A Deficiency of Natural Deduction

Natural languages are ill-suited to express mathematical reasoning. The purpose of an artificial language-predicate logic-with a deduction system is to enable us to express reasoning clearly and concisely. In this article we give two examples illustrating how Gentzen's system of 'natural de-duction' [2] fails to meet the case. In [1], the following problem is stated and solved: Given two caskets, gold and silver, one of which contains a portrait of a lady. Both caskets bear an inscription: on the gold casket is written "The portrait is not in here", and on the silver one "Exactly one of these inscriptions is true". Which of the two caskets contains the portrait? The problem is solved by proving that the portrait is in the gold casket. The proof is conducted in the formal system of natural deduction and comprises 29 steps. Here is another formal proof: let G and S be the inscriptions on the gold and silver caskets, respectively. Then we write S formally as S-~G and derive = true = (definition of S) s-(associativity of-} = { reflexivity of-} true =-,G = {identity element of = } ~G = {definition of G} "The portrait is in the gold casket". This proof is 5 steps long. Although the proof in [1] is a bit longer than necessary-the present author knows of a proof of 21 steps-, the difference in length between the two proofs illustrates a defect of Gentzen's system: it does not handle equivalence efficiently. Equivalence of A and B can only be expressed by something like (A =* B) A (B =* A) or (A A B) V (~A A ~B); since A and B occur twice, these formulae are twice as long as desirable. In case of more than