Question

a system with will provide absurd equation

Answer 1

Answer:

They are the same. Any formula or derivation which implies or is equivalent to something of the form B∧¬B for some formula B is considered as a contradiction or an absurd.

Note 1: B∧¬B means "B and (not B)".

The Proof by Contradiction can be stated as the following metatheorem:

Let A be a closed formula. Then Γ⊢A if and only if Γ+¬A is inconsistent.

Note 2: Γ⊢A means that with the set of assumptions Γ I can prove the formula A.

Note 3: in most natural deduction systems The Proof by Contradiction is the name of an inference rule and not a metatheorem, in that case, the proposition that I stated is normally used implicitly to prove the Deduction Theorem.

So, in order to prove A with a set of assumptions Γ you can add ¬A as a new assumption and find any contradiction (it doesn't have to be A∧¬A necessarily, anything in that form will do the trick), as Γ+¬A is inconsistent if and only if it contains a contradiction (don't worry about this "being inconsistent" thing, for now you can consider my previous statement as the definition of inconsistency).

One of the most notorious examples of a proof by contradiction (seeing it as the metatheorem that I stated and not the rule of inference) is the proof that 2–√ is an irrational number and, to do it, you suppose that it isn't, i.e., 2–√=p/q with p,q∈Z and gcd(p,q)=1 and you have a contradiction when you get that ¬gcd(p,q)=1: on one hand you have that gcd(p,q)=1 and, at the same time, ¬gcd(p,q)=1 on the other hand, which means (gcd(p,q)=1)∧(¬gcd(p,q)=1) (a contradiction). So, with that, you conclude that 2–√ is indeed irrational, because if it wasn't, you get an absurd. You can check the complete proof here:

Answer 2

Step-by-step explanation:

BBB

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