Table of Contents
A Student's Prayer for Language Proofs and Logic ....................................................................................... 4
Chapter 1: Atomic Sentences................................................................................................................... 5
Introduction.............................................................................................................................................. 5
Subsections and Key Points: ................................................................................................................ 5
- 1.2 Predicate Symbols............................................................................................................................. 5
- 1.3 Atomic Sentences: ............................................................................................................................ 6
- 1.4 General First-Order Languages ......................................................................................................... 6
Combined Quick Notes Summary: ............................................................................................................ 6
Chapter 2: The Logic of Atomic Sentences ................................................................................................... 8
Introduction .............................................................................................................................................. 8
Subsections and Key Points: ..................................................................................................................... 8
- 2.1 Valid and Sound Arguments ............................................................................................................. 8
- 2.2 Methods of Proof .............................................................................................................................. 8
- 2.3 Formal Proofs .................................................................................................................................... 9
- 2.4 Constructing Proofs in Fitch .............................................................................................................. 9
Chapter 3: Boolean Connectives ................................................................................................................. 10
Introduction ............................................................................................................................................ 10
Subsections and Key Points: ................................................................................................................... 10
- 3.1 Negation (¬): ................................................................................................................................... 10
- 3.2 Conjunction (∧): .............................................................................................................................. 10
- 3.3 Disjunction (∨): ............................................................................................................................... 10
- 3.4 Remarks on the Game .................................................................................................................... 11
- 3.5 Ambiguity and Parentheses ............................................................................................................ 11
- 3.6 Equivalent Expressions: .................................................................................................................. 11
- 3.7 Translation ...................................................................................................................................... 11
Combined Quick Notes Summary for Chapters 2 and 3: ........................................................................ 12
Chapter 4: Conjunctive and Disjunctive Normal Forms .............................................................................. 13
Introduction ............................................................................................................................................ 13
Subsections and Key Points: ................................................................................................................... 13
- 4.1 Converting to Normal Forms: ......................................................................................................... 13
- 4.2 Equivalence and Simplification: ...................................................................................................... 13
, - 4.3 Chain of Equivalences ..................................................................................................................... 14
- 4.4 Applications of Normal Forms: ....................................................................................................... 14
Chapter 5: Methods of Proof for Boolean Logic ......................................................................................... 15
Introduction ............................................................................................................................................ 15
Subsections and Key Points: ................................................................................................................... 15
- 5.1 Valid Inference Steps ...................................................................................................................... 15
- 5.2 Proof by Cases................................................................................................................................. 16
- 5.3 Proof by Contradiction: ................................................................................................................... 16
- 5.4 Informal Proof Techniques ............................................................................................................. 16
Combined Quick Notes Summary for Chapters 4 and 5: ........................................................................ 16
Chapter 6: Formal Proofs and Boolean Logic.............................................................................................. 18
Introduction ............................................................................................................................................ 18
Subsections and Key Points .................................................................................................................... 18
- 6.1 Conjunction Rules ........................................................................................................................... 18
- 6.2 Disjunction Rules............................................................................................................................. 18
- 6.3 Negation Rules: ............................................................................................................................... 18
- 6.4 The Proper Use of Subproofs .......................................................................................................... 19
- 6.5 Strategy and Tactics ........................................................................................................................ 19
- 6.6 Proofs without Premises: ................................................................................................................ 19
Chapter 7: Conditionals .............................................................................................................................. 20
Introduction ............................................................................................................................................ 20
Subsections and Key Points: ................................................................................................................... 20
- 7.1 Material Conditional (→): ............................................................................................................... 20
7.3 Conversational Implicature ............................................................................................................... 20
- 7.4 Truth-functional Completeness: ..................................................................................................... 21
Combined Quick Notes Summary for Chapters 6 and 7: ........................................................................ 21
- Conjunction Rules: ............................................................................................................................ 21
- Disjunction Rules: ............................................................................................................................. 21
- Negation Rules: ................................................................................................................................. 21
- Proper Use of Subproofs:.................................................................................................................. 21
- Strategy and Working Backwards: .................................................................................................... 21
- Biconditional (↔):............................................................................................................................ 22
Chapter 8: Conditionals and Biconditionals in Proofs................................................................................. 22
, Introduction ............................................................................................................................................ 22
Subsections and Key Points: ................................................................................................................... 22
