1. For a deductive system $\text{L}$, the property that for every disjunctive $\text{L}$-theorem $\phi \vee \psi$, either $\phi$ is theorem of $\text{L}$ or $\psi$ is theorem of $\text{L}$. Most notably, intuitionistic logic enjoys the disjunction property.

2. For a theory $T$ and deductive system $\text{L}$, the property holding of $T$ when $T{⊢}_{\text{L}}\phi \vee \psi$ only if $T⊢{\phi }_{\text{L}}$ or $T⊢{\psi }_{\text{L}}$. The disjunction property is most commonly discussed in the context of intuitionistic logic, e.g., the intuitionistic Heyting Arithmetic $\text{HA}$ enjoys the disjunction property.