Logical Thinking Becomes Automatic

Abstract

In the first phase of AI research, the search for general problem-solving methods was successful at least in formal logic. A mechanical procedure was given to prove the logical truth of formulas. The procedure could also be carried out by a computer program and introduced automatic proving in computer science.

The basic idea is easy to understand. In algebra, letters x, y, z… are used by arithmetic operations such as add (+) or subtract (−). The letters serve as spaces (variables) to insert numbers. In formal logic, propositions are represented by variables A, B, C…, which are connected by logical connectives such as (∧), “or” (∨)′ “if-then” (→), “not” (¬). The propositional variables serve as blanks to use statements that are either true or false. For example, the logical formula A ∧ B, by using the true statements 1 + 3 = 4 for A and 4 = 2 + 2 for B, is transformed into the true statement 1 + 3 = 4 ∧ 4 = 2 + 2. In arithmetic, this leads to the true conclusion 1 + 3 = 4 ∧ 4 = 2 + 2→1 + 3 = 2 + 2. But, in general, the conclusion A ∧ B → C is not true? On the other hand, the conclusion A ∧ B → A logically generally valid, since for the insertion of any true or false statements for A and B there is always a true overall statement.