After searching this site for the key words "unique parsing" "parsing" and "unique parsing lemma," and finding nothing really applicable to me (or as far as I could tell), I decided to post the question. I'm not a math person, or logic.. so if anybody can even just tell me what on earth is this question even asking us to do... that'd be appreciated. If you have further insight on how to approach solving this, then that'd be great.
Prove the unique parsing lemma for first-order predicate logic (which includes identity and complex terms). That is, let ∝ be a well-formed formula of first-order predicate logic. Prove that exactly one of the following holds:
(i) ∝ is atomic.
(ii) ∝ has the form ~β for some well-formed formula β.
(iii) ∝ has the form (β -> γ ) for some well-formed formulae β and γ.
(iii) ∝ has the form (β & γ) for some well-formed formulae β and γ.
(iv) ∝ has the form (β v γ) for some well-formed formulae β and γ.
(v) ∝ has the form ∀vβ for some variable v and some well-formed formula β.
(v) ∝ has the form ∃vβ for some variable v and some well-formed formula β.
This result - more commonly called "unique readability" - basically says that there's only one way to interpret a well-formed formula. To see what could go wrong in principle, suppose we didn't include parentheses in our syntax for first-order logic. Then we'd wind up (for example) with formulas which look like $$a\wedge b\vee c.$$ It's unclear what this means: is it "$(a\wedge b)\vee c$" or "$a\wedge (b\vee c)$"? We could resolve this by adding some PEMDAS-like rules for order of operations, but it winds up being more convenient to just use a syntactical construction which prevents these issues from arising in the first place.
The phrasing of the lemma reflects the idea of evaluating a formula by breaking it into its constituents until we get "all the way down" to atomic formulas; the fact that exactly one case holds for any given formula tells us that there's never any ambiguity as to how to perform this breaking-down process.