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Propositional Logics The Semantic Foundations of Logic

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ISBN-10: 053455847X

ISBN-13: 9780534558475

Edition: 2nd 2001 (Revised)

Authors: Richard L. Epstein, Walter A. Carnielli

List price: $101.95
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Description:

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.
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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