Summary: Algebraic proofs of cut elimination
April 5, 2001
Algebraic proofs of the cut-elimination theorems for classical and intu-
itionistic logic are presented, and are used to show how one can sometimes
extract a constructive proof and an algorithm from a proof that is noncon-
structive. A variation of the double-negation translation is also discussed:
if is provable classically, then ¬(¬)nf
is provable in minimal logic,
denotes the negation-normal form of . The translation is used
to show that cut-elimination theorems for classical logic can be viewed as
special cases of the cut-elimination theorems for intuitionistic logic.
The cut-elimination theorems for classical and intuitionistic logic are a mainstay
of proof theory, and with good reason. Even when it comes to pure first-order
logic, cut-elimination is a remarkably powerful tool, allowing one to extract
additional information from derivations in a wide range of axiomatic theories.
For the classical case, there is a simple, nonconstructive route to proving