Combining Texts

All the ideas for 'Realism in Mathematics', 'On Second-Order Logic' and 'The Limits of Contingency'

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22 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
18851 Pairing (with Extensionality) guarantees an infinity of sets, just from a single element [Rosen]
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
8755 Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
14249 Boolos reinterprets second-order logic as plural logic [Boolos, by Oliver/Smiley]
10830 Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems [Boolos]
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
10829 A sentence can't be a truth of logic if it asserts the existence of certain sets [Boolos]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
10832 '∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed [Boolos]
5. Theory of Logic / K. Features of Logics / 4. Completeness
10834 Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences [Boolos]
5. Theory of Logic / K. Features of Logics / 6. Compactness
13841 Why should compactness be definitive of logic? [Boolos, by Hacking]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
10833 Many concepts can only be expressed by second-order logic [Boolos]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10718 A natural number is a property of sets [Maddy, by Oliver]
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
8756 Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17733 We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins]
9. Objects / A. Existence of Objects / 4. Impossible objects
18852 A Meinongian principle might say that there is an object for any modest class of properties [Rosen]
10. Modality / A. Necessity / 5. Metaphysical Necessity
18849 Metaphysical necessity is absolute and universal; metaphysical possibility is very tolerant [Rosen]
18850 'Metaphysical' modality is the one that makes the necessity or contingency of laws of nature interesting [Rosen]
18858 Sets, universals and aggregates may be metaphysically necessary in one sense, but not another [Rosen]
18857 Standard Metaphysical Necessity: P holds wherever the actual form of the world holds [Rosen]
18856 Non-Standard Metaphysical Necessity: when ¬P is incompatible with the nature of things [Rosen]
10. Modality / A. Necessity / 6. Logical Necessity
18848 Something may be necessary because of logic, but is that therefore a special sort of necessity? [Rosen]
10. Modality / B. Possibility / 3. Combinatorial possibility
18855 Combinatorial theories of possibility assume the principles of combination don't change across worlds [Rosen]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / a. Conceivable as possible
18853 A proposition is 'correctly' conceivable if an ominiscient being could conceive it [Rosen]
26. Natural Theory / D. Laws of Nature / 4. Regularities / b. Best system theory
18854 The MRL view says laws are the theorems of the simplest and strongest account of the world [Rosen]