Logic DeMYSTiFied by Boutelle, Tony -- Read -- Imperial Library of Trantor

Tables for Basic Operations

Table for Logical Negation Table for Conjunction Tables for Logical Disjunction Table for Logical Implication Table for Logical Equivalence A Quick Proof Precedence Examples of Precedence

Proofs Using Truth Tables

Reversing the Order of a Conjunction Grouping of Conjunctions Reversing the Order of an Inclusive Disjunction Grouping of Inclusive Disjunctions Reversing the Order of an Implication Ungrouping the Negation of a Conjunction Ungrouping the Negation of an Inclusive Disjunction Regrouping with Mixed Operations Truth Tables Have Power!

Quiz

Chapter 2 Propositional Logic

A Formal System

Propositional Formulas (PFs) Is This Mathematics? Contradiction Sequents Backing Up the Argument Assumption (A) Double Negation (DN) Conjunction Introduction (^I) Conjunction Elimination (^E) Disjunction Introduction (∨l) Disjunction Elimination (∨E) Conditional Proof (CP) and Theorems Biconditional Introduction (↔l) Biconditional Elimination (↔E) Substitution

Laws

Law of Contradiction Law of Excluded Middle Law of Double Negation Commutative Laws Associative Laws Law of Implication Reversal DeMorgan’s Laws Distributive Laws Interderivability (╬), Deriving Laws, and Theorem Introduction (TI)

Quiz

Chapter 3 Predicate Logic

Symbolizing Sentence Structure

Formal Predicates Formal Subjects: Names Formal Subjects: Variables Predicate Sentence Formulas More Predicate Sentences Identity (=) Building More Complex Formulas

Quantified Statements

Some versus All The Existential Quantifier (∃) The Universal Quantifier (∀) The Universe Multiple Quantifiers Translating the Quantifiers Laws of Quantifier Transformation Logical Relationships: The Square of Opposition

Well-Formed Formulas Properties of Two-Part Relations

Symmetry Asymmetry Antisymmetry Nonsymmetry Reflexivity Irreflexivity Nonreflexivity Transitivity Intratransitivity Nontransitivity Equivalence Relations

Predicate Proofs

Existential Introduction (∃I) Existential Elimination (∃E) Universal Introduction (∀I) Universal Elimination (∀E) Identity Introduction (=I) Identity Elimination (=E) Conventions for Predicate Proofs

Syllogisms

A Sample Syllogism Allowed Propositions Terms and Arguments Conversion Laws Classifying Syllogisms: Mood and Figure Testing Syllogistic Arguments Valid Syllogisms Other Kinds of Syllogism

Quiz

Chapter 4 A Boot Camp for Rigor

Definitions

Elementary Terms Line Segment Half-Open Line Segment Open Line Segment Length of Line Segment Closed-Ended Ray Open-Ended Ray Point of Intersection Collinear Points Coplanar Points Coincident Lines Collinear Line Segments and Rays Transversal Parallel Lines Parallel Line Segments Parallel Rays Angle Measure of Angle Straight Angle Supplementary Angles Right Angle Complementary Angles Perpendicular Lines, Line Segments, and Rays Triangle

Similar and Congruent Triangles

Direct Similarity Inverse Similarity Direct Congruence Inverse Congruence Two Crucial Facts Two More Crucial Facts

Axioms

The Two-Point Axiom The Extension Axiom The Right Angle Axiom The Parallel Axiom The Side-Side-Side (SSS) Axiom The Side-Angle-Side (SAS) Axiom The Angle-Side-Angle (ASA) Axiom The Side-Angle-Angle (SAA) Axiom

Some Proofs at Last

Alternate Interior Angles

Quiz

Chapter 5 Fallacies, Paradoxes, and Revelations

The Probability Fallacy

Belief Parallel Worlds, Fuzzy Worlds We Must Observe

Weak and Flawed Reasoning

“Proof” by Example Begging the Question Hasty Generalization Misuse of Context Circumstance Fallacies with Syllogisms Fun with Silliness Inductive Reasoning

Simple Paradoxes

A Wire around the Earth Direct-Contradiction Paradox Who Shaves Hap? Arrow Paradox The Frog and the Wall A Geometry Trick A “proof” that −1 = 1 Wheel Paradox

Classical Paradoxes

Execution Paradox A Two-Pronged Defense Saloon Paradox Barbershop Paradox Shark Paradox

Russell and Gödel

Sets Two Special Sets The Paradox Professor N’s Machine What Did You Prove?

Quiz

Chapter 6 Strategies for Proofs

How Does a Theory Evolve?

Definitions Elementary Terms Axioms Euclid’s Postulates A Denial Propositions

Theorems

A Classical Theorem Lemmas Corollaries Proofs, Truth, and Beauty

Techniques

Deductive Reasoning What’s the Universe? WeakTheorems Demonstrating a Weak Theorem Strong Theorems Demonstrating a Strong Theorem Reductio ad Absurdum Revisited Mathematical Induction

Cause, Effect, and Implication

Correlation and Causation Complications

Quiz

Chapter 7 Boolean Algebra

New Symbols for Old Operations

The NOT Operation (−) The AND Operation (×) The OR Operation (+) Boolean Implication (⇒) Boolean Equivalence (=)

Truth Tables, Boolean Style

Truth Table for Boolean Negation (NOT) Truth Table for Boolean Multiplication (AND) Truth Table for Boolean Addition (OR) Truth Table for Boolean Implication (IF/THEN) Truth Table for Boolean Equality (IFF)

Basic Boolean Laws

Precedence Contradiction Law of Double Negation Commutative Laws Associative Laws Law of the Contrapositive DeMorgan’s Law for Products DeMorgan’s Law for Sums Distributive Law

Quiz

Chapter 8 The Logic of Sets

Set Fundamentals

To Belong, or Not to Belong Listing the Elements The Empty Set Finite or Infinite? Sets within Sets

Venn Diagrams

People and Numbers Subsets Proper Subsets Congruent Sets Disjoint Sets Overlapping Sets

Set Intersection

Intersection of Two Congruent Sets Intersection with the Null Set Intersection of Two Disjoint Sets Intersection of Two Overlapping Sets

Set Union

Union of Two Congruent Sets Union with the Null Set Union of Two Disjoint Sets Union of Two Overlapping Sets

Quiz

Chapter 9 The Logic of Machines

Numeration Systems

Decimal Binary Octal Hexadecimal

Digital Circuits

Positive versus Negative Logic Logic Gates Black Boxes Forms of Binary Data Clocks Counters

Digital Signals

Bits Bytes Baud Examples of Data Speed Analog-to-Digital Conversion Digital-to-Analog Conversion Serial versus Parallel Digital Signal Processing Digital Color

Quiz

Chapter 10 Reality Remystified

The Illogic of Time

The Light-Beam Conundrum Synchronize Your Watches! When Is a Second Not a Second? Traveling into the Future Traveling into the Past The Twin Paradox

The Illogic of Matter and Space

Particles without End A Mad Professor’s Monologue Order from Randomness The Ultimate State of Order Hyperspace Hypospace Time-Space Time as a Dimension Infinity-Space The Dwindling-Displacement Effect

The Illogic of Chaos

Was Andrew “Due”? Slumps and Spurts Correlation, Coincidence, or Chaos? Scale-Recurrent Patterns The Maximum Unswimmable Time The Butterfly Effect Scale Parallels The Malthusian Model A Bumpy Ride What Is Randomness?

Final Exam Answers to Quizzes and Final Exam Suggested Additional Reading Index

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