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Science Forum Index » Logic Forum » Is "existence" a predicate?
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Posted: Sun Apr 13, 2008 10:47 pm |
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Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
Thanks,
S. |
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| Guest |
Posted: Sun Apr 13, 2008 10:54 pm |
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On Apr 14, 4:47 am, sanchopanch...@web.de wrote:
Quote: Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
Thanks,
S.
Existence is a predicate but, it is not a primary predicate such as
"having a leg".
x exists =df EF(Fx).
Gx -> x exists. If there is a primary predicate that x has, then, x
exists. |
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| Guest |
Posted: Mon Apr 14, 2008 2:01 am |
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On 14 Apr., 10:54, holden_o...@yahoo.ca wrote:
Quote: On Apr 14, 4:47 am, sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
Thanks,
S.
Existence is a predicate but, it is not a primary predicate such as
"having a leg".
x exists =df EF(Fx).
Gx -> x exists. If there is a primary predicate that x has, then, x
exists.
Thanks for the answer. I would be very pleased if you give me a
reference for such things like "primary predicate".
Thanks you,
S- |
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| LauLuna |
Posted: Mon Apr 14, 2008 3:56 am |
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On Apr 14, 2:01 pm, sanchopanch...@web.de wrote:
Quote: On 14 Apr., 10:54, holden_o...@yahoo.ca wrote:
On Apr 14, 4:47 am, sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
Thanks,
S.
Existence is a predicate but, it is not a primary predicate such as
"having a leg".
x exists =df EF(Fx).
Gx -> x exists. If there is a primary predicate that x has, then, x
exists.
Thanks for the answer. I would be very pleased if you give me a
reference for such things like "primary predicate".
Thanks you,
S-- Hide quoted text -
- Show quoted text -
I'm not sure holden's definition of exists is the best. Formally,
satisfying a predicate is not enough for existing. Supose that the
second order predicate
EF (Fx)
were equivalent to the existence predicate.
Since F is any predicate, we can take F = ~EP (Px)
Then, if that characterization of existence is correct, all
nonexistent objects (such as the square circle) would satisfy F; hence
they would exist.
I think it preferable:
x exists =: Ey (y=x)
As for existence as a very special kind of predicate, I'd say the
discussion goes back at least to Kant's assessment of the ontological
argument in the KrV. If we can include existence in the essence of an
object, we can trivially prove its existence. If we can include
existence in a concept as a any other feature, such as having wings or
being rational, we could prove the existence of whatever.
E.g. take the concept of 'existent unicorn'. An existent unicorn
cannot fail to exist. And an existent unicorn is a unicorn;
consequently, unicorns exist.
Kant proposed the distinction between material (or 'real') predicates
and formal predicates, existence being a formal one. That's why modern
logic took existence to be a logical constant (a quantifier) rather
than a predicate; at most existence can be a predicate constructed
with just logical constants.
It seems clear to me that existence is not the sort of predicate that
could contribute to the definition of a concept.
Regards |
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| Guest |
Posted: Mon Apr 14, 2008 7:04 am |
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On Apr 14, 9:36 am, David C. Ullrich <dullr...@sprynet.com> wrote:
Quote: On Mon, 14 Apr 2008 01:47:21 -0700 (PDT), sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
A predicate is supposed to divide the universe into two classes,
the things that satisfy the predicate and the things that don't.
Existence doesn't do this, because _everything_ exists!
Yes, everything exists. If you want to claim otherwise you
have to prove the existence of something that does not exist,
and that's going to be hard.
The way language is used it often sounds like it's talking about
things that do not exist, but that's just a problem with the way
language is used.
Thanks,
S.
David C. Ullrich
I don't agree.
If we allow 'things' to include described objects as well as existent
objects,
then there are non-existent things, For example the described object,
the present king of France,
does not exist.
There is no primary predicate (property) that is true of the present
king of France.
ie. ~EF(F(the present king of France)) means that it does not exist. |
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Posted: Mon Apr 14, 2008 7:44 am |
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On Apr 14, 9:56 am, LauLuna <laureanol...@yahoo.es> wrote:
Quote: On Apr 14, 2:01 pm, sanchopanch...@web.de wrote:
On 14 Apr., 10:54, holden_o...@yahoo.ca wrote:
On Apr 14, 4:47 am, sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
Thanks,
S.
Existence is a predicate but, it is not a primary predicate such as
"having a leg".
x exists =df EF(Fx).
Gx -> x exists. If there is a primary predicate that x has, then, x
exists.
Thanks for the answer. I would be very pleased if you give me a
reference for such things like "primary predicate".
Thanks you,
S-- Hide quoted text -
- Show quoted text -
I'm not sure holden's definition of exists is the best. Formally,
satisfying a predicate is not enough for existing. Supose that the
second order predicate
EF (Fx)
were equivalent to the existence predicate.
Since F is any predicate, we can take F = ~EP (Px)
Then, if that characterization of existence is correct, all
nonexistent objects (such as the square circle) would satisfy F; hence
they would exist.
I said that F, G, etc. are primary predicates (properties).
Compound predicates such as ~EP(Px) are not properties (primary
predicates).
Quote:
I think it preferable:
x exists =: Ey (y=x)
Within the context of FOPL x=x or Ey(x=y) or will work.
But in that case, because Ax(x=x) is an axiom, everything exists is a
theorem,
as David Ullrich said.
Quote:
As for existence as a very special kind of predicate, I'd say the
discussion goes back at least to Kant's assessment of the ontological
argument in the KrV. If we can include existence in the essence of an
object, we can trivially prove its existence. If we can include
existence in a concept as a any other feature, such as having wings or
being rational, we could prove the existence of whatever.
We prove the existence of x by asserting that it has at least one
property.
If x has the property F, then x exists.
Existence is not a property of things.
Quote:
E.g. take the concept of 'existent unicorn'. An existent unicorn
cannot fail to exist. And an existent unicorn is a unicorn;
consequently, unicorns exist.
Nonsense. The description 'an existent unicorn' does not exist at all.
Dictionary.com Unicorn:
-noun 1. a mythical creature resembling a horse, with a single horn in
the center of its forehead: often symbolic of chastity or purity.
Because mythical creatures do not exist by definition, unicorns cannot
exist.
The described object 'an existent unicorn' is a contradictiory term.
Quote:
Kant proposed the distinction between material (or 'real') predicates
and formal predicates, existence being a formal one. That's why modern
logic took existence to be a logical constant (a quantifier) rather
than a predicate; at most existence can be a predicate constructed
with just logical constants.
ExFx, says that there is some existent x such that Fx, within
classical predicate logic,
but, if we extend the variable x to include no-existent things,
then Ex does not say there exists an x.
Ex says there is at least one instance of the variable x such that,
whether the values of the variable x exist or not.
The quantifier cannot represent the predicate of existence.
Quote:
It seems clear to me that existence is not the sort of predicate that
could contribute to the definition of a concept.
Regards- Hide quoted text -
- Show quoted text - |
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| David C. Ullrich |
Posted: Mon Apr 14, 2008 8:36 am |
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Guest
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On Mon, 14 Apr 2008 01:47:21 -0700 (PDT), sanchopancho80@web.de wrote:
Quote: Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
A predicate is supposed to divide the universe into two classes,
the things that satisfy the predicate and the things that don't.
Existence doesn't do this, because _everything_ exists!
Yes, everything exists. If you want to claim otherwise you
have to prove the existence of something that does not exist,
and that's going to be hard.
The way language is used it often sounds like it's talking about
things that do not exist, but that's just a problem with the way
language is used.
David C. Ullrich |
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Posted: Mon Apr 14, 2008 9:47 am |
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Guest
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<sanchopancho80@web.de> skrev i melding
news:aa42f3a7-431e-49e8-bfb4-2a60205f2873@x41g2000hsb.googlegroups.com...
Quote: Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
Hi,
You might consider the difference between a predicate (a part of a
proposition),
and a property (an aspect or a feature or a trait ... some quality of an
entity).
