …, such that the nth sentence features “complicated behavior” loosely defined—such as the presence of many independent scope readings, themselves featuring many non-trivially incomparable pairs.
Solution: Consider a collection of disjoint groups of people G1, G2, …, Gn, … and the following sentences (with embeddings bracketed to resolve syntactic ambiguity):
- There are exactly two G1s.
- There are exactly two G2s [who talked to exactly two G1s].
- There are exactly two Gns [who talked to exactly two G(n-1)s [ … [who talked to exactly two G1s] … ]].
In pursuit of the somewhat contrived task put forward by this game, I’ll undertake to characterize the scope readings available to the the arbitrary sentence n indicated above. This characterization will proceed in several stages.
Recall from the above that each scope reading available in a given sentence corresponds uniquely to a partial order on the set of that sentence’s determiner phrases; the set of all such partial orders in turn is in one-to-one correspondence with set of all graphs with nodes the sentence’s determiner phrases which are directed acyclic and transitively reduced. Associating to each determiner phrase exactly two Gks the node , we thus must characterize sentence n‘s scope readings among the directed acyclic transitively reduced graphs on the n vertices . I’ll begin with an inductive characterization:
Claim 1 (Inductive characterization of admissible scope readings). The graphs associated to sentence n‘s scope readings are characterized among the directed acyclic transitively reduced graphs on n vertices as exactly those which can be built using the following inductive process:
- Define by setting and then following exactly one of the following two procedures:
Briefly, the placeholder variable r keeps track of what we may think of as the tree’s root node; at each stage, we may either append a new root or build upon a chain of nodes trailing from the existing root.
Proof. These steps clearly generate directed acyclic transitively reduced graphs. There are now two things to show here: That each generated graph represents an admissible scope reading, and that these are the only such.
The idea is essentially, again, an inductive one. I’ll take for granted here the perhaps subtle fact that sentence n is given meaning—and its scope ambiguities resolved—in an “inside-out” manner (in other words, its syntax tree is semantically analyzed using a bottom-up traversal). The content of the claim, then, is that the noun phrase
exactly two Gks who talked to exactly two G(k-1)s … who talked to exactly two G1s.
can be given meaning only by inductively giving meaning to
exactly two G(k-1)s … who talked to exactly two G1s.
and then using exactly one of the following two procedures:
- Let exactly two Gks outscope the entire noun phrase exactly two G(k-1)s … who talked to exactly two G1s.
- Let exactly two Gks be outscoped by the single determiner phrase exactly two G(k-1)s.
That these modifications are sound (the first implication above) is essentially a linguistic fact, and must be proven “by linguistic intuition”: all I can do is provide examples. That the opposite implication holds—and these are the only acceptable steps—is again a consequence of the bottom-up semantic analysis of a sentence: any other steps would intercede within the existing phrase, which has already been given meaning in the bottom-up traversal.
For example, let’s consider in particular the sentence 3:
- There are exactly two G3s [who talked to exactly two G2s [who talked to exactly two G1s]].
Claim 1 is powerful enough for us to exhaustively characterize the scope readings of sentence 3. In the following diagram, I display the directed acyclic transitively reduced graph corresponding to each of this sentence’s 4 scope readings. Under each graph, I describe a situation under which that graph’s scope reading is true and none of the others are. As a visual aid, in this diagram and those that follow, edges adjoined using procedure A. above will be drawn horizontally and to the left, while edges adjoined using procedure B. above will be drawn upwards and to the left.
Interestingly, the fourth panel above exhibits our first instance of non-trivial incomparability (between nodes 1 and 3). We will discuss this more below.
For further visual aid, I’ll indicate a few of the admissible scope readings on a graph with 10 nodes. Here, I just provide the scope graphs:
That Claim 1 describes all and only the acceptable scope readings is, at the end of the day, a matter of linguistic intution—I don’t have the tools to say more!
It may be of interest, now, to characterize the graphs generated by Claim 1 among the directed acyclic transitively reduced graphs on n nodes in a way that is not inductive, but employs instead a “closed form”: a set of conditions that one may explicitly test on any given graph. This closed form is given below.
I’ll write to signify that and are connected by an edge, and I’ll write to indicate that there exists a path from node to node . We now have:
Claim 2 (Closed form characterization of admissible scope readings). A directed acyclic transitively reduced graph on n vertices is generated using the inductive procedure of Claim 1, and thus represents an available scope reading of sentence n, if and only if it is connected and in addition satisfies each of the following three conditions:
- If for nodes , then the boolean expression is true.
- If for nodes for which , then .
- Every node is covered by at most one other node.
One can check, for example, that the graphs in the diagrams above satisfy these conditions.
Proof. Since you’ve made it this far, I may once again disappoint you here. I have in fact taken the pains to prove this theorem myself. The proof, on the other hand, was an immense, and fairly tedious, exercise in subdividing into cases and manipulating boolean expressions. If somebody bothers to deem this work valuable, then I will be happy to write out a rigorous proof!
In the mean time, I will make a few statements. The forward direction of this proof (Claim 1 → Claim 2) is easier; it involves simply demonstrating that the inductive procedures of Claim 1 preserve the conditions of Claim 2. Note first that the base case (, a graph with one node and no arrows) clearly satisfies the conditions of Claim 2. One must then show that, provided satisfies the claim’s conditions, does too. This itself involves handling the options k. A. and k. B. separately; for each such option, the conditions 1., 2., and 3. of Claim 3 must be verified independently. (This is not very hard to do.)
The opposite implication is trickier. We must show that any graph not constructible using the procedures of Claim 1 is either disconnected or fails to satisfy at least one of the conditions of Claim 2. My idea here is to use a sort of converse induction. Fix a directed acyclic transitively reduced graph on the nodes which is not constructible using the procedures of Claim 1. Note first G can be assembled in a step-by-step procedure whereby on step k the node is added together with all of the edges in the sets and . Moreover, on at least one of these steps, the edges added must fail to conform to the requirements of Claim 1, or else G would be generable using these procedures after all. It’s enough to show, then, that if our graph violates Claim 1’s procedures at any one of these stages, the ultimate graph G must fail at least one of the conditions of Claim 2. This involves a fair amount of sophisticated checking, and I will set aside that task here.
These sentences satisfy the loosely defined conditions put forth by Game 3. For one, fairly many of the partial orders on n elements are of the form specified by Claim 2—exactly 2n-1 of them, it would seem, as appending each non-initial node represents a choice between exactly two options. We have thus provided a second, and roundabout, solution to Game 2. (Counting the overall number of partial orders on n elements is much harder). Meanwhile, as c-command relations obtain between all the determiner phrases in sentence n, in any admissible partial order any two nodes between which no order relation exists are non-trivially incomparable—and of these, too, there are many. The analogous fact was not true of our first solution to Game 2.
It would be interesting, of course, to establish more general results regarding the availability of scope readings—for instance, to formulate a general procedure whereby an arbitrary syntax tree is mapped onto the set of its admissible scope readings. In particular, one could discover sentences for which a great many possible scope readings obtain among its determiner phrases, or for which few obtain.
Game 4: Construct a family of sentences 1, …, n, …, such that the nth sentence has n determiner phrases among which all, or many, or few (or no?) scope orderings are available.
- Kripke, Speaker’s Reference and Semantic Reference
- Fromkin, et. al. Linguistics: An Introduction to Linguistic Theory
- Jong C. Park. Quantifier Scope, Lexical Semantics, and Surface Structure Constituency