| Autore |
Pollard Stephen
|
| Pubbl/distr/stampa |
Cham, Switzerland : , : Springer, , [2022]
|
| Descrizione fisica |
1 online resource (355 pages)
|
| Disciplina |
511.324
|
| Collana |
Synthese library
|
| Soggetto topico |
Algebraic logic
|
| ISBN |
9783031056710
9783031056703
|
| Formato |
Materiale a stampa  |
| Livello bibliografico |
Monografia |
| Lingua di pubblicazione |
eng
|
| Nota di contenuto |
Intro -- Preface -- Translator's Introduction -- 0.1 The Lehrbuch Project -- 0.2 Doing Algebra with Classes -- 0.3 From Boole to Boolean Algebra -- References -- Contents -- 1 Numbers -- 1.1 The Mathematical Sciences -- 1.2 Numbers in General -- 1.3 What Makes Counting Possible? -- 1.4 When Do We Count? -- 1.5 The Emergence or Construction of the Natural Numbers -- 1.6 The Appellation of a Number -- 1.7 The Concept of Multiplicity -- 1.8 Numbers as Measures of Numerousness -- 1.9 Numbers as Rules -- 1.10 The General Purpose of Numbers -- 1.11 Cardinal and Ordinal Numbers -- 1.12 Independence of Numbers from the Ordering Imposed by the Counting Process -- 1.13 A Single Axiom -- 1.14 A Fundamental Proposition -- 1.15 Calculation: Expressions -- 1.16 Relations: Equations and Inequalities -- 1.17 Propositions About Equations -- 1.18 More on Equations -- 1.19 Ambiguous Expressions -- 1.20 Logical Subordination -- 1.21 Substitution -- 1.22 Parentheses -- 1.23 Literal and Numerical Numbers -- 1.24 Use of Letters Justified: Principles of Nomenclature -- 1.25 Use of Letters: Motivations and Benefits -- 1.26 Analytic and Synthetic Equations -- 1.27 Theorems, Rules -- 1.28 Constant and Variable Numbers, Functions -- 1.29 Conclusion -- References -- 2 The Three Direct Operations -- 2.1 Independent Treatment of Addition: Concepts and Terminology -- 2.2 First Law of Addition -- 2.3 Second Law of Addition -- 2.4 The Two Laws Combined -- 2.5 Addendum on Inequalities -- 2.6 Recursive Treatment of Addition: Number and Sum -- 2.7 The Associative Law for Trinomials -- 2.8 The Associative Law for Polynomials -- 2.9 The Commutative Law for Binomials -- 2.10 The Commutative Law for Polynomials -- 2.11 Independent Treatment of Multiplication: Basic Concepts -- 2.12 The Commutative Law of Multiplication -- 2.13 The Associative Law of Multiplication.
2.14 Combination and Extension of the Two Laws -- 2.15 Expanded Definition of Product -- Dirichlet's Proof of the Fundamental Theorem -- 2.16 The Distributive Laws -- 2.17 Extension of the Distributive Laws -- Multiplying Out and Factoring -- 2.18 Fusion of the Distributive Laws in the Rule for Multiplying Polynomials -- 2.19 Toward an Alternative Treatment of the Foregoing -- 2.20 Additional Properties of Inequalities -- 2.21 Recursive Treatment of Multiplication: Concept and Laws -- 2.22 Another Proof of the Fundamental Theorem -- 2.23 Independent Treatment of Elevation: Potentiation and Exponentiation -- 2.24 Laws Governing These Operations -- the Iteration Law -- 2.25 The Second Law -- 2.26 The Third Law -- 2.27 Addendum on Inequalities -- 2.28 Recursive Treatment of Elevation -- 2.29 Review -- Iteration -- References -- 3 The Four Inverse Operations -- 3.1 Introduction -- Inversion -- 3.2 Inversion of Univocal Operations: Subtraction -- 3.3 Division -- 3.4 Roots and Logarithms -- 3.5 Overview of the Operations, Expressions, and Operands -- 3.6 The Transposition Rules -- 3.7 Application to the Solution of Pure Equations -- 3.8 Necessity of the New Terminology -- 3.9 Concise Definition of the Outputs of Inverse Operations -- Checking Inverse Operations -- 3.10 Realizability of the Inverse Operations -- 3.11 Univocity of the Inverse Operations -- 3.12 Special Cases (Singularities) -- 3.13 Additional Fundamental Properties -- Using an Inverse to Check a Direct Operation -- 3.14 Another View of Transposition -- 3.15 The Final Fundamental Property -- Using an Inverse Operation to Check an Inverse Operation -- 3.16 Simplifying Equations -- 3.17 Inversion of Ambiguous Operations: Non-commutative Multiplication -- 3.18 Operations on Ambiguous Expressions -- 3.19 Relations Between Expressions Involving a Community of Values.
