Using only the rules of inference and the logical equivalences, show that the following argument is valid. You may assume that all the premises given are true.
Premises:
∧
→
s ∨ ( → )
¬ ∧
¬ ∧
Show: ¬p
So I've attempted the solution but I'm stuck and not sure what steps I should take next. Any input or help would be appreciated.
My solution so far:
1) ∧
2) →
3) s ∨ ( → )
4) ¬ ∧
5) ¬ ∧
6) ¬ by simplification (4)
7) ( → ) by disjunctive syllogism (3, 6)
8) ( → q) by hypothetical syllogism (2, 7)
9) ¬ by simplification (5)
10) ¬p by modus tollens (8, 9)
I'm trying to get rid of 1) and 6) left but I'm out of formulas I can think of to use. I thought maybe I could use addition somehow, but I'm not quite sure how that formula works. Like would that allow me to add any variable from the alphabet that I want or can I only use the variables that have already been defined?
Any thoughts or ideas on what steps I could take next would be appreciated.
$$\dfrac{\dfrac{\lower{1.5ex}{r\to q}~~\dfrac{\lower{1.5ex}{s\lor (p\to r)}~~\dfrac{\lnot s\land t}{\lnot s}{\tiny\text{simplification}}}{p\to r}{\tiny\text{disjunctive syl.}}}{p\to q}{\tiny\text{hypothetical syl.}}~~\dfrac{\lnot q\land u}{\lnot q}{\tiny\text{simplificaton}}}{\lnot p}{\tiny\text{modus tolens}}$$
You have correctly proven that $r \to q, s\lor (p\to r),\lnot s\land t,\lnot q\land u\vdash \lnot p$ and that was all you needed to do.
As the commenters have stated, the fact that you do not need premise $u\land t$ to derive $\lnot p$ is not a problem. It is just a redundant premise. You may have as many redundant premises as you wish.
$$m\lor n~,~u\land t~,~ r \to q~, s\lor (p\to r)~,\lnot s\land t~,\lnot q\land u\vdash \lnot p$$