Classical and Nonclassical Logics

Name: Classical and Nonclassical Logics | An Introduction to the Mathematics of Propositions
Brand: Princeton University Press
Price: 127.99 EUR
Availability: OnlineOnly

An Introduction to the Mathematics of Propositions

Eric Schechter(Autor*in)

Princeton University Press

1. Auflage

Erschienen am 6. Oktober 2020

520 Seiten

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978-0-691-22014-7 (ISBN)

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Inhalt

Cover Page
Half-title Page
Title Page
Copyright Page
Contents
A. Preliminaries
1. Introduction for teachers
Purpose and intended audience
Topics in the book
Why pluralism?
Feedback
Acknowledgments
2. Introduction for students
Who should study logic?
Formalism and certification
Language and levels
Semantics and syntactics
Historical perspective
Pluralism
Jarden's example (optional)
3. Informal set theory
Sets and their members
Russell's paradox
Subsets
Functions
The Axiom of choice (optional)
Operations on sets
Venn diagrams
Syllogisms (optional)
Infinite sets
4. Topologies and interiors (postponable)
Topologies
Interiors
Generated topologies and finite topologies (optional)
5. English and informal classical logic
Language and bias
Parts of speech
Semantic values
Disjunction (or)
Conjunction (and)
Negation (not)
Material implication
Cotenability fusion and constants (postponable)
Methods of proof
Working backwards
Quantifiers
Induction
Induction examples (optional)
6. Definition of a formal language
The alphabet
The grammar
Removing parentheses
Denned symbols
Prefix notation (optional)
Variable sharing
Formula schemes
Order preserving or reversing subformulas (postponable)
B. Semantics
7. Definitions for semantics
Interpretations
Functional interpretations
Tautology and truth preservation
8. Numerically valued interpretations
The two-valued interpretation
Fuzzy interpretations
Two integer-valued interpretations
More about comparative logic
More about Sugihara's interpretation
9. Set-valued interpretations
Powerset interpretations
Hexagon interpretation (optional)
The crystal interpretation
church's diamond (optional)
10. Topological semantics (postponable)
Topological interpretations
Examples
Common tautologies
Nonredundancy of symbols
Variable sharing
Adequacy of finite topologies (optional)
Disjunction property (optional)
11. More advanced topics in semantics
common tautologies
Images of interpretations
Dugundji formulas
C. Basic syntactics
12. Inference systems
13. Basic implication
Assumptions of basic implication
A few easy derivations
Lemmaless expansions
Detachmental corollaries
Iterated implication (postponable)
14. Basic logic
Further assumptions
Basic positive logic
Basic negation
Substitution principles
D. One-formula extensions
15. Contraction
Weak contraction
Contraction
16. Expansion and positive paradox
Expansion and mingle
Positive paradox (strong expansion)
Further consequences of positive paradox
17. Explosion
18. Fusion
19. Not-elimination
Not-elimination and contrapositives
Interchangeability results
Miscellaneous consequences of notelimination
20. Relativity
E. Soundness and major logics
21. Soundness
22. constructive axioms: avoiding not-elimination
Constructive implication
Herbrand-Tarski Deduction Principle
Basic logic revisited
Soundness
Nonconstructive axioms and classical logic
Glivenko's Principle
23. Relevant axioms: avoiding expansion
Some syntactic results
Relevant deduction principle (optional)
Soundness
Mingle: slightly irrelevant
Positive paradox and Classical logic
24. Fuzzy axioms: avoiding contraction
Axioms
Meredith's chain proof
Additional notations
Wajsberg logic
Deduction principle for Wajsberg logic
25. Classical logic
Axioms
Soundness results
Independence of axioms
26. Abelian logic
F. Advanced results
27. Harrop's principle for constructive logic
Meyer's valuation
Harrop's principle
The disjunction property
Admissibility
Results in other logics
28. Multiple worlds for implications
Multiple worlds
Implication models
Soundness
Canonical models
Completeness
29. completeness via maximality
Maximal unproving sets
Classical logic
Wajsberg logic
Constructive logic
Non-finitely-axiomatizable logics
References
Symbol list
Index

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