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]].

Click on the Idea Number for the full details    |     back to texts     |     expand these ideas


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 / 4. 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
A partial ordering becomes 'total' if any two members of its field are comparable
A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets
'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
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
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
Model theory studies how set theory can model sets of sentences
Model theory is mostly confined to first-order theories
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 less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers
There are at least as many infinite cardinals as transfinite ordinals (because they will map)
The axiom of infinity with separation gives a least limit ordinal ω
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
Cantor: there is no size between naturals and reals, or between a set and its power set
If set theory is consistent, we cannot refute or prove the Continuum Hypothesis
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 / 2. 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 / B. Meaning / 8. Meaning through Sentences
Leibniz was the first modern to focus on sentence-sized units (where empiricists preferred word-size)