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Introduction to Mathematical Logic
Book · May 2017
DOI: 10.13140/RG.2.2.30866.66245
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Some of the authors of this publication are also working on these related projects:
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Philosophy of Modeling in 1870s: a Tribute to Hans Vaihinger View project
Karlis Podnieks
University of Latvia
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Introduction to Mathematical
Logic
Hyper-textbook for students
by Vilnis Detlovs, Dr. math.,
and Karlis Podnieks, Dr. math.
University of Latvia
This work is licensed under a Creative Commons License and is
copyrighted © 2000-2017 by us, Vilnis Detlovs and Karlis Podnieks.
Sections 1, 2, 3 of this book 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.
Vilnis Detlovs. Memorial Page
In preparation – forever (however, since 2000, used successfully in a real logic course
for computer science students).
This hyper-textbook contains links to:
Wikipedia, the free encyclopedia;
MacTutor History of Mathematics archive
of the University of St Andrews;
MathWorld of Wolfram Research.
2
Table of Contents
References..........................................................................................................3
1. Introduction. What Is Logic, Really?.............................................................4
1.1. Total Formalization is Possible!..............................................................5
1.2. Predicate Languages.............................................................................10
1.3. Axioms of Logic: Minimal System, Constructive System and Classical
System..........................................................................................................27
1.4. The Flavor of Proving Directly.............................................................40
1.5. Deduction Theorems.............................................................................43
2. Propositional Logic......................................................................................53
2.1. Proving Formulas Containing Implication only...................................54
2.2. Proving Formulas Containing Conjunction..........................................55
2.3. Proving Formulas Containing Disjunction...........................................57
2.4. Formulas Containing Negation – Minimal Logic.................................59
2.5. Formulas Containing Negation – Constructive Logic..........................64
2.6. Formulas Containing Negation – Classical Logic................................66
2.7. Constructive Embedding. Glivenko's Theorem....................................69
2.8. Axiom Independence. Using Computers in Mathematical Proofs........71
3. Predicate Logic.............................................................................................86
3.1. Proving Formulas Containing Quantifiers and Implication only..........86
3.2. Formulas Containing Negations and a Single Quantifier.....................89
3.3. Proving Formulas Containing Conjunction and Disjunction................98
3.4. Replacement Theorems.........................................................................99
3.5. Constructive Embedding.....................................................................106
4. Completeness Theorems (Model Theory)..................................................115
4.1. Interpretations and Models.................................................................115
4.2. Classical Propositional Logic − Truth Tables.....................................129
4.3. Classical Predicate Logic − Gödel's Completeness Theorem.............138
4.4. Constructive Propositional Logic – Kripke Semantics.......................158
5. Normal Forms. Resolution Method............................................................178
5.1. Prenex Normal Form..........................................................................180
5.2. Skolem Normal Form.........................................................................188
5.3. Conjunctive and Disjunctive Normal Forms......................................193
5.4. Clause Form........................................................................................197
5.5. Resolution Method for Propositional Formulas..................................203
5.6. Herbrand's Theorem............................................................................211
5.7. Resolution Method for Predicate Formulas........................................217
6. Miscellaneous.............................................................................................230
6.1. Negation as Contradiction or Absurdity.............................................230
3
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)
Podnieks K. [1997] What is Mathematics: Gödel's Theorem and Around.
1997-2015 (available online, Russian version available).
4
1. Introduction. What Is Logic, Really?
WARNING! In this book,
predicate language is used as a synonym of first order language,
formal theory – as a synonym of formal system, deductive system,
constructive logic – as a synonym of intuitionistic logic,
algorithmically solvable – as a synonym of recursively solvable,
algorithmically enumerable – as a synonym of recursively enumerable.
What is logic?
See also Factasia Logic by Roger Bishop Jones.
In a sense, logic represents the most general means of reasoning used by
people and computers.
Why are means of reasoning important? Because any body of data may
contain not only facts visible directly. For example, assume the following data:
the date of birth of some person X is January 1, 2000, and yesterday,
September 14, 2010 some person Y killed some person Z. Then, most likely, X
did not kill Z. This conclusion is not represented in our data directly, but can
be derived from it by using some means of reasoning – axioms (“background
knowledge”) and rules of inference. For example, one may use the following
statement as an axiom: “Most likely, a person of age 10 can´t kill anybody”.
There may be means of reasoning of different levels of generality, and of
different ranges of applicability. The above “killer axiom” represents the
lowest level – it is a very specific statement. But one can use laws of physics
to derive conclusions from his/her data. Theories of physics, chemistry,
biology etc. represent a more general level of means of reasoning. But can
there be means of reasoning applicable in almost every situation? This – the
most general – level of means of reasoning is usually regarded as logic.
Is logic absolute (i.e. unique, predestined) or relative (i.e. there is more than
one kind of logic)? In modern times, an absolutist position is somewhat
inconvenient – you must defend your “absolute” concept of logic against
heretics and dissidents, but very little can be done to exterminate these people.
They may freely publish their concepts on the Internet.
So let us better adopt the relativist position, and define logic(s) as any
common framework for building theories. For example, the so-called
absolute geometry can be viewed as a common logic for both the Euclidean
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