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Author: Tegiri NenashiTegiri Nenashi Date: Sep 6, 2008 11:52
Observation: all quantuifiers correspond to some binary algebraic
operation. Consider:
1. Sigma summation and integral are iterative forms of binary plus.
2. Pi-capital product is iterative form of multiplication.
3. Lattice supremum is iterative form binary meet.
5. Lattice infinum is iterative form binary join.
6. Existential and universal quatifiers are lattice supremum and
infinum.
Now, there must be a binary operation that lambda is iterative form as
well?
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Author: Mariano Suárez-AlvarezMariano Suárez-Alvarez Date: Sep 6, 2008 14:06
On Sep 6, 3:52Â pm, Tegiri Nenashi gmail.com> wrote:> Observation: all quantuifiers correspond to some binary algebraic
> operation. Consider:
>
> 1. Sigma summation and integral are iterative forms of binary plus.
> 2. Pi-capital product is iterative form of multiplication.
> 3. Lattice supremum is iterative form binary meet.
> 5. Lattice infinum is iterative form binary join.
> 6. Existential and universal quatifiers are lattice supremum and
> infinum.
>
> Now, there must be a binary operation that lambda is iterative form as
> well?
No.
-- m
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Author: galathaeagalathaea Date: Sep 6, 2008 15:31
On Sep 6, 11:52 am, Tegiri Nenashi gmail.com> wrote:> Observation: all quantuifiers correspond to some binary algebraic
> operation. Consider:
>
> 1. Sigma summation and integral are iterative forms of binary plus.
> 2. Pi-capital product is iterative form of multiplication.
> 3. Lattice supremum is iterative form binary meet.
> 5. Lattice infinum is iterative form binary join.
> 6. Existential and universal quatifiers are lattice supremum and
> infinum.
>
> Now, there must be a binary operation that lambda is iterative form as
> well?
if i understand what you mean by lambda quantifier
(as in the standard form on which eval's work through application
e.g. lambda calculus lambdas
with their well known universal properties)
then i would expect the exponential to be the corresponding operation
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Author: Tegiri NenashiTegiri Nenashi Date: Sep 6, 2008 18:03
On Sep 6, 3:31Â pm, galathaea gmail.com> wrote:> On Sep 6, 11:52 am, Tegiri Nenashi gmail.com> wrote:
>>> Observation: all quantuifiers correspond to some binary algebraic
>> operation. Consider:
>
>> 1. Sigma summation and integral are iterative forms of binary plus.
>> 2. Pi-capital product is iterative form of multiplication.
>> 3. Lattice supremum is iterative form binary meet.
>> 5. Lattice infinum is iterative form binary join.
>> 6. Existential and universal quatifiers are lattice supremum and
>> infinum.
>
>> Now, there must be a binary operation that lambda is iterative form as
>> well?
>
> if i understand what you mean by lambda quantifier
> (as in the standard form on which eval's work through application
> Â e.g. lambda calculus lambdas
> Â Â with their well known universal properties) ...
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Author: galathaeagalathaea Date: Sep 6, 2008 18:34
On Sep 6, 6:03 pm, Tegiri Nenashi gmail.com> wrote:> On Sep 6, 3:31 pm, galathaea gmail.com> wrote:
>
>
>>> On Sep 6, 11:52 am, Tegiri Nenashi gmail.com> wrote:>
>>> Observation: all quantuifiers correspond to some binary algebraic
>>> operation. Consider:
>
>>> 1. Sigma summation and integral are iterative forms of binary plus.
>>> 2. Pi-capital product is iterative form of multiplication.
>>> 3. Lattice supremum is iterative form binary meet.
>>> 5. Lattice infinum is iterative form binary join.
>>> 6. Existential and universal quatifiers are lattice supremum and
>>> infinum.
>
>>> Now, there must be a binary operation that lambda is iterative form as
>>> well?
> ...
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Author: Tegiri NenashiTegiri Nenashi Date: Sep 6, 2008 19:14
On Sep 6, 6:34Â pm, galathaea gmail.com> wrote:> On Sep 6, 6:03 pm, Tegiri Nenashi gmail.com> wrote:
>>> On Sep 6, 3:31 pm, galathaea gmail.com> wrote:>>> if i understand what you mean by lambda quantifier
>>> (as in the standard form on which eval's work through application
>>> Â e.g. lambda calculus lambdas
>>> Â Â with their well known universal properties)
>>> then i would expect the exponential to be the corresponding operation
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Author: Tegiri NenashiTegiri Nenashi Date: Sep 6, 2008 19:18
On Sep 6, 7:14Â pm, Tegiri Nenashi gmail.com> wrote:> On Sep 6, 6:34Â pm, galathaea gmail.com> wrote:>> why do you think an operation must be associative?
>
> Because all the other analogies (sum, product, and predicate logic
> quantifiers) have this property?
Better argument may be that if operation is not associative, then,
when applied iteratively the result is no longer well defined. Sure
with non associative operation '*' the expression a * b * c is
ambiguous!
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Author: galathaeagalathaea Date: Sep 6, 2008 20:10
On Sep 6, 7:14 pm, Tegiri Nenashi gmail.com> wrote:> On Sep 6, 6:34 pm, galathaea gmail.com> wrote:>> On Sep 6, 6:03 pm, Tegiri Nenashi gmail.com> wrote:>>> On Sep 6, 3:31 pm, galathaea gmail.com> wrote:>>>> if i understand what you mean by lambda quantifier
>>>> (as in the standard form on which eval's work through application
>>>> e.g. lambda calculus lambdas
>>>> with their well known...
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Author: Dirk ThierbachDirk Thierbach Date: Sep 6, 2008 23:50
Tegiri Nenashi gmail.com> wrote:> Observation: all quantuifiers correspond to some binary algebraic
> operation. Consider:
Or more precisely: Four commonly known "big operations" (which are not
quantifiers in the strict sense) happen to express "folds" (or
reductions, or whatever you like to call them) based on binary
operations. Forall- and existential quantification can be seen like
this, too, but only if you reduce them to operations on truth-values.
But these are not "all quantifiers" in the sense of "all 'big' operations".
> 1. Sigma summation and integral are iterative forms of binary plus.
The integral is more complicated, because it involves (among other
things) a limit. Also, the integral without bounds behaves a bit
differently (doesn't reduce to a single value).
And what about differentiation? It's the inverse operation of
integration, so what binary operation should it correspond to? :-)
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Author: galathaeagalathaea Date: Sep 7, 2008 03:04
On Sep 6, 11:50 pm, Dirk Thierbach
wrote:
> Tegiri Nenashi gmail.com> wrote:>> Observation: all quantuifiers correspond to some binary algebraic
>> operation. Consider:
>
> Or more precisely: Four commonly known "big operations" (which are not
> quantifiers in the strict sense) happen to express "folds" (or
> reductions, or whatever you like to call them) based on binary
> operations. Forall- and existential quantification can be seen like
> this, too, but only if you reduce them to operations on truth-values.
>
> But these are not "all quantifiers" in the sense of "all 'big' operations".
>
>> 1. Sigma summation and integral are iterative forms of binary plus.
>
> The integral is more complicated, because it involves (among other
> things) a limit. Also, the integral without bounds behaves a bit
> differently (doesn't reduce to a single value).
> ...
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