Ideas from 'On Second-Order Logic' by George Boolos [1975], by Theme Structure

[found in 'Logic, Logic and Logic' by Boolos,George [Harvard 1999,0-674-53767-x]].

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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Boolos reinterprets second-order logic as plural logic [Oliver/Smiley]
Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems
5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
A sentence can't be a truth of logic if it asserts the existence of certain sets
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
'∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed
5. Theory of Logic / K. Features of Logics / 4. Completeness
Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
5. Theory of Logic / K. Features of Logics / 6. Compactness
Why should compactness be definitive of logic? [Hacking]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Many concepts can only be expressed by second-order logic