Nobody can stop you from using anything you want as a predicate in
formulating any proposition;what follows from that, is another question.
When ascribing properties, you are leaving the field of logic and language
and moving into ontology.
After 2000 years on debate, I think the verdict was that properties is
something that things "have",
while existing is what they "do" (and having and doing just like that is
what they "are", their identity).
Hence, being is not a regular property of which you can say that things
"have it" or not; they either "do" exist or not, and that is a different
kind of statement than what they are like. THAT they are, as opposed to WHAT
they are.
HTH
T |
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| Tron |
Posted: Mon Apr 14, 2008 9:49 am |
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Hi,
That's why modern
logic took existence to be a logical constant (a quantifier) rather
than a predicate; at most existence can be a predicate constructed
with just logical constants.
It seems clear to me that existence is not the sort of predicate that
could contribute to the definition of a concept.
---
Detailed discussion in Frege's postscipt to his "Dialogue with Pünjer",
in his posthumous papers.
T |
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Posted: Tue Apr 15, 2008 4:04 am |
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On Apr 15, 7:34 am, David C. Ullrich <dullr...@sprynet.com> wrote:
Quote: On Mon, 14 Apr 2008 10:04:31 -0700 (PDT), holden_o...@yahoo.ca wrote:
On Apr 14, 9:36 am, David C. Ullrich <dullr...@sprynet.com> wrote:
On Mon, 14 Apr 2008 01:47:21 -0700 (PDT), sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
A predicate is supposed to divide the universe into two classes,
the things that satisfy the predicate and the things that don't.
Existence doesn't do this, because _everything_ exists!
Yes, everything exists. If you want to claim otherwise you
have to prove the existence of something that does not exist,
and that's going to be hard.
The way language is used it often sounds like it's talking about
things that do not exist, but that's just a problem with the way
language is used.
Thanks,
S.
David C. Ullrich
I don't agree.
We know that.
If we allow 'things' to include described objects as well as existent
objects,
then there are non-existent things, For example the described object,
the present king of France,
does not exist.
Yes, if we allow the meaning of the word "things" to include things
which are not things then things change. There is no such thing
as the present king of France. _calling_ it "the described object"
does not change the fact that there is no such thing.
The description 'the present king of France' does exist, but it does
not have a referent.
It has sense but no reference.
Quote:
Saying that the present king of France does not exist is of course
exactly an example of what I meant when I said that the way
we use language sometimes seems to contradict the fact that
everything exists. The sentence really means that for every x,
x is not the present king of France.
Yes, ~Ex(x = (the present king of France)) is true, as is ~((the
present king of France)=(the present king of France)).
Quote: There is no primary predicate (property) that is true of the present
king of France.
ie. ~EF(F(the present king of France)) means that it does not exist.
This has always seemed like one of the stranger of your assertions.
If we are going to talk about the present king of France, why do
we not say that he satisfies the predicate "is the king of a country"?
Because it is clearly false.
(the present king of France) is a king, is false.
(the present king of France) is present, is false.
(the present king of France) is a Frenchman, is false.
(the present king of France) has existence, is false.
(the present king of France) has non-existence, is false.
There is no property that (the present king of France) has.
Proof:
F(the present king of France) <-> Ey(Ax(x=y <-> x is a present king of
France) & Fy)
F(the present king of France) <-> Ey(Ax(x=y <-> (x is a present king
of France & there is no present king of France) & Fy)
(x is a present king of France & there is no present king of France),
is a contradiction.
F(the present king of France) <-> Ey(Ax(x=y <-> contradiction) & Fy).
But, Ax(x=y <-> contradiction) <-> ~Ex(x=y).
But, Ex(x=y) is a theorem.
therefore,
F(the present king of France) <-> contradiction.
ie. ~(F(the present king of France)), for all F.
~EF(F(the present king of France), is a theorem.
There is no property that the present king of France has. ie. it does
not exist.
~(Fx) <-> (~F)x, iff, x exists.
ie. ~(F(the present king of France)) <-> (~F)(the present king of
France), is false.
