1. The thesis satisfied by a particular deductive system $\text{L}$ with a negation $\neg$ when $\text{L}$ is explosive, that is, when $\text{L}$ satisfies the property that for all sets of formulae $\text{Γ}\cup \left\{\phi ,\psi \right\}$:

• • $\text{Γ},\phi ,\neg \phi {⊢}_{\text{L}}\psi$

In other words, the thesis that the only $\text{L}$-theory is the trivial theory. Paraconsistent logics may be defined as deductive systems with a negation $\neg$ that fail to satisfy the principle of explosion in this sense. Because the inference corresponding to all formulae following from inconsistency or contradiction, the inference is frequently described as ex contradictione quodlibet or ECQ.

The principle of explosion is distinct from but related to the principle of noncontradiction, which asserts that no proposition is both true and false. When the truth of $\neg \phi$ is independent from the truth of $\phi$—as is found in many non-deterministic semantics for logics of formal inconsistency—that $\phi \wedge \neg \phi$ is true merely entails that both $\phi$ and $\neg \phi$ are true and not that either of the conjuncts is false.

2. Where $\to$ is a conditional connective, the axiom scheme:

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

Because the principle has been falsely attributed to philosopher John Duns Scotus (1265–1308), the principle is sometimes known as ‘pseudo Scotus’.