Understanding Symbolic Logic

Preface
Table of Suggested Courses
Explanation of Grading of Exercises
Part ONE: Truth-Functional Logic
1. Preliminaries
1.0. Scope of Text
1.1. Symbolic Logic; the Logistic Method
2. Truth-functional Sentence Connectives (I)
2.0. Conjunction
2.1. Disjunction or Alternation
2.2. Negation
3. Exercises
4. Language Schema P
4.0. Vocabulary
4.1. Formation Rules
4.2. Use and Mention; Quotation Marks and Corners
5. Exercises
6. Semantics of Language Schema P
6.0. Semantics
6.1. Truth Tables
6.2. Interpreting Language Schema P
7. Exercises
8. Logical Truth and Analyticity
8.0. Logical Truth and Logical Falsehood
8.1. Analytic and Synthetic Sentences
9. Exercises
10. Abbreviation and Equivalence
10.0. Some Abbreviative Conventions
10.1. Equivalence
10.2. Tolerably Ambiguous Abbreviations
11. Exercises
12. Functional Completeness
12.0. Mutual Entailment, Equivalence, and Expressive Power
12.1. Functional Completeness of the Tilde, Wedge, and Dot
13. Truth-functional Sentence· Connectives (II)
13.0. Truth-functional Sentence Connectives; a Second Look
13.1. Redundancy of the Tilde, Wedge, and Dot
13.2. Sheffer's Stroke
14. Implication and Equivalence
14.0. The Conditional
14.1. The Biconditional
14.2. Implication; Consequence Relation
14.3. Short-cut Test for Validity and Implication
14.4. More Abbreviative Conventions
14.5. Equivalence and Implication
15. Exercises
16. Normal Forms; Duality
16.0. Normal Forms
16.1. Reduction to Normal Form
16.2. Simple Disjunctive Normal Form
16.3. Duality
17. Exercises
18. Boolean Equations; Electrical Circuits
18.0. Boolean Equations
18.1. Design of Electrical Circuits
19. Exercises
20. Application of Formalized Languages to the Logical Analysis of Natural Languages
20.0. Proving Correctness of Arguments
20.1. Proving Incorrectness of Arguments
20.2. Rendering Logical Structure Explicit
20.3. Bringing Logical Form to the Surface
20.4. The Semantics of Atomic Sentences
21. Exercises
22. Functional Incompleteness
22.0. Mathematical Induction
22.1. Strong Mathematical Induction
22.2. Functional Incompleteness of the Dot and Wedge
23. Exercises
24. Alternative Notations
24.0. Trivial Alternatives
24.1. Nontrivial Alternative: Polish Notation
25. Exercises
Part TWO: Axiomatization of Truth-Functional Logic
26. Axiomatic System of Truth-Functional Logic
26.1. Basic Concepts of Axiomatics
27. Exercises
28. Metatheory of System P (I)
28.0. Consistency of System P
28.1. Independence of Axioms and Rules
28.2. Independence and Consistency
29. Exercises
30. Metatheory of System P (II)
30.0. The Deduction Theorem
30.1. Some Key Theorem Schemata
30.2. Maximal Consistent Classes
30.3. Completeness Theorem
30.4. Compactness Theorem and Concluding Remarks
31. Exercises
Part THREE: Sentential Modal Logic
32. Truth Tables and Modal Logic
32.0. Motivation
32.1. Actual and Possible Truth-Value Outcomes
32.2. Language Schema M
32.3. Full and Partial Truth Tables
32.4. Fundamental Truth Tables Revisited
33. Exercises
34. Validity in LSM
34.0. Value Assignments; Plenary Sets of Truth Tables
34.1. Validity
34.2. Relation of LSP to LSM
34.3. Analytic Truth, Logical Truth, Entailment, Implication, and Equivalence
35. Exercises
36. Truth-Tabular Connectives
36.1. The Singulary Truth-Tabular Connectives
36.2. N-ary Truth-Tabular Connectives: Strict Implication
36.3. Strict Equivalence; Compatibility; the Star
37. Exercises
38. Functional Completeness; Reduction of Modal WFFs
38.0. Functional Completeness
38.1. Reduction of Modal Wffs
38.2. The Six Modalities
39. Exercises
40. Axiomatic Modal Logic
40.0. Primitive Basis of the System S5
