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Preface and Acknowledgements | |

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The Basic Assumptions of Propositional Logic | |

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What is Logic? | |

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Propositions | |

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Sentences, propositions, and truth | |

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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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Form and Content | |

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Propositional Logic and the Basic Connectives | |

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Exercises for Sections E and F | |

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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 Section G | |

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Classical Propositional Logic - PC - | |

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The Classical Abstraction and the Fregean Assumption | |

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Truth-Functions and the Division of Form and Content | |

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Models | |

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Exercises for Sections A--C | |

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Validity and Semantic Consequence | |

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Tautologies | |

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Semantic Consequence | |

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Exercises for Section D | |

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The Logical Form of a Proposition | |

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On logical form | |

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Criteria of formalization | |

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Other propositional connectives | |

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Examples of Formalization | |

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Ralph is a dog or Dusty is a horse and Howie is a cat Therefore: Howie is a cat | |

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Ralph is a dog and George is a duck and Howie is a cat | |

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Ralph is a dog or he's a puppet | |

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Ralph is a dog if he's not a puppet | |

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Ralph is a dog although he's a puppet | |

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Ralph is not a dog because he's a puppet | |

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Three faces of a die are even numbered Three faces of a die are not even numbered Therefore: Ralph is a dog | |

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Ken took off his clothes and went to bed | |

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The quotation marks are signals for you to understand what I mean; they are not part of the realization | |

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Every natural number is even or odd | |

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If Ralph is a dog, then Ralph barks Ralph barks Therefore: Ralph is a dog | |

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Suppose {s[subscript n]} is monotonic. Then {s[subscript n]} converges if and only if it is bounded | |

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Dedekind's Theorem | |

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Exercises for Sections E and F | |

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Further Abstractions: The Role of Mathematics in Logic | |

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Induction | |

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Exercises for Sections G and H | |

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A Mathematical Presentation of PC | |

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The formal language | |

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Exercises for Section J.1 | |

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Models and the semantic consequence relation | |

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Exercises for Section J.2 | |

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The truth-functional completeness of the connectives | |

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The choice of language for PC | |

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Normal forms | |

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The principle of duality for PC | |

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Exercises for Sections J.3-J.6 | |

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The decidability of tautologies | |

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Exercises for Section J.7 | |

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Some PC-tautologies | |

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Exercises for Sections J.8 | |

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Formalizing the Notion of Proof | |

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Reasons for formalizing | |

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Proof, syntactic consequence, and theories | |

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What is a logic? | |

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Exercises for Section K | |

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An Axiomatization of PC | |

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The axiom system | |

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Exercises for Section L.1 | |

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A completeness proof | |

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The Strong Completeness Theorem for PC | |

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Exercises for Sections L.2 and L.3 | |

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Derived rules: substitution | |

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Exercises for Section L.4 | |

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Other Axiomatizations and Proofs of Completeness of PC | |

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History and Post's proof | |

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A constructive proof of the completeness of PC | |

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Exercises for Sections M.1 and M.2 | |

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Schema vs. the rule of substitution | |

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Independent axiom systems | |

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Proofs using rules only | |

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Exercises for Sections M.3--M.5 | |

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An axiomatization of PC in L([characters not reproducible], [right arrow], [logical and], [logical or]) | |

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An axiomatization of PC in L([characters not reproducible], [logical and]) | |

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Exercises for Sections M.6 and M.7 | |

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Classical logic without negation: the positive fragment of PC | |

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Exercises for Section M.8 | |

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The Reasonableness of PC | |

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Why classical logic is classical | |

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The paradoxes of PC | |

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Relatedness Logic: The Subject Matter of a Proposition - S and R - | |

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An Aspect of Propositions: Subject Matter | |

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The Formal Language | |

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Properties of the Primitive: Relatedness Relations | |

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Subject Matter as Set-Assignments | |

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Exercises for Sections A--D | |

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Truth and Relatedness-Tables | |

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The Formal Semantics for S | |

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Models based on relatedness relations | |

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Models based on subject matter assignments | |

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Non-symmetric relatedness logic, R | |

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Exercises for Sections E and F | |

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Examples | |

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If the moon is made of green cheese, then 2 + 2 = 4 | |

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2 + 2 = 4 Therefore: If the moon is made of green cheese, then 2 + 2 = 4 | |

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If Ralph is a dog and if 1 = 1 then 1 = 1, then 2 + 2 = 4 or 2 + 2 [not equal] 4 | |

