Is it valid to make an assumption that directly contradicts a given premise?
For example, if I want to deduct the proposition
$$¬(p→q) ⊢ p∧¬q$$
I'd like to assume $p→q$, so I can falsify things based on the assumption at any given time, but I'm not sure if it's even allowed to continue assumptions once they're evident as contradictions.
Yes, you can assume $p → q$, but you will not go very far ...
1) $\lnot (p → q)$ premise
2) $p→q$ --- assumed [a]
3) $\bot$ --- from 1) and 2)
4) $\lnot (p→q)$ --- from 2) and 3) by $\lnot$-introduction, discharging [a]
and we are back to the start.
What we need is :
1) $\lnot (p → q)$ premise
2) $\lnot (p \land \lnot q)$ --- assumed [a]
3) $p$ assumed [b]
4) $\lnot q$ --- assumed [c]
5) $p \land \lnot q$ --- from 3) and 4) by $\land$-introduction
6) $\bot$ --- form 2) and 5)
7) $\lnot \lnot q$ --- from 4) and 6) by $\lnot$-introduction, discharging [c]
8) $q$ --- from 7) by Double Negation
9) $p \rightarrow q$ --- from 3) and 8) by $\rightarrow$-introduction, discharging [b]
10) $\bot$ --- from 1) and 9)
11) $\lnot \lnot (p \land \lnot q)$ --- from 2) and 10) by $\lnot$-introduction, discharging [a]