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From: "Dmitry A. Kazakov" <mailbox@dmitry-kazakov.de>
Subject: Re: Preventing type extensions
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Date: Fri, 24 Sep 2010 18:27:46 +0200
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Date: 2010-09-24T18:27:44+02:00
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On Fri, 24 Sep 2010 16:30:29 +0300, Niklas Holsti wrote:

> Dmitry A. Kazakov wrote:
>> On Tue, 21 Sep 2010 19:11:33 +0200, Georg Bauhaus wrote:
>> 
>>> On 21.09.10 18:25, Dmitry A. Kazakov wrote:
>>>
>>>> horse /= animal
>>> It will help immensely, I think, at least those of us who do
>>> not do type theory on an every day basis, if we had a glossary
>>> of terms for interpreting expression like "horse /= animal".
>> 
>> Gosh, it was no more than just this: if horse is an animal that does not
>> imply they are interchangeable. Compare: "all horses have hoofs" and "all
>> animals have hoofs."
>> 
>> is-a /= is
>> member /= set
>> 
>> Moreover:
>> 
>> is-a /= substitutable (greeting to the LSP!)
>> 
>> Compare: "some animals have tentacles" and "some horses have tentacles."
> 
> Those are not properties of "horses" and "animals", but properties of 
> the *set* of horses and the *set* of animals.

horses is set of horses.

> The property "all horses have hooves" expresses a universal property of 
> horses: if X is a horse, then X has hooves. The statement "some horses 
> have tentacles" says nothing about any given horse X.

Both are predicates:

forall x in horses, x has hooves
exists x in animals, x has tentacles

[ if you dropped existential quantification, you would not be able to
describe functions returning values ]

-- 
Regards,
Dmitry A. Kazakov
http://www.dmitry-kazakov.de