Symbolic Logic

Description

This book provides a comprehensive introduction to the essential elements of standard (classical) symbolic logic. Key topics covered include:  ·           The characteristic nature and scope of logic as a discipline ·           The construction of a series of distinctly named formal languages suitable for formal translation ·           Semantic models ·           The construction of decision procedures ·           The execution of proof-theoretic arrangements like natural deduction and proof-sequent systems The book covers both the semantics and proof theory of the standard sentential (propositional) logic and predicate (first-order) logic. Other topics covered include: parsing trees, extraction of alternative notations (for instance, Polish notation), Fitch-style proof-theory, sequent and ‘tree’ proof systems, comparisons and contrasts with intuitionistic logic, and presentations of predicate logic models. An ancillary chapter on elements of set theory is conveniently placed at the end and includes insights into the Zermelo-Fraenkel systematization of set theory. The philosophy of logic is also explored. Exercises in the text provide instruction on mathematical induction for the construction of formula, tests for the well-formedness of Polish notation, and functional completeness. Symbolic Logic is essential reading for all philosophy students taking intermediate level formal logic courses and will also appeal to diligent first year students of logic. The text is replete with exercises on both the formal machinery and the philosophical aspects of logic. Contents Chapter 1: What Logic Studies 1.1 General Characteristics of Logic 1.2 Logical Meaning, Logic-Words and Logical Form 1.2.1 Exercises 1.3 Sentences and Meanings 1.3.1 Meaning and Truth Conditions 1.3.2 Exercises 1.4 Arguments 1.5 Consistency 1.6 Logical Truths/Falsehoods and Analytic Sentences 1.6.1 Exercises Chapter 2: Concepts of Deductive Reasoning 2.1 Argument Validity 2.1.1 Exercises 2.2 Consistency 2.2.1 Exercises 2.3 Logical Status of a Sentence 2.3.1 Exercises Chapter 3: Formal Logic of Sentences, Sentential Logic (also called Sentential Logic and Statement Logic) 3.1 Formal Languages: Variations, Extensions, and Deviations 3.2 Grammar of our Formal Language of Sentential Logic: ∑ 3.2.1 Scope of a Connective Symbol and the Major Connective Symbol 3.2.2 Use of Parentheses 3.2.3 Well-Formedness 3.2.4 The Polish Notation 3.2.5 Exercises 3.3 Parsing Trees of Well-Formed Formulas of ∑: ℑ(∑) 3.3.1 Extracting the Polish Notation of the Formula from the Parsing Tree 3.3.2 Exercises Chapter 4: Sentential Logic Languages ∑ 4.1 Definitions of the Connectives by an Equational Method 4.1.1 Definition of Connectives by the Truth Table Method 4.1.2 Semantic Analysis 4.1.3 Exercises 4.2 Computation of Truth Values of Well-Formed Formulas of ∑ 4.2.1 Computation by a Quasi-Algebraic Method 4.2.2 Computation by a Diagrammatic Method: Semantic Computation Trees 4.2.3 Computation Under Incomplete Information 4.2.4 Computation by Truth Tables 4.2.5 Exercises 4.3 A System of Truth Tables for ∑: ∑⊞ 4.3.1 Application of the Truth Table as a Decision Procedure: Validity, Logical Status of Sentences, Consistency, Relations 4.3.2 Partial Truth Table: ∑⊞p 4.3.3 Short Truth Table Method (Quick Computation Method): ∑⊞𝑠 4.3.4 Referring to the Truth Table to Prove Metalogical Theses 4.3.5 Range of a Well-Formed Formula and Logical Consequence 4.3.6 Exercises 4.4 