# 'Any heavyweight can defeat any lightweight.' : How do you bracket the Nested Conditional?

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 ) ]

Even after reading this, Nested Conditionals still confuse me.
I bolded the only 2 changes. Why is it wrong to translate 1 as 3 below?
I pursue only intuition; please do not answer with formal proofs or Truth Tables.

1. (x) [ (y) ( Hx ⊃ Ly ) ⊃ Dxy ]
• Why would you translate it in that way?
– user2953
Commented Jan 5, 2016 at 22:36
• @Keelan My problem is that I am confused (and do not know) which way is correct, for Nested Conditionals.
– user8572
Commented Jan 5, 2016 at 22:38
• Now it is wrong; as already discussed at lenght : A ⊃ (B ⊃ C) and (A ⊃ B) ⊃ C are not equivalent: see truth table. Commented Jan 5, 2016 at 22:38
• @MauroALLEGRANZA Thanks. Yes, I remember, but as I wrote above, I still am entirely confused how to select which.
– user8572
Commented Jan 5, 2016 at 22:39
• A good rule of thumb would be: if the sentence isn't grammatically awkward, use A ⊃ (B ⊃ C). Here, the sentence ("if x is heavyweight then if y is lightweight then x can defeat y") isn't too awkward. The other way around would be "if it is true that if x is heavyweight then y is lightweight, then x can defeat y" which is much more awkward.
– user2953
Commented Jan 5, 2016 at 22:43

We can use a step-by-step approach.

We have already learned that : "Every human is mortal" is formalized as :

(x) ( Hx ⊃ Mx ).

Now conisder [1.] : "Any heavyweight can defeat any lightweight" and firstly "approximate" it as : "Any heavyweight is a defeater".

This statement has the same logical form of the previous example, i.e.:

(x) ( Hx ⊃ Defx ).

Now consider : "x is a defeater of any lightweight". We can rewrite it as "Any lightweight is a defeated by x",i.e.:

(y) ( Ly ⊃ Dxy ).

Now you have only to replace "x is a defeater" [i.e. Defx] with "x is a defeater of any lightweight" [i.e. (y) ( Ly ⊃ Dxy )] to get :

(x) ( Hx ⊃ (y) ( Ly ⊃ Dxy ) ).

What makes your version wrong has nothing to do with the quantifiers. As Mauro pointed out, the general form (A ⊃ B) ⊃ C is NOT equivalent to A ⊃ (B ⊃ C). What you want is (A & B) ⊃ C.

With that in mind, you could write this

(x) [ (y) ( Hx & Ly ) ⊃ Dxy ]