Mathematically speaking, what is a contradiction?

The world of predicate logic interests me, especially how it provides a foundation for understanding the logic behind many mathematical proofs. It is interesting to know how the negation, contrapositive and inverse are defined with respect to some implication  A \Rightarrow B   ( A \wedge \neg B, \neg B \Rightarrow \neg A  and   B \Rightarrow A respectively). What got me thinking about predicate logic again was when I asked myself, “What is a contradiction?”

My big Collins Dictionary and Thesaurus defines ‘contradict’ as “to declare the opposite of (a statement) to be true” (verbatim). But, this leaves some room for debate as the meaning of the word “opposite” is not logically clear. Is the negation true? Is the inverse true? My reasoning says that a contradiction of the above implication is defined as  A \Rightarrow \neg B . In a less formal way (and also less strongly),  A \not \Rightarrow B .

Let me briefly pause here for the sake of those unfamiliar with the symbols I have already used. \neg denotes ‘not’, \Rightarrow denotes ‘implies’, \wedge denotes ‘and’, and A and B are symbols which represent a statement, such as “dogs are black” or “black animals are dogs”.…