3.20 Linkage of Propositions of All Types by Univocal and Ambiguous Operations -- 3.21 Accommodating Ambiguous Operations -- 3.22 Using Univocal/Ambiguous Operations to Transpose Univocal/Ambiguous Expressions -- 3.23 What Inferences Are Permitted and Which of Them Are Reversible? -- 3.24 Substitutions Involving Ambiguous Expressions -- Diagonals and Functions -- References -- 4 Rules for Linking All Our Operations -- 4.1 The Road Ahead -- 4.2 Rules for Transforming Algebraic Expressions: Resolution and Reduction -- 4.3 Additional Rules: Transitions from One Logarithmic System to Another -- 4.4 Rules for Connecting First-Order Operations with One Another: A Précis of Those Rules -- 4.5 Proof and Detailed Formulation of the Propositions -- 4.6 Another Proof and More Propositions -- 4.7 Alternative Proofs -- 4.8 Overview of the Preceding Propositions -- 4.9 Rules for Connecting Second-Order Operations with One Another -- 4.10 Overview of the Propositions -- 4.11 Rules for Connecting First- and Second-Order Operations with One Another -- 4.12 The First Group of Third-Order Rules -- 4.13 The Second Group of Third-Order Rules -- 4.14 The Third Group of Third-Order Rules -- 4.15 Review and Final Remarks -- 4.16 Addendum on Inequalities -- 4.17 More on Inequalities -- 4.18 Conventions Involving Parentheses -- 4.19 First Convention -- 4.20 Second Convention -- 4.21 Identification of Expressions and Operands -- 4.22 Linking up More than Three Numbers: Four Numbers -- 4.23 Arbitrarily Many Numbers -- 4.24 Conclusion -- 4.25 The Formal Approach to Algebra -- 4.26 The Elementary Assumptions -- 4.27 Separating the Elementary Assumptions -- 4.28 A Comprehensive Algorithm for Pure Associative Operations -- 4.29 More Properties of Pure Associative Operations -- 4.30 Alternatives to the Associative Law -- 4.31 Type-Two Algorithms: Conjugation.
4.32 Type-Three Algorithms -- 4.33 Type-Four Algorithms -- 4.34 Type-Five Algorithms: Cyclic Multiplication -- 4.35 Type-Six Algorithms: The Pure Third-Order Laws -- 4.36 Completeness of the Cycle of Formulas -- 4.37 Division of the 100 Equations of the Second Class into Elementary Cycles -- 4.38 Division of the Equations of the First and Third Classes into Elementary Cycles -- 4.39 Specific Third-Order Laws -- 4.40 Consequences of the Distributive Laws for All Three Orders -- 4.41 Review and Final Thoughts -- References -- 5 The Operations of the Logical Calculus -- 5.1 Introduction -- 5.2 Five Operations -- 5.3 Axioms -- 5.4 Solution of a Boolean Problem -- 5.5 The Inverse Operations -- 5.6 Note on the Operations of the Logical Calculus -- References -- Index.
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| Record Nr. | UNINA-9910734862203321 |
Info
Ernst Schröder on algebra and logic / / Stephen Pollard
