Combining Texts

All the ideas for 'works', 'The Epic of Gilgamesh' and 'Philosophy of Logic'

expand these ideas     |    start again     |     specify just one area for these texts


68 ideas

2. Reason / B. Laws of Thought / 3. Non-Contradiction
If you say that a contradiction is true, you change the meaning of 'not', and so change the subject [Quine]
3. Truth / F. Semantic Truth / 2. Semantic Truth
Talk of 'truth' when sentences are mentioned; it reminds us that reality is the point of sentences [Quine]
3. Truth / H. Deflationary Truth / 1. Redundant Truth
Truth is redundant for single sentences; we do better to simply speak the sentence [Quine]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
We can eliminate 'or' from our basic theory, by paraphrasing 'p or q' as 'not(not-p and not-q)' [Quine]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
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 [Cantor, by Hart,WD]
Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
My logical grammar has sentences by predication, then negation, conjunction, and existential quantification [Quine]
5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
Maybe logical truth reflects reality, but in different ways in different languages [Quine]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Quine rejects second-order logic, saying that predicates refer to multiple objects [Quine, by Hodes]
Quantifying over predicates is treating them as names of entities [Quine]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Excluded middle has three different definitions [Quine]
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
Quantification theory can still be proved complete if we add identity [Quine]
5. Theory of Logic / F. Referring in Logic / 1. Naming / f. Names eliminated
Names are not essential, because naming can be turned into predication [Quine]
5. Theory of Logic / G. Quantification / 1. Quantification
Universal quantification is widespread, but it is definable in terms of existential quantification [Quine]
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
You can't base quantification on substituting names for variables, if the irrationals cannot all be named [Quine]
Some quantifications could be false substitutionally and true objectually, because of nameless objects [Quine]
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
Putting a predicate letter in a quantifier is to make it the name of an entity [Quine]
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
A sentence is logically true if all sentences with that grammatical structure are true [Quine]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
There are infinite sets that are not enumerable [Cantor, by Smith,P]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox
Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
Cantor took the ordinal numbers to be primary [Cantor, by Tait]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner]
Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD]
Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten]
Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS]
Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Pure mathematics is pure set theory [Cantor]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Cantor says that maths originates only by abstraction from objects [Cantor, by Frege]
8. Modes of Existence / B. Properties / 12. Denial of Properties
Predicates are not names; predicates are the other parties to predication [Quine]
9. Objects / A. Existence of Objects / 1. Physical Objects
A physical object is the four-dimensional material content of a portion of space-time [Quine]
9. Objects / E. Objects over Time / 4. Four-Dimensionalism
Four-d objects helps predication of what no longer exists, and quantification over items from different times [Quine]
10. Modality / B. Possibility / 8. Conditionals / b. Types of conditional
Some conditionals can be explained just by negation and conjunction: not(p and not-q) [Quine]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend]
18. Thought / E. Abstraction / 2. Abstracta by Selection
Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor]
19. Language / A. Nature of Meaning / 8. Synonymy
Single words are strongly synonymous if their interchange preserves truth [Quine]
19. Language / D. Propositions / 6. Propositions Critique
It makes no sense to say that two sentences express the same proposition [Quine]
There is no rule for separating the information from other features of sentences [Quine]
We can abandon propositions, and just talk of sentences and equivalence [Quine]
19. Language / F. Communication / 5. Pragmatics / a. Contextual meaning
A good way of explaining an expression is saying what conditions make its contexts true [Quine]
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
The gods alone live forever with Shamash. The days of humans are numbered. [Anon (Gilg)]
27. Natural Reality / C. Space / 3. Points in Space
Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg]
28. God / A. Divine Nature / 2. Divine Nature
Only God is absolutely infinite [Cantor, by Hart,WD]