1
Scope is the part a word governs
A negation governs what follows inside its brackets, and a quantifier governs the formula it binds. Moving a negation across a quantifier, or a quantifier across another, changes which formula the sentence is.
Lesson · Mind, Language and Meaning · Language
Scope Ambiguity and Negation
A negation can sit outside a description or inside it, and two quantifiers can come in either order. Each placement is a different formula with its own value. Evaluate “the King is not bald” and “everyone loves someone” under both scopes in a single model, and find the models in which the readings agree.
scope: what a ¬ or a quantifier covers
¬ before the F, or after it: two readings
∀x ∃y against ∃y ∀x: two readings
one sentence, two values, one model
∃y ∀x implies ∀x ∃y, not the reverse
One sentence can hold two formulas
Core ideas
What this lesson establishes
The last lesson turned a description into a quantified claim. A quantified claim has a scope, and a sentence with two scope-taking parts can be placed in more than one order.
2
English leaves scope to be recovered
“The King is not bald” and “everyone loves someone” each fit more than one formula. The words do not say which, and a hearer chooses by context.
3
Two readings can have two values
In one model the readings of one sentence can differ in value, which is a sharp test that they are not the same reading.
The material
Two scopes, one model
Negation and a description first, then two quantifiers, then when the readings agree.
Definition · Scope
The scope of a negation or a quantifier is the part of the formula it applies to. A sentence is scope ambiguous when its words fit two formulas that differ only in scope.
Take “the King is not bald”, in the model of the previous lesson in which nobody is a king. There are two places to put the negation. Outside the description, it denies the whole claim: it is not the case that there is one king and he is bald. Inside, it is part of what is said of the king: there is one king and he is not bald.
Evaluate the wide one. The claim that the king is bald is False, as in the previous lesson, because nobody is a king. The negation of False is True, so with the negation outside the sentence is True. Evaluate the narrow one. The expansion now ends in ¬Bald(x), and still opens by asserting a king, which there is not, so the sentence is False. The two readings, in the same model, have opposite values.
reading formula value
-------------------------------------------------
wide ¬ (the king is bald) True
narrow (the king) is not bald False
That settles one thing about the ambiguity: it is not a difference of style. Someone who says “the King of France is not bald” and means the wide reading is saying what is true, that the claim of baldness is false. Someone who means the narrow reading is saying something false, since it adds that there is a king. Russell used the distinction to defend the law of excluded middle, which the sentence seemed to threaten: either the king is bald or he is not, and neither holds if there is no king, unless the second is read with the negation outside.
The same thing happens with two quantifiers, and the order question of “Quantifiers and Their Order” in Arguments and Validity returns in a model you can type. Take “everyone loves someone”. One reading is ∀x ∃y Loves(x, y): each person loves a person, who may differ. The other is ∃y ∀x Loves(x, y): there is one person everyone loves. Put three people in a ring, a loving b, b loving c and c loving a. The first reading is True; each one loves someone. The second is False; nobody is loved by all three. The sentence in English is the same, and the two formulas are not equivalent.
They are related, though. If there is one person everyone loves, then each person loves someone, namely that one. So ∃y ∀x implies ∀x ∃y, and the ring shows that the implication does not run back. In a model where a, b and c all love c, both readings are True, with c as the witness for the second.
Readings can also agree. If the King is a real person and he is not bald, both the wide negation and the narrow reading come out True. A description that denotes something gives the two placements one value, and the negation only separates them where the description fails. That is why the ambiguity went unnoticed for so long: in ordinary cases it makes no difference.
The lab evaluates the formulas, and it does not choose between them. Which one a speaker meant is a question about the speaker, and the lab has no access to it; the lesson's claim is only that a careful hearer should ask.
Interactive
Truth in a model
A first-order sentence evaluated over a domain you can edit
Method
Resolving a scope ambiguity
Five steps, for a negation and for a pair of quantifiers alike.
1
Find the scope-taking parts
Look for a negation and for each quantifier or description. A sentence with only one of them cannot be ambiguous in this way.
2
Write each order as a formula
Put the negation outside, then inside. Put one quantifier first, then the other. Each placement is its own formula.
3
Evaluate each in the model
Use the same model for both. If they give the same value, this model does not tell the readings apart.
4
Find a model that separates them
Change an extension until the two values differ. An empty description separates the negations, and a ring of love separates the quantifiers.
5
Say which reading was meant
State why: the context, or what the speaker would have to be committed to. The lab cannot choose.
Placing the scope
Score 0 / 0Choose the answer you can justify.
The King is not bald, and the ring of love
Each sentence has two formulas. The model for the first has no king; the model for the second is a ring of three.
no king: ¬ (the King is bald)
the King is bald is False, so the negation is True
no king: the King is not bald
clause 1, something is a king: no x, False
ring: ∀x ∃y Loves(x, y)
a loves b, b loves c, c loves a: True
ring: ∃y ∀x Loves(x, y)
no y is loved by a, b and c together: False
One sentence of English, in one model, takes both True and False depending on the formula. A reader who asked only whether the sentence is true would have had no way to answer.
Common mistakes
1
Believing a sentence has one logical formThe same English sentence, “the King is not bald”, has a wide and a narrow reading. In the model with no king the lab gives True for the first and False for the second. Taking one form to be the form is a choice, and in ordinary cases, where the description denotes, it costs nothing because the two readings agree.
2
Reading “everyone loves someone” as saying there is one belovedThat is the second order, and in a ring of three it is False while the first order is True. Someone who hears the first and answers “but who?” has supplied the second.
3
Moving a negation across a quantifier as if it were free¬∀x A and ∀x ¬A differ, as ¬∃x A and ∃x ¬A do. In the model with no king, shifting the negation from outside the description to inside changed True to False. Moving it needs a rule, and the rule exchanges ∀ and ∃.
Check your understanding
✓
Finish when you can give both readings and a model that separates them.
Given an ambiguous sentence with a negation or two quantifiers, you should be able to write each formula, evaluate both in one model, and change the model until their values differ.
Note: Scope is one of several sources of ambiguity, and linguists argue about which readings a sentence has. The lesson treats the readings as given and teaches how to tell them apart.