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Propositions and Propositional Logic | |

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Logic | |

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Propositions | |

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Propositions and agreements | |

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Other views of propositions | |

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Words and Propositions as Types | |

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Propositions in English | |

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Exercises for Sections A-D | |

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The Basic Connectives of Propositional Logic | |

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A Formal Language for Propositional Logic | |

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Defining the formal language | |

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Realizations: semi-formal English | |

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Exercises for Sections E and F | |

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Classical Propositional Logic | |

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The Classical Abstraction and truth-functions | |

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Models | |

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Validity and semantic consequence | |

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Determining whether a wff is a tautology | |

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Examples of Formalization | |

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Exercises for Sections G and H | |

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Relatedness Logic | |

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The subject matter of a proposition | |

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Relatedness relations | |

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Subject matter as the content of a proposition | |

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Models | |

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An Overview of Semantics for Propositional Logics | |

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Exercises for Sections J and K | |

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The Internal Structure of Propositions | |

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Things, the World, and Propositions | |

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Names and Predicates | |

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Propositional Connectives | |

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Variables and Quantifiers | |

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Compound Predicates and Quantifiers | |

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The Grammar of Predicate Logic | |

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Exercises | |

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A Formal Language for Predicate Logic | |

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A Formal Language | |

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The Unique Readability of Wffs | |

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The Complexity of Wffs | |

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Free and Bound Variables | |

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The Formal Language and Propositions | |

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Exercises | |

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Semantics | |

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Syntax vs. Semantics as a Basis for Logic | |

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Atomic Propositions | |

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Names | |

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A name picks out at most one thing | |

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A name picks out at least one thing | |

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Predicates | |

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A predicate applies to an object | |

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Predications involving relations | |

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Other conceptions of predicates and predications | |

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How many predicates are there? | |

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Naming, Pointing, and What There Is | |

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Agreements | |

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Naming, pointing, and descriptions | |

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Avoiding names completely? | |

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Forms of pointing: what there is | |

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Exercises for Sections A-E | |

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The Universe of a Realization | |

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The Self-Reference Exclusion Principle | |

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Models | |

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The assumptions of the realization: Form and Meaningfulness | |

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Interpretations: assignments of references and valuations | |

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The Fregean Assumption and The Division of Form and Content | |

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The truth-value of a complex proposition | |

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Truth in a model | |

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Logics, Validity, Semantic Consequence | |

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Exercises for Sections F-J | |

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Summary Chapters II-IV.J | |

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Tarski's Definition of Truth | |

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Eliminating semantic terms: Convention T | |

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Other logics, other views of truth | |

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Extensionality | |

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Intensional predicates | |

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The Extensionality Restriction | |

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Quantification and intensional predicates | |

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Languages without names | |

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Models in which every object is named | |

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Inconsistent predications and quantification | |

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Other Interpretations of the Quantifiers and the Use of Variables | |

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A current variation on Tarski's definition | |

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The substitutional interpretation | |

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Naming all elements of the universe at once | |

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Surveying all interpretations of the name symbols | |

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Exercises for Sections K-M | |

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The Logical Form of a Proposition | |

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Rewriting English Sentences | |

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Common Nouns as Subject and Object | |

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Relative quantification: [for all] | |

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Relative quantification: [exist] | |

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Nouns into Predicates | |

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Adjectives | |

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Indexicals | |

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Adverbs | |

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Tenses | |

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Collections and Qualities | |

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Mass Terms | |

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Aristotelian Logic | |

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Formalizations Relative to Formal Assumptions | |

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Analysis vs. formalization | |

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Extending the scope of predicate logic | |

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Formalizing a notion | |

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Criteria of Formalization | |

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Examples of Formalization | |

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Exercises | |

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Identity | |

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Identity | |

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The Equality Predicate | |

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The Interpretation of '=' in a Model | |

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The Identity of Indiscernibles | |

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The Predicate Logic Criterion of Identity (p.l.c.i.) | |

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The p.l.c.i. vs. the implicit identity of the universe | |

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The p.l.c.i. and names | |

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Validity | |

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Is the Equality Predicate Syncategorematic? | |

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Exercises | |

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Quantifiers | |

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The Order of Quantifiers | |

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[for all]x[exist]y and [exist]y[for all]x | |

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[for all]x[exist]y and [exist]x[exist]y | |

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Superfluous quantifiers | |

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The Scope of Quantifiers: Substituting One Variable for Another | |

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Names, Quantifiers, and Existence | |

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Is '--exists' a Predicate? | |

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Quantifying Over a Finite Universe: [for all] as Conjunction, [exist] as Disjunction | |

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Modeling Other Quantifiers | |

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Positive quantifiers: 'there are at least n' | |

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Negative quantifiers: 'there are at most n', 'no', 'nothing' | |

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Exact quantifiers: 'there are exactly n' | |

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Quantifications we can't model | |

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Relative Quantification | |

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Nouns into Predicates revisited | |

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Formalizations involving the same quantifier | |

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Formalizations involving mixtures of quantifiers | |

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Examples of Formalization | |

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Exercises | |

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Descriptive Names | |

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Descriptive Names: A Problem in Formalization | |

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Descriptive Names Relative to Formal Assumptions | |

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Russell's Method of Eliminating Descriptive Names from Atomic Propositions | |

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Eliminating All Names? | |

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Examples of Formalization | |

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Exercises | |

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Functions | |

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Name-Makers | |

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Functions | |

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A definition | |

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Terms | |

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The value of a function | |

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Functions compared to predicates | |

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A Formal Language with Function Symbols and Equality | |

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Realizations and Truth in a Model | |

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Partial Name-Makers | |

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Russell's abstraction operator | |

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The [varepsilon]-operator | |

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Examples of Formalization | |

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Exercises | |

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Quantifying Over Predicates: Second-Order Logic | |

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Quantifying over Predicates? | |

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Predicates and Things | |

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Predicate Variables and their Interpretation: Avoiding Self-Reference | |

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Predicate variables | |

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The interpretation of predicate variables | |

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Note: Higher-order logics | |

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A Formal Language for Second-Order Logic: L[subscript 2] | |

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Realizations | |

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Identifying Predicates with Collections of n-tuples of the Universe | |

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Exercises for Sections A-F | |

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Models | |

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Examples of Formalization | |

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Exercises for Sections G and H | |

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Predicates as Things: Reducing General Second-Order Logic to First-Order Logic | |

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One universe for predicates and individuals | |

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The translation | |

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Proof that the mapping preserves consequences | |

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Does the reduction preserve meaning? | |

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Quantifying over Functions | |

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Why quantify over functions? | |

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A formal language: L[subscript 2F] | |

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Realizations and models | |

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The difficulty of reducing quantification over functions to first-order logic | |

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Many-Sorted Languages | |

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Exercises for Sections J-L | |

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Language, the World, and Predicate Logic | |

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The World | |

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The Template Analogy | |

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Eliminating Natural Languages? | |

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Predicate Logic as a Model of or Guide to Reasoning | |

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Appendices | |

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The Notion of Thing in Predicate Logic | |

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What There Is: Restrictions on the Universe of a Realization | |

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Primitives and Assumptions of Predicate Logic | |

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Formalization: Criteria and Agreements | |

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Bibliography | |

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Index of Examples | |

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Index of Notation | |

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Index | |