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Rules of Inference, Logic and Symbology

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Designate each simple statement with a letter. Then write down the compound statements using the following rules(modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, addition, simplification, or resolution), arrive at the conclusion. I've done all but the latter.

If you send an email then I will write the program
If you do not send an email then I will sleep
If I go to sleep then I will be refreshed
If I do not write the program then I will be refreshed.

p -> q
~q -> r
r -> s
~q -> s

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Symbolic Logic is applied.

Prove the following argument:

It is false that both Arthur is not anxious and Billy is not boisterous
If Xavier is difficult and Billy is boisterous then Penelope is a prude
If Xavier is not difficult then Arthur is anxious
Arthur is not anxious
Therefore, it is false that if Xavier is difficult then Penelope is not a prude.

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