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Natural Deduction 

Natural Deduction

An Introduction to Logic with Real Arguments, a Little History, and Some Humour

Written by: Richard T.W. Arthur

Publication Date: May 25, 2011
452pp • Paperback / PDF / ePub
ISBN: 9781551111827 / 1551111829

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Richard Arthur's Natural Deduction provides a wide-ranging introduction to logic. In lively and readable prose, Arthur presents a new approach to the study of logic, one that seeks to integrate methods of argument analysis developed in modern "informal logic" with natural deduction techniques. The dry bones of logic are given flesh by unusual attention to the history of the subject, from Pythagoras, the Stoics, and Indian Buddhist logic, through Lewis Carroll, Venn, and Boole, to Russell, Frege, and Monty Python.


"This excellent text covers all the standard topics and more. Its real strength lies in the clarity and humour of exposition and in the richness of examples and exercises. The illustrations are invariably interesting, since often they are related to current events or the history of philosophy and science or are drawn from Monty Python. The last of these provides several memorable fallacies. Arthur's Natural Deduction is one of the finest introductions to logic available today." - James Robert Brown, University of Toronto

"Richard Arthur's engagingly written Natural Deduction introduces symbolic logic as something that grows naturally out of everyday reasoning. The treatment of category logic in this book, based on an elegant notation created by Lewis Carroll, is original and insightful.  I also like Arthur's suggestion that there is a link between logic and humour: the same mental muscles that enable one to see the point of a joke must also be those that enable one to see a logical connection. His exercises are ingenious and thought-provoking. This unique book is superb and I recommend it highly." - Kent A. Peacock, University of Lethbridge

Richard T.W. Arthur is Professor of Philosophy at McMaster University.

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Table of Contents: [Back to Top]

Preface for Students

Preface for Instructors



Chapter One: Arguments


Identifying Arguments

  1. Inference Indicators
  2. Explanations
  3. Implicit Arguments
  4. Enthymemes

Natural Arguments

  1. Argument and Inference
  2. Techniques of Diagramming

Chapter Two: Validity


  1. Defining Validity
  2. Soundness

Argument Forms and Formal Validity

Evaluating Natural Arguments


Chapter Three: Statements and Conditionals

Statements and Compounds

  1. Statements
  2. Compounds
  3. Statement Operators

Conditional Statements

Modus Ponens

  1. Argument Form and Substitution Instance
  2. Affirming the Consequent

Chapter Four: Negation

Symbolizing Negations

  1. Negations
  2. Contradictories

Modus Tollens

  1. Modus Tollens and Double Negation
  2. Denying the Antecedent

Inference and Implication

Chapter Five: Conjunction

Symbolizing Conjunctions

Rules of Inference for Conjunction

Evaluating Extended Arguments

Chapter Six: Disjunction

Symbolizing Disjunctions

Rules of Inference for Disjunctions

  1. Disjunctive Syllogism
  2. Disjunction
  3. De Morgan's Laws

Chapter Seven: Conditional Proof

More on Symbolizing

  1. Disjunctions in Conditionals
  2. 'Unless'
  3. 'Otherwise,' 'Else'

More Rules Involving Conditionals

  1. Conditional Proof and Supposition
  2. The Hypothetical Syllogism

Supposition in Natural Argument

Chapter Eight: Biconditionals

Necessary and Sufficient Conditions

  1. 'Only if'
  2. Necessary and Sufficient Conditions


  1. Symbolizing
  2. Conversational Implicature
  3. Rules of Inference

Chapter Nine: Dilemmas


Natural Dilemmas

Chapter Ten: Reductio Arguments

Reductio ad Absurdum

Natural Reductio Arguments

Chapter Eleven: Review and Consolidation

Rules of Inference

  1. Rules of Inference and Equivalence Rules
  2. Proof Strategies

Derived Rules

Chapter Twelve: SL as a Formal System

Rules of Formation

  1. Symbols, Formulas, and Wffs
  2. Soundness and Completeness

Sequents, Theorems, and Axioms

  1. Sequents and Theorems
  2. Axioms and the Propositional Calculus

Chapter Thirteen: Truth Tables

Truth Tables and Statements

  1. Truth Tables
  2. Material Implication
  3. Tautologies, Contradictions, and Contingent Statements
  4. Logical Equivalence

Truth Tables and Validity

  1. The Full Truth Table Method
  2. Invalid Argument Forms

The Brief Truth Table Method

Chapter Fourteen: Truth Trees for SL

Truth Trees

  1. The Truth Tree Method
  2. Decomposition Rules

Soundness and Completeness

  1. Tautologies, Contradictions, and Logical Equivalence
  2. Soundness and Completeness


Chapter Fifteen: Syllogistic Logic

Category Logic

  1. Aristotle's Logic
  2. A-, E-, I-, and O-Statements
  3. Ambiguous Statements

Carroll Diagrams

  1. Carroll's Diagrams
  2. Existence and Non-Existence
  3. Conversion

Evaluating Validity of Syllogisms

Chapter Sixteen: Universal Quantification

Universal and Singular Statements

  1. Universal Quantification
  2. 'Only' and 'Nothing but'
  3. Singular Statements and Individual Names

Rules of Inference: UI and UG

Chapter Seventeen: Existential Quantification

Particular Statements

  1. Existential Quantification

Rules of Inference

  1. Existential Instantiation
  2. Existential Generalization
  3. Proof Strategy

Chapter Eighteen: Advanced Class Logic

Arguments with More than 3 Predicates

  1. Carroll Diagrams for 4 or 5 Categories
  2. Sorites

Existential Import

  1. On Giving Universal Statements Existential Import
  2. Penevalid Arguments
  3. Non-Emptiness of the UD

Chapter Nineteen: Asyllogistic Logic

More on Symbolizing

  1. Non-Classical Statements
  2. 'Any'

Asyllogistic Proofs: QN

Predicate Logic as a Formal System

  1. Symbols, Formulas, and Wffs
  2. Propositional Functions and Quantifier Scope

Chapter Twenty: Relational Logic

The Logic of Relations

  1. Relations
  2. Symbolizing Relations
  3. Nested Quantifiers
  4. Relational Proofs

Properties of Binary Relations

  1. Transitivity, Symmetry, and Reflexivity
  2. Equivalence Relations

Chapter Twenty-One: Logic with Identity

Identity and Quantity

  1. Symbolizing Identities and Quantities
  2. Russell's Theory of Definite Descriptions

Inferences Involving Identity

  1. The Rule of Inference SI
  2. Properties of Identity

Ordering Relations

Chapter Twenty-Two: Relational Arguments

More on Symbolizing Relational Statements

  1. A Method for Symbolizing
  2. Prenex Forms

Relational Arguments

  1. Arguments beyond the Scope of Traditional Logic
  2. Ambiguities and the Quantifier Scope Fallacy

Chapter Twenty-Three: Truth Trees for PL

Predicate Logic Truth Trees

  1. Truth Tree Rules from Statement Logic
  2. Additional Truth Tree Rules for Quantifications
  3. Negated Quantifier Decomposition Rules
  4. Effective Completeness

Trees for Relational Logic and Identity

  1. Truth Tree Rules in Relational Logic
  2. Additional Truth Tree Rules for Identity and Diversity

Chapter Twenty-Four: Other Logics

Second Order Logic

Modal Logic

Deontic Logic

Quantum Logic

Intuitionistic Logic

Appendix 1: The Paradoxes of Material Implication

Appendix 2: A Little History: Consequentiae

Appendix 3: Logic Diagrams



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Natural Deduction

2011 • 452pp • Paperback • 9781551111827 / 1551111829

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