# Proof using Rule of Disjunctive Amplification

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• hotvette
In summary, the conversation discusses the rule of disjunctive amplification and its application to logical propositions. It is noted that the rule can be applied by replacing any proposition with its negative and still remain valid. An example of this is shown using the rule of Modus Ponens. The conversation concludes by stating that this application follows the law of substitution.
hotvette
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use of negation in disjunctive amplification
Book shows a proof where a conclusion is reached of: ##\neg r##. The next step says ##\neg r \lor \neg s## using the rule of disjunctive amplification. The rule of disjunctive amplification as I know it is ##p \implies p \lor q##. I don't see how from this you can also say ##\neg p \implies \neg p \lor \neg q##. I can see that the truth table is a tautology so I know it's true, I just don't see how to get there.

Is this as simple as letting ##p = \neg r## and ##q = \neg s##?

hotvette said:
Is this as simple as letting ##p = \neg r## and ##q = \neg s##?
Yes.

Really? So this means I can take any of the logic Rules and just replace anything with its negative and vice versa and it is still valid? Example of M. Ponens ##[p \land (p \implies q] \implies q## can be written as ##[\neg p \land (\neg p \implies \neg q] \implies \neg q##? The book makes no mention of this. I wonder how we are expected to know...

##p## is any proposition, including ##\neg q## (kind of. Technically it follows the law of substitution, but functionally it’s the same thing).

Think through it carefully. If you have ##\neg p## and ##\neg p \implies \neg q##, why wouldn’t you have ##\neg q##?

Sure, that's just M. Ponens. Makes sense, thanks!

## 1. What is the Rule of Disjunctive Amplification?

The Rule of Disjunctive Amplification is a logical rule that states that if either of two statements, known as disjuncts, is true, then the disjunction of those statements is also true. In other words, if A or B is true, then A or B is true.

## 2. How is the Rule of Disjunctive Amplification used in proofs?

The Rule of Disjunctive Amplification is often used in proofs to establish the truth of a disjunction. It allows us to conclude that at least one of the disjuncts is true, without specifying which one. This can be useful in proving more complex statements.

## 3. What are the conditions for using the Rule of Disjunctive Amplification?

In order to use the Rule of Disjunctive Amplification, there must be a disjunction, or "or" statement, present in the premises of the proof. Additionally, at least one of the disjuncts must be known to be true or have been previously proven to be true.

## 4. Can the Rule of Disjunctive Amplification be used in both directions?

Yes, the Rule of Disjunctive Amplification can be used in both directions. This means that if we know that A or B is true, we can conclude that either A is true or B is true. Similarly, if we know that either A is true or B is true, we can conclude that A or B is true.

## 5. Are there any limitations to using the Rule of Disjunctive Amplification?

Yes, there are limitations to using the Rule of Disjunctive Amplification. It can only be used when there is a disjunction present in the premises, and at least one of the disjuncts is known to be true. Additionally, it cannot be used to prove a statement that is not a disjunction.

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