The logical framework

Definition: Statement

A statement is a declarative sentence that is either true, denoted by or false, denoted by .

Definition: Logical operations

For any statements , we define the following statements:

pronounced "not " is if , otherwise .
pronounced " and " is if both and , otherwise .
pronounced " or " is if either or , otherwise .
pronounced " implies " is if and not , otherwise .
pronounced " if and only if " is if either both and , or neither nor , otherwise . If , we call and logically equivalent.

The definition of deserves a small discussion. This definition feels weird since we are not used to " implies " being a logical statement in the same sense as " and ", but of course, it is either true or false, so it is a logical statement. What we need to keep in mind is that we are often not interested in knowing if , but rather in proving . So we often know that the statement is true, without knowing whether or are true. Then if we succeed in proving , we also know .

Remark

We can define

Theorem: Contrapositive

For any two statements , Proof:
We have

So proving that can be done by proving that , which is often useful. The statement is called the contrapositive of the statement , and the theorem above thus shows that these are logically equivalent.