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If John loves Mary, then Mary has 2 apples If Mary has 2 apples, then 2 + 2 = 4 Therefore: If John loves Mary, then 2 + 2 = 4 | |

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If Don squashed a duck and Don drives a car, then a duck is dead Therefore: If Don squashed a duck, then if Don drives a car, then a duck is dead | |

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Exercises for Section G | |

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Relatedness Logic Compared to Classical Logic | |

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The decidability of relatedness tautologies | |

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Every relatedness tautology is a classical tautology | |

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Classical tautologies that aren't relatedness tautologies | |

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Exercises for Section H | |

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Functional Completeness of the Connectives and the Normal Form Theorem for S | |

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Exercises for Section J | |

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Axiomatizations | |

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S in L([characters not reproducible], [right arrow]) | |

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Exercises for Section K.1 | |

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S in L([characters not reproducible], [right arrow], [Lambda]) | |

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R in L([characters not reproducible], [right arrow], [Lambda]) | |

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Substitution | |

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The Deduction Theorem | |

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Exercises for Section K.2-K.5 | |

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Historical Remarks | |

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A General Framework for Semantics for Propositional Logics | |

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Aspects of Sentences | |

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Propositions | |

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The logical connectives | |

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Two approaches to semantics | |

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Exercises for Section A | |

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Set-Assignment Semantics | |

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Models | |

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Exercises for Section B.1 | |

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Abstract models | |

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Semantics and logics | |

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Semantic and syntactic consequence relations | |

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Exercises for Sections B.2-B.4 | |

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Relation-Based Semantics | |

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Exercises for Section C | |

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Semantics Having a Simple Presentation | |

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Some Questions | |

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Simply presented semantics | |

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The Deduction Theorem | |

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Functional completeness of the connectives | |

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Representing the relations within the formal language | |

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Characterizing the class of relations in terms of schema | |

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Translating other semantics into the general framework | |

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Decidability | |

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Extensionally equivalent propositions and the rule of substitution | |

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On the Unity and Division of Logics | |

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Quine on deviant logical connectives | |

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Classical vs. nonclassical logics | |

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A Mathematical Presentation of the General Framework with the assistance of Walter Carnielli | |

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Languages | |

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Formal set-assignment semantics | |

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Formal relation-based semantics | |

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Exercises for Sections G.1-G.3 | |

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Specifying semantic structures | |

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Set-assignments and relations for SA | |

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Relations for RB | |

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Wholly intensional connectives | |

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Truth-default semantic structures | |

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Exercises for Sections G.4-G.6 | |

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Tautologies of the general framework | |

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Valid deductions of the general framework | |

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Examples | |

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The subformula property | |

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Axiomatizing deductions in L([characters not reproducible], [right arrow], [Lambda]) | |

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Deductions in languages containing disjunction | |

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Exercises for Sections G.7-G.8 | |

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Dependence Logics - D, Dual D, Eq, DPC - | |

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Dependence Logic | |

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The consequent is contained in the antecedent | |

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The structure of referential content | |

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Set-assignment semantics | |

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Relation-based semantics | |

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Exercises for Sections A.1-A.4 | |

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The decidability of dependence logic tautologies | |

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Examples of formalization | |

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Ari doesn't drink Therefore: If Ari drinks, then everyone drinks | |

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Not both Ralph is a dog and cats aren't nasty Therefore: If Ralph is a dog, then cats are nasty | |

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If Ralph is a bachelor, then Ralph is a man | |

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If dogs barks and Juney is a dog, then Juney barks Therefore: If dogs bark, then if Juney is a dog, then Juney barks | |

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If dogs bark, then Juney barks If Juney barks, then a dog has scared a thief Therefore: If dogs bark, then a dog has scared a thief | |

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If Ralph is a dog, then Ralph barks Therefore: If Ralph doesn't bark, then Ralph is not a dog | |

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Ralph is not a dog because he's a puppet | |

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Dependence logic tautologies compared to classical tautologies | |

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The functional completeness of the connectives | |

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Exercises for Sections A.5-A.8 | |

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An axioms system for D | |

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Exercises for Section A.9 | |

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History | |

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Dependence-Style Semantics | |

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Exercises for Section B | |

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Dual Dependence Logic, Dual D | |

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Exercises for Section C | |

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A Logic of Equality of Contents, Eq | |

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Motivation | |

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Set-assignment semantics | |

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Characterizing Eq-relations | |

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An axiom system for Eq | |

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Exercises for Section D | |

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A Syntactic Comparison of D, Dual D, Eq, and S | |

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Content as Logical Consequences | |

