Source: p 498, A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

Any heavyweight can defeat any lightweight.
[1.] (x) [ Hx ⊃ (y)(Ly ⊃ Dxy) ]

I bolded the only change. Why is it wrong to translate 1 as 2 below?
I pursue only intuition; please do not answer with formal proofs or Truth Tables.

  1. (x) (y) [ Hx ⊃ (Ly ⊃ Dxy) ]
  • 2
    It is not wrong; both are equivalent. Jan 5 '16 at 22:36
  • 1
    Hurley's articulation has the advantage of narrowing the domain scope down to the relevant part but you can trivially do so. There's no y in Hx ⊃ so you can move the domain scope in or out (you cannot do the same to (x) because it is also used in the right hand part.
    – virmaior
    Jan 5 '16 at 23:01
  • I've noticed you have "I pursue only intuition; please do not answer with formal proofs or Truth Tables" in your question, but ask specifically about a formal notation. In this case, your search for only intuition has lead you to ask "why" to a false statement. My recommendation would be to either learn the formal proofs, or rephrase the question. You could have asked "What is the logical difference between 'any heavyweight can defeat any lightweight' and 'given any two fighters, x, and y, the combination of x is a heavyweight and y is a lightweight and x defeats y is true'"
    – Cort Ammon
    Feb 5 '16 at 17:50
  • In this case, the answer Mauro gives is a fundamental rule for how quantifiers may be rearranged. Perhaps, a third option for your question, might be "intuitively, why is ϕ → ∀x ψ equivalent to : ∀x (ϕ → ψ)"
    – Cort Ammon
    Feb 5 '16 at 17:56

There are rules for moving all the quantifiers upfront, in order to rewrite a formula in an equivalent Prenex Normal Form.

The rules are based on several provable equivalences; in particular :

ϕ → ∀x ψ is equivalent to : ∀x (ϕ → ψ)

provided that :

the quantified variable x is not free in ϕ.


To put it in as plain language as possible:

A quantifier only affects its own variable. If a sentence can be broken down into smaller integral parts, and only one of those parts has the variable, you can move the quantifier between that part and the whole without changing the meaning.

In general, you keep the quantifier with the smaller section because it makes it more clear what it affects.

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