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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 | |