# How to translate this statement into a mathematical one(using appropriate quantifiers)?

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).

• 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 $\exists m \forall n (n < m)$. May 25, 2020 at 10:01

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 $$d∈D$$.
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.