"No formula can be represented in the form (F . G), where F and G are formulas and is a binary connective, in more than one way.
By representing a formula in the form ¬F or (F . G) we start “parsing” it. The assertion of the previous problem shows that a formula can be parsed in only one way. From now on, we will abbreviate formulas of the form (F . G) by dropping the outermost parentheses in them. We will also agree that ≡ has a lower binding power than the other binary connectives. For instance,
p ∨ q ≡ p ⊃ r
will be viewed as shorthand for
((p ∨ q) ≡ (p ⊃ r)).
Finally, for any formulas F1, F2, . . . , Fn (n > 2),
F1 ∧ F2 ∧ · · · ∧ Fn
will stand for
(· · ·(F1 ∧ F2) ∧ · · · ∧ Fn).
The abbreviation F1 ∨ F2 ∨ · · · ∨ Fn will be understood in a similar way."
I understood it ,but i could not prove it with form of induction.
I know that, In another useful form of induction, we check that all atoms and 0-place connectives have property P, and that the property is preserved when a new formula is formed using a unary or binary connective. More precisely, we show that
• every atom has property P,
• both 0-place connectives have property P,
• if a formula F has property P then so does ¬F,
• for any binary connective , if formulas F and G have property P then so does
(F . G).
Then we can conclude that property P holds for all formulas.
Do F and G, if they are formulas and not abbreviations, both have the property that they can get represented in only one way?
If you can't tell, what about all of the subformulas {F$_1$ ... F$_n$} of F and {G$_1$ ... G$_n$} of G? Can they get represented in only one way?
If you can't tell that, can you tell that all of the subformulas of F$_1$, all the subformulas of F$_2$ ... all the subformulas of F$_n$ and similarly the subformulas of G$_1$ ... those of G$_n$ can only get represented in one way?
Do you eventually end up with letters if you repeat the above? And if you do, then can you build up F and G via the formation rules?
Remember, the only objects which qualify as formulas consist of strings which satisfies the formation rules (for that language). Nothing else is a formula.