[om-list] Classes vs. Sets

[Mark Butler]

Classes vs. Sets

For those of you paying attention, I made a mistake yesterday when I 
described any singular English expression as a class.  Certainly a 
singleton class can be constructed from any expression, but a class is 
not the same as the members of the class. That seemingly natural 
assumption leads to subtle logical errors.

A class is a concept that names a group of things with shared 
properties.  Its defining feature is a membership predicate.  You can 
make a class out of any set by making the class predicate be membership 
in that set, but that is a rather unnatural way of defining a class.  
Normally a class predicates substantive properties of its members, 
rather than arbitrary ones.

A set, on the other hand, is an unordered group of distinct elements.  
All plural expressions are sets.  Mathematics makes a distinction 
between an object and a singleton set containing that object, but 
English normally does not.  In English, sets of unknown cardinality are 
always marked plural, but as soon as they are known to be singular, the 
plural indicator is removed, syntactically removing the only common 
distinction between a set containing one object and the object itself.   
That is equivalent to the substitution rule  "a = { a }", where "a" is a 
single set member.  

A consequence of the latter rule is the ambiguity in a singular abstract 
term like "apple" - are we referring to the concept "apple" or an 
instance "apple".  "I like apple" is a statement about a class 
indicating preference for the qualities entailed in "apple".  "I like 
apples" is a statement about a set, expressing preference for actual 
apples, not just the flavor apple. 

"I would like an apple" is an indication of desire for an instance of a 
class. "I would like one of the apples", on the other hand, is an 
indication of desire for a member of a set.

Note that again, as I described yesterday, a number operating on a set 
produces a set of all the subsets of that set with the specified 
cardinality, much like a generalized power set operator.  "three" in 
"three apples" is not a multiplication, but rather an indication that 
any three members of the set of apples will satisfy.  A number can only 
be interpreted as a strict multiplier if the operand is an 
indistinguishable unit, like a "second" or a "meter".

 - Mark
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