File : pdf, 4 MB, 179 pages
by Michal Walicki
TOC
A. Logic – patterns of reasoning
A1. Reductio ad absurdum
A2. Aristotle
A3. Other patterns and later developments
B. Logic – a language about something
B1. Early semantic observations and problems
B2. The Scholastic theory of supposition
B3. Intension vs extension
B4. Modalities
C. Logic – a symbolic language
C1. The “universally characteristic language”
D. 19th and 20th Century – mathematization of logic
D1. George Boole
D2. Gottlob Frege
D3. Set theory
D4. 20th century logic
E. Modern Symbolic Logic
E1. Formal logical systems : syntax
E2. Formal semantics
E3. Computability and Decidability
Part I. Basic Set Theory
1. Sets, Functions, Relations
11. Sets and Functions
12. Relations
13. Ordering Relations
14. Infinities
2. Induction
21. Well-Founded Orderings
211. Inductive Proofs on Well-founded Orderings
22. Inductive Definitions
221. “1-1″ Definitions
222. INductive Definitions and Recursive Programming
223. Proofs by Structural Induction
23. Transfinite Induction [optional]
3. Turing Machines
31. Alphabets and Languages
32. Turing Macines
321. Composing Turing machines
322. Alternative representation of TMs [optional]
33. Universal Turing Machine
34. Decidability and the Halting Problem
Part III. Statement Logic
4. Syntax and Proof Systems
41. Axiomatic Systems
42. Syntax of SL
43. The axiomatic system of Hilbert’s
44. Natural Deduction system
45. Hilbert vs ND
46. Provable Equivalence of formulae
47. Consistency
48. The axiomatic system of Gentzen’s
481 Decidability of the axiomatic systems for SL
482 Gentzen’s rules for abbreviated connectives
49. Some proof techniques
5. Semantics of SL
51. Semantics of SL
52. Semantic properties of formulae
53. Abbreviations
54. Sets, Proporitions, and Boolean Algebras
541. Laws
542. Sets and SL
543. Boolean Algebras
6. Soundness and Completeness
61. Adequate Sets of Connectives
62. DNF, CNF
63. Soundness
64. Completeness
641. Some Applications of Soundness and Completeness
Part IV. Predicate Logic
7. Syntax and Proof System of FOL
71. Syntax of FOL
72. Scope of Quantifiers, Free Variables,
Substitution
73. Proof Systems
74. Gentzen’s system for FOL
8. Semantics
81. Semantics of FOL
82. Semantic properties of formulae
83. Open vs closed formulae
9. More Semantics
91. Prenex operations
92. A few bits of Model Theory
93. “Syntactic” semantic and Computations
931. Reachable structures and Term structures
932. Herbrand’s theorem
933. Horn clauses and logic programming
10. Soundness, Completeness
11. Identity and Some Consequences
111. FOL with Identity
112. A few more bits of Model Theory
113. Semi-Decidability and Undecidability of FOL
114. Why is First-Order Logic “First Order”?
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