Are there any contingent existences?
o_ho...@my-deja.com
x=y->.Fx<->Fy.
Existence is then defined; E!y defined Ex(x=y), where E! is the
existence predicate.
But Ex(x=y) is a theorem, see; for example, *13.17 Principia
Mathematica.
In modal propositional logic it is an axiom that; if p is a theorem
then, necessary p, []p, is a theorem.
Therefore Ex(x=y)->[]Ex(x=y), that is, if y exists it necessarily
exists.
Therefore []Ex(x=y) is a theorem. That is Ay[]Ex(x=y) is a theorem.
There are no contingent existences!?
It's necessary that (Bill clinton exists)? Surely this cannot be
asserted.
What has gone wrong here, in your opinion?
Sent via Deja.com
http://www.deja.com/
slightly...@my-deja.com
knowledge in matters formal og logical:
> Existence is then defined; E!y defined Ex(x=y), where E! is the
> existence predicate.
E!, here, is not formally speaking an existence predicate. Existence
cannot be treated as a predicate - what this definition says is that
existence is implied when predicating. If I claim that (for any x it
holds that x is P) what I say is that (if an x were to exists it would
be P in a domain where any x is P). Logic is not concerned with the
factual existence of x. Logic is only concerned with 'substances' and
properties. Treating existence as a property is a logical no-no.
> But Ex(x=y) is a theorem, see; for example, *13.17 Principia
> Mathematica.
> In modal propositional logic it is an axiom that; if p is a theorem
> then, necessary p, []p, is a theorem.
And this seems to be the problem. In this step it would seem that you
apply a rule of predicate identity to predicated existence. This rule
of identity states that if two objects share the exact same properties
they are identical -- discounting of course the illegal property of
existence (it is simply implied). Therefore the rest doesn't really
hold, I think.
> Therefore Ex(x=y)->[]Ex(x=y), that is, if y exists it necessarily
> exists.
> Therefore []Ex(x=y) is a theorem. That is Ay[]Ex(x=y) is a theorem.
> There are no contingent existences!?
> It's necessary that (Bill clinton exists)? Surely this cannot be
> asserted.
Now, this is not a trivial matter. Existence and its place in the noble
art of reasoning has been the concern of philosophers for ages. The
heart of the matter is of course that logic is not concerned with
empirical fact -- with the actual existence of the logical entities.
Logic is broadly speaking the art of stipulating rules for hypothetical
reasoning: That if one condition obtain certain other conditions must
also obtain.
On the subject of logic there are actually several ways of interpreting
the very nature of logic (and mathematics and similar symbolic,
abstract reasoning). There are two basic philosophical ways of treating
logic:
1) That logic is descriptive of either
a) certain psychological mechanisms that control our reasoning
and/or
b) certain formal syntactic-structural mechanisms in language
itself.
2) The study of logic is the study of certain necessary components in
epistemology -- constitutive of the human representational
faculties. It is either
a) the study of the way in which our epistemological faculties
necessarily structures the world or
b) the study of the structure of reality itself.
According to 1a) sentences in logic and mathematics can be reduced to
the way we *conventionally* treat the concept of knowledge. A logical
derivation, for example, is the verbalisation or symbolization of
certain expected psychological/cognitive states in a person that has
the knowledge that s -- namely that this person also has the knowledge
that r (where r follows logically from s). Logic is then the study of a
necessary -- albeit conventional and contingent -- tool for performing
adequately in cultural, social, linguistic and even physical contexts.
1b) performs a similar reduction of the sentences of logic and
mathematics. According to 1b) a sentence in logic is descriptive of the
necessary *deep structure* of human language -- a distinction is made
here between the surface-structure of language (grammar and syntax) and
this alleged deep structure that is the common denominator of the many
different languages.
