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15.6: Conclusion

You have worked hard trying to understand these proofs of soundness and completeness. I too have worked hard, first in understanding them and then in my efforts to write them up in a clear and accessible form. Working on the strength of the presentations of others, I will be very happy if I have made some small contribution to improving the accessibility of soundness and completeness and if I have avoided both horns of the dilemma of too much complication versus inaccuracies in the proofs. Whatever I have accomplished, I am sure that my presentation can be improved. I welcome your comments and suggestions. In the meantime, you should not be discouraged if you have found part I1 of this text to be very difficult. Soundness and completeness are substantial mathematical results. If you understand them only in a fragmentary way, you can greatly improve your grasp by patiently going over these chapters again.


In reviewing this chapter, be sure you have a firm grasp on the following:

  1. Interpretation
  2. Unnamed Object
  3. s-Variant Truth of an Existentially Quantified Sentenced)
  4. Truth of a Universally Quantified Sentence
  5. Infinite Tree
  6. Infinite Semantic Tableau Derivation
  7. Downward correctness of a Truth Tree Rule
  8. Upward correctness of a Truth Tree Rule
  9. Interpretation Represented by an Infinite Truth Tree Branch
  10. Restrictions on the Derivation Rule for VI
  11. Restrictions on the Derivation Rule for 3E
  12. Terminal Interpretation in a Semantic Tableau Derivation
  13. Closed Semantic Tableau Derivation
  14. Compactness for Predicate Logic
  15. Interpretation for a Language with Identity
  16. Interpretation for a Language with Function Symbols


  • Paul Teller (UC Davis). The Primer was published in 1989 by Prentice Hall, since acquired by Pearson Education. Pearson Education has allowed the Primer to go out of print and returned the copyright to Professor Teller who is happy to make it available without charge for instructional and educational use.

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