Ideas from 'The Evolution of Logic' by William D. Hart [2010], by Theme Structure

[found in 'The Evolution of Logic' by Hart,W.D. [CUP 2010,978-0-521-74772-1]].

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1. Philosophy / C. History of Philosophy / 4. Later European Philosophy / c. Eighteenth century philosophy
We are all post-Kantians, because he set the current agenda for philosophy
1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / d. Philosophy as puzzles
The problems are the monuments of philosophy
1. Philosophy / F. Analytic Philosophy / 1. Analysis
To study abstract problems, some knowledge of set theory is essential
3. Truth / C. Correspondence Truth / 2. Correspondence to Facts
Tarski showed how we could have a correspondence theory of truth, without using 'facts'
3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do
3. Truth / F. Semantic Truth / 2. Semantic Truth
A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...'
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Set theory articulates the concept of order (through relations)
Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
∈ relates across layers, while ⊆ relates within layers
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
Cantor's Theorem: for any set x, its power set P(x) has more members than x
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Without the empty set we could not form a∩b without checking that a and b meet
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
In the modern view, foundation is the heart of the way to do set theory
Foundation Axiom: an nonempty set has a member disjoint from it
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
With the Axiom of Choice every set can be well-ordered
We can choose from finite and evident sets, but not from infinite opaque ones
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
Not every predicate has an extension, but Separation picks the members that satisfy a predicate
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
If we accept that V=L, it seems to settle all the open questions of set theory
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Na´ve logical sets
Na´ve set theory has trouble with comprehension, the claim that every predicate has an extension
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The iterative conception may not be necessary, and may have fixed points or infinitely descending chains
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
'Well-ordering' must have a least member, so it does the natural numbers but not the integers
Von Neumann defines α<β as α∈β
A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets
A partial ordering becomes 'total' if any two members of its field are comparable
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Maybe sets should be rethought in terms of the even more basic categories
5. Theory of Logic / G. Quantification / 3. Objectual Quantification
The universal quantifier can't really mean 'all', because there is no universal set
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory studies how set theory can model sets of sentences
Model theory is mostly confined to first-order theories
Models are ways the world might be from a first-order point of view
Modern model theory begins with the proof of Los's Conjecture in 1962
5. Theory of Logic / K. Features of Logics / 6. Compactness
First-order logic is 'compact': consequences of a set are consequences of a finite subset
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
The Burali-Forti paradox is a crisis for Cantor's ordinals
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
The machinery used to solve the Liar can be rejigged to produce a new Liar
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
Von Neumann treated cardinals as a special sort of ordinal
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton
The axiom of infinity with separation gives a least limit ordinal ω
There are at least as many infinite cardinals as transfinite ordinals (because they will map)
The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
19th century arithmetization of analysis isolated the real numbers from geometry
Descartes showed a one-one order-preserving match between points on a line and the real numbers
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
We can establish truths about infinite numbers by means of induction
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
If set theory is consistent, we cannot refute or prove the Continuum Hypothesis
Cantor: there is no size between naturals and reals, or between a set and its power set
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Mathematics makes existence claims, but philosophers usually say those are never analytic
7. Existence / C. Structure of Existence / 8. Stuff / a. Pure stuff
Mass words do not have plurals, or numerical adjectives, or use 'fewer'
12. Knowledge Sources / A. A Priori Knowledge / 2. Self-Evidence
Fregean self-evidence is an intrinsic property of basic truths, rules and definitions
12. Knowledge Sources / A. A Priori Knowledge / 11. Denying the A Priori
The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori
18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
The Fregean concept of GREEN is a function assigning true to green things, and false to the rest
19. Language / A. Nature of Meaning / 7. Meaning Holism / a. Sentence meaning
Leibniz was the first modern to focus on sentence-sized units (where empiricists preferred word-size)