Introduction                       <------  User Comment (blue)

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We want to "prove" that if is sunny, then it is not raining.

We will make the following assumptions:

(a) If it is sunny, then it is not cloudy.

(b) If it is raining, then it is cloudy.

Logical Symbols

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=> means "implies"

~  means "not"

Proof

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Suppose that, if it is sunny, then is it not cloudy.

1     Sunny => ~Cloudy            <------   Line#, Statement  (black)

      Premise                            <------   Rule                          (gray)

Suppose further that, if it is raining, then it is cloudy.

2     Raining => Cloudy

      Premise

     Now, prove that, if it is raining, then it cannot be sunny.

     First, we suppose that it is raining.

      3     Raining

            Premise

     Then, by our second assumption, it must also be cloudy.

      4     Cloudy

            Detach, 2, 3

     Taking the contrapostive of the our first assumption, we obtain...

      5     ~~Cloudy => ~Sunny

            Contra, 1

     Removing the double negation...

      6     Cloudy => ~Sunny

            Rem DNeg, 5

     Since it must be cloudy, it must also not be sunny.

      7     ~Sunny

            Detach, 6, 4

As required, we conclude that, if it is raining, then if must not be sunny.

8     Raining => ~Sunny

      4 Conclusion, 3