A form of inference according to which for some proposition $\phi$, if even the falsity of $\phi$ entails its truth, then the truth of $\phi$ can be inferred. Formally, consequentia mirabilis can be represented by the axiom scheme:

• • $\left(\neg \phi \to \phi \right)\to \phi$

where $\neg$ and $\to$ are the negation and conditional connectives respectively. (Or, assuming the principle of double negation, $\left(\phi \to \neg \phi \right)\to \neg \phi$.)

Consequentia mirabilis can be recognized as an instance of proof by cases where the truth and falsity of $\phi$ exhausts all cases. Its justification may be described by observing that $\phi$ is trivially true in the first case, i.e., the case in which it is true, entailing that if the case in which $\phi$ is false (or $\neg \phi$ is true) is also one in which $\phi$ is true, then $\phi$ is true in all cases. This makes clear the duality between consequentia mirabilis and its negative counterpart reductio ad absurdum, according to which if the assertion that $\phi$ is true entails its own falsehood, then one is licensed to reject $\phi$.