 Skip to content

# Propositional Logics The Semantic Foundations of Logic

## Edition: 2nd 2001 (Revised)

### Authors: Richard L. Epstein, Walter A. Carnielli

List price: \$101.95 30 day, 100% satisfaction guarantee!
Out of stock
what's this?
Rush Rewards U
Members Receive:  You have reached 400 XP and carrot coins. That is the daily max!
This book is a survey and overview of the major systems of propositional logics, answering the question, "If logic is the right way to reason, why are there so many logics?" Each system is presented both formally and with philosophical motivation in Epstein's renowned clear style.
Customers also bought

### Book details

List price: \$101.95
Edition: 2nd
Copyright year: 2001
Publisher: Wadsworth
Publication date: 7/25/2000
Binding: Paperback
Pages: 525
Size: 7.50" wide x 9.00" long x 1.25" tall
Weight: 1.848
Language: English

 Preface and Acknowledgements The Basic Assumptions of Propositional Logic What is Logic? Propositions Sentences, propositions, and truth Other views of propositions Words and Propositions as Types Propositions in English Exercises for Sections A--D Form and Content Propositional Logic and the Basic Connectives Exercises for Sections E and F A Formal Language for Propositional Logic Defining the formal language Realizations: semi-formal English Exercises for Section G Classical Propositional Logic - PC - The Classical Abstraction and the Fregean Assumption Truth-Functions and the Division of Form and Content Models Exercises for Sections A--C Validity and Semantic Consequence Tautologies Semantic Consequence Exercises for Section D The Logical Form of a Proposition On logical form Criteria of formalization Other propositional connectives Examples of Formalization Ralph is a dog or Dusty is a horse and Howie is a cat Therefore: Howie is a cat Ralph is a dog and George is a duck and Howie is a cat Ralph is a dog or he's a puppet Ralph is a dog if he's not a puppet Ralph is a dog although he's a puppet Ralph is not a dog because he's a puppet Three faces of a die are even numbered Three faces of a die are not even numbered Therefore: Ralph is a dog Ken took off his clothes and went to bed The quotation marks are signals for you to understand what I mean; they are not part of the realization Every natural number is even or odd If Ralph is a dog, then Ralph barks Ralph barks Therefore: Ralph is a dog Suppose {s[subscript n]} is monotonic. Then {s[subscript n]} converges if and only if it is bounded Dedekind's Theorem Exercises for Sections E and F Further Abstractions: The Role of Mathematics in Logic Induction Exercises for Sections G and H A Mathematical Presentation of PC The formal language Exercises for Section J.1 Models and the semantic consequence relation Exercises for Section J.2 The truth-functional completeness of the connectives The choice of language for PC Normal forms The principle of duality for PC Exercises for Sections J.3-J.6 The decidability of tautologies Exercises for Section J.7 Some PC-tautologies Exercises for Sections J.8 Formalizing the Notion of Proof Reasons for formalizing Proof, syntactic consequence, and theories What is a logic? Exercises for Section K An Axiomatization of PC The axiom system Exercises for Section L.1 A completeness proof The Strong Completeness Theorem for PC Exercises for Sections L.2 and L.3 Derived rules: substitution Exercises for Section L.4 Other Axiomatizations and Proofs of Completeness of PC History and Post's proof A constructive proof of the completeness of PC Exercises for Sections M.1 and M.2 Schema vs. the rule of substitution Independent axiom systems Proofs using rules only Exercises for Sections M.3--M.5 An axiomatization of PC in L([characters not reproducible], [right arrow], [logical and], [logical or]) An axiomatization of PC in L([characters not reproducible], [logical and]) Exercises for Sections M.6 and M.7 Classical logic without negation: the positive fragment of PC Exercises for Section M.8 The Reasonableness of PC Why classical logic is classical The paradoxes of PC Relatedness Logic: The Subject Matter of a Proposition - S and R - An Aspect of Propositions: Subject Matter The Formal Language Properties of the Primitive: Relatedness Relations Subject Matter as Set-Assignments Exercises for Sections A--D Truth and Relatedness-Tables The Formal Semantics for S Models based on relatedness relations Models based on subject matter assignments Non-symmetric relatedness logic, R Exercises for Sections E and F Examples If the moon is made of green cheese, then 2 + 2 = 4 2 + 2 = 4 Therefore: If the moon is made of green cheese, then 2 + 2 = 4 If Ralph is a dog and if 1 = 1 then 1 = 1, then 2 + 2 = 4 or 2 + 2 [not equal] 4 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 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 Exercises for Section G Relatedness Logic Compared to Classical Logic The decidability of relatedness tautologies Every relatedness tautology is a classical tautology Classical tautologies that aren't relatedness tautologies Exercises for Section H Functional Completeness of the Connectives and the Normal Form Theorem for S Exercises