(the present king of France) has existence. or (the present king of
France) has non-existence. ..is a contradiction.
(the present king of France) has existence. or ~((the present king of
France) has existence). ..is a tautology.
Quote:
David C. Ullrich- Hide quoted text -
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| Gc |
Posted: Tue Apr 15, 2008 4:40 am |
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On 14 huhti, 16:36, David C. Ullrich <dullr...@sprynet.com> wrote:
Quote: On Mon, 14 Apr 2008 01:47:21 -0700 (PDT), sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
A predicate is supposed to divide the universe into two classes,
the things that satisfy the predicate and the things that don't.
Existence doesn't do this, because _everything_ exists!
What about P(x) = {x: x = x)? This is a predicate, but it is always
true. |
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| David C. Ullrich |
Posted: Tue Apr 15, 2008 6:34 am |
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Guest
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On Mon, 14 Apr 2008 10:04:31 -0700 (PDT), holden_owen@yahoo.ca wrote:
Quote: On Apr 14, 9:36 am, David C. Ullrich <dullr...@sprynet.com> wrote:
On Mon, 14 Apr 2008 01:47:21 -0700 (PDT), sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
A predicate is supposed to divide the universe into two classes,
the things that satisfy the predicate and the things that don't.
Existence doesn't do this, because _everything_ exists!
Yes, everything exists. If you want to claim otherwise you
have to prove the existence of something that does not exist,
and that's going to be hard.
The way language is used it often sounds like it's talking about
things that do not exist, but that's just a problem with the way
language is used.
Thanks,
S.
David C. Ullrich
I don't agree.
We know that.
Quote: If we allow 'things' to include described objects as well as existent
objects,
then there are non-existent things, For example the described object,
the present king of France,
does not exist.
Yes, if we allow the meaning of the word "things" to include things
which are not things then things change. There is no such thing
as the present king of France. _calling_ it "the described object"
does not change the fact that there is no such thing.
Saying that the present king of France does not exist is of course
exactly an example of what I meant when I said that the way
we use language sometimes seems to contradict the fact that
everything exists. The sentence really means that for every x,
x is not the present king of France.
Quote: There is no primary predicate (property) that is true of the present
king of France.
ie. ~EF(F(the present king of France)) means that it does not exist.
This has always seemed like one of the stranger of your assertions.
If we are going to talk about the present king of France, why do
we not say that he satisfies the predicate "is the king of a country"?
David C. Ullrich |
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| LauLuna |
Posted: Wed Apr 16, 2008 4:24 am |
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Guest
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On Apr 14, 7:44 pm, holden_o...@yahoo.ca wrote:
Quote: On Apr 14, 9:56 am, LauLuna <laureanol...@yahoo.es> wrote:
On Apr 14, 2:01 pm, sanchopanch...@web.de wrote:
On 14 Apr., 10:54, holden_o...@yahoo.ca wrote:
On Apr 14, 4:47 am, sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
Thanks,
S.
Existence is a predicate but, it is not a primary predicate such as
"having a leg".
x exists =df EF(Fx).
Gx -> x exists. If there is a primary predicate that x has, then, x
exists.
Thanks for the answer. I would be very pleased if you give me a
reference for such things like "primary predicate".
Thanks you,
S-- Hide quoted text -
- Show quoted text -
I'm not sure holden's definition of exists is the best. Formally,
satisfying a predicate is not enough for existing. Supose that the
second order predicate
EF (Fx)
were equivalent to the existence predicate.
Since F is any predicate, we can take F = ~EP (Px)
Then, if that characterization of existence is correct, all
nonexistent objects (such as the square circle) would satisfy F; hence
they would exist.
I said that F, G, etc. are primary predicates (properties).
Compound predicates such as ~EP(Px) are not properties (primary
predicates).
I think it preferable:
x exists =: Ey (y=x)
Within the context of FOPL x=x or Ey(x=y) or will work.
But in that case, because Ax(x=x) is an axiom, everything exists is a
theorem,
as David Ullrich said.