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Exercises for Section F | |

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Modal Logics - S4, S5, S4Grz, T, B, K, QT, MSI, ML, G, G* - | |

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Implication, Possibility, and Necessity | |

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Strict implication vs. material implication | |

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Possible worlds | |

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Necessity | |

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Different notions of necessity: accessibility relations | |

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Exercises for Section A | |

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The General Form of Possible-World Semantics for Modal Logics | |

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The formal framework | |

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Possibility and necessity in the formal language | |

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Exercises for Section B | |

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Semantic Presentations | |

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Logical necessity: S5 | |

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Semantics | |

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Semantic consequence | |

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Iterated modalities | |

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Syntactic characterization of the class of universal frames | |

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Some rules valid in S5 | |

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Exercises for Section C.1 | |

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K--all accessibility relations | |

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Exercises for Section C.2 | |

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T, B, and S4 | |

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Decidability and the Finite Model Property | |

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Exercises for Section C.4 | |

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Examples of Formalization | |

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If roses are red, then sugar is sweet | |

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If Ralph is a bachelor, then Ralph is a man | |

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If the moon is made of green cheese, then 2 + 2 = 4 | |

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If Juney was a dog, then surely it's possible that Juney was a dog | |

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If it's possible that Juney was a dog, then Juney was a dog | |

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It's not possible that Juney was a dog and a cat | |

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If it is necessary that Juney is a dog, then it is necessary that it is necessary that Juney is a dog | |

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If this paper is white, it must necessarily be white | |

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If Hoover was elected president, then he must have received the most votes Hoover was elected president Therefore: Hoover must have received the most votes | |

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A sea fight must take place tomorrow or not. But it is not necessary that it should take place tomorrow; neither is it necessary that it should not take place. Yet it is necessary that it either should or should not take place tomorrow | |

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It is contingent that US $1 bills are green | |

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It is possible for Richard L. Epstein to print his own bank notes | |

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If there were no dogs, then everyone would like cats | |

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It is permissible but not obligatory to kill cats | |

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A dog that likes cats is possible | |

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Example 2 of Chapter II.F is possible | |

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Ralph knows that Howie is a cat | |

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Example 18 is not possible | |

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Exercises for Section D | |

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Syntactic Characterizations of Modal Logics | |

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The general format | |

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Defined connectives | |

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PC in the language of modal logic | |

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Normal modal logics | |

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Axiomatizations and completeness theorems in L([characters not reproducible], [logical and], [square]) | |

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Axiomatizations and completeness theorems in L([characters not reproducible], [right arrow], [logical and]) | |

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Exercises for Sections E.1-E.3 | |

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Consequence relations | |

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Without necessitation, [characters not reproducible] | |

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With necessitation, [characters not reproducible]bL[superscript square] | |

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Exercises for Section E.4 | |

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Quasi-normal modal logics | |

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Exercises for Section F | |

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Set-Assignment Semantics for Modal Logics | |

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Semantics in L([characters not reproducible], [right arrow], [logical and]) | |

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Modal semantics of implication | |

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Weak modal semantics of implication | |

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Semantics in L([characters not reproducible], [logical and], [square]) | |

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Exercises for Sections G.1 and G.2 | |

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Connections of meanings in modal logics: the aptness of set-assignment semantics | |

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S5 | |

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Exercises for Section G.4 | |

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S4 in collaboration with Roger Maddux | |

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Exercises for Section G.5 | |

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T | |

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B | |

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Exercises for Sections G.6 and G.7 | |

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The Smallest Logics Characterized by Various Semantics | |

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K | |

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QT and quasi-normal logics | |

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The logic characterized by modal semantics of implication | |

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Exercises for Section H | |

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Modal Logics Modeling Notions of Provability | |

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'[square]' read as 'it is provable that' | |

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S4Grz | |

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G | |

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G* | |

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Intuitionism - Int and J - | |

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Intuitionism and Logic | |

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Exercises for Section A | |

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Heyting's Formalization of Intuitionism | |

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Heyting's axiom system Int | |

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Kripke semantics for Int | |

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Exercises for Section B | |

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Completeness of Kripke Semantics for Int | |

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Some syntactic derivations and the Deduction Theorem | |

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Completeness theorems for Int | |

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Exercises for Sections C.1 and C.2 | |

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On completeness proofs for Int, and an alternate axiomatization | |

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Exercises for Section C.3 | |

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Translations and Comparisons with Classical Logic | |

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Translations of Int into modal logic and classical arithmetic | |

