Logically equivalent formulas and contradiction

Question

$\lnot A \Rightarrow A$ , is a contradiction.

But $\lnot A \Rightarrow A$ is logically equivalent to $A\lor A$. Does it mean that $A\lor A$ always give contradiction?

Answer 1

You're mistaking $\neg (A \Rightarrow A)$ with $(\neg A) \Rightarrow A$:

• $\neg (A \Rightarrow A)$ is false no matter whether $A$ is true or false.
• $(\neg A) \Rightarrow A$ is logically equivalent to $A \vee A$, which is itself logically equivalent to $A$. Thus it is true when $A$ is true and false when $A$ is false.

They are not the same thing. Unless you have some kind of binding convention, the expression $\neg A \Rightarrow A$ is ambiguous, and you should insert parentheses.

Answer 2

A proof by contradiction is a proof in which you make an hypothesis $\neg A$ and get $A$ as a consequence. As you said it means that $\neg A \Rightarrow A$ which is equivalent to $A$.

Finally it is a proof as you have either $A$ or $\neg A$ and in fact the only possible case is $A$.

Answer 3

I would like to add what @clive answered.

The statement ¬ A ⇒ A which reduces to A is more formally called a Contingency