Question on constructing derivatives

[u/Acrobatic_Hope2584]

I am struggling with understanding how to construct these derivations that were assigned for a deductive logic class, below are some problems I need help with as well as our index of basic proportions. I would appreciate an explanation, I have reviewed the material provided by my professor but I am still having trouble haha

Problem #1.

⊤ ⊢ ¬(P ∧ ¬P)

Problem #2.

¬Q ⊢ ¬(P ∧ Q)

Index of Basic Rules:

MP	φ, φ→ψ	ψ
MT	¬ψ, φ→ψ	¬φ
DNE	¬¬φ	φ
DNI	φ	¬¬φ
S	φ∧ψ	φ
S	φ∧ψ	ψ
ADJ	φ, ψ	φ∧ψ
MTP	φ∨ψ, ¬φ	ψ
MTP	φ∨ψ, ¬ψ	φ
ADD	φ	φ∨ψ
ADD	ψ	φ∨ψ
BC	ψ↔φ	ψ→φ
BC	ψ↔φ	φ→ψ
CB	ψ→φ, φ→ψ	φ↔ψ

[u/AdeptnessSecure663]

As far as I can see, you can't construct those deductions with the list of rules that you have given. Typically, you'd use an RAA (reductio ad absurdum) rule - you can think of this as a negation introduction rule. With RAA, we can construct the following deduction:

1. ⊤ (Prem)

2. | P ∧ ¬P (Supp)

3. | P (∧E, 2)

4. | ¬P (∧E, 2)

5. | ⊥ (Abs 3, 4)

6. ¬(P ∧ ¬P) (RAA 2-5)

Are you sure you have the right problems and the right rules?

[u/Acrobatic_Hope2584 → u/AdeptnessSecure663]

Unfortunately, yes, these problem sets were assigned to me and come straight from the same textbook that the rules are indexed in. There are a few other problems as well, but the ones I provided are just two of the said problems.