Existential Quantification

Variables in mathematics, such as x, y, a, b, c, etc., are quantified with phrases such as "for all x", "for any z", "for every z", "there is at least one a", "there is some b", "there is an x", "there is a w", "for at least one j", etc. "There is at least one", "there is some", "there is an", "there is a", "for at least one", and similar phrases mean the same thing: there is at least one object with the property being described and MAYBE more. No claim is being made about there being more than one such object: there might be exactly one, there might be more than one.

Multiple "There Is" Phrases

Consider these two sentences: "There is an x and there is a y such that x+y=y" and "There is a y and there is an x such that x+y=y". These mean the same thing. Order is not important with multiple "there is" phrases. This is NOT true for mixtures of "there is" and "for all" phrases.

"Not" Applied To A "There Is" Sentence
Consider what it means for "there is at least one x such that P(x)" to be true, where P(x) is some sentence about x. For some object x, P(x) is true; maybe only one; maybe more than one. If the sentence is false, it means that, for every possible x, P(x) is false. So, not[there is at least one x such that P(x)] means the same as { for all x, [not(P(x))] }.

 

"For All" Mixed With "There Exists": Order Is Important!

Compare these two sentences: "For all x, there exists at least one y such that Q(x,y)" and "there is at least one y such that, for all x, Q(x,y)". Here Q(x,y) is some sentence about x and y, such as "x+y=0". The first of these means that for all x, there is a y WHICH IS ALLOWED TO VARY WITH EACH x which makes Q(x,y) true. So, for example, "for all x, there exists at least one y such that x+y=0" is true because y=-x makes it true.

The sentence "there is at least one y such that, for all x, Q(x,y)" means that there exists one y which does the job for all x of making Q(x,y) true. This is a much tougher statement to make true! Notice that "there is at least one y such that, for all x, x+y=0" is false for the real numbers---there y has to be -x and must vary with x which this sentence does not allow.

Of course the tougher statement implies the easier. Consider "there is a z such that, for all x, x+z=x". This is true for real numbers, because z=0 plays that role for ALL x. It is likewise true that, "for all x, there is at least one z such that x+z=x." While z is allowed to vary with x to make this last sentence true, it does not have to vary---z=0 still works for each and every x! To summarize this paragraph, the following is always true:

If [there is at least one y such that, for all x, Q(x,y)], then [for all x, there is at least one y such that Q(x,y)].
Warning! The converse is usually false.