In predicate logic, can the form of a translation of an existentially quantified sentence be used with a universally quantified sentence?
That's a lot of words for a simple question.
But here's what I'm wondering.
Say we have the sentence
"There is at least one brown horse"
With an unrestricted domain of discourse that would translate to
∃x[Bx & Hx]
If we have the sentence "All horses are brown"
∀x[Hx → Bx]
That translation seems to say
"(∀x[) What I say next will apply to every instance of the thing I'll talk about" "The thing I'm talking about (x) is a horse (H)"(Hx). "Whenever the thing you're talking about is a horse, it's also the case that (→) the thing you're talking about is a thing that's brown(Bx)" (→ Bx)
Doesn't '∀x' seem redundant. Doesn't it follow from H → B that all horses are brown? Like R → W (if you have rain, then you have water). If that weren't the case, you wouldn't be able to write H → B.
Even if we wanted to say something about x, whatever it is, couldn't we write: (x → H) → B or (x → H) & (x → B)
Couldn't you also write ∀x[Hx → Bx] as ∀x[Hx & Bx]?
And isn't ∀x[Hx & Bx] written just the same as ∃x[Bx & Hx], but with a universal quantifier?
So why is my text saying I should write ∀x[Hx → Bx]?