Re: [ontolog-forum] Is there something I missed?

[Александр Шкотин <alex.shkotin@xxxxxxxxx>]

John,

 

in axiomatic theory there is a small group of initial entities (predicates, constants, functions) we may say mutualy defined by axioms.

This is why circles and loops in definitions are very important - we can find initial entities.

 

And for example in NBG set theory: there is only one initial entity - binary predicate 'in'.

 

Alex

2009/2/5 John F. Sowa <sowa@xxxxxxxxxxx>

Azamat,

That is not only true, it is inevitable:

 > You must know such common things that most dictionary definitions
 > are ending in loops.

In mathematics, the field in which the most precise definitions are
written, there are very few "closed form" definitions that define
some entity type X as a structure composed of previously defined types.

Instead, the axioms of mathematical systems are usually stated as
relationships among terms, none of which can be defined independently.
It would be extremely unlikely that the real world in all its
complexity should happen to be so simple that all our conceptions
of it could be defined by the simplest form of definition.

For a more realistic view, see the _Web of Belief_ by Quine and Ullian:

   http://socialistica.lenin.ru/analytic/txt/q/quine_1.htm
   The Web of Belief

Following is an article that discusses the Quine-Ullian view and relates
it to other methods of definition:

   http://plato.stanford.edu/entries/justep-coherence/
   Coherentist Theories of Epistemic Justification

An excerpt:

   "Quine's web of belief metaphor -- according to which our beliefs
   form an interconnected web in which the structure hangs or falls
   as a whole."

No term in that web can be defined by itself.  They are all defined
by the pattern of relationships to other terms.  This view has
strong similarities to Wittgenstein's language games (which probably
had a strong influence on Quine and Ullian).

John

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