As for existence as a very special kind of predicate, I'd say the
discussion goes back at least to Kant's assessment of the ontological
argument in the KrV. If we can include existence in the essence of an
object, we can trivially prove its existence. If we can include
existence in a concept as a any other feature, such as having wings or
being rational, we could prove the existence of whatever.
We prove the existence of x by asserting that it has at least one
property.
If x has the property F, then x exists.
We can prove nothing by simply asserting. But I repeat, representing
existence by predicate possession is inconsistent unless there are a
clear definition of what predicates serve for the prurpose and what
not. For example, you suggest the predicate 'x=x' does not do the job
either.
What are the primary predicates?
On the other hand, what's the problem with proving that everything
exists? Quantifiers are usually deemed to range over a domain of
existing objects!
Regards
Quote: Existence is not a property of things.
E.g. take the concept of 'existent unicorn'. An existent unicorn
cannot fail to exist. And an existent unicorn is a unicorn;
consequently, unicorns exist.
Nonsense. The description 'an existent unicorn' does not exist at all.
That's exactly what I'm saying. |
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| Gc |
Posted: Wed Apr 16, 2008 4:57 am |
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Guest
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On 16 huhti, 17:24, LauLuna <laureanol...@yahoo.es> wrote:
Quote: On Apr 14, 7:44 pm, holden_o...@yahoo.ca wrote:
On Apr 14, 9:56 am, LauLuna <laureanol...@yahoo.es> wrote:
On Apr 14, 2:01 pm, sanchopanch...@web.de wrote:
On 14 Apr., 10:54, holden_o...@yahoo.ca wrote:
On Apr 14, 4:47 am, sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
Thanks,
S.
Existence is a predicate but, it is not a primary predicate such as
"having a leg".
x exists =df EF(Fx).
Gx -> x exists. If there is a primary predicate that x has, then, x
exists.
Thanks for the answer. I would be very pleased if you give me a
reference for such things like "primary predicate".
Thanks you,
S-- Hide quoted text -
- Show quoted text -
I'm not sure holden's definition of exists is the best. Formally,
satisfying a predicate is not enough for existing. Supose that the
second order predicate
EF (Fx)
were equivalent to the existence predicate.
Since F is any predicate, we can take F = ~EP (Px)
Then, if that characterization of existence is correct, all
nonexistent objects (such as the square circle) would satisfy F; hence
they would exist.
I said that F, G, etc. are primary predicates (properties).
Compound predicates such as ~EP(Px) are not properties (primary
predicates).
I think it preferable:
x exists =: Ey (y=x)
Within the context of FOPL x=x or Ey(x=y) or will work.
But in that case, because Ax(x=x) is an axiom, everything exists is a
theorem,
as David Ullrich said.
As for existence as a very special kind of predicate, I'd say the
discussion goes back at least to Kant's assessment of the ontological
argument in the KrV. If we can include existence in the essence of an
object, we can trivially prove its existence. If we can include
existence in a concept as a any other feature, such as having wings or
being rational, we could prove the existence of whatever.
We prove the existence of x by asserting that it has at least one
property.
If x has the property F, then x exists.
We can prove nothing by simply asserting. But I repeat, representing
existence by predicate possession is inconsistent unless there are a
clear definition of what predicates serve for the prurpose and what
not. For example, you suggest the predicate 'x=x' does not do the job
either.
Isn`t x=x a two place relation? So it is satisfied by some pairs of
objects (namely (a,a) and some not (a,b): a=/y. Therefore it does
divide the pairs of objects of the proper universe in two distinct
classes. |
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| Guest |
Posted: Wed Apr 16, 2008 7:17 am |
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On Apr 16, 8:21 am, David C. Ullrich <dullr...@sprynet.com> wrote:
Quote: On Tue, 15 Apr 2008 07:04:11 -0700 (PDT), holden_o...@yahoo.ca wrote:
On Apr 15, 7:34 am, David C. Ullrich <dullr...@sprynet.com> wrote:
On Mon, 14 Apr 2008 10:04:31 -0700 (PDT), holden_o...@yahoo.ca wrote:
On Apr 14, 9:36 am, David C. Ullrich <dullr...@sprynet.com> wrote:
On Mon, 14 Apr 2008 01:47:21 -0700 (PDT), sanchopanch...@web.de wrote:
Hello,
I have read somewhere that "existence" wouldn't be a predicate in the
way e.g. "having a leg" is a predicate. Does anyone have a good and
actual reference on that or liks me to tell why?
