A similarly important concept as equivalent propositions is that of contradictory propositions. We have seen that in a general formal language, a contradiction in an expression $\phi$ which is always invalid, i.e. for any interpretation $I$, the corresponding valuation function $[[\phi]]_I$ is false. In the semantics of propositional logic , contradictions are exactly the same concept.
Contradictions are extraordinarily useful tools when we want to find the right conclusion by using the exclusion principle. If we manage to exclude all conclusions because they are invalid and there is only one conclusion left - the last possibility must be valid.
We will now learn some important examples of contradictions.