for Section J Axiomatizations S in L([characters not reproducible], [right arrow]) Exercises for Section K.1 S in L([characters not reproducible], [right arrow], [Lambda]) R in L([characters not reproducible], [right arrow], [Lambda]) Substitution The Deduction Theorem Exercises for Section K.2-K.5 Historical Remarks A General Framework for Semantics for Propositional Logics Aspects of Sentences Propositions The logical connectives Two approaches to semantics Exercises for Section A Set-Assignment Semantics Models Exercises for Section B.1 Abstract models Semantics and logics Semantic and syntactic consequence relations Exercises for Sections B.2-B.4 Relation-Based Semantics Exercises for Section C Semantics Having a Simple Presentation Some Questions Simply presented semantics The Deduction Theorem Functional completeness of the connectives Representing the relations within the formal language Characterizing the class of relations in terms of schema Translating other semantics into the general framework Decidability Extensionally equivalent propositions and the rule of substitution On the Unity and Division of Logics Quine on deviant logical connectives Classical vs. nonclassical logics A Mathematical Presentation of the General Framework with the assistance of Walter Carnielli Languages Formal set-assignment semantics Formal relation-based semantics Exercises for Sections G.1-G.3 Specifying semantic structures Set-assignments and relations for SA Relations for RB Wholly intensional connectives Truth-default semantic structures Exercises for Sections G.4-G.6 Tautologies of the general framework Valid deductions of the general framework Examples The subformula property Axiomatizing deductions in L([characters not reproducible], [right arrow], [Lambda]) Deductions in languages containing disjunction Exercises for Sections G.7-G.8 Dependence Logics - D, Dual D, Eq, DPC - Dependence Logic The consequent is contained in the antecedent The structure of referential content Set-assignment semantics Relation-based semantics Exercises for Sections A.1-A.4 The decidability of dependence logic tautologies Examples of formalization Ari doesn't drink Therefore: If Ari drinks, then everyone drinks Not both Ralph is a dog and cats aren't nasty Therefore: If Ralph is a dog, then cats are nasty If Ralph is a bachelor, then Ralph is a man If dogs barks and Juney is a dog, then Juney barks Therefore: If dogs bark, then if Juney is a dog, then Juney barks 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 If Ralph is a dog, then Ralph barks Therefore: If Ralph doesn't bark, then Ralph is not a dog Ralph is not a dog because he's a puppet Dependence logic tautologies compared to classical tautologies The functional completeness of the connectives Exercises for Sections A.5-A.8 An axioms system for D Exercises for Section A.9 History Dependence-Style Semantics Exercises for Section B Dual Dependence Logic, Dual D Exercises for Section C A Logic of Equality of Contents, Eq Motivation Set-assignment semantics Characterizing Eq-relations An axiom system for Eq Exercises for Section D A Syntactic Comparison of D, Dual D, Eq, and S Content as Logical Consequences Exercises for Section F Modal Logics - S4, S5, S4Grz, T, B, K, QT, MSI, ML, G, G* - Implication, Possibility, and Necessity Strict implication vs. material implication Possible worlds Necessity Different notions of necessity: accessibility relations Exercises for Section A The General Form of Possible-World Semantics for Modal Logics The formal framework Possibility and necessity in the formal language Exercises for Section B Semantic Presentations Logical necessity: S5 Semantics Semantic consequence Iterated modalities Syntactic characterization of the class of universal frames Some rules valid in S5 Exercises for Section C.1 K--all accessibility relations Exercises for Section C.2 T, B, and S4 Decidability and the Finite Model Property Exercises for Section C.4 Examples of Formalization If roses are red, then sugar is sweet If Ralph is a bachelor, then Ralph is a man If the moon is made of green cheese, then 2 + 2 = 4 If Juney was a dog, then surely it's possible that Juney was a dog If it's possible that Juney was a dog, then Juney was a dog It's not possible that Juney was a dog and a cat If it is necessary that Juney is a dog, then it is necessary that it is necessary that Juney is a dog If this paper is white, it must necessarily be white 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 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 It is contingent that US \$1 bills are green It is possible for Richard L. Epstein to print his own bank notes If there were no dogs, then everyone would like cats It is permissible but not obligatory to kill cats A dog that likes cats is possible Example 2 of Chapter II.F is possible Ralph knows that Howie is a cat Example 18 is not possible Exercises for Section D Syntactic Characterizations of Modal Logics The general format Defined connectives PC in the language of modal logic Normal modal logics Axiomatizations and completeness theorems in L([characters not reproducible], [logical