Introduction Rule for Conditionals

Question

I have a quick question regarding the introduction rule for the conditional: do you always need to proceed from an assumption or could you start from a premise? If so, then how would you go about it?

I need to prove the following formula using natural deduction: ¬Q ∧ ¬P ˫ ¬P → (R → ¬Q)

I've also made a quick proof.

1 (1) ¬Q ∧ ¬P       P 

2 (2) ¬P            A

3 (3) R             A

1 (4) ¬Q            1∧E

1 (5) R → ¬Q        1,3 →I

1 (6) ¬P → (R →¬Q)  2, 5 →I

I'm just wondering whether I have to assume ¬P in (2), when I already have it in (1)? Would it mean that ¬P is perfectly superfluous for the argument? That would seem to follow from the paradoxes of material implication (e.g B ˫ A → B).

Answer 1

It depends on the proof system and how exactly it has formally defined its formal inference rules.

If your particular proof system insists on making an explicit distinction between premises and assumptions, and if it insists on having the $\rightarrow I$ rule only use assumptions for the antecedent, then no, you can't use a premise.

But, in terms of pure logic, it should of course work just as well whether you use an assumption or a premise for a conditional proof: in both cases, the assumption base will tell you where the derived statement comes from. Indeed, most systems really don't make any explicit difference between premises and assumptions. So then your proof could be done like this:

1 (1) ¬Q ∧ ¬P       A 

1 (2) ¬P            1∧E

2 (3) R             A

1 (4) ¬Q            1∧E

1 (5) R → ¬Q        3,4 →I (you had a little mistake in line numbers here)

1 (6) ¬P → (R → ¬Q) 2,5 →I

Since line 6 shows that ¬P → (R → ¬Q) is a logical consequence of ¬Q ∧ ¬P, you can conclude ¬Q ∧ ¬P ˫ ¬P → (R → ¬Q)