A predicate is supposed to divide the universe into two classes,
the things that satisfy the predicate and the things that don't.
Existence doesn't do this, because _everything_ exists!
Yes, everything exists. If you want to claim otherwise you
have to prove the existence of something that does not exist,
and that's going to be hard.
The way language is used it often sounds like it's talking about
things that do not exist, but that's just a problem with the way
language is used.
Thanks,
S.
David C. Ullrich
I don't agree.
We know that.
If we allow 'things' to include described objects as well as existent
objects,
then there are non-existent things, For example the described object,
the present king of France,
does not exist.
Yes, if we allow the meaning of the word "things" to include things
which are not things then things change. There is no such thing
as the present king of France. _calling_ it "the described object"
does not change the fact that there is no such thing.
The description 'the present king of France' does exist, but it does
not have a referent.
It has sense but no reference.
Of course the description exists! That has no bearing on the
question of whether everything exists - the description does
in fact exist.
I didn't say that the description didn't exist. And I didn't
say that the description does not "have sense". What
I said was that there's no such thing as the present
king of France. There isn't. So it's not a counterexample
to my assertion that everything exists.
What you mean to say, imo, is that there is no existent object
described as 'the present king of France'.
And we all agree within the context of First Order Predicate Logic.
If we only allow existent objects as values of our variables,
then it is not a surprise that we can conclude that all values of our
variables must exist.
That everything exists, can only be asserted within the context of
FOPL.
There are other logics in which this is not the case.
All of truth is relative to the system that decides it.
Is the Russell class a thing for you?
Is the universal set a thing for you?
Surely the answers are dependent on which system of decision you are
using.
Quote:
Saying that the present king of France does not exist is of course
exactly an example of what I meant when I said that the way
we use language sometimes seems to contradict the fact that
everything exists. The sentence really means that for every x,
x is not the present king of France.
Yes, ~Ex(x = (the present king of France)) is true,
as is ~((the
present king of France)=(the present king of France)).
There is no primary predicate (property) that is true of the present
king of France.
ie. ~EF(F(the present king of France)) means that it does not exist.
This has always seemed like one of the stranger of your assertions.
If we are going to talk about the present king of France, why do
we not say that he satisfies the predicate "is the king of a country"?
Because it is clearly false.
(the present king of France) is a king, is false.
(the present king of France) is present, is false.
(the present king of France) is a Frenchman, is false.
(the present king of France) has existence, is false.
(the present king of France) has non-existence, is false.
There is no property that (the present king of France) has.
Proof:
F(the present king of France) <-> Ey(Ax(x=y <-> x is a present king of
France) & Fy)
F(the present king of France) <-> Ey(Ax(x=y <-> (x is a present king
of France & there is no present king of France) & Fy)
(x is a present king of France & there is no present king of France),
is a contradiction.
F(the present king of France) <-> Ey(Ax(x=y <-> contradiction) & Fy).
But, Ax(x=y <-> contradiction) <-> ~Ex(x=y).
But, Ex(x=y) is a theorem.
therefore,
F(the present king of France) <-> contradiction.
ie. ~(F(the present king of France)), for all F.
~EF(F(the present king of France), is a theorem.
There is no property that the present king of France has. ie. it does
not exist.
~(Fx) <-> (~F)x, iff, x exists.
ie. ~(F(the present king of France)) <-> (~F)(the present king of
France), is false.
(the present king of France) has existence. or (the present king of
France) has non-existence. ..is a contradiction.
(the present king of France) has existence. or ~((the present king of
France) has existence). ..is a tautology.
David C. Ullrich- Hide quoted text -
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David C. Ullrich- Hide quoted text -
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