# Questions tagged [natural-deduction]

For questions concerning natural deduction, a formal proof system studied in proof theory. A natural deduction proof starts with a set of premises and applies introduction and elimination rules to arrive at the conclusion. This tag is not specific to any particular logic, classical or intuitionistic, propositional or allowing quantifiers.

504 questions
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### how to prove C when given A ∨ (B ∧ C) and A → C [on hold]

can somebody help me to prove this using natural deduction fitch style: A ∨ (B ∧ C), A → C ∴ C here, what i got so far and i dont know if i am on the right track:
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### Natural deduction: predicate logic proof (Prenex form)

I'm pretty familiar with proofs in propositional logic, but not so much with predicate logic. I'm trying to prove the following (which can be used during construction of prenex normal form). If $x$ ...
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### Natural deduction proof - is this correct?

I don't know of any means to check my work, can anyone point out if they're any mistakes?
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### Proving ∀x(A(x) ∨ B) → ∀xA(x) ∨ B, with x is not in B, by natural deduction

how can prove ∀x(A(x) ∨ B) → ∀xA(x) ∨ B where x is not in B using natural deduction. i am not sure how should use for all introduction rule here. any help wpuld be highly appreciate. Cheers
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### How to prove ~($\forall$x Q(x)) is logically equivalent to $\exists$x(~Q(x)) using natural deduction for first order logic

I am thinking of assuming Q(x1) and then deriving to reach to a contradiction but I have not been able to do so.
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### Relationship between sequent calculus and Hilbert systems, natural deduction, etc

I am trying to learn the basics of logic and I'm confused on how these proof systems work together. The big ones I see are Hilbert style, and then Gentzen style which includes natural deduction, and ...
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### Find a natural deduction proof to show ∃x∃y (S(x,y) ∨ S(y,x)) ⊢ ∃x∃y S(x,y) by predicate logic.

I'm trying to prove $\exists x \exists y (S(x,y) \lor S(y,x)) \vdash \exists x \exists y S(x,y)$ in natural deduction, and I have already applied existential elimination to get $S(x_0,y_0)$, with $x_0$...
Let $\Phi = \left \{ \alpha, \beta, \gamma \right \}$ be a set of three well-formed formulas. To show $\Phi$ is inconsistent, should I use deduction to show that $\Phi \vdash \phi$ for all \$\phi \in \...