Ideas of George Boolos, by Theme

[American, 1940 - 1996, Professor of Philosophy at MIT.]

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4. Formal Logic / F. Set Theory ST / 1. Set Theory
The logic of ZF is classical first-order predicate logic with identity
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
A few axioms of set theory 'force themselves on us', but most of them don't
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Do the Replacement Axioms exceed the iterative conception of sets?
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
The use of plurals doesn't commit us to sets; there do not exist individuals and collections
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Na´ve logical sets
Na´ve sets are inconsistent: there is no set for things that do not belong to themselves
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects)
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
Does a bowl of Cheerios contain all its sets and subsets?
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Boolos reinterprets second-order logic as plural logic
Monadic second-order logic might be understood in terms of plural quantifiers
Boolos showed how plural quantifiers can interpret monadic second-order logic
Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems
Any sentence of monadic second-order logic can be translated into plural first-order logic
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 / D. Assumptions for Logic / 4. Identity in Logic
Identity is clearly a logical concept, and greatly enhances predicate calculus
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 / G. Quantification / 5. Second-Order Quantification
Second-order quantifiers are just like plural quantifiers in ordinary language, with no extra ontology
5. Theory of Logic / G. Quantification / 6. Plural Quantification
We should understand second-order existential quantifiers as plural quantifiers
Plural forms have no more ontological commitment than to first-order objects
5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
Boolos invented plural quantification
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?
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
Infinite natural numbers is as obvious as infinite sentences in English
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / f. Uncountable infinities
Mathematics and science do not require very high orders of infinity
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2nd-order
Many concepts can only be expressed by second-order logic
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
Mathematics isn't surprising, given that we experience many objects as abstract
7. Existence / D. Theories of Reality / 10. Ontological Commitment / b. Commitment of quantifiers
First- and second-order quantifiers are two ways of referring to the same things
8. Modes of Existence / D. Universals / 1. Universals
It is lunacy to think we only see ink-marks, and not word-types
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
I am a fan of abstract objects, and confident of their existence
9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
We deal with abstract objects all the time: software, poems, mistakes, triangles..
18. Thought / D. Concepts / 6. Abstract Concepts / g. Abstracta by equivalence
An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect