where can I read the whole proof of impossibility of proving Law of Excluded middle without proof by contradiction in Natural deduction?

Question

Intuitionistic logic can't prove as many sentences as classical logic, for example: Peirce's Law; Reductio Ad Absurdum; Double Negation Elimination; and Tertium Non Datur, which are all equivalent in classical propositional calculus and they can't be proved in intuitionistic systems.

I can prove their equivalences in some famous system, but I can't prove that we can't prove all of them in intuitionistic proof system especially without using semantics.

I would like to prove this fact only using syntactic properties, i.e. proving these formulas are not in the set of provable sentences or proving that there aren't any proof of them in the set of proofs of intuitionistc proof system, without interpretation into some semantics: Kripke models or Heyting algebra.

Where can I read this topic? is there any great book for this proof?

Answer 1

Intuitionistic logic has the so-called disjunction property: A v B is derivable as a theorem only if one of A or B is itself derivable as a theorem. So that rules out excluded middle generally holding (as it isn't the case, for any A, that one of A or ¬A must be a theorem). [Added: as I now see Z.A.K. explains in his linked comment!]

So what you are looking for is a non-semantic, proof-theoretic, demonstration of the disjunction property. You'll find a nice one in that modern classic, Negri and von Plato's Structural Proof Theory: it's Theorem 2.5.2, where they show an elegant sequent calculus for intuitionistic propositional logic has the disjunction property.