I remember that last year a few students got confused by this question, and I’m afraid a few more will be this year. The problem is that it's difficult to parse what "the formula" refers to in answer C.
If "the formula" refers to (ϕ ∧ ψ) (grammatically the most plausible interpretation), I am not sure how to envision how it can both be a conjunction and a disjunction at the same time. Perhaps that is the intended clue of the exercise (the solution in the appendix suggests as much): there is no choice of ϕ and ψ that can yield something exactly equal to (p∨¬p). But in that case I find it’s still worded in a confusing way, because the wording seemingly presupposes that it’s possible to find such an equivalence. Because the answer is presented the way it is, students feel as if they’re not supposed to investigate whether it is possible or not, but rather that it’s okay to just assume it is and deduce from there whether it’s a correct answer (like you’re able to do for answers a and b). If you do assume it to be possible to rewrite (ϕ ∧ ψ) as (p∨¬p), it ought to be true that (ϕ ∧ ψ) is in fact a logical truth.
The impossibility of finding such an equivalence also tempts students to think that, though grammatically less plausible, “the formula” might refer to either ϕ or ψ (or both), in which case it’s obviously possible that (ϕ ∧ ψ) be a logical truth, namely if the other conjunct is also a tautology, like in answer B. (For reference: answer C is supposed to be wrong according to the solutions)
Also, what does it mean to be "of the form of"? That can have multiple interpretations, among which: being literally the same formula (which i think is the intended reading), or being a formula that is merely equivalent. That also adds to the confusion, as it's definitely possible to come up with formulas that are equivalent to (p∨¬p), such as (trivially) ϕ = (p∨¬p) and ψ = (p∨¬p).
Do you remember what you had in mind with this answer?
Self study question 5.5.2 asks:
I remember that last year a few students got confused by this question, and I’m afraid a few more will be this year. The problem is that it's difficult to parse what "the formula" refers to in answer C.
If "the formula" refers to (ϕ ∧ ψ) (grammatically the most plausible interpretation), I am not sure how to envision how it can both be a conjunction and a disjunction at the same time. Perhaps that is the intended clue of the exercise (the solution in the appendix suggests as much): there is no choice of ϕ and ψ that can yield something exactly equal to (p∨¬p). But in that case I find it’s still worded in a confusing way, because the wording seemingly presupposes that it’s possible to find such an equivalence. Because the answer is presented the way it is, students feel as if they’re not supposed to investigate whether it is possible or not, but rather that it’s okay to just assume it is and deduce from there whether it’s a correct answer (like you’re able to do for answers a and b). If you do assume it to be possible to rewrite (ϕ ∧ ψ) as (p∨¬p), it ought to be true that (ϕ ∧ ψ) is in fact a logical truth.
The impossibility of finding such an equivalence also tempts students to think that, though grammatically less plausible, “the formula” might refer to either ϕ or ψ (or both), in which case it’s obviously possible that (ϕ ∧ ψ) be a logical truth, namely if the other conjunct is also a tautology, like in answer B. (For reference: answer C is supposed to be wrong according to the solutions)
Also, what does it mean to be "of the form of"? That can have multiple interpretations, among which: being literally the same formula (which i think is the intended reading), or being a formula that is merely equivalent. That also adds to the confusion, as it's definitely possible to come up with formulas that are equivalent to (p∨¬p), such as (trivially) ϕ = (p∨¬p) and ψ = (p∨¬p).
Do you remember what you had in mind with this answer?