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Proof Writing
You should be familiar with the operations \(\vee,\) \(\wedge,\) \(\neg\) and \(\rightarrow\) from mathematical logic for this lesson.

Simple Argument Forms

The following forms of arguments are valid. In other words, they are tautologies.

  1. If \(P\) and \(Q\) are both true, then we can conclude \(P\) is true. \[P \wedge Q \rightarrow P\] Example: I want to know if it is hot outside so I check the weather. The weather report says it is hot and cloudy. Therefore, I know it is hot.
  2. If \(P\) and \(Q\) are both true, then we can conclude \(Q\) is true. \[P \wedge Q \rightarrow Q\] Example: I want to know if it is raining, so I check the weather. The weather report says it is cold and raining. Therefore, I know it is raining.
  3. If \(P\) is true, then we can conclude \(P\) or \(Q\) is true. \[P \rightarrow P \vee Q\] Example: I order a vegetable lovers pizza. My friend asks if the pizza will have vegetables or pepperonis. I can answer yes.
  4. If \(P\) or \(Q\) is true, and \(P\) is not true, then \(Q\) is true.\[(P \vee Q) \wedge \neg P \rightarrow Q\] Exmaple: My professor said he will meet students at the coffee shop or in his office. She is not in the coffee shop. She must be in her office.
  5. If \(P\) is true and \(P\) implies \(Q\), we can conclude \(Q\) is true.\[(P \wedge (P \rightarrow Q)) \rightarrow Q\] Example: It is hot outside. If it is hot outside, then my ice sculpture will melt. My ice sculpture is melting.
  6. If \(P\) implies \(Q\) and \(Q\) is false, we can conclude that \(P\) is false.\[((P \rightarrow Q) \wedge \neg Q) \rightarrow \neg P\] Example: If it is hot outside my ice sculpture will melt. My ice sculpture is not melting. It must not be hot outside.
  7. If \(P\) implies \(Q\) and \(Q\) imples \(R,\) then we can conclude that \(P\) imples \(R.\)\[((P \rightarrow Q) \wedge (Q \rightarrow R)) \rightarrow (P \rightarrow R)\] Example: If I buy an expensive car then I will have to make high monthly payments. If I have to make high monthly payments I will be stressed out. So, if I buy this car I will be stressed out.
  8. If \(P\) or \(Q\) is true, \(P\) implies \(R,\) and \(Q\) implies \(R,\) then \(R\) must be true.\[((P \vee Q) \wedge (P \rightarrow R) \wedge (Q \rightarrow R)) \rightarrow R\] Example: I know the dessert served tonight will be chocolate or strawberry. If it is chocolate, I will like it. If it is strawberry, I will like it. So, I will like the dessert they serve tonight.

Step by step argument

Sometimes you have to combine the rules to draw a conclusion.

Suppose I know the following are true:

From this information, can I conclude \(P\)?

By rule 5: \[(R \wedge (R \rightarrow P \vee Q)) \rightarrow (P \vee Q)\] So, \(P \vee Q\) is true.

By rule 4: \[(P \vee Q) \wedge \neg Q \rightarrow P\] Therefore, we can conclude \(P.\)

The following is a tautology: \[(R \rightarrow P \vee Q) \wedge (\neg Q) \wedge R \rightarrow P\]



You usually cannot apply the rules to show that you cannot draw a conclusion.

Suppose I know the following are true:

From this information, can I conclude \(P\)?

I cannot, because if \(Q = F\) and \(R = T,\) then \(P = T\) or \(P = F\) make all the statements true. I cannot conclude \(P.\)

This statement is not a tautology: \[(P \vee Q \rightarrow R) \wedge (\neg Q) \wedge R \rightarrow P\]