Ideas from 'Parts of Classes' by David Lewis [1991], by Theme Structure
[found in 'Parts of Classes' by Lewis,David [Blackwell 1991,0-631-17656-x]].
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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18395
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Sets are mereological sums of the singletons of their members [Armstrong]
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15496
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We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
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15500
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Classes divide into subclasses in many ways, but into members in only one way
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15499
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A subclass of a subclass is itself a subclass; a member of a member is not in general a member
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
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15498
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We can accept the null set, but there is no null class of anything
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15503
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We needn't accept this speck of nothingness, this black hole in the fabric of Reality!
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15502
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There are four main reasons for asserting that there is an empty set
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
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15497
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We can replace the membership relation with the member-singleton relation (plus mereology)
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15506
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If we don't understand the singleton, then we don't understand classes
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15511
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If singleton membership is external, why is an object a member of one rather than another?
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15513
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Maybe singletons have a structure, of a thing and a lasso?
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
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15507
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Set theory has some unofficial axioms, generalisations about how to understand it
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10191
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Set theory reduces to a mereological theory with singletons as the only atoms [MacBride]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
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15508
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If singletons are where their members are, then so are all sets
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15514
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A huge part of Reality is only accepted as existing if you have accepted set theory
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15523
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Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it
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5. Theory of Logic / G. Quantification / 6. Plural Quantification
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15525
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Plural quantification lacks a complete axiom system
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15518
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I like plural quantification, but am not convinced of its connection with second-order logic
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6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
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15524
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Zermelo's model of arithmetic is distinctive because it rests on a primitive of set theory
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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15517
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Giving up classes means giving up successful mathematics because of dubious philosophy
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6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
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15515
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To be a structuralist, you quantify over relations
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7. Existence / A. Nature of Existence / 2. Types of Existence
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15520
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Existence doesn't come in degrees; once asserted, it can't then be qualified
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7. Existence / C. Structure of Existence / 8. Stuff / a. Pure stuff
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15501
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We have no idea of a third sort of thing, that isn't an individual, a class, or their mixture
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15504
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Atomless gunk is an individual whose parts all have further proper parts
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8. Modes of Existence / B. Properties / 11. Properties as Sets
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15516
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A property is any class of possibilia
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9. Objects / C. Structure of Objects / 5. Composition of an Object
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14748
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The many are many and the one is one, so they can't be identical
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6129
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Lewis affirms 'composition as identity' - that an object is no more than its parts [Merricks]
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9. Objects / C. Structure of Objects / 8. Parts of Objects / b. Sums of parts
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15512
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In mereology no two things consist of the same atoms
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15519
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Trout-turkeys exist, despite lacking cohesion, natural joints and united causal power
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15521
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Given cats, a fusion of cats adds nothing further to reality
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15522
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The one has different truths from the many; it is one rather than many, one rather than six
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9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
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10566
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Lewis prefers giving up singletons to giving up sums [Fine,K]
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14244
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Lewis only uses fusions to create unities, but fusions notoriously flatten our distinctions [Oliver/Smiley]
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10660
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A commitment to cat-fusions is not a further commitment; it is them and they are it
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12. Knowledge Sources / B. Perception / 2. Qualities in Perception / a. Qualities in perception
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15509
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Some say qualities are parts of things - as repeatable universals, or as particulars
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