(Ebook) Logic Without Borders 1st Edition by Asa Hirvonen, Juha Kontinen, Roman Kossak, Andres Villaveces ISBN 9781614517726 161451772X

(Ebook) Logic Without Borders 1st Edition by Asa Hirvonen, Juha Kontinen, Roman Kossak, Andres Villaveces ISBN 9781614517726 161451772X

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ISBN 10: 161451772X
ISBN 13: 9781614517726
Author: Asa Hirvonen, Juha Kontinen, Roman Kossak, Andres Villaveces

(Ebook) Logic Without Borders 1st Edition Table of contents:

On the “Logic without Borders” Point of View

1 Introduction

2 First Episode: Model Theory

2.1 A Remark of Sacks

3 Second Episode: The Symbiosis between Model Theory and Set Theory

4 Third Episode: The Entanglement of Second Order Logic with Set Theory

5 Conclusion

Bibliography

Arrow’s Theorem by Arrow Theory

1 Introduction

1.1 What Arrow’s Theorem Says

2 A ‘standard’ account of Arrow’s theorem

2.1 Preference Relations

2.2 Social choice situations

2.3 Proof of Arrow’s Theorem

2.3.1 Decisiveness, Neutrality and Monotonicity

2.3.2 The Ultrafilter of Decisive Sets

2.4 Arrow’s Theorem

3 Categorical Formulation of Arrow’s Theorem

3.1 Categorical formulation of UD

3.2 Categorical formulation of IIA

3.3 Categorical formulation of P

3.4 Categorical Formulation of Arrow’s Theorem

3.5 An Analogous Result in Type Theory

3.5.0.1 Question

3.5.0.2 Answer

3.5.0.3 Sketch

3.5.0.4 Exercise

4 Categorical perspective on the proof of Arrow’s Theorem

4.1 Neutrality as Naturality

4.2 The Factorization Theorem

5 Discussion

Bibliography

How Big Should the Monster Model Be? - To Jouko Väänänen on his 60th birthday

1 The two programs

1.1 Set Theoretic Background

1.2 Grothendieck Universes and Wiles proof

1.3 Monster models in Model theory

2 Hanf Numbers and Monster models

3 Three Examples

3.1 Is replacement needed?

3.2 Bounded orbits of types

3.3 Saturation and omission

4 What kind of conditions imply only weak set theory is needed to prove a proposition?

Bibliography

Modal Logic in the Modal Sense of Modality - for Jouko Väänänen

1 Modal Logic & Border-Crossing

2 Modality Lost

3 Modality Regained

4 Necessity and Convention Reconsidered

5 Realism and Pragmatism about Modality

6 The Varied Sorrows of Modal Logic

7 Realism, Pragmatism, and the Modal Paradox

Note

Lindström’s Theorem for Positive Logics, a Topological View

1 Introduction

2 The elementary topology

3 Topological regularity

4 An intrinsic characterization of the elementary topology

5 Lindström’s theorem revisited

6 Compactness and ultraproducts

7 Imperfect information logic as [0,1]-valued logic

8 Final remarks

Bibliography

Model Theory of Fields With Operators – a Survey - For Jouko, on the occasion of his 60th birthday

