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!