1. The semantic thesis according to which no proposition may be both true and false simultaneously, i.e., the thesis that there exist no truth value gluts. The principle is closely related to a thesis concerning logical consequence, the principle of explosion, according to which from inconsistent premisses, arbitrary formulae may be inferred. The principle of explosion is frequently explained by appeal to the principle of non-contradiction, e.g., if validity is construed as the necessary preservation of truth, then because premisses $\left\{\phi ,\neg \phi \right\}$ can never jointly be true, then truth is vacuously preserved for any $\psi$.

The denial of the principle of non-contradiction—i.e., to suggest that it is possible that a proposition may be both true and false—is closely related to the stronger position of dialetheism, according to which there are such propositions.

2. The thesis that logics ought to be non-contradictory in the sense that it counts no inconsistencies as theorems. More formally, the principle is satisfied by a particular deductive system $\text{L}$ with a negation $\neg$ when $\text{L}$ enjoys the following property:

• • There is a set of formulae $\text{Γ}$ (possibly empty) such that for all formulae $\phi$, either $\text{Γ}{⊬}_{\text{L}}\phi$ or $\text{Γ}{⊬}_{\text{L}}\neg \phi .$

The overwhelming majority of non-classical logics are weaker than classical logic, that is, they license no inferences or theorems not already classically valid and must satisfy the principle of non-contradiction in this sense. Hence, deductive systems that fail to satisfy the principle in this sense are rare.