Logic Works: A Rigorous Introduction to Formal Logic

Cover
Half Title
Endorsement
Title Page
Copyright Page
Table of Contents
Preface
Instructors’ Preface
Acknowledgements
Symbol Summary
1 Introduction to the Study of Logic
	1.1 Demonstration and Interpretation
	1.2 Deductive and Inductive Demonstrations
	1.3 The Principle of Noncontradiction
	1.4 Abstraction, Variables, and Formalization; Logical and Nonlogical Elements; Formal Contradiction
	1.5 A Fundamental Problem
	1.6 Chapter Outline
	Technical Appendix: Elements of a Theory of Demonstrative Logic
	Notes
	References
Part I Sentential Logic
	2 Vocabulary and Syntax
		2.1 Introduction
		2.2 Conventions
		2.3 Syntactic Demonstrations and Trees
		2.4 Scope; Main Connective and Immediate Components; Named Forms
		2.5 Formal Properties
		Note
	3 Semantics
		3.1 Semantics for ⊥ and the Sentence Letters
		3.2 Semantics for the Connectives
		3.3 Semantics for Compound Sentences
			3.3.1 Extensional Meaning
			3.3.2 Intensional Meaning
		3.4 Intensional Concepts
		Appendix Expressive Adequacy; Disjunctive Normal Form; The Lean Language
		Notes
	4 Formalization
		4.1 Looseness of Fit
			4.1.1 Formalization of Sentences as Sentence Letters
			4.1.2 Formalization of Connective Expressions
		4.2 Conditional Sentences of English
		4.3 Necessary Conditions
		4.4 Sufficient Conditions
		4.5 Necessary and Sufficient Conditions; The Principle of Charity
		4.6 Formalizing Necessary and Sufficient Conditions
		4.7 Exceptions and Strong Exceptions
		4.8 Disjunction
		4.9 Negations and Conjunctions
		4.10 Punctuation
		4.11 Limits of Formalization
		4.12 Formalizing Demonstrations
		Notes
		References
	5 Working With SL Semantics
		5.1 Identifying and Verifying Interpretations
		5.2 Demonstrating That There Is No Interpretation
		5.3 Demonstrating General Principles
		5.4 Falsifying General Claims
		5.5 Relations Between Intensional Concepts; Models; Entailment; Biconditional Proof
		Appendix Alternatives to Bivalence
			Appendix 5.1 Supervaluations
			Appendix 5.2 Three-Valued Logic
			Appendix 5.3 Paraconsistent Logic
		Notes
		References
	A-1: Advanced Topics Concerning SL Semantics
		A-1.1 Mathematical Induction
		A-1.2 Bivalence
		A-1.3 Extensionality
		A-1.4 Compactness
		Note
		References
	6 Derivations
		6.1 DL: A Lean Derivation System
		6.2 Strategies for Doing Derivations in DL
		6.3 Ds: A Derivation System for SL
		6.4 Strategies for Doing Derivations in Ds
		6.5 Extensions of Ds; Bracket Free Notation
			6.5.1 Systematic Overview; Adequacy of DL
			6.5.2 Metatheorems and Derived Rules for Ds
			6.5.3 Substitution Principles
			6.5.4 Disjunctive Normal Form
			6.5.5 Relations Between Intensional Concepts
			6.5.6 Bracket Free Notation
		6.6 Intuition and “Intuitionism”: Derivation in Intuitionistic Logic
		Notes
		References
	A-2: Advanced Topics Concerning the Soundness and Completeness of Ds
		A-2.1 Soundness
		A-2.2 Corollary Results
		A-2.3 Henkin Completeness
		A-2.4 Demonstration of the Lindenbaum Lemma
		A-2.5 Demonstration of Lemma 2
		A-2.6 Demonstration of Lemma 3
		A-2.7 Corollary Results
