The statement I'd like to translate into a mathematical one is

"Every American has a dream".

Let $A$ and $D$ denote the set of all Americans and the set of all dreams, respectively, and $P(a,d)$ denote the proposition "American $a$ has a dream $d$". The mathematically equivalent statement I've deduced is $$\forall a\in A.\exists d\in D.P(a, d)$$

However, I suspect the above statement implies that for every American there exists a common dream $d$ such that $P(a,d)$ holds true. I would like to know how to rectify this error(if there is one).

  • 1
    $\begingroup$ Correct: "forall a there is a d..." does not mean that the d is the same for all a. To state that the d is the same, you have to write "there is a d for all a...". Compare $\forall n \exists m (n < m)$ and $\exists m \forall n (n < m)$. $\endgroup$ May 25, 2020 at 10:01

1 Answer 1


The statement you wrote is completely right, and the answer you initially accepted was completely wrong (and has been deleted). If you need any help in understanding quantifiers intuitively, you may want to first read this post introducing game semantics (but note that in that post "prover" is used in the layman sense of "verifier"), and then read the following description of game semantics of quantifiers:

  1. The verifier of a sentence of the form "$∀a{∈}A\ ( P(a) )$" must let the refuter first choose any arbitrary $a∈A$ and then verify $P(a)$ no matter what $a∈A$ was chosen.
  2. The verifier of a sentence of the form "$∃d{∈}D\ ( Q(d) )$" must first choose some $d∈D$ and then verify $Q(d)$ for that chosen $d∈D$.

If the verifier can win no matter what the refuter does, then the sentence is true.

In your example, the verifier of "$∀a{∈}A\ ∃d{∈}D\ ( P(a,d) )$" must let the refuter make the first move in choosing an $a∈A$, and then verify "$∃d{∈}D\ ( P(a,d) )$" no matter what $a$ was chosen. But since the verifier makes the second move in choosing some $d∈D$, the verifier can choose this $d$ based on the refuter's first move (i.e. based on $a$). That is why "Every American has a dream." corresponds to this sentence.

In contrast, the verifier of "$∃d{∈}D\ ∀a{∈}A\ ( P(a,d) )$" must make the first move in choosing some $d∈D$, before the refuter makes the second move in choosing an $a∈A$. You can see easily that the verifier can win only if there is a single choice of $d∈D$ that defeats every possible choice of $a∈A$. That is why "All Americans have a common dream." corresponds to this sentence and not the other one.

  • $\begingroup$ Note that the established syntax of quantifiers in mathematical logic is already 100% precise. Please do not attempt to 'improve' on it unless you know what you are doing. Also, it is worth noting that one of the most common logical errors is to accidentally swap quantifiers. The game semantics should make clear why "∃∀" is stronger (i.e. harder to win) than "∀∃", and Mauro has provided in a comment an example where the "∀∃" version is true while the "∃∀" version is false. $\endgroup$
    – user21820
    May 31, 2020 at 9:20

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