Combining Texts

All the ideas for 'fragments/reports', 'Higher-Order Logic' and 'The Law of Peoples'

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40 ideas

2. Reason / A. Nature of Reason / 1. On Reason
Parmenides was much more cautious about accepting ideas than his predecessors [Simplicius on Parmenides]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The axiom of choice is controversial, but it could be replaced [Shapiro]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Some say that second-order logic is mathematics, not logic [Shapiro]
If the aim of logic is to codify inferences, second-order logic is useless [Shapiro]
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Logical consequence can be defined in terms of the logical terminology [Shapiro]
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
Second-order variables also range over properties, sets, relations or functions [Shapiro]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
7. Existence / A. Nature of Existence / 3. Being / a. Nature of Being
Being is not divisible, since it is all alike [Parmenides]
No necessity could produce Being either later or earlier, so it must exist absolutely or not at all [Parmenides]
Being must be eternal and uncreated, and hence it is timeless [Parmenides]
7. Existence / A. Nature of Existence / 3. Being / d. Non-being
The realm of necessary non-existence cannot be explored, because it is unknowable [Parmenides]
There is no such thing as nothing [Parmenides]
7. Existence / A. Nature of Existence / 3. Being / f. Primary being
Parmenides at least saw Being as the same as Nous, and separate from the sensed realm [Parmenides, by Plotinus]
7. Existence / B. Change in Existence / 1. Nature of Change
All our concepts of change and permanence are just names, not the truth [Parmenides]
8. Modes of Existence / B. Properties / 11. Properties as Sets
Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
9. Objects / E. Objects over Time / 1. Objects over Time
Something must be unchanging to make recognition and knowledge possible [Aristotle on Parmenides]
10. Modality / A. Necessity / 5. Metaphysical Necessity
The first way of enquiry involves necessary existence [Parmenides]
10. Modality / A. Necessity / 8. Transcendental Necessity
Necessity sets limits on being, in order to give it identity [Parmenides]
11. Knowledge Aims / B. Certain Knowledge / 4. The Cogito
Thinking implies existence, because thinking depends on it [Parmenides]
12. Knowledge Sources / B. Perception / 1. Perception
Parmenides treats perception and intellectual activity as the same [Theophrastus on Parmenides]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
Only reason can prove the truth of facts [Parmenides]
24. Political Theory / A. Basis of a State / 1. A People / c. A unified people
Rawls rejected cosmopolitanism because it doesn't respect the autonomy of 'peoples' [Rawls, by Shorten]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / e. The One
People who say that the cosmos is one forget that they must explain movement [Aristotle on Parmenides]
The one is without any kind of motion [Parmenides]
There could be movement within one thing, as there is within water [Aristotle on Parmenides]
The one can't be divisible, because if it was it could be infinitely divided down to nothing [Parmenides, by Simplicius]
Defenders of the One say motion needs the void - but that is not part of Being [Parmenides, by Aristotle]
Reason sees reality as one, the senses see it as many [Aristotle on Parmenides]
Reality is symmetrical and balanced, like a sphere, with no reason to be greater one way rather than another [Parmenides]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / f. Ancient elements
He taught that there are two elements, fire the maker, and earth the matter [Parmenides, by Diog. Laertius]
27. Natural Reality / A. Classical Physics / 1. Mechanics / a. Explaining movement
It is feeble-minded to look for explanations of everything being at rest [Aristotle on Parmenides]
27. Natural Reality / C. Space / 1. Void
The void can't exist, and without the void there can't be movement or separation [Parmenides, by Aristotle]
27. Natural Reality / D. Time / 3. Parts of Time / a. Beginning of time
What could have triggered the beginning [of time and being]? [Parmenides]
27. Natural Reality / E. Cosmology / 1. Cosmology
He was the first to discover the identity of the Morning and Evening Stars [Parmenides, by Diog. Laertius]
He was the first person to say the earth is spherical [Parmenides, by Diog. Laertius]