Re: [ontolog-forum] Universal Basic Semantic Structures

[Chris Menzel <chris.menzel@xxxxxxxxx>]

On Sat, Sep 29, 2012 at 8:05 PM, William Frank <williamf.frank@xxxxxxxxx> wrote:

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?

It doesn't imply anything to me. It is just a simple fact about how the subset relation is introduced into set theory.

 

arithmetic has the operations plus and minus and times and successor.  Of course, most texts define all of them from sucessor. 

You are quite mistaken. It is impossible to define addition in terms of successor and multiplication in terms of addition and successor. You appear to be mistaking the usual recursive axioms for those operators for definitions. They are not. Full arithmetic requires all three operations. (Interestingly, exponentiation can be defined in terms of addition and multiplication.)

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.

I have no idea what you are talking about.

-chris

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