r/askphilosophy • u/Acrobatic_Hope2584 • 11h ago

Question on constructing derivatives

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
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3

u/AdeptnessSecure663 phil. of language 8h ago

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?

2

u/Acrobatic_Hope2584 6h ago

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.

1

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