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Re: [ontolog-forum] Axiomatic ontology

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Thu, 9 Oct 2008 14:17:00 -0500
Message-id: <B9A931F3-646B-4676-83D2-3A6340CC26E5@xxxxxxxx>
On Oct 9, 2008, at 6:59 AM, Rob Freeman wrote:
> Pat,
> ...
> You say for the case of "cardinality in logical interpretations", at  
> least, the question has been looked into, and the answer was that  
> "the number of possible relations over a universe is a much larger  
> cardinality than the universe ..., but this doesn't matter because  
> the logic isn't obliged to consider all of these relations, only  
> enough to provide denotations for all the relation-naming terms in  
> the language."
> That argument strikes me as completely circular.    (01)

Nonetheless, it is not.    (02)

> Who says how many relations are "enough to provide denotations for  
> all the relation-naming terms in the language"?    (03)

How many are enough is clearly and precisely determined by the  
language in question together with its semantics.  At a minimum, in a  
classical semantics, a denotation must be assigned to every relation- 
naming term in the language.  Depending on the language/semantics in  
question, this could require as little as just one relation -- absent  
any further constraints, every relation-naming term could denote that  
one relation.  More typically, however, the language+semantics  
together require more than this.  Notably, in the language L2 of  
second-order logic with abstraction terms, for every formula  
A(x1,...,xn) of the language there is a relation-denoting expression  
[\lambda x1...xn A(x1,...,xn)] whose extension comprises exactly those  
n-tuples that satisfy A(x1,...,xn).  Given the presence of the usual  
array of boolean operators in the language -- notably, negation,  
conjunction, and disjunction -- this means that the class of relations  
(viewed extensionally) has at least to be closed under complementation  
and finite unions and intersections.  In standard second-order  
semantics this constraint is satisfied by the fact that the class of  
(n-place) relations is required to be the entire power set of the  
class of n-tuples of elements of the domain.  So in every classical  
interpretation I of L2, the number of n-place relations, for a given  
n, is exactly 2^(m*n), where m is the number of individuals in the  
domain of I.  If, however, we interpret a second-order language by the  
so-called "general semantics" of Henkin, this constraint can be  
satisfied by the minimal closure under the above operations, relative  
to some initial assignment to the primitive predicates of the  
language.  And for a countable domain, that is still countable.    (04)

More generally, then, the answer to your question "who [better: what]  
sez how many relations are enough to provide denotations for all the  
relation-naming terms in the language?" is: "The language and its  
semantics."  And how many relations are needed, for a given language 
+semantics, can be determined quite precisely.    (05)

> The number of relations we need is exactly the question at issue.    (06)

Vide above.    (07)

> This is again the question when I ask "whether you can ignore sets and
> work only with categories."    (08)

That looks like a very different question indeed.    (09)

> The distinction at the end you "completely fail to follow" is yet  
> another a restatement of this question whether all these possible  
> sets "matter"/"are distinct in meaningful ways".    (010)

I fail to follow as well if, as it seems, this is supposed to be a  
Deep Question.  For surely, say, the set of men and the set of women  
are meaningfully distinct, no?  And surely the union and intersection  
of those sets are meaningfully distinct, aren't they?    (011)

Chris Menzel    (012)

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