As a sentential logic, intuitionistic logic plus the law of the excluded middle gives classical logic.


Is there a logical law that is consistent with intuitionistic logic but inconsistent with classical logic?



Alternatives to the law of the excluded middle
No, every consistent propositional logic that extends intuitionistic logic is a sublogic of classical logic. (That’s why consistent superintuitionistic logics are also called *intermediate* logics.)


To see this, assume that a logic $L\supseteq\mathbf{IPC}$ proves a formula $\phi(p\_1,\dots,p\_n)$ that is not provable in $\mathbf{CPC}$. Then there exists an assignment $a\_1,\dots,a\_n\in\{0,1\}$ such that $\phi(a\_1,\dots,a\_n)=0$. Being a logic, $L$ is closed under substitution; thus, it proves the substitution instance $\phi'$ of $\phi$ where we substitute each variable $p\_i$ with $\top$ or $\bot$ according to $a\_i$. But already intuitionistic logic can evaluate variable-free formulas, in the sense that it proves each to be equivalent to $\top$ or to $\bot$ in accordance with its classical value. Thus, $\mathbf{IPC}$ proves $\neg\phi'$, which makes $L$ inconsistent.


