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All the ideas for 'fragments/reports', 'Language,Truth and Logic' and 'Understanding the Infinite'

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76 ideas

1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined
Philosophy is a department of logic [Ayer]
1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / e. Philosophy as reason
Philosophers should abandon speculation, as philosophy is wholly critical [Ayer]
1. Philosophy / E. Nature of Metaphysics / 7. Against Metaphysics
Humeans rejected the a priori synthetic, and so rejected even Kantian metaphysics [Ayer, by Macdonald,C]
1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
Critics say analysis can only show the parts, and not their distinctive configuration [Ayer]
1. Philosophy / G. Scientific Philosophy / 3. Scientism
Philosophy deals with the questions that scientists do not wish to handle [Ayer]
3. Truth / H. Deflationary Truth / 2. Deflationary Truth
We cannot analyse the concept of 'truth', because it is simply a mark that a sentence is asserted [Ayer]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Those who reject infinite collections also want to reject the Axiom of Choice [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
The Power Set is just the collection of functions from one collection to another [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Replacement was immediately accepted, despite having very few implications [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The iterative conception of set wasn't suggested until 1947 [Lavine]
The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Mathematical proof by contradiction needs the law of excluded middle [Lavine]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Every rational number, unlike every natural number, is divisible by some other number [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
The infinite is extrapolation from the experience of indefinitely large size [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
The intuitionist endorses only the potential infinite [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory will found all of mathematics - except for the notion of proof [Lavine]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Maths and logic are true universally because they are analytic or tautological [Ayer]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionism rejects set-theory to found mathematics [Lavine]
7. Existence / D. Theories of Reality / 1. Ontologies
Positivists regard ontology as either meaningless or stipulated [Ayer, by Robinson,H]
11. Knowledge Aims / B. Certain Knowledge / 1. Certainty
Only tautologies can be certain; other propositions can only be probable [Ayer]
11. Knowledge Aims / C. Knowing Reality / 2. Phenomenalism
Logical positivists could never give the sense-data equivalent of 'there is a table next door' [Robinson,H on Ayer]
Material things are constructions from actual and possible occurrences of sense-contents [Ayer]
12. Knowledge Sources / A. A Priori Knowledge / 4. A Priori as Necessities
We could verify 'a thing can't be in two places at once' by destroying one of the things [Ierubino on Ayer]
12. Knowledge Sources / A. A Priori Knowledge / 5. A Priori Synthetic
Whether geometry can be applied to reality is an empirical question outside of geometry [Ayer]
12. Knowledge Sources / A. A Priori Knowledge / 7. A Priori from Convention
By changing definitions we could make 'a thing can't be in two places at once' a contradiction [Ayer]
12. Knowledge Sources / A. A Priori Knowledge / 8. A Priori as Analytic
To say that a proposition is true a priori is to say that it is a tautology [Ayer]
12. Knowledge Sources / B. Perception / 4. Sense Data / a. Sense-data theory
Positivists prefer sense-data to objects, because the vocabulary covers both illusions and perceptions [Ayer, by Robinson,H]
12. Knowledge Sources / B. Perception / 7. Causal Perception
Causal and representative theories of perception are wrong as they refer to unobservables [Ayer]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
The main claim of rationalism is that thought is an independent source of knowledge [Ayer]
12. Knowledge Sources / D. Empiricism / 1. Empiricism
Empiricism lacked a decent account of the a priori, until Ayer said it was entirely analytic [O'Grady on Ayer]
All propositions (especially 'metaphysics') must begin with the senses [Ayer]
My empiricism logically distinguishes analytic and synthetic propositions, and metaphysical verbiage [Ayer]
12. Knowledge Sources / D. Empiricism / 4. Pro-Empiricism
It is further sense-experience which informs us of the mistakes that arise out of sense-experience [Ayer]
12. Knowledge Sources / D. Empiricism / 5. Empiricism Critique
Empiricism, it is said, cannot account for our knowledge of necessary truths [Ayer]
14. Science / C. Induction / 2. Aims of Induction
The induction problem is to prove generalisations about the future based on the past [Ayer]
14. Science / C. Induction / 3. Limits of Induction
We can't use the uniformity of nature to prove induction, as that would be circular [Ayer]
15. Nature of Minds / A. Nature of Mind / 4. Other Minds / b. Scepticism of other minds
Other minds are 'metaphysical' objects, because I can never observe their experiences [Ayer]
15. Nature of Minds / A. Nature of Mind / 4. Other Minds / c. Knowing other minds
A conscious object is by definition one that behaves in a certain way, so behaviour proves consciousness [Ayer]
16. Persons / B. Nature of the Self / 5. Self as Associations
If the self is meaningful, it must be constructed from sense-experiences [Ayer]
16. Persons / B. Nature of the Self / 7. Self and Body / a. Self needs body
Two experiences belong to one self if their contents belong with one body [Ayer]
Empiricists can define personal identity as bodily identity, which consists of sense-contents [Ayer]
17. Mind and Body / A. Mind-Body Dualism / 8. Dualism of Mind Critique
The supposed 'gulf' between mind and matter is based on the senseless concept of 'substances' [Ayer]
19. Language / A. Nature of Meaning / 5. Meaning as Verification
A sentence is factually significant to someone if they know how to verify its proposition [Ayer]
Factual propositions imply (in conjunction with a few other premises) possible experiences [Ayer]
Tautologies and empirical hypotheses form the entire class of significant propositions [Ayer]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / c. Ethical intuitionism
Moral intuition is worthless if there is no criterion to decide between intuitions [Ayer]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / h. Expressivism
Ayer defends the emotivist version of expressivism [Ayer, by Smith,M]
To say an act is wrong makes no further statement about it, but merely expresses disapproval [Ayer]
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius]
27. Natural Reality / G. Biology / 3. Evolution
Archelaus said life began in a primeval slime [Archelaus, by Schofield]
28. God / A. Divine Nature / 4. Divine Contradictions
A person with non-empirical attributes is unintelligible. [Ayer]
28. God / B. Proving God / 2. Proofs of Reason / b. Ontological Proof critique
When we ascribe an attribute to a thing, we covertly assert that it exists [Ayer]
28. God / C. Attitudes to God / 5. Atheism
If theism is non-sensical, then so is atheism. [Ayer]
29. Religion / D. Religious Issues / 1. Religious Commitment / c. Religious Verification
The 'truths' expressed by theists are not literally significant [Ayer]