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