		A-2.8 Post/Hilbert–Ackermann Completeness
	7 Reduction Trees
		7.1 Method and Strategies
		7.2 Using Trees to Determine Derivability
			7.2.1 Theorems; Inconsistent and Contingent Sentences
			7.2.2 Demonstrations
			7.2.3 Interderivability
		7.3 Theory and Definitions
		Appendix Trees for Three-Valued and Paraconsistent Logic
		Notes
		References
	A-3: Advanced Topics Concerning the Soundness and Completeness of Ts
		A-3.1 Soundness of Ts
		A-3.2 Completeness of Ts
		A-3.3 Decidability of Ts
		A-3.4 Tree Conversion; Completeness and Decidability of Ds
Part II Modal Sentential Logic
	8 Vocabulary, Syntax, Formalization, and Derivations
		8.1 Vocabulary and Syntax
		8.2 Formalization
		8.3 Derivations
		Notes
	9 Semantics and Trees for Modal and Intuitionistic Sentential Logic
		9.1 Semantics for Modal Sentential Logic
			9.1.1 Discovering Interpretations
			9.1.2 Demonstrating That There Is No Interpretation
		9.2 Reduction Trees for Modal Sentential Logic
		9.3 Semantics for Intuitionistic Sentential Logic
		9.4 Reduction Trees for Intuitionistic Sentential Logic
		References
	A-4: Advanced Topics Concerning the “Soundness” and “Completeness” of Dm and Tm
		A-4.1 “Soundness” of Dm
		A-4.2 Completeness of Tm
		A-4.3 Tree Conversions
		A-4.4 Adequacy of Dm and Tm
		Notes
		References
	Part III Predicate Sentential Logic
		10 Vocabulary, Syntax, Formalization, and Derivations
			10.1 English Predication
			10.2 Simple Terms
				10.2.1 Vocabulary and Syntax of Predicate Sentential Logic
				10.2.2 Formalization
				10.2.3 Derivations
			10.3 Complex Terms
				10.3.1 Functional Terms
				10.3.2 Vocabulary and Syntax of PLf; Formalization
				10.3.3 Dpf
				10.3.4 Definite Descriptions
				10.3.5 Vocabulary and Syntax of PL1; Formalization
	11 Semantics and Trees
		11.1 Interpretations
			11.1.1 Domains
			11.1.2 Names
			11.1.3 Predicates and Satisfaction
			11.1.4 Identity
		11.2 Valuation Rules
		11.3 Working With the Semantics
		11.4 Tp
		11.5 Semantics for Functional Terms
		11.6 Tpf
			11.6.1 Systematic Paths
			11.6.2 Decidability
		11.7 Semantics for PL1
			11.7.1 Variable Assignments
			11.7.2 Denotation for Variables
			11.7.3 Satisfaction
			11.7.4 Denotation for Proper Descriptions
			11.7.5 Truth Conditions
			11.7.6 Denotation for Improper Descriptions
			11.7.7 Free Description Theory
		Notes
		References
	A-5: Advanced Topics for PSL
		A-5.1 Extensionality and Variance
			A-5.1.1 Extensionality
			A-5.1.2 Variance
				A-5.1.2.1 The Variant Interpretation Principle
				A-5.1.2.2 The Variant Name Principle
		A-5.2 Soundness of Dp
		A-5.3 Completeness of Tp
		A-5.4 Tree Conversion; Soundness of Tp; Completeness of Dp
		Note
Part IV Quantified Predicate Logic
	12 Vocabulary, Syntax, and Formalization
		12.1 Informal Vocabulary and Syntax
			12.1.1 Symbols for Objects
			12.1.2 Symbols for Quantities of Objects
			12.1.3 Informal Syntax
		12.2 Formal Vocabulary and Syntax
		12.3 Formalizing English Sentences in Quantified Predicate Logic
			12.3.1 Simply Quantified Sentences; Scope
			12.3.2 Multiply Quantified Sentences; Scope Ambiguity
