Re: [ontolog-forum] Universal Basic Semantic Structures

[William Frank <williamf.frank@xxxxxxxxx>]

On Sat, Sep 29, 2012 at 7:03 PM, Chris Menzel <chris.menzel@xxxxxxxxx> wrote:

On Fri, Sep 28, 2012 at 11:54 AM, John F Sowa <sowa@xxxxxxxxxxx> wrote:

MW

> I do not see set theory and mereology as alternatives that you choose
> one from for your ontology, rather I see them as being appropriate
> in different circumstances. One of the tests I use to determine which
> is appropriate is whether I am or could be interested in the weight
> of the collection. Sets are abstract and so do not have a weight.
> A mereological sum on the other hand does.

That's a very good, one-paragraph summary of the difference.  Formally,
I would emphasize that set theory has two operators (subset and isIn),
but mereology has only one operator (partOf).

John, virtually every text on set theory presents the axioms with just a single binary predicate "∈" for membership. The subset relation is always defined; using "isin" instead of "∈":

What does this imply to you?

arithmetic has the operations plus and minus and times and successor.  Of course, most texts define all of them from sucessor.  Interesting and profound, but it remains the case that all three are part of arithmetic.   Not just the ones you choose in your axoimatization to be primitive.     Instead, they can all be defined from +  and 1, if one chose.  There was a time before successor was discovered.  When it was, did the others go away?  In propositional logic, we can use all the natural deduction operators, and after defining the inference rules for them all, prove many equivelences, or instead, take our pick of a pair such as not and or, or use nor or nand only as primitives.    I have found that some people, depending on what course they happen to have taken, believe that if a then b "really is" not a or b, etc.

  (iff (subset x y) (forall z) (if (isin z x) (isin z y))).

 

I agree that you can use set theory for those purposes.  And set

theory can be used in conjunction with continuous math in physics.
Furthermore, the method of partitioning and projection by Bittner
et al. can be adapted just as easily to set theory as to mereology.

I doubt that is true except in the sense that one can model mereology in set theory (e.g., as a certain type of algebra, depending on the mereological theory in question).

-chris

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-- 
William Frank

413/376-8167


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