Deduction Systems

1. Front Matter

   Pages i-xii

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2. Introduction

   • Rolf Socher-Ambrosius, Patricia Johann

   Pages 1-7

3. Mathematical Preliminaries

   • Rolf Socher-Ambrosius, Patricia Johann

   Pages 8-23

4. Syntax of First-order Languages

   • Rolf Socher-Ambrosius, Patricia Johann

   Pages 24-35

5. Semantics of First-order Languages

   • Rolf Socher-Ambrosius, Patricia Johann

   Pages 36-45

6. The Gentzen Calculus G

   • Rolf Socher-Ambrosius, Patricia Johann

   Pages 46-64

7. Normal Forms and Herbrand’s Theorem

   • Rolf Socher-Ambrosius, Patricia Johann

   Pages 65-90

8. Resolution and Unification

   • Rolf Socher-Ambrosius, Patricia Johann

   Pages 91-131

9. Improving Deduction Efficiency

   • Rolf Socher-Ambrosius, Patricia Johann

   Pages 132-164

10. Resolution in Sorted Logic

    • Rolf Socher-Ambrosius, Patricia Johann

    Pages 165-198

11. Back Matter

    Pages 199-206

    PDF

The idea of mechanizing deductive reasoning can be traced all the way back to Leibniz, who proposed the development of a rational calculus for this purpose. But it was not until the appearance of Frege's 1879 Begriffsschrift-"not only the direct ancestor of contemporary systems of mathematical logic, but also the ancestor of all formal languages, including computer programming languages" ([Dav83])-that the fundamental concepts of modern mathematical logic were developed. Whitehead and Russell showed in their Principia Mathematica that the entirety of classical mathematics can be developed within the framework of a formal calculus, and in 1930, Skolem, Herbrand, and Godel demonstrated that the first-order predicate calculus (which is such a calculus) is complete, i. e. , that every valid formula in the language of the predicate calculus is derivable from its axioms. Skolem, Herbrand, and GOdel further proved that in order to mechanize reasoning within the predicate calculus, it suffices to Herbrand consider only interpretations of formulae over their associated universes. We will see that the upshot of this discovery is that the validity of a formula in the predicate calculus can be deduced from the structure of its constituents, so that a machine might perform the logical inferences required to determine its validity. With the advent of computers in the 1950s there developed an interest in automatic theorem proving.

• Syntax
• automated deduction
• calculus
• complexity
• logic
• proving
• semantics

• FB Elektrotechnik und Informatik, Fachhochschule Ostfriesland, Emden, Germany

  Rolf Socher-Ambrosius

• Pacific Software Research Center, Oregon Graduate Institute, Beaverton, USA

  Patricia Johann

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