			12.3.3 Negations of Quantified Claims; Duality; The Square of Opposition; “Any”
			12.3.4 Formalizing Relations Between Predicates
			12.3.5 A, E, I, and O Sentences; Existential Import
			12.3.6 Predicate Descriptions; Changing Scope
			12.3.7 Quantities and Superlatives
			12.3.8 Definite Descriptions
			12.3.9 Bare Existence; Limits of Formalization
		Notes
		References
	13 Derivations
		13.1 Dq
		13.2 Extensions of Dq
			13.2.1 Functional Terms
			13.2.2 Intuitionistic Logic
			13.2.3 Free Logic
			13.2.4 Free Description Theory
		Notes
		References
	14 Trees and Tree Model Semantics for Quantified Predicate Logic
		14.1 Rules
		14.2 Method
		14.3 Tree Model Semantics
		14.4 Extensions of Tq
			14.4.1 Functional Terms
			14.4.2 Semantics and Trees for Intuitionistic Logic
			14.4.3 Semantics and Trees for Free Logic
			14.4.4 Semantics and Trees for Free Description Theories
		Notes
		References
	15 Semantics for QPL Without Mixed Multiple Quantification
		15.1 Objectual Semantics
		15.2 Denotation
			15.2.1 Variable Assignments
			15.2.2 Names
		15.3 Satisfaction
			15.3.1 Satisfaction Conditions for Predicate and Identity Formulas
			15.3.2 Satisfaction Conditions for ., Zero-Place Predicates, and Connective Compounds
			15.3.3 Satisfaction Conditions for Singly Quantified Formulas
		15.4 Truth
		15.5 Working With the Semantics
			15.5.1 Discovering Interpretations
			15.5.2 Discovering Contradictions
		15.6 Demonstrating General Principles
			15.6.1 Extensionality
			15.6.2 Variance
		Notes
	16 Semantics for QPL With Mixed Multiple Quantification
		16.1 Variants On Variable Assignments; Denotation of Variables
		16.2 Satisfaction Conditions for Quantified Formulas
		16.3 (P) and (=) Applications
		16.4 Truth Conditions for Sentences
		16.5 Working With the Semantics
			16.5.1 Order of List Items
			16.5.2 Embellishing the Variant List
			16.5.3 Describing the Model at a More Abstract Level
			16.5.4 Avoiding Inversion
		Appendix Demonstration of the Exclusivity Principle
	A-6: Advanced Topics for QPL
		A-6.1 Extensionality and Variance
			A-6.1.1 Name Extensionality
			A-6.1.2 Variable Extensionality
			A-6.1.3 Formula Extensionality
			A-6.1.4 Variance
		A-6.2 Soundness of Dq
		A-6.3 Completeness of Tq
		A-6.4 Tree Conversion; Soundness of Tq; Completeness of Dq
		Appendix Quantified Modal Logic
		A-6.A.1 Objects and Worlds
			A-6.A.1.1 Counterpart Theory
			A-6.A.1.2 Haecceity Theory
		A-6.A.2 Names and Predicates
		A-6.A.3 Quantifier Domains and the Barcan Formulas
		A-6.A.4 Derivation and Tree Rules
		A-6.A.5 Substances
		Reference
	17 Higher-Order Logic
		17.1 Vocabulary and Syntax
		17.2 Formalization; Definitions of Higher-Order Predicates
		17.3 Syntax II: Instances
		17.4 Derivations
		17.5 Semantics
		17.6 Trees and Incompleteness
		Notes
		References
	Main Appendix: Rule Summaries
		1 Foundational Definitions
		2 Intensional Concepts
		3 Formation Rules
			(Def Sentence)
			Informal Notational Conventions
			(Def Instance)
		4 Sentential Valuation Rules
		5 Formulaic and Free Valuation Rules
		6 Derivation Rules
		7 Tree Rules
			Rules for Intuitionistic Sentential Trees
Index