Automated Theorem Proving by Wolfgang Bibel (auth.)

By Wolfgang Bibel (auth.)

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0. A matrix F is called in normal form if maxrt:n (F) 6 (r) " 2 • The elements of a matrix in normal form are also called clauses , denoted by c, d, e. A clause is called a Horn clause if it contains at most one literal of the form (l,P) for P E pO • matrix is called a Horn matrix form, and all its clauses are Horn clauses. A i f it is in normal o With the relation established between matrices and formulas, these definitions also apply to the represented formulas. But as we said before, this goes without explicit mentioning.

THE SEMANTICS OF PROPOSITIONAL LOGIC In the previous section we have introduced the language of propositional logic in two forms. In the view taken in this book it consists of the set of matrices while in its traditional form it consists of the set of formulas. Both forms are so closely related, however, that in most cases we may even ignore the difference and identify the two concepts. Any such matrix corresponds to a compound statement in natural language as has been illustrated in the first chapter.

Therefore (F,Sn-l) r- is a connection deduction. Formally, this argument is again an induction (on exercise. n-1) which is left to the reader For this deduction the induction as an easy (of the hypo~thesis main induction) may be applied since n-1

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