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Translations of classical logic into Int | |

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Axiomatizations of classical logic relative to Int | |

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Exercises for Section D | |

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Set-assignment Semantics for Int | |

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The semantics | |

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Observations and refinements of the set-assignment semantics | |

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Exercises for Section E.1 and E.2 | |

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Bivalence in intuitionism: the aptness of set-assignment semantics | |

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The Minimal Calculus J | |

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The minimal calculus | |

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Kripke-style semantics | |

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An alternate axiomatization | |

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Kolmogorov's axiomatization of intuitionistic reasoning in L([characters not reproducible], [right arrow]) | |

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Exercises for Sections F.1-F.4 | |

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Set-assignment semantics | |

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Exercises for Section F.5 | |

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Many-Valued Logics - L[subscript 3], L[subscript n], L[subscript N], K[subscript 3], G[subscript 3], G[subscript n], G[subscript N], S5 - | |

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How Many Truth-Values? | |

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History | |

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Hypothetical reasoning and aspects of propositions | |

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A General Definition of Many-Valued Semantics | |

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The Lukasiewicz Logics | |

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The 3-valued logic L[subscript 3] | |

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The truth-tables and their interpretation | |

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Exercises for Section C.1.a | |

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A finite axiomatization of L[subscript 3] | |

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Exercises for Section C.1.b | |

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Wajsberg's axiomatization of L[subscript 3] | |

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Set-assignment semantics for L[subscript 3] | |

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Exercises for Section C.1.d | |

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The logics L[subscript n] and L[subscript N] | |

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Generalizing the 3-valued tables | |

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An axiom system for L[subscript N] | |

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Set-assignment semantics for L[subscript N] | |

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Exercises for Section C.2 | |

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Kleene's 3-Valued Logic | |

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The truth-tables | |

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Set-assignment semantics | |

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Exercises for Section D | |

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Logics Having No Finite-Valued Semantics | |

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General criteria | |

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Infinite-valued semantics for the modal logic S5 | |

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The Systems G[subscript n] and G[subscript N] | |

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Exercises for Section F | |

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A Method for Proving Axiom Systems Independent | |

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Exercises for Section G | |

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Paraconsistent Logic: J[subscript 3] | |

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Paraconsistent Logics | |

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The Semantics of J[subscript 3] | |

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Motivation | |

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The truth-tables | |

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Exercises for Sections B.2 | |

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Definability of the connectives | |

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The Relation Between J[subscript 3] and Classical Logic | |

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Exercises for Section C | |

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Consistency vs. Paraconsistency | |

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Definitions of completeness and consistency for J[subscript 3] theories | |

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The status of negation in J[subscript 3] | |

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Axiomatizations of J[subscript 3] | |

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As a modal logic | |

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As an extension of classical logic | |

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Exercises for Section E | |

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Set-Assignment Semantics for J[subscript 3] | |

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Truth-Default Semantics | |

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Exercises for Sections F and G | |

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Translations Between Logics | |

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Syntactic translations | |

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A formal notion of translation | |

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Exercises for Section A.1 | |

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Examples | |

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Exercises for Section A.2 | |

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Logics that cannot be translated grammatically into classical logic | |

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Exercises for Section A.3 | |

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Translations where there are no grammatical translations: R [two-head right arrow] PC and S [two-head right arrow] PC | |

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Exercises for Section A.4 | |

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Semantically faithful translations | |

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A formal notion of semantically faithful translation | |

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Exercises for Section B.1 | |

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Examples of semantically faithful translations | |

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The archetype of a semantically faithful translation: Int [right arrow, hooked] S4 | |

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The translations of PC into Int | |

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Exercises for Section B.4 | |

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The translation of S into PC | |

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Different presentations of the same logic and strong definability of connectives | |

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Exercises for Section B.6 | |

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Do semantically faithful translations preserve meaning? | |

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The Semantic Foundations of Logic | |

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Concluding Philosophical Remarks | |

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Summary of Logics | |

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Classical Logic, PC | |

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Relatedness and Dependence Logics S, R, D, Dual D, Eq, DPC | |

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Classical Modal Logics S4, S5, S4Grz, T, B, K, QT, MSI, ML, G, G* | |

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Intuitionistic Logics, Int and J | |

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Many-Valued Logics L[subscript 3], L[subscript n], L[subscript N], K[subscript 3], G[subscript 3], G[subscript n], G[subscript N], Paraconsistent J[subscript 3] | |

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Bibliography | |

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Glossary of Notation | |

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Index of Examples | |

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Index | |