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Introduction to Mathematical Logic, Edition 2021
Abstract
NEW EDITION 2021: more and better motivations, chapter about tableaux method added, improved treatment of resolution method. [[[[[]]]]]Textbook for students in mathematical logic. Part 1. CONTENTS. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux method. Resolution method. Herbrand's theorem. [[[[[[]]]]]] For Part 2 see my book at https://www.researchgate.net/publication/306112247_What_is_Mathematics_Godel's_Theorem_and_Around
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this report. These are: 1. The one literal rule also known as the unit rule
52 2.1. Proving Formulas Containing Implication only
- ................................................................................. . Propositional Logic
Propositional Logic......................................................................................52
2.1. Proving Formulas Containing Implication only...................................52
2.2. Proving Formulas Containing Conjunction..........................................53
2.3. Proving Formulas Containing Disjunction...........................................56
2.4. Formulas Containing Negation -Minimal Logic.................................58
2.5. Formulas Containing Negation -Constructive Logic..........................63
2.6. Formulas Containing Negation -Classical Logic................................65
2.7. Constructive Embedding: Glivenko's Theorem....................................69
2.8. Axiom Independence. Using Computers in Mathematical Proofs........71
Russian translation available: "Kib. sbornik (novaya seriya)
- J A Robinson
J. A. Robinson. A machine-oriented logic based on the resolution principle, "Jour. Assoc.
Comput. Mach.", vol.12, N1, January 1965, pp.23-41 (available online, Russian translation
available: "Kib. sbornik (novaya seriya)", 7, 1970, pp.194-218)
Computational Logic: Memories of the Past and Challenges for the Future. Computational Logic -CL
- J A Robinson
J. A. Robinson. Computational Logic: Memories of the Past and Challenges for the Future.
Computational Logic -CL 2000, First International Conference, London, UK, 24-28 July,
2000, Proceedings, Springer, Lecture Notes in Computer Science, 2000, Vol. 1861, pp. 1-24
(online copy).
├ C', then, c) If [L 1 -L 11
- Now
Now, if [L 1 -L 8, f→B, MP]:├ C', then,
c) If [L 1 -L 11, MP]:├ C, then [L 1 -L 9, MP]: B 1,..., B n ├ C, where hypotheses
are instances of the axioms L 10 and L 11. Return to case (b). Q.E.D.
Herbrand completed his Ph.D. thesis in 1929. In the same 1929 Gödel completed his doctoral dissertation about completeness (see Section 4.3). In fact, Herbrand's method allows proving of Gödel's Completeness Theorem, but he (Herbrand) "did not notice it
Unlike the proof presented below, the original proof of Herbrand's Theorem does not depend
on Gödel's Completeness Theorem (or Model Existence Theorem). Herbrand completed his
Ph.D. thesis in 1929. In the same 1929 Gödel completed his doctoral dissertation about
completeness (see Section 4.3). In fact, Herbrand's method allows proving of Gödel's
Completeness Theorem, but he (Herbrand) "did not notice it". Why? See
Samuel R. Buss. On Herbrand's Theorem. "Lecture Notes in Computer Science", Vol. 960,
1995, Springer-Verlag, pp.195-209 (available online).