Order of universal and existential quantifier

Question

In mathematics we use the universal and existential quantifiers (represented symbolically by ∀ and ∃, respectively) to make our lives easier. We can also use them in English. From a logical standpoint, these two sentences mean different things:

1. There exists a car for all people such that condition A is satisfied.
or
2. For all people there exists a car such that condition A is satisfied.

(Sentence 1 specifies the existence of one (not necessarily unique) car for the whole world satisfying A; sentence 2 means that each person can find a (possibly different) car satisfying A.)
However, when I use these constructions, I find that people either misunderstand me or do not understand the difference between the two. It seems that these types of sentences are often fraught with ambiguity in real-world communication, as I have discovered by reading these questions.
Nonetheless, this is an important distinction! For instance, a relativist might say "For all people there exists a god that they believe in" (meaning everyone believes in at least one god from the set of gods), but she would likely disagree with the idea that "There exists a god for all people that they believe in" (meaning everyone believes in the same set of gods, which set has size at least 1).
Question: Is there some way to phrase these two sentences that emphasizes the difference between the constructions? The problem is that using for all and there exists is the proper way to express the desired meaning, but it is very confusing. (Just read the comments below to see how confusing it is!)
Idea: I thought of clearly specifiying the "one"-ness in sentence 1 and to emphasize the "choice" in sentence 2, but I don't know if that is clear enough.

bib · Accepted Answer

The structure of your two examples does establish the different meanings.

In the first example,

There exists a car for all people such that condition A is satisfied.

the phrase for all people is an adjectival modifier explaining car. The placement of the phrase immediately following car helps establish that.

In the second example,

For all people there exists a car such that condition A is satisfied.

the phrase For all people is an adverbial modifier explaining exists. While it does not immediately precede or follow exists, it is in closer proximity to the verb than it is to car.  It could also be expressed as

There exists, for all people, a car such that condition A is satisfied.

Having said that, these distinctions are subtle and are easily misunderstood. To avoid misinterpretation, it would be better to be more explicit, such as

There exists [a single/one] car for all people such that condition A is satisfied.
  For [each person] there exists a [particular] car such that condition A is satisfied.

TrevorD · Answer

Something along the lines of:

There is a single car model that will satisfy condition A for everyone.
  There are enough choices of car models available such that condition A can be satisfied for everyone.

[I realise that I may have misinterpreted your meanings or intentions, particularly because the nature of the condition is unknown, but I hope this might give some ideas for rephrasing your statements.]

John Lawler · Answer

English is not logic. Logic is in fact a skeletal model of some parts of language, but it is only a skeleton, and the bones can be articulated in a number of ways. The Universal and Existential quantifiers are the only two used in philosophy, logic, and mathematics, but they are not the only ones in natural language, which does not share the minimalist esthetics of mathematics. Being evolved and alive, language bristles with variations on everything, some of which survive to the next generation.
Quantifiers are one such type of variation. As you point out, the linear order of quantifiers in a logical proposition is criterial for determining their relative scopes. I.e,

(∃y: BOOK (y)) (∀x: BOY (x)) READ (x, y)
("there exists some y which is a book, such that for every x which is a boy, x reads y")
(∀x: BOY (x)) (∃y: BOOK (y)) READ (x, y)
("for every x which is a boy, there exists some y which is a book, such that x reads y")

are each unambiguous, and display both of the readings of Every boy read some book.
However, the syntax of natural language outranks the artificial syntax of logic; words that have quantification built in are also usually modifiers of one sort or another, and word order is quite fixed in English syntax, whatever the requirements of logic might prefer.
One of the results is that there are syntactic rules in English that move pieces of sentences around without preserving their original order or precedence.
One of the rules that does this is Quantifier-Float, which produces
The boys all read some book
from
All the boys read some book.
Another is Passive, which replaces the subject NP with the object NP:
Some book was read by all the boys.
As can be seen, word order is not the same as quantifier order.
Another result is that virtually any English sentence with two Operators (anything with a focus and scope -- typically Modals, Negatives, Quantifiers) in it will be ambiguous, because their scopes can become entangled.  I.e,

more than one type of quantifier (like Every boy read some book)
or
any quantifier plus a negative (like All the boys didn't leave)
or
any negative plus a modal (like She may not go tonight)

One of the ways to get around this is with qualified quantifiers, like (∃y: BOOK (y)).
Such that is the phrase one uses for this, but again, this is an idiomatic relative clause construction and has its own syntax; you can't just put such that clauses any old where.

John Visosky · Answer

I think the whole thing would be clearer if a car were replaced by some car in both statements. I believe it retains your mathematical rigour while conveying the intended meaning to the “average” reader.

Answered by John Visosky on July 27, 2021

Order of universal and existential quantifier

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