40.1. Relationship to System P; Consistency of S5
40.2. Deduction Theorem; Key Theorem Schemata
40.3. Completeness Theorem for S5
40.4. The System S5' (Completeness)
40.5. The System S5' (Consistency)
41. Exercises
Part FOUR: Quantification Theory
42. Atomic Analysis
42.0. Molecular Analysis versus Atomic Analysis
42.1. Singular Terms
42.2. Predicates and Circled Numerals
42.3. Transparent versus Opaque Predicates
42.4. Individual Variables and Predicate Variables
43. Exercises
44. Semantics of Atomic WFFs
44.0. Semantics of Individual Variables
44.1. Semantics of Predicate Variables
44.2. Semantics of Atomic Wffs
45. Exercises
46. Quantifiers
46.0. Existential and Universal Quantifiers
46.1. Grammar of LSQ
46.2. Free and Bound Variables
46.3. Interpretations; Minimal Interpretations
46.4. Inductive Definition of the Value of a Wff Under a Minimal Interpretation
46.5. Applying the Inductive Definition
47. Exercises
48. Model Theory
48.0. Models: Satisfiability and Validity
48.1. Inflation Theorem
48.2. Löwenheim Theorem; Spectrum Problem
48.3. Generalized Inflation Theorem; Löwenheim-Skolem Theorem
48.4. Implication and Truth-Functional Implication; Equivalence
49. Exercises
50. Logical Analysis of English Discourse
50.0. Logical Truth and the Empty Domain
50.1. Universes of Discourse of English Statements
50.2. Translating English into Q-Languages
51. Exercises
52. Quine's System of Natural Deduction (I)
52.0. Instances
52.1. Natural Deduction Systems versus Logistic Systems
52.2. Rule of Premiss
52.3. Rule of Truth Functions
52.4. Rule of Universal Instantiation
52.5. Rule of Existential Generalization
52.6. Rule of Conditionalization
52.7. The Five Soundness-Preserving Rules
53. Exercises
54. Quine's System of Natural Deduction (II)
54.0. Conservative Instances
54.1. Rule of Universal Generalization
54.2. Rule of Existential Instantiation
54.3. Rationale Behind Universal Generalization and Existential Instantiation
54.4. Finished Deductions; Proofs; Metatheorems
54.5. Proof of the Consistency Theorem
55. Exercises
56. Applying the Natural Deduction System
56.0. Deductive Strategies
56.1. Time-Savers
56.2. Identity
56.3. Postulate Systems; Calculus of Individuals
56.4. Completeness Theorem for System QI
57. Exercises
58. Proof of the Completeness Theorem for System Q
58.0. Corollaries of the Skolem-Gödel Theorem
58.1. Maximal, Consistent, ω-Complete Classes
58.2. Proof of the Lemma of Section 58.0
59. Quantification with Function Variables
59.0. Function Variables; Terms
59.1. Natural Deduction Rules for System QIF
59.2. Peano Arithmetic
60. Exercises
61. Decision Problems and Incompleteness
61.0. Decidability; Church's Thesis
61.1. Church's Theorem
61.2. Gödel Incompleteness Theorem
62. Special Cases of the Decision Problem
62.0. Special Cases
62.1. Syllogisms
62.2. Reduction of Decision Problems; Prenex Normal Forms
63. Exercises
Appendixes
A. Set Theory
Introduction to Set Theory
B. Semantic Tableaux
Semantic Tableaux for Truth-Functional Logic
Semantic Tableaux for Quantificational Logic
C. Alternative Proof of the Completeness of System P
D. Alternative Proof of the Compactness Theorem for System P
E. Alternative Proof of the Completeness of System Q
F. Alternative Approaches to the Semantics of Quantifiers
G. Quantification Theory with Modality
Kripke's 1959 Semantics; LSQ-M
System Q-M; Natural Deduction System of Quantification Theory with Modality
Alternative Semantics for Quantification Theory with Modality (KA)
Alternative Semantics for Quantification, Theory with Modality (KB)
H. Tense Logic
I. Logistic System of Quantification Theory
Index
ABC
DE
F
GHIJK
LM
NOP
QR
S
T
UV
W