Combining Texts

All the ideas for 'The Sign of Four', 'Fact, Fiction and Forecast (4th ed)' and 'Higher-Order Logic'

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18 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10301 The axiom of choice is controversial, but it could be replaced [Shapiro]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
10588 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
10298 Some say that second-order logic is mathematics, not logic [Shapiro]
10299 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
10300 Logical consequence can be defined in terms of the logical terminology [Shapiro]
5. Theory of Logic / C. Ontology of Logic / 4. Logic by Convention
12394 If the result is bad, we change the rule; if we like the rule, we reject the result [Goodman]
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
10290 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
10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
10296 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
10294 Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro]
8. Modes of Existence / B. Properties / 11. Properties as Sets
10591 Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / a. Dispositions
14292 Dispositions seem more ethereal than behaviour; a non-occult account of them would be nice [Goodman]
14. Science / C. Induction / 1. Induction
22200 If you eliminate the impossible, the truth will remain, even if it is weird [Conan Doyle]
14. Science / C. Induction / 5. Paradoxes of Induction / a. Grue problem
18749 Goodman argued that the confirmation relation can never be formalised [Goodman, by Horsten/Pettigrew]
17646 Goodman showed that every sound inductive argument has an unsound one of the same form [Goodman, by Putnam]
26. Natural Theory / D. Laws of Nature / 3. Laws and Generalities
4794 We don't use laws to make predictions, we call things laws if we make predictions with them [Goodman]