Expressive adequacy help

Question

I'm learning about functional incompleteness/expressive adequacy, and I want to show the following:

(i) → cannot be defined in terms of ∨ alone.

(ii) ∧ cannot be defined in terms of → alone.

I understand the basic idea of what I'm trying to do here: I need to show a property one connective has cannot be expressed using only the properties of another.

My attempts: (i) Initially, I wanted to say that any formula of ∧ will be false when a structure assigns F to all letters in the sentence; and no formula that expresses → can have this property.

I realized this isn't true though: (P → Q) → Q will be false when every letter is false.

A similar issue arises in my attempt at (ii):

Some formula of → must come out as true when a structure assigns F to all letters. No formula that expresses ∧ can have this property.

Again, while the second sentence has this property, the first does not.

So I'm stuck here. Any help would be greatly appreciated!

Answer 1

For i): P → Q is F when P is T and Q is F. However, if you have an expression that contains at least one P, and otherwise only Q's and ∨'s, then that expression will always evaluate to T, and so cannot be F when P is T and Q is F. This means that you cannot put any P in the expression to capture P → Q. But if you have is Q's and ∨'s, then you can't capture the fact that when P is F and Q is F, P → Q is T, since for such an expression, it will always evluate to F.

Can you construct a similar argument for ii)?