Can a premise imply contradictory statements?

Question

Can a premise imply contradictory statements? Can two contradictory premises imply the same conclusion? Determine the answers to these questions by doing the following.

1. Prove or disprove: the following is a contradiction $(p\rightarrow q)\land(p\rightarrow\neg q)$

2. Prove or disprove: the following is a contradiction $(p\rightarrow q)\land(\neg p\rightarrow q)$

I'm assuming that I can prove these to be true, because the premise of an implication and its conclusion do not have to be related in any fashion, but I still don't understand how to answer these questions.

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Question

• Where should I start to better understand this concept of contradiction?

Answer 1

"Can a premise imply contradictory statements?"

A false premise implies anything.

"Can two contradictory premises imply the same conclusion?"

Anything implies a true conclusion.

(1) and (2): truth tables are allowed?

Answer 2

You could make a truth table. It's not difficult for just two variables.

But to get some intuitive notion for their satisfiability, you can see that the first statement is the entire premise behind proof by contradiction.

For the second statement, you can notice that if a tautology (such as $p\vee\neg p$) implies a statement $q$, then $q$ must be true. This should help you see that the second is satisfiable.

Answer 3

Some hints, assuming you are using natural deduction, not truth tables:

In (1), what do you obtain if you assume $p$ is true?

In (2), either $p$ is true or $p$ is false by the law of the excluded middle. What do you obtain in either case?