How to prove that the law of the excluded middle is necessary? This is a follow up question to this answer by Carl Mummert to the question whether every proof with contradiction can also be proved without contradiction. As Carl Mummert pointed out, there are proofs in classical logic which cannot be proved with intuitionistic logic, i.e. which need the law of the excluded middle.
Is there a way or method to show, that a theorem can just be shown with the law of the excluded middle?
Because in minimal logic there is also no principle of explosion: Is it possible to show/prove, that any proof of a theorem needs the principle of explosion?
 A: Do you mean theorems which cannot be proven in intuitionistic logic, or classical proofs of theorems that do not have a mere translation to intuitionistic logic? For an example of the latter, there is the classic elementary proof that "there exist $a$,$b$, both irrational, so that $a^b$ is rational". This considers $a=b=\sqrt{2}$, then says $a^b$ is either rational or irrational and splits the problem into cases, without saying which case actually applies. As I recall the relevant case has actually been identified since, so an intuitionistic proof of this result does exist.
An example of the former is relevant to smooth infinitesimal analysis: one cannot show that $x^2=0$ implies $x=0$ under intuitionistic logic, because trichotomy does not hold without the law of excluded middle. 
A: Make the law of the excluded middle ApNp into your only axiom for classical logic, and select suitable rules of transformation and replacement.  This gets done in Appendix D of the textbook Elementary Symbolic Logic by Ulrich and Gustason.  Consequently, all theorems in classical logic become consequences of the law of the excluded middle.
One could also make the principle of explosion CKpNpq, or CpCNpq, or CNpCpq into one's only axiom if one has suitable rules. 
