Validity of the conclusion using logic rules

Question

a) u V w

b) ~w

c) q →s

d) u → ~p

e) ~p → (r ^ ~s)

∴ ~q

Answer 1

a) u V w

sol.

Rule of Addition: given u, conclude u V w

b) ~w

sol.

Modolus tollens rules

U---->W, ~ q, therefore, ~ p

c) q →s

Sol;

Hypothesis rule

d) u → ~p

Sol.

Contradiction Rules

e) ~p → (r ^ ~s)

∴ ~q

Disjunctive Rules

Answer 2

So from premise (a) and (b) you can infer 'u' by disjunctive syllogism (https://philosophy.fandom.com/wiki/Disjunctive_Syllogism). Now that we have 'u', we can get '~p' from our 'u' and premise (d) by modus ponens (some books call this conditional elimination). Then using '~p' we can get 'r & ~s' from (e) again by modus ponens. Then we break apart that conjunction giving us '~s' by conjunction elimination. Finally, we use that '~s' and (c) to give us '~q' by modus tollens. QED!

1. u v w A
2. ~w A
3. q --> s A
4. u --> ~p A
5. ~p --> (r & ~s) A
6. u 1,2 disjunctive syllogism
7. ~p 4,6 MP
8. r & ~s 5,7 MP
9. ~s 8, conjunction elim.

———

~q 3,9 MT

EDIT: sorry the spacing got messed up, but hopefully it's still legible. After each premise I give the numbers of the lines that premise was derived from and the rule used. The lines with an A are the assumptions given in the problem.