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
10482

The logic of ZF is classical firstorder predicate logic with identity

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10492

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
18192

Do the Replacement Axioms exceed the iterative conception of sets? [Maddy]

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
7785

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. Naïve logical sets
10485

Naïve 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
10484

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
13547

Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects) [Potter]

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10699

Does a bowl of Cheerios contain all its sets and subsets?

5. Theory of Logic / A. Overview of Logic / 7. SecondOrder Logic
14249

Boolos reinterprets secondorder logic as plural logic [Oliver/Smiley]

10830

Secondorder logic metatheory is settheoretic, and secondorder validity has settheoretic problems

10225

Monadic secondorder logic might be understood in terms of plural quantifiers [Shapiro]

10736

Boolos showed how plural quantifiers can interpret monadic secondorder logic [Linnebo]

10780

Any sentence of monadic secondorder logic can be translated into plural firstorder logic [Linnebo]

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

5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
10697

Identity is clearly a logical concept, and greatly enhances predicate calculus

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

5. Theory of Logic / G. Quantification / 5. SecondOrder Quantification
13671

Secondorder quantifiers are just like plural quantifiers in ordinary language, with no extra ontology [Shapiro]

5. Theory of Logic / G. Quantification / 6. Plural Quantification
10267

We should understand secondorder existential quantifiers as plural quantifiers [Shapiro]

10698

Plural forms have no more ontological commitment than to firstorder objects

5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
7806

Boolos invented plural quantification [Benardete,JA]

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

5. Theory of Logic / K. Features of Logics / 6. Compactness
13841

Why should compactness be definitive of logic? [Hacking]

6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
10491

Infinite natural numbers is as obvious as infinite sentences in English

6. Mathematics / A. Nature of Mathematics / 4. The Infinite / f. Uncountable infinities
10483

Mathematics and science do not require very high orders of infinity

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / e. Peano arithmetic 2ndorder
10833

Many concepts can only be expressed by secondorder logic

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
10490

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
10700

First and secondorder quantifiers are two ways of referring to the same things

8. Modes of Existence / D. Universals / 1. Universals
10488

It is lunacy to think we only see inkmarks, and not wordtypes

9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
10487

I am a fan of abstract objects, and confident of their existence

9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
10489

We deal with abstract objects all the time: software, poems, mistakes, triangles..

18. Thought / E. Abstraction / 7. Abstracta by Equivalence
8693

An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect
