Combining Texts

All the ideas for 'fragments/reports', 'Higher-Order Logic' and 'Inexpressible Properties and Propositions'

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18 ideas

3. Truth / H. Deflationary Truth / 3. Minimalist Truth
Instances of minimal truth miss out propositions inexpressible in current English [Hofweber]
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 / 4. Substitutional Quantification
Quantification can't all be substitutional; some reference is obviously to objects [Hofweber]
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
Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [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]
8. Modes of Existence / B. Properties / 1. Nature of Properties
Since properties have properties, there can be a typed or a type-free theory of them [Hofweber]
8. Modes of Existence / B. Properties / 11. Properties as Sets
Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
19. Language / F. Communication / 6. Interpreting Language / a. Translation
Holism says language can't be translated; the expressibility hypothesis says everything can [Hofweber]
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]