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
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10482
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The logic of ZF is classical first-order predicate logic with identity
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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10492
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A few axioms of set theory 'force themselves on us', but most of them don't
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
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18192
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Do the Replacement Axioms exceed the iterative conception of sets? [Maddy]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
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7785
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The use of plurals doesn't commit us to sets; there do not exist individuals and collections
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
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10485
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Naïve sets are inconsistent: there is no set for things that do not belong to themselves
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
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10484
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The iterative conception says sets are formed at stages; some are 'earlier', and must be formed first
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
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13547
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Limitation of Size is weak (Fs only collect is something the same size does) or strong (fewer Fs than objects) [Potter]
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4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
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10699
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Does a bowl of Cheerios contain all its sets and subsets?
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
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14249
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Boolos reinterprets second-order logic as plural logic [Oliver/Smiley]
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10830
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Second-order logic metatheory is set-theoretic, and second-order validity has set-theoretic problems
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10225
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Monadic second-order logic might be understood in terms of plural quantifiers [Shapiro]
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10736
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Boolos showed how plural quantifiers can interpret monadic second-order logic [Linnebo]
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10780
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Any sentence of monadic second-order logic can be translated into plural first-order logic [Linnebo]
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5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
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10829
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A sentence can't be a truth of logic if it asserts the existence of certain sets
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5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
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10697
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Identity is clearly a logical concept, and greatly enhances predicate calculus
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5. Theory of Logic / G. Quantification / 2. Domain of Quantification
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10832
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'∀x x=x' only means 'everything is identical to itself' if the range of 'everything' is fixed
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5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
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13671
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Second-order quantifiers are just like plural quantifiers in ordinary language, with no extra ontology [Shapiro]
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5. Theory of Logic / G. Quantification / 6. Plural Quantification
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10267
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We should understand second-order existential quantifiers as plural quantifiers [Shapiro]
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10698
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Plural forms have no more ontological commitment than to first-order objects
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5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
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7806
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Boolos invented plural quantification [Benardete,JA]
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5. Theory of Logic / K. Features of Logics / 4. Completeness
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10834
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Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
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5. Theory of Logic / K. Features of Logics / 6. Compactness
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13841
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Why should compactness be definitive of logic? [Hacking]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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10491
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Infinite natural numbers is as obvious as infinite sentences in English
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
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10483
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Mathematics and science do not require very high orders of infinity
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
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10833
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Many concepts can only be expressed by second-order logic
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6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
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10490
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Mathematics isn't surprising, given that we experience many objects as abstract
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7. Existence / D. Theories of Reality / 10. Ontological Commitment / b. Commitment of quantifiers
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10700
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First- and second-order quantifiers are two ways of referring to the same things
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8. Modes of Existence / D. Universals / 1. Universals
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10488
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It is lunacy to think we only see ink-marks, and not word-types
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9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
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10487
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I am a fan of abstract objects, and confident of their existence
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9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
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10489
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We deal with abstract objects all the time: software, poems, mistakes, triangles..
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18. Thought / E. Abstraction / 7. Abstracta by Equivalence
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8693
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An 'abstraction principle' says two things are identical if they are 'equivalent' in some respect
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