1. (model-theoretic completeness) The property of a theory $T$ holding whenever $T$ is the theory of a model $\mathfrak{M}$ with respect to a deductive system $\text{L}$, i.e., whenever there exists an $\text{L}$-model $\mathfrak{M}$ such that $T=T{h}_{\text{L}}\left(\mathfrak{M}\right)$.

2. (negation completeness) Describes theories $T$ when for all formulae $\phi$ and a unary negation connective $\neg$, the following holds:

either $\phi \in T$ or $\neg \phi \in T.$

3. For a logic $\text{L}$, the property that a corresponding semantic consequence relation ${\models }_{\text{L}}$ enjoys with respect to a syntactic consequence relation ${⊢}_{\text{L}}$ when the extension of ${\models }_{\text{L}}$ is appropriately ‘included’ in that of ${⊢}_{\text{L}}$. When this inclusion is such that all sentences that are valid according to ${\models }_{\text{L}}$ are also theorems according to ${⊢}_{\text{L}}$ (i.e., whatever is logically true is provable), it is said that ${\models }_{\text{L}}$ is weakly complete with respect to ${⊢}_{\text{L}}$. When the consequence relations for a logic $\text{L}$ enjoy the stronger inclusion that whenever $\Gamma {\models }_{\text{L}}\phi$, also $\Gamma {⊢}_{\text{L}}\phi$ (i.e., $\Gamma {\models }_{\text{L}}\phi$ entails that $\Gamma {⊢}_{\text{L}}\phi$), it is said that ${\models }_{\text{L}}$ is strongly complete with respect to ${⊢}_{\text{L}}$.