Kanger's choices in automated reasoning

Automated deduction, or automated theorem proving is a branch of science that deals with automatic search for a proof. The contribution of Kanger to automated deduction is well-recognized. His monograph 1957] introduced a calculus LC, which was one of the rst calculi intended for automated proof-search. His article 1963] was later republished as Kanger 1983] in the collection of \classical papers on computational logic". Kanger's 1963] (and also 1959]) calculi used some interesting features that have not been noted for a number of years, and the importance of which in the area of automated deduction has been recognized only much later. Kanger 1963] gives no proofs and uses very succinct presentation. Automated deduction is an area in which very subtle changes in deenitions and assertions may lead to inconsistent conclusions. Kanger's 1963] area was theorem proving in sequent calculi with equality and function symbols. Most papers published in this area before 1995 contained serious mistakes, except for Kanger's. Now, when we are equipped with the impressive amount of techniques developed in this area, we are amazed with the incredible intuition of Kanger that allowed him to choose elegant, interesting (and correct) solutions among many possible choices. This article explains these choices and their place in modern automated deduction. 1 j = ` The title of this section j = ` is the logo of the Association for Logic Programming: truth is equivalent to provability. The equivalence of validity and provability for classical logic was proved by GG odel 1930] and is known as GG odel's completeness theorem. The notions of truth and validity in logic are formulated as semantical properties, while the notion of provability is deened in a purely syntactical way, so there seems to be a gap between the two notions. In 1955{1957 several new proofs of GG odel's completeness theorem appeared Beth 1955, Hintikka 1955, Sch utte 1956, Kanger 1957] in which model theory and proof theory were connected in a very natural manner. They are based 1