- 8.1 Rules for Conditionals: ................................................................................................................ 22
- 8.2 Rules for Biconditionals: ............................................................................................................. 23
Chapter 9: Quantifiers and Their Scope ...................................................................................................... 24
Subsections and Key Points .................................................................................................................... 24
- 9.1 Universal Quantifiers (∀): ........................................................................................................... 24
- 9.2 Existential Quantifiers (∃): .......................................................................................................... 24
- 9.3 Quantifier Scope and Boundaries: .............................................................................................. 25
Combined Quick Notes Summary for Chapters 8 and 9: .................................................................... 25
Chapter 10: The Logic of Quantifiers .......................................................................................................... 26
Introduction ............................................................................................................................................ 26
Subsections and Key Points: ................................................................................................................... 26
- 10.1 Tautologies and Quantification................................................................................................. 26
- 10.2 Valid Inference with Quantifiers ............................................................................................... 27
- 10.3 Multiple Quantifiers .................................................................................................................. 27
Chapter 11: Nested Quantifiers .................................................................................................................. 28
Introduction ............................................................................................................................................ 28
Subsections and Key Points: ................................................................................................................... 28
Chapter 12: Methods of Proof for Quantifiers ........................................................................................... 30
Introduction ............................................................................................................................................ 30
Subsections and Key Points: ................................................................................................................... 30
- 12.2 Universal Generalization ........................................................................................................... 30
- 12.3 Combining Quantifier Proofs .................................................................................................... 30
Chapter 13: Formal Proofs and Quantifiers ................................................................................................ 32
Introduction ............................................................................................................................................ 32
Subsections and Key Points: ................................................................................................................... 32
- 13.1 Existential Quantifier Rules ....................................................................................................... 32
- 13.2 Universal Quantifier Rules ........................................................................................................ 33
Combined Quick Notes Summary for Chapters 10 to 13 ........................................................................ 33
, A Student's Prayer for Language Proofs and Logic
Dear Logic Gods,
As I embark on this journey through the realms of truth and contradiction,
Grant me clarity in my premises and wisdom in my conclusions.
Help me see through the fog of fallacies,
And may my arguments be as sound as a well-formed formula.
If I encounter a paradox,
Let it be one of those that leads to deep insights,
Not just confusion or a head-scratching "Why?!"
And if I find myself lost in a sea of quantifiers,
Please send a friendly tutor or a clear textbook my way.
In moments of doubt, remind me:
Just because it’s not intuitive,
Doesn't mean it’s not true!
And if all else fails, let me at least remember this:
A conclusion drawn from a false premise
Is still a great excuse for a snack break!
Amen.
A Student's Prayer for Language Proofs and Logic ....................................................................................... 4
Chapter 1: Atomic Sentences................................................................................................................... 5
Introduction.............................................................................................................................................. 5
Subsections and Key Points: ................................................................................................................ 5
- 1.2 Predicate Symbols............................................................................................................................. 5
- 1.3 Atomic Sentences: ............................................................................................................................ 6
- 1.4 General First-Order Languages ......................................................................................................... 6
Combined Quick Notes Summary: ............................................................................................................ 6
Chapter 2: The Logic of Atomic Sentences ................................................................................................... 8
Introduction .............................................................................................................................................. 8
Subsections and Key Points: ..................................................................................................................... 8
- 2.1 Valid and Sound Arguments ............................................................................................................. 8
- 2.2 Methods of Proof .............................................................................................................................. 8
- 2.3 Formal Proofs .................................................................................................................................... 9
- 2.4 Constructing Proofs in Fitch .............................................................................................................. 9
Chapter 3: Boolean Connectives ................................................................................................................. 10
Introduction ............................................................................................................................................ 10
Subsections and Key Points: ................................................................................................................... 10
- 3.1 Negation (¬): ................................................................................................................................... 10
- 3.2 Conjunction (∧): .............................................................................................................................. 10
- 3.3 Disjunction (∨): ............................................................................................................................... 10
- 3.4 Remarks on the Game .................................................................................................................... 11
- 3.5 Ambiguity and Parentheses ............................................................................................................ 11
- 3.6 Equivalent Expressions: .................................................................................................................. 11
- 3.7 Translation ...................................................................................................................................... 11
Combined Quick Notes Summary for Chapters 2 and 3: ........................................................................ 12
Chapter 4: Conjunctive and Disjunctive Normal Forms .............................................................................. 13
Introduction ............................................................................................................................................ 13
Subsections and Key Points: ................................................................................................................... 13
- 4.1 Converting to Normal Forms: ......................................................................................................... 13
- 4.2 Equivalence and Simplification: ...................................................................................................... 13
, - 4.3 Chain of Equivalences ..................................................................................................................... 14
- 4.4 Applications of Normal Forms: ....................................................................................................... 14