2a) and 1a) actually appear to be similar since they both entail the
premise that logic is descriptive of certain faculties that are
necessary components in our epistemological relation to the world. But
this likeness is only skin-deep. 2a) stipulates that logic is a meta-
physical/transcendental study whereas 1a) states that logic (in a
manner of speaking) is a meta-psychological/social study. According to
2a) (incidentally a view that Kant embraces -- indeed a view that is
central to his validation of synthetical sentences a priori) the
logical derivation of r from s finds its possibility and validation in
the way certain relevant cognitive mechanisms operates when a subject
internally represents the world. 1a), on the other hand, stipulates
that the logical derivation of r from s is possible because certain
contingent conventions for 'correct thinking' demands it.
I'm not really sure what 2b) entails. Perhaps someone could conjure up
an interesting philosophical case story based on this notion of logic?
And ... Oh ... Sorry. I really went overboard here, didn't I? Please
let me know if you find my treatment of the problem of contingent and
necessary existences to be satisfactory. It's really not my particular
area ...
Michael
o_ho...@my-deja.com
slightly...@my-deja.com wrote:
> Well ... One central thing has gone wrong here to the best of my
> knowledge in matters formal og logical:
>
> > Existence is then defined; E!y defined Ex(x=y), where E! is the
> > existence predicate.
>
> E!, here, is not formally speaking an existence predicate. Existence
> cannot be treated as a predicate - what this definition says is that
> existence is implied when predicating. If I claim that (for any x it
> holds that x is P) what I say is that (if an x were to exists it would
> be P in a domain where any x is P). Logic is not concerned with the
> factual existence of x. Logic is only concerned with 'substances' and
> properties. Treating existence as a property is a logical no-no.
Nonsense, 'x exists' is a statement containing 'x' as subject and
'exists' as Predicate! QED.
>Existence cannot be treated as a predicate
Why not. I just did.
>what this definition says is that
> existence is implied when predicating.
E!y = Ex(x=y) Df. has no reference to implication nor to predicating.
What the hell are you talking about?
slightly...@my-deja.com
> Nonsense, 'x exists' is a statement containing 'x' as subject and
> 'exists' as Predicate! QED.
'x is blue', within modal logic, states that 'if this x were to exist
it would be blue in a domain where x is blue'. 'x exists', within modal
logic, states that 'if this x were to exist it would exist in a domain
where x exists'. And you ask: What is wrong with this? I will try to
reply more formally sound than in my previous email.
Consider the sentence: E!x(Fx). This sentence states that 'there is one
and only one x that has the property F'. If I were to treat F as the
property of existence the sentence would go like this: 'there exists
just one x that exists'. This is the equivalent of the modal logical
sentence: E!x(E!x). And this sentence is illegal, logically speaking.
The syntax is wrong. What the sentence states is that 'there is one and
only one x such that there is one and only one x', which is an empty
sentence, devoid of logical meaning of any kind.
But what has really gone wrong, then?
The df. E!y = Ex(x=y) indeed says nothing of predication. And there is
a reason for this. Modal logic does not contain an existence predicate -
instead it contains an existence quantifier. It seems that you have
treated this existence quantifier as a predicate within predicate
logic. Even though it is at times called an existence predicate it is
not a predicate within predicate logic.
I seems to me, then, that you have made two mistakes:
1) You treated the existence quantifier as a predicate within predicate
logic - thereby allowing the use of the identity rules within predicate
logic. Even though it's fairly clever it's none the less a logically
unsound step.
and
2) You treated necessary p as if it was a predicate in predicate logic
(p is necessary?!). This is also a logically unsound assumption.
I look forward to hearing your reply,
o_ho...@my-deja.com
slightly...@my-deja.com wrote:
>
> > Nonsense, 'x exists' is a statement containing 'x' as subject and
> > 'exists' as Predicate! QED.
>
> 'x is blue', within modal logic, states that 'if this x were to exist
> it would be blue in a domain where x is blue'. 'x exists', within
modal
> logic, states that 'if this x were to exist it would exist in a domain
> where x exists'. And you ask: What is wrong with this? I will try to
> reply more formally sound than in my previous email.
>
> Consider the sentence: E!x(Fx). This sentence states that 'there is
one
> and only one x that has the property F'.
It seems that you have a previous interpretation of the expression
E!x(Fx), where did you get that from?
