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Public lectures: What is mathematics? - What is science? Science is modeling!

Description Logics and Reasoners (in Latvian)

Any comments are welcome - e-mail to Karlis.Podnieks@mii.lu.lv

My favorite (printed) textbook on mathematical logic, since many years:
"Introduction to Mathematical Logic", by Elliott Mendelson

Vilnis Detlovs. Memorial Page (in Latvian)

Introduction to Mathematical Logic

Hyper-textbook for students
by Vilnis Detlovs, Dr. math.,
and Karlis Podnieks, Dr. math.
University of Latvia

Sections 1, 2, 3 represent an extended translation of the corresponding chapters of the book: V.Detlovs, Elements of Mathematical Logic, Riga, University of Latvia, 1964, 252 pp. ( in Latvian). With kind permission of Dr. Detlovs.
In preparation - forever (however, already for 8 years, used successfully in a real logic course for computer science students).
Creative Commons License This work is licensed under a Creative Commons License and is copyrighted © 2000-2008 by  us, Vilnis Detlovs and Karlis Podnieks.


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Table of Contents

1. Introduction. What is logic, really?

  • 1.1. Total formalization is possible!
  • 1.2. Predicate languages
  • 1.3. Axioms of logic: minimal system, constructive system and classical system
  • 1.4. The flavour of proving directly
  • 1.5. Deduction theorems
    •

    2. Propositional logic

  • 2.1. Proving formulas containing implication only
  • 2.2. Proving formulas containing conjunction
  • 2.3. Proving formulas containing disjunction
  • 2.4. Formulas containing negation - minimal logic
  • 2.5. Formulas containing negation - constructive logic
  • 2.6. Formulas containing negation - classical logic
  • 2.7. Constructive embedding. Glivenko's theorem
  • 2.8. Axiom independence. Using computers in mathematical proofs
    •

    3. Predicate logic

  • 3.1. Proving formulas containing quantifiers and implication only
  • 3.2. Formulas containing negations and a single quantifier
  • 3.3. Proving formulas containing conjunction and disjunction
  • 3.4. Replacement theorems
  • 3.5. Constructive embedding
    •

    4. Completeness theorems (model theory)

  • 4.1. Interpretations and models
  • 4.2. Classical propositional logic - truth tables
  • 4.3. Classical predicate logic - Goedel's completeness theorem
  • 4.4. Constructive propositional logic - Kripke semantics
  • 4.5. Constructive predicate logic - Kripke semantics
    •

    5. Normal forms. Resolution method - with Proofs!

  • 5.1. Prenex normal form
  • 5.2. Conjunctive and disjunctive normal forms
  • 5.3. Skolem normal form
  • 5.4. Clause form
  • 5.5. Resolution method for propositional formulas - with proofs!
  • 5.6. Herbrand's theorem - with proofs!
  • 5.7. Resolution method for predicate formulas - with proofs!
    •

    6. Complexity and unsolvability

  • 6.1. Classical propositional logic - complexity
  • 6.2. Classical predicate logic - unsolvability
  • 6.3. Constructive propositional logic - complexity
  • 6.2. Constructive predicate logic - unsolvability
  • ...
    •

    7. Miscellaneous

  • 7.1. Negation as contradiction or absurdity
  • 7.2. Finite interpretations - Trakhtenbrot's theorem
  • 7.3. Principle of duality
  • 7.4. Set algebra
  • 7.5. Switching circuits
  • 7.6. Kolmogorov interpretation
  • 7.7. Markov' s principle
    •

     

    8. References

    Hilbert D., Bernays P. [1934]

    Grundlagen der Mathematik. Vol. I, Berlin, 1934, 471 pp. (Russian translation available)

    Kleene S.C. [1952]

    Introduction to Metamathematics. Van Nostrand, 1952 (Russian translation available)

    Kleene S.C. [1967]

    Mathematical Logic. John Wiley & Sons, 1967 (Russian translation available)

    Mendelson E. [1997]

    Introduction to Mathematical Logic. Fourth Edition. International Thomson Publishing, 1997, 440 pp. (Russian translation available)

     

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    mathematical logic, tutorial, what is logic, logic, mathematical, online, hyper-text, web, book, textbook, teaching, learning, study, student, Podnieks, Karlis, Detlovs, Vilnis, introduction, students, hypertext, text, hyper, free, download