The rule of inference for a deductive system $\text{L}$ according to which for any two formulae $\phi$ and $\psi$ that are logically equivalent in $\text{L}$, one can replace any instance of $\phi$ appearing in a formula with an instance of $\psi$ without loss of generality. Let $\phi ⊣{⊢}_{\text{L}}\psi$ represent the sentence

• • $\phi {⊢}_{\text{L}}\psi$ and $\psi {⊢}_{\text{L}}\phi$

and let $\xi \left[\psi /\phi \right]$ represent any formula $\xi$ in which one or more instances of $\phi$ is replaced by an instance of $\psi$. Then intersubstitutivity can be expressed as

$\frac{\xi }{\xi \left[\psi /\phi \right]}$ when $\phi ⊣{⊢}_L\psi$

While intersubstitutivity is a very natural inference, intersubstitutivity fails in many non-classical logics. In the three-valued logic ${\text{K}}_3$ of Stephen Cole Kleene (1909–1994), for example, the following holds:

• • $p\wedge \neg p⊣{⊢}_{{\text{K}}_3}q\wedge \neg q$

i.e., $p\wedge \neg p$ and $q\wedge \neg q$ are provably equivalent, and

• • $\neg \left(p\wedge \neg p\right){⊢}_{{\text{K}}_3}\neg \left(p\wedge \neg p\right)$

holds as well. However, $q\wedge \neg q$ cannot be substituted for $p\wedge \neg p$ within the scope of the negation, i.e.,

• • $\neg \left(p\wedge \neg p\right){⊬}_{{\text{K}}_3}\neg \left(q\wedge \neg q\right)$

Semantically, this is explained because $p$ may be true in a model while $q$ is a truth value gap, in which case $p\vee \neg p$ will be true and $q\vee \neg q$ will be a gap. In such a model, $\neg \left(p\wedge \neg p\right)$ will be true although $\neg \left(q\wedge \neg q\right)$ will not.