How to do indirect proof (reductio ad absurdum) using natural deduction for modal logic?

Does the deduction theorem fail for modal logic?

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems.

Is proof by contradiction reductio ad absurdum?

There is in mathematics a powerful method of proof known as “reductio ad absurdum” (Latin phrase: “reducing to absurdity”) or commonly referred to as “proof by contradiction”. Its reasoning is based on the fact that given a mathemati- cal statement S, either S is true or else not-S (negation of S) is true.

Which method is based on principle of reductio ad absurdum?

One of the oldest solutions is to introduce a new proof method, traditionally called “reductio ad absurdum”, which means a reduction to absurdity. This method is also often called an “indirect proof” or “indirect derivation”.
8.2 Indirect proofs.

P ¬P (P ^ ¬P)

What is RAA proof?

Reductio Ad Absurdum (RAA) is a proof technique that takes advantage of our newly found ability to introduce any assumption into a proof at any time (with the proviso that we properly discharge the assumption).

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How do you use reductio ad absurdum?

Examples of Reductio Ad Absurdum: In a location where there is a sign saying not to pick the flowers, a small child says to his mother, “It’s just one flower.” Mother responds, “Yes, but if everyone who came by picked just one flower, there would be none left.”

Is reductio ad absurdum a logical fallacy?

Arguments that use universals such as, “always”, “never”, “everyone”, “nobody”, etc., are prone to being reduced to absurd conclusions. The fallacy is in the argument that could be reduced to absurdity — so in essence, reductio ad absurdum is a technique to expose the fallacy. Logical Form: Assume P is true.

What is reductio ad absurdum examples?

An example of reductio ad absurdum would be a an anti-Equal Rights advocate claiming that anyone in favor of an Equal Right Amendment must be in favor of killing babies. The reductio ad absurdum fallacy is similar to the straw person fallacy.

How do you write indirect proofs?

Indirect Proofs

  1. Assume the opposite of the conclusion (second half) of the statement.
  2. Proceed as if this assumption is true to find the contradiction.
  3. Once there is a contradiction, the original statement is true.
  4. DO NOT use specific examples. Use variables so that the contradiction can be generalized.

How do you prove a contradiction?

To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequences of this are not possible. That is, the consequences contradict either what we have just assumed, or something we already know to be true (or, indeed, both) – we call this a contradiction.

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What are the three types of proofs?

There are many different ways to go about proving something, we’ll discuss 3 methods: direct proof, proof by contradiction, proof by induction. We’ll talk about what each of these proofs are, when and how they’re used.

Which of the following occurs with a direct proof?

Which of the following occurs with a direct proof? A conditional statement is proven. A series of statements are made. Statements are supported by known facts and definitions.