Russell, *14 of Principia Mathematica defines existence for described
objects thus: E!(the x:Fx) = EyAx(x=y<->Fx) Df. *14.02
>If I were to treat F as the
> property of existence the sentence would go like this: 'there exists
> just one x that exists'. This is the equivalent of the modal logical
> sentence: E!x(E!x).
At this point, description theory as in Russell *14, has nothing to do
with modal logic.
'there exists just one x that exists'. means E!(the x:E!x),
..not E!x(E!x)
E!(the x:E!x)<->EyAx(x=y<->E!x) But E!x is true for all x,
E!(the x:E!x)<->EyAx(x=y), but EyAx(x=y) is contradictory.
Therefore ~(E!(the x:E!x)) ie. the existence predicate is not unique,
there is more than one individual which exists!
Your E!x(E!x) is not well formed.
> And this sentence is illegal, logically speaking.
> The syntax is wrong. What the sentence states is that 'there is one
and
> only one x such that there is one and only one x', which is an empty
> sentence, devoid of logical meaning of any kind.
>
> But what has really gone wrong, then?
>
> The df. E!y = Ex(x=y) indeed says nothing of predication. And there is
> a reason for this. Modal logic does not contain an existence
predicate -
There is no Modal Logic here either.
> instead it contains an existence quantifier. It seems that you have
> treated this existence quantifier as a predicate within predicate
> logic. Even though it is at times called an existence predicate it is
> not a predicate within predicate logic.
'The existential quantifier' is never treated as the predicate 'exists'.
> I seems to me, then, that you have made two mistakes:
>
> 1) You treated the existence quantifier as a predicate within
predicate
> logic - thereby allowing the use of the identity rules within
predicate
> logic. Even though it's fairly clever it's none the less a logically
> unsound step.
Nonsense again, at no time did I consider 'there is some y such that' a
predicate.
ExFx says there is some x such that Fx.
E!x says x exists.
E!(the x:Fx) says that there is only one x such that Fx.
> and
>
> 2) You treated necessary p as if it was a predicate in predicate logic
> (p is necessary?!). This is also a logically unsound assumption.
Nonsense again, the necessity operator is indeed a predicate of the
proposition in much the same way as 'it is false that'. '[]p' is a
statement in which 'p' is the subject and 'it is necessary that' is its
predicate. A very logically sound assumption!
Try again!
slightly...@my-deja.com
closely ... No philosopher should ever be too proud to start all
over ...
I think I now have the correct interpretation of your argument. Broadly
speaking you are correct - but what does your argument actually say? It
does not - as you seem to think - say that Bill Clinton exists with
necessity. Have a look:
I think that the problem arises because of a misconception of the
notion of necessity. Modalities are not concerned with existence - only
indirectly. Modalities are only applicable to truths and falsities.
What your sentence 'Ex(x=y)->[]Ex(x=y)' says is not that if y exists it
necessarily exists. Instead it says something to the effect of: If it
is true that y exists then it is necessarily true that y exists (the
notion of []p should be read 'it is necessarily true that p' and ~[]p
should be read 'it is necessarily false that p').
Now what does the sentence Ay[]Ex(x=y) entail with regards to this
notion of necessity?
Ay[]Ex(x=y): For all y (in existence) it is necessarily true that there
are some y (in existence) where x is y.
Listen carefully now:
For any shoe to exist it must necessarily be true that there is at
least one object x that is a shoe.
For any man Bill Clinton to exist it must necessarily be true there is
at least one object x that is Bill Clinton.
And this is, I think, the correct approach to your problem. What was
needed was a more adequate notion of necessity than the one implied in
your argument. But please feel free to complicate matters with a
different notion of necessity.
I look forward to your reply,
Michael
o_ho...@my-deja.com
slightly...@my-deja.com wrote:
> Starting from scratch after having reviewed your arguments a bit more
> closely ... No philosopher should ever be too proud to start all
> over ...
Agreed.
> I think I now have the correct interpretation of your argument.
>Broadly
> speaking you are correct - but what does your argument actually say?
