How are statements laid out in deductive proof?

Question

so I am doing deductive proof, I know this contains a lot of laws that you need to be familiar with. And I have all that I need here. However, I don't known when to use more than one premise. For example say:

---

1: A ⇒ B Premise
2: C ⇒ D Premise
3: B ∨ D ⇒ E Premise
4: ¬E Premise
5: ¬(B ∨ D) from 3 & 4 modus tollens
Here what I don't understand is when it says 3 & 4 modus tollens, How is 3 & 4 actually laid out before applying modus tollens?
Also the modus tollens rule says: ¬p ∧ (p ⇒ q) ⇒ ¬p
So how can it be applied to 3 & 4?

Answer 1

Obviously, in a derivation, when we apply a rule, this rule must use (one or more) formulas already present in the derivation.

Thus, if we refer to lines in the derivation by numbers, the simple rule must be:

in line $n$ apply Rule X to formulas in lines $i$ and $j$, with $i,j < n$.

With Modus tollens we can refer either to a rule :

$P\to Q,\lnot Q\vdash \lnot P$,

or to a tautology :

$((P\to Q)\land \neg Q)\to \neg P$.

In the first case, your line 5 is fine: it follows from lines 3) and 4) by MT.

In the second case, to be "formal", we have to interpose some intermediate steps :

4') $((B \lor D) \to E)\land \neg E$ --- from 3) and 4) by Conjunction

4'') $[((B \lor D) \to E)\land \neg E] \to \neg ((B \lor D) \to E)$ --- tautology : Modus Tollens

to get :

5) $\neg (B \lor D) \to E$ --- from 4') and 4'') by Modus ponens.