1 Introduction

2 Basic results

2.1 Differential fields of characteristic 0

2.2 Several derivations

2.3 Geometric axiomatisation

2.4 Other theories of differential fields of characteristic 0

2.5 More on DCF0

2.6 Differential fields of positive characteristic

2.7 Separably closed fields of positive characteristic

2.8 Hasse-Schmidt derivations

2.9 Derivations of the Frobenius

2.10 Difference fields

2.11 Mixing operators: the commutative case

2.12 A remark

2.13 A general framework for fields with operators

3 Stability theoretic notions

3.1 Algebraic closure, independence

3.2 Zilber’s conjecture – the trichotomy

3.3 Two generalisations of modularity

3.4 Properties of one-basedness

3.5 Some other structures which satisfy the dichotomy

3.6 Some words on proofs of the dichotomy

3.7 A few words on the CBP

3.8 Definable groups

Bibliography

Some Aspects of the Ramsey Theory of Real Numbers

1 Introduction

2 Ramsey’s Theorem

3 Sublattices of P(ℕ)

4 Homogeneity in terms of products of finite sets

5 The Ramsey property parametrized by products of finite sets.

6 Independence of the Ramsey property from the polarized property.

7 Independence of the sublattice property from the polarized property.

Bibliography

The Singular World of Singular Cardinals

1 Introduction

2 Tree embeddings

3 Universal graphs

4 Conclusion

Bibliography

Logical Nihilism

1 Introduction

2 The law of excluded middle

3 Logic imposed and logic discovered

4 Conclusion

Bibliography

The Doxastic Interpretation of Team Semantics

1 Introduction

2 Belief Models

3 Atoms and First Order Formulas

4 Belief Updates

5 Adjoints

6 Minimal updates

7 Quantifiers

8 Conclusion

Bibliography

The Size of a Formula as a Measure of Complexity

1 Introduction

2 A Game for Propositional Logic

3 The Propositional Complexity of Parity

4 A Game for Predicate Logic

A Game for Existential Formulas

5 The Existential Complexity of Non-Emptiness of Boolean Combinations

6 The Existential Complexity of the Length of Linear Order

Bibliography

Notes on the History of Scope

1 Early resolutions of scope ambiguities

2 Possible anticipations of scope

3 Frege’s introduction of quantifier scope

4 Grammars 1: Stemmata

5 Grammars 2: Phrase markers

6 Ibn Sīnā opens up the field of logic

7 Negations and quantifiers

8 The semantics of quantifiers

9 Obstacles to scope

Bibliography

Universal Structures with Forbidden Homomorphisms

1 Introduction

1.0.0.5 Lifts and shadows.

1.0.0.6 Homogenization.

1.0.0.7 Our motivation and results.

2 Regular families of structures and

2.0.0.8 Example.

2.0.0.9 Example.

3 Construction of universal structures

4 Non-existence of universal structures for classes Forbh().

4.0.0.10 Example.

5 Homomorphism dualities and constraint satisfaction problems

5.0.0.11 Remark.

Bibliography

Counting Measure and Forking in Finite Models

1 Counting measure and forking

2 Examples

Bibliography

The Model Theory of Generic Cuts

1 Preliminaries

2 Generic cuts

3 Saturation

4 Model completeness

5 Existential closure

Bibliography

On Natural Deduction in Dependence Logic

1 Introduction

2 Preliminaries

3 Sentences versus open formulas

3.1 Generalizing completeness for open formulas

3.2 The case of independence logic

Bibliography

Infinitary Methods in Finite Model Theory

1 Introduction

2 Preliminaries

3 An easy inexpressibility result – connectivity

4 Locality Theorems

4.1 Gaifman locality

4.2 Hanf locality

5 Ultraproducts and directed reachability

Bibliography

Saturating the Random Graph with an Independent Family of Small Range - Dedicated to Jouko Väänänen on the occasion of his 60th birthday

1 Introduction

1.1 Keisler’s order and model theory

1.1.1 What is Keisler’s order and why is it model-theoretically interesting?

1.2 Prolegomena to our main theorem

1.2.1 What avenues of investigation does this suggest for simple theories?

1.3 Translations between set theory and model theory

1.4 Set-theoretic theorems and aspects of this program

2 Preliminaries

3 Main Theorem

Appendix

Bibliography

Constructive Realism in Mathematics

1 Introduction

2 Philosophy of Mathematics: Three Main Problems

3 Abstract Entities: Three Main Schools

4 Popper’s Three Worlds

5 Mathematics and World 3

6 Mathematical Knowledge

7 Conclusion

Bibliography

The Twin Continua of Inductive Methods - Dedicated to our good friend Jouko

1 Introduction

2 Carnap’s Continuum of Inductive Methods

3 Another Continuum

4 Two Rationalities?

5 Conclusion

Bibliography

A.E.C. with Not Too Many Models - Dedicated to Jouko Väänänen honouring his 60th birthday

1 Introduction to the subject

2 Introduction to the paper

2.1 Motivation/Content

2.2 Discussion

2.3 What is Done

2.4 Recalling Definitions and Notation

3 More on Templates

4 Approximation to EM models

5 Concluding Remarks

Bibliography

Pursuing Logic without Borders

1 Introduction

2 Generalized quantifiers

3 Härtig quantifier

4 Set theory

5 Trees, games and models

6 Dependence logic

7 Second order logic

8 Friends

Bibliography

A Radio Interview with Jouko Väänänen

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