phrase requests - Order of universal and existential quantifier


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.



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





  1. There exists [a single/one] car for all people such that condition A is satisfied.




  2. For [each person] there exists a [particular] car such that condition A is satisfied.





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