1. With respect to deductive systems $\text{L}$ and ${\text{L}}^{\prime }$ in which the language of $\text{L}$ is a subset (not necessarily proper) of the language of ${\text{L}}^{\prime }$, describes ${\text{L}}^{\prime }$ when for any set of formulae $\Gamma \cup \left\{\phi \right\}$ in the language of $\text{L}$, the following holds:

• • $\Gamma {\models }_{{\text{L}}^{\prime }}\phi$ only if $\Gamma {\models }_{\text{L}}\phi$

That is, ${\text{L}}^{\prime }$ is in agreement with $\text{L}$ with respect to the more modest language. As an example, the modal logic $\text{K}$ is a conservative extension of classical logic because all novel theorems of $\text{K}$ contain an occurrence of the necessity operator $\square$. On the other hand, classical logic is not a conservative extension of intuitionistic logic, as many classical theorems in the language of intuitionistic logic are not intuitionistically valid, e.g., Peirce’s law:

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

2. With respect to theories $T$ and $T^{\prime }$ for a deductive system $\text{L}$ in which the language of $T^{\prime }$ is richer than that of $T$, describes $T^{\prime }$ when the restriction of $T^{\prime }$ to the language of $T$ is precisely $T$. In other words, when ${ℒ}_T$ is the language of $T$, a theory $T^{\prime }$ is a conservative extension of $T$ when $T^{\prime }\cap {ℒ}_T=T$. As a property of theories, restricted notions of conservativity are frequently introduced, especially on the basis of the complexity of first-order formulae. For example, the classical theory $\text{PA}+\neg \text{Con}\left(\text{PA}\right)$, i.e., the deductive closure of Peano Arithmetic together with the sentence asserting the inconsistency of $\text{PA}$, is ${\text{Π}}_1$ conservative over $\text{PA}$, that is, every ${\text{Π}}_1$ sentence provable in $\text{PA}+\neg \text{Con}\left(\text{PA}\right)$ is already provable in $\text{PA}$.