This book is based on the notes for a course in logic given by Paul Halmos. This book retains the spirit and purpose of those notes, which was to show that logic can (and perhaps should) be viewed from an algebraic perspective.Propositional logic and monadic predicate calculus-predicate logic with a single quantifier-are the principal topics treated. The connections between logic and algebra are carefully explained. The key notions and the fundamental theorems are elucidated from both a logical and algebraic perspective.The final section gives a unique and illuminating algebraic treatment of the theory of syllogisms-perhaps the oldest branch of logic, and a subject that is neglected in most modern logic texts.The presentation is aimed at a broad audience-mathematics amateurs, students, teachers, philosophers, linguists, computer scientists, engineers, and professional mathematicians. All that is required of the reader is an acquaintance with some of the basic notions encountered in a first course in modern algebra. In particular, no prior knowledge of logic is assumed. The book could serve equally well as a fireside companion and as a course text.
Author(s): Paul Halmos
Publisher: MAA
Year: 2009
Language: English
Pages: 149
Cover
Title Page
Table of Contents
Preface
What is logic?
1. To count or to think
2. A small alphabet
3. A small grammar
4. A small logic
5. What is truth?
6. Motivation of the small language
7. All mathematics
Propositional calculus
8. Propositional symbols
9. Propositional abbreviations
10. Polish notation
11. Language as an algebra
12. Concatenation
13. Theorem schemata
14. Formal proofs
15. Entailment
16. Logical equivalence
17. Conjunction
18. Algebraic identities
Boolean algebra
19. Equivalence classes
20. Interpretations
21. Consistency and Boolean algebra
22. Duality and commutativity
23. Properties of Boolean algebras
24. Subtraction
25. Examples of Boolean algebras
Boolean universal algebra
26. Subalgebras
27. Homomorphisms
28. Examples of homomorphisms
29. Free algebras
30. Kernels and ideals
31. Maximal ideals
32. Homomorphism theorem
33. Consequences
34. Representation theorem
Logic via algebra
35. Pre-Boolean algebras
36. Substitution rule
37. Boolean logics
38. Algebra of the propositional calculus
39. Algebra of proof and consequence
Lattices and infinite operations
40. Lattices
41. Non-distributive lattices
42. Infinite operations
Monadic predicate calculus
43. Propositional functions
44. Finite functions
45. Functional monadic algebras
46. Functional quantifiers
47. Properties of quantifiers
48. Monadic algebras
49. Free monadic algebras
50. Modal logics
51. Monadic logics
52. Syllogisms
Index