A System of Natural Deduction (Proof Method) for ∑: ∑∎ 4.4.1 Grammar of ∑∎ Rules for ⌜∙⌝: Conjunction and Simplification Rules for ⌜⊃⌝: Modus Ponens, Modus Tollens, Hypothetical Syllogism, Conditional Proof An Additional Rule for ⌜⊃⌝: Conditional Proof Method Rules for ⌜∨⌝: Disjunctive Syllogism, Addition, Constructive Dilemma Incurring and Discharging Assumptions: CD+ and CP Rules for ⌜~⌝: Indirect Proof Method (IP) and Double Negation (DN) Replacement (Two-Directional, Equivalence, Equivalential) Rules Strategies and Patterns in Derivations Exercises 4.5 Other Natural Deduction Systems 4.5.1 Fitch-Type Natural Deduction System: ∑|| Formal Language and Mechanics for ∑↙↓↘ 4.5.2 Translations from English into ∑ (also called Formalizations, Symbolizations) 4.5.3 Simple and Compound Sentences 4.5.4 Observations Regarding the Disjunction Symbol Exercises Chapter 5: Formal Predicate Logic (also called First-Order Logic) ∏ 5.1 Grammar of our Formal Language of Predicate Logic: ∏ 5.1.1 Exercises 5.2 Monadic Predicate Logic: The Formal Language ∏μ 5.3 # Standard Logic and Existential Commitment 5.4 Expanding ∏μ to ∏ρ=/∏μπφ=: Polyadic (or Relational) Predicate Logic with Function Symbols and the Identity Symbol 5.5 Parsing Trees of Well-Formed Formulas of ∏πφ=: ℑ(∏πφ=) 5.5.1 Exercises Chapter 6: Translations from English into ∏πφ= (also called Symbolizations, Formalizations) 6.1 # Tips for Translation (Symbolization, Formalization) 6.1.1 Multiply Quantified Statements 6.1.2 Quantifier Extractions and Relettering 6.1.3 Translations of Numerical Statements and Definite Descriptions 6.1.4 Exercises Chapter 7: Semantic Models for ∏: ∏⧉ 7.1 ∞ Countermodels with Infinite Domains, ∏⧉∞ 7.1.1 Domains with All Named Objects: ∏⧉⌹ 7.1.2 # Prenex Formulas 7.1.3 Domains with Unnamed Objects: ∏⧉⌻ 7.1.4 Exercises Chapter 8: Proof-Theoretical System for Predicate Logic: ∏πφ= 8.1 A System of Natural Deduction for ∏πφ=: ∏πφ=∎ 8.1.1 ∃I: Rule for Introduction of the Existential Quantifier Symbol (Existential Generalization) 8.1.2 ∀I: Rule for Introduction of the Universal Quantifier Symbol (Universal Generalization) 8.1.3 ∀E: Rule for Elimination of the Universal Quantifier Symbol (Universal Instantiation) 8.1.4 ∃E: Rule for Elimination of the Existential Quantifier Symbol (Existential Generalization) 8.1.5 # Restrictions for Introduction and Elimination of Nested Quantifier Symbols 8.1.6 Rules for Interchange of the Quantifier Symbols (Replacement Rule) 8.1.7 =I: Rule for Introduction of the Identity Symbol 8.1.8 =E: Rule for Elimination of the Identity Symbol 8.1.9 Rules for Function Symbols 8.1.10 Intuitionistic versus Classical Predicate Logic 8.1.11 Exercises 8.2 A Tree System for Polyadic Predicate Logic: ∏ρ=↙↓↘ 8.2.1 Exercises Chapter 9: Definite Descriptions: ∏πφ=⍳ 9.1 Exercises Chapter 10: Basics of Set Theory 10.1 Definitions: Set, Membership, Distinguished Types of Sets; Ways of Defining Sets; Theoretical Issues About Sets 10.1.1 Exercises 10.2 Subsethood, Power Set, Cardinality 10.2.1 Exercises 10.3 Ordered Pairs and Cartesian Products 10.3.1 Exercises 10.4 Set-Theoretic Operations 10.4.1 Examples 10.4.2 Exercises 10.5 The Zermelo-Fraenkel Systematization of Set Theory 10.6 Use of Truth Tables in Set Theory 10.6.1 Exercises 10.7 Relations and Functions; Inverses and Relative Products of Relations; Converses, Inverses and Compositions of Functions 10.7.1 Characteristics of Relations 10.7.2 Functions 10.7.3 Exercises Glossary References Index