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!

  • You are fighting a lost cause because all propositional logic formulas have three main connectors: the NOT, And, as well as exclusive OR. The other symbols are variations of those three mentioned. All implications have equivalent propositions using the OR and the AND. I doubt you can get around this. You need to understand propositions and not focus on Sentences. Deductive reasoning does not use sentence meanings. Only propositional meanings are considered. Keep in mind the same proposition can be expressed with different words. The meaning of what is expressed does not change or vary.
    – Logikal
    Jun 12 '18 at 18:39

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)?

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