and], [square]) Axiomatizations and completeness theorems in L([characters not reproducible], [right arrow], [logical and]) Exercises for Sections E.1-E.3 Consequence relations Without necessitation, [characters not reproducible] With necessitation, [characters not reproducible]bL[superscript square] Exercises for Section E.4 Quasi-normal modal logics Exercises for Section F Set-Assignment Semantics for Modal Logics Semantics in L([characters not reproducible], [right arrow], [logical and]) Modal semantics of implication Weak modal semantics of implication Semantics in L([characters not reproducible], [logical and], [square]) Exercises for Sections G.1 and G.2 Connections of meanings in modal logics: the aptness of set-assignment semantics S5 Exercises for Section G.4 S4 in collaboration with Roger Maddux Exercises for Section G.5 T B Exercises for Sections G.6 and G.7 The Smallest Logics Characterized by Various Semantics K QT and quasi-normal logics The logic characterized by modal semantics of implication Exercises for Section H Modal Logics Modeling Notions of Provability '[square]' read as 'it is provable that' S4Grz G G* Intuitionism - Int and J - Intuitionism and Logic Exercises for Section A Heyting's Formalization of Intuitionism Heyting's axiom system Int Kripke semantics for Int Exercises for Section B Completeness of Kripke Semantics for Int Some syntactic derivations and the Deduction Theorem Completeness theorems for Int Exercises for Sections C.1 and C.2 On completeness proofs for Int, and an alternate axiomatization Exercises for Section C.3 Translations and Comparisons with Classical Logic Translations of Int into modal logic and classical arithmetic Translations of classical logic into Int Axiomatizations of classical logic relative to Int Exercises for Section D Set-assignment Semantics for Int The semantics Observations and refinements of the set-assignment semantics Exercises for Section E.1 and E.2 Bivalence in intuitionism: the aptness of set-assignment semantics The Minimal Calculus J The minimal calculus Kripke-style semantics An alternate axiomatization Kolmogorov's axiomatization of intuitionistic reasoning in L([characters not reproducible], [right arrow]) Exercises for Sections F.1-F.4 Set-assignment semantics Exercises for Section F.5 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 - How Many Truth-Values? History Hypothetical reasoning and aspects of propositions A General Definition of Many-Valued Semantics The Lukasiewicz Logics The 3-valued logic L[subscript 3] The truth-tables and their interpretation Exercises for Section C.1.a A finite axiomatization of L[subscript 3] Exercises for Section C.1.b Wajsberg's axiomatization of L[subscript 3] Set-assignment semantics for L[subscript 3] Exercises for Section C.1.d The logics L[subscript n] and L[subscript N] Generalizing the 3-valued tables An axiom system for L[subscript N] Set-assignment semantics for L[subscript N] Exercises for Section C.2 Kleene's 3-Valued Logic The truth-tables Set-assignment semantics Exercises for Section D Logics Having No Finite-Valued Semantics General criteria Infinite-valued semantics for the modal logic S5 The Systems G[subscript n] and G[subscript N] Exercises for Section F A Method for Proving Axiom Systems Independent Exercises for Section G Paraconsistent Logic: J[subscript 3] Paraconsistent Logics The Semantics of J[subscript 3] Motivation The truth-tables Exercises for Sections B.2 Definability of the connectives The Relation Between J[subscript 3] and Classical Logic Exercises for Section C Consistency vs. Paraconsistency Definitions of completeness and consistency for J[subscript 3] theories The status of negation in J[subscript 3] Axiomatizations of J[subscript 3] As a modal logic As an extension of classical logic Exercises for Section E Set-Assignment Semantics for J[subscript 3] Truth-Default Semantics Exercises for Sections F and G Translations Between Logics Syntactic translations A formal notion of translation Exercises for Section A.1 Examples Exercises for Section A.2 Logics that cannot be translated grammatically into classical logic Exercises for Section A.3 Translations where there are no grammatical translations: R [two-head right arrow] PC and S [two-head right arrow] PC Exercises for Section A.4 Semantically faithful translations A formal notion of semantically faithful translation Exercises for Section B.1 Examples of semantically faithful translations The archetype of a semantically faithful translation: Int [right arrow, hooked] S4 The translations of PC into Int Exercises for Section B.4 The translation of S into PC Different presentations of the same logic and strong definability of connectives Exercises for Section B.6 Do semantically faithful translations preserve meaning? The Semantic Foundations of Logic Concluding Philosophical Remarks Summary of Logics Classical Logic, PC Relatedness and Dependence Logics S, R, D, Dual D, Eq, DPC Classical Modal Logics S4, S5, S4Grz, T, B, K, QT, MSI, ML, G, G* Intuitionistic Logics, Int and J 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] Bibliography Glossary of Notation Index of Examples Index