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Questions tagged [natural-deduction]

12 questions
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Explanation of implication-introduction rule

I read in Proofs and Types by Girard et alii. the following excerpt that talks about the calculus of natural deduction: Now a sentence at a leaf (of the deduction tree) can be dead, when it no longer plays an active part in the proof. Dead…
user1868607
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Predicate Logic - Natural Deduction; Assumptions about exists-elimination

I am stuck on how to progress with this proof; i cannot see my next move. The task is to show $S \to \exists x Q(x) \vdash \exists x (S \to Q(x))$ using natural deduction for predicate logic. My first attempt was the following; 1: $S \to \exists x…
Robert
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Elimination of duplicate premise

Suppose that I have a proof tree starting with some statement |- B in a sequent calculus, leading to two premises/leaves |- A. Is it always possible to transform such a proof tree into another proof tree, which only contains one leaf |- A only using…
aiquita
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Formal Logic - Natural deduction: Problem with assumptions about exists-negation

I'm stuck on how to progress with this proof, despite I have tried, I cannot see the next move. Given this proof without predicate: So far what I've accomplished: My idea is, as I can't see any other option using (-(Sv(P->Q)) as the first…
Jesús Díaz Castro
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Natural deduction: understanding bottom elimination (¬e)

I am new to natural deduction and upon reading about various methods online, I came across the rule of bottom-elimination in the following example. I do not understand the step in line 10. Upon inspection, my initial thought would be that the…
Newbie123
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Natural deduction proof: distributivity of existential quantification

In a current exam-prep exercise, we were tasked to prove the following formula using natural deduction of first-order logic: $(\exists x. P \lor Q) \rightarrow P \lor (\exists x.Q)$ for arbitrary $P,Q$ where $P$ does not have free occurences of…
iMrFelix
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Proving the validity of a sequent using Modus Tollens

Problem: Prove $p \rightarrow (q \vee r), \neg q, \neg r \vdash \neg p$ using Modus Tollens. I need to prove the validity of the above sequent by using natural deduction. Initially, I didn't read the entire problem, and went the long way by doing an…
Smiley
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Algorithm for automatic construction of natural deduction proofs

I was wondering if there exists any algorithm for automatic construction of nautral deduction proofs. I'm interested in propositional logic and first order logic. If there is no algoritm, can you provide some proof of this fact? PD0: I'm not…
Jay Jay
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Natural deduction proof without any predicate

I am gaving trouble proving a natural deduction proof when there is no predicate given. Only conclusion is given. I understand the rules of elimnations, inclusions, IPs and others but I having trouble applying them when no predicate is given. The…
Arth Patel
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Some explanations needed about Negation in Gentzen's Natural Deduction rules

I'm beginning to have an understanding thanks to some videos relating to "Proposition as Types". But, I don't come from a theoretical CS background, so maybe I'm blocked probably a bit by notation... Given that I have understood, Conjunction,…
Stephane Rolland
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Prove with natural deduction

Prove P ∨ Q, Q ∨ R, P → ¬R |- Q with natural deduction. P ∨ Q, premiss Q ∨ R, premiss P → ¬R, premiss ... ... Conclusion, Q I dont know how to properly solve this question? I know somewhat how to solve similar questions like below, but on this…
phuck
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Prove (p → ¬q) is equivalent to ¬(p ∧ q)

I need to prove the above sequent using natural deduction. I did the first half already i.e. I proved $(p\rightarrow\neg q)\rightarrow \neg (p \wedge q)$, but I'm stuck on where to start for the reverse i.e. proving $\neg (p \wedge q) \rightarrow…
Smiley
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