Chapter 5: Methods of Proof for Boolean Logic ......................................................................................... 15
Introduction ............................................................................................................................................ 15
Subsections and Key Points: ................................................................................................................... 15
- 5.1 Valid Inference Steps ...................................................................................................................... 15
- 5.2 Proof by Cases................................................................................................................................. 16
- 5.3 Proof by Contradiction: ................................................................................................................... 16
- 5.4 Informal Proof Techniques ............................................................................................................. 16
Combined Quick Notes Summary for Chapters 4 and 5: ........................................................................ 16
Chapter 6: Formal Proofs and Boolean Logic.............................................................................................. 18
Introduction ............................................................................................................................................ 18
Subsections and Key Points .................................................................................................................... 18
- 6.1 Conjunction Rules ........................................................................................................................... 18
- 6.2 Disjunction Rules............................................................................................................................. 18
- 6.3 Negation Rules: ............................................................................................................................... 18
- 6.4 The Proper Use of Subproofs .......................................................................................................... 19
- 6.5 Strategy and Tactics ........................................................................................................................ 19
- 6.6 Proofs without Premises: ................................................................................................................ 19
Chapter 7: Conditionals .............................................................................................................................. 20
Introduction ............................................................................................................................................ 20
Subsections and Key Points: ................................................................................................................... 20
- 7.1 Material Conditional (→): ............................................................................................................... 20
7.3 Conversational Implicature ............................................................................................................... 20
- 7.4 Truth-functional Completeness: ..................................................................................................... 21
Combined Quick Notes Summary for Chapters 6 and 7: ........................................................................ 21
- Conjunction Rules: ............................................................................................................................ 21
- Disjunction Rules: ............................................................................................................................. 21
- Negation Rules: ................................................................................................................................. 21
- Proper Use of Subproofs:.................................................................................................................. 21
- Strategy and Working Backwards: .................................................................................................... 21
- Biconditional (↔):............................................................................................................................ 22
Chapter 8: Conditionals and Biconditionals in Proofs................................................................................. 22
, Introduction ............................................................................................................................................ 22
Subsections and Key Points: ................................................................................................................... 22
- 8.1 Rules for Conditionals: ................................................................................................................ 22
- 8.2 Rules for Biconditionals: ............................................................................................................. 23
Chapter 9: Quantifiers and Their Scope ...................................................................................................... 24
Subsections and Key Points .................................................................................................................... 24
- 9.1 Universal Quantifiers (∀): ........................................................................................................... 24
- 9.2 Existential Quantifiers (∃): .......................................................................................................... 24
- 9.3 Quantifier Scope and Boundaries: .............................................................................................. 25
Combined Quick Notes Summary for Chapters 8 and 9: .................................................................... 25
Chapter 10: The Logic of Quantifiers .......................................................................................................... 26
Introduction ............................................................................................................................................ 26
Subsections and Key Points: ................................................................................................................... 26
- 10.1 Tautologies and Quantification................................................................................................. 26
- 10.2 Valid Inference with Quantifiers ............................................................................................... 27
- 10.3 Multiple Quantifiers .................................................................................................................. 27
Chapter 11: Nested Quantifiers .................................................................................................................. 28
Introduction ............................................................................................................................................ 28
Subsections and Key Points: ................................................................................................................... 28
Chapter 12: Methods of Proof for Quantifiers ........................................................................................... 30
Introduction ............................................................................................................................................ 30
Subsections and Key Points: ................................................................................................................... 30
- 12.2 Universal Generalization ........................................................................................................... 30
- 12.3 Combining Quantifier Proofs .................................................................................................... 30
Chapter 13: Formal Proofs and Quantifiers ................................................................................................ 32
Introduction ............................................................................................................................................ 32
Subsections and Key Points: ................................................................................................................... 32
- 13.1 Existential Quantifier Rules ....................................................................................................... 32
- 13.2 Universal Quantifier Rules ........................................................................................................ 33
Combined Quick Notes Summary for Chapters 10 to 13 ........................................................................ 33
, A Student's Prayer for Language Proofs and Logic
Dear Logic Gods,
As I embark on this journey through the realms of truth and contradiction,
Grant me clarity in my premises and wisdom in my conclusions.
Help me see through the fog of fallacies,
And may my arguments be as sound as a well-formed formula.
If I encounter a paradox,
Let it be one of those that leads to deep insights,
Not just confusion or a head-scratching "Why?!"
And if I find myself lost in a sea of quantifiers,
Please send a friendly tutor or a clear textbook my way.
In moments of doubt, remind me:
Just because it’s not intuitive,
Doesn't mean it’s not true!
And if all else fails, let me at least remember this:
A conclusion drawn from a false premise
Is still a great excuse for a snack break!
Amen.