>It
> does not - as you seem to think - say that Bill Clinton exists with
> necessity. Have a look:
>
> I think that the problem arises because of a misconception of the
> notion of necessity. Modalities are not concerned with existence -
>only
> indirectly. Modalities are only applicable to truths and falsities.
Yes, modalities like necessity and possibility, apply to statements
including existence statements.
> What your sentence 'Ex(x=y)->[]Ex(x=y)' says is not that if y exists
>it
> necessarily exists. Instead it says something to the effect of: If it
> is true that y exists then it is necessarily true that y exists (the
> notion of []p should be read 'it is necessarily true that p'
Yes.
> and ~[]p
> should be read 'it is necessarily false that p').
No, []~p or ~<>p means 'it is necessarily false that p'.
> Now what does the sentence Ay[]Ex(x=y) entail with regards to this
> notion of necessity?
>
> Ay[]Ex(x=y): For all y (in existence) it is necessarily true that > >
>there
> are some y (in existence) where x is y.
>
> Listen carefully now:
>
> For any shoe to exist it must necessarily be true that there is at
> least one object x that is a shoe.
No way! A proposition is necessary if we can prove it's truth by some
deductive system. We accepted the usual axiom: if p is a theorem
(deductively proven) then it is necessary.
It is necessary that there is at least one object x that is a shoe,
cannot be proven. Indeed it is contingent on the time frame in which
there are shoes, eg. it was not true 1000000 years ago, and not true
1000000 years in the future.
No contingent statements are necessary. To say of a fact that it is
necessary is contradictory.
The same applies to Bill Clinton.
slightly...@my-deja.com
> No, []~p or ~<>p means 'it is necessarily false that p'.
Oh, well ...
> > For any shoe to exist it must necessarily be true that there is at
> > least one object x that is a shoe.
> No way! A proposition is necessary if we can prove it's truth by some
> deductive system. We accepted the usual axiom: if p is a theorem
> (deductively proven) then it is necessary.
Yes, we did. It states: If p is a theorem then it is necessarily
*true*. Necessarily true for any theorem, then, means that within the
body of axioms - constitutive for any particular logical system - that
sentence cannot be false. It is true because of the constitution of the
system - henceforth: Necessarily true.
What might be wrong here is that any true theorem p entails necessarily
true p. However, any necessarily true sentence p does not entail a true
theorem p. Our particular necessarily true sentence []Ex(x=y) is
clearly not a theorem. It is however not a faulty use of the
term 'necessary' because of this:
Imagine a logical system (like the one we move about now) consisting of
the axioms constitutive of the system + one additional sentence 'y
exists' - not derived or deduced from the axioms, but simply
stipulated. Given this state of logical affairs it would seem
reasonable to state that []Ex(x=y) is valid.
> It is necessary that there is at least one object x that is a shoe,
> cannot be proven.
I do not claim that this sentence can be proven. Again, it seems that
you forget the stipulation or assumption that 'y exists' - I claim that
the sentence 'For any object y to exist it must be necessarily true
that there exists an object x that is y' can be proven. Indeed, I claim
that you have exactly proven this sentence. And cleverly so, I might
add.
But you have actually proven even more: You have proven the
*theorem*: 'It is necessarily true that for any object y to exist it
must be necessarily true that there exists an object x that is y' -
this sentence: [](Ay[]Ex(x=y)) - following your own reasoning. And, I
conjecture, this sentence might be said to be constitutive of our
particular body of logic or merely derivative thereof (It is a theorem,
hence necessarily true).
> Indeed it is contingent on the time frame in which
> there are shoes, eg. it was not true 1000000 years ago, and not true
> 1000000 years in the future.
> No contingent statements are necessary. To say of a fact that it is
> necessary is contradictory.
> The same applies to Bill Clinton.
I don't care about shoes or Bill Clinton. The sentence, you have
proven, can be refrased as follow: For any POSSIBLE object y, possesing
a finite number of properties, to exist it must be necessarily true (or
simply necessary) that there exists at least one object x that posseses
the exact same finite number of properties.
It would seem to me that you think this is the same as saying: 'it is
necessarily true that there exists one object x'. It isn't. It is,
however, the same as saying that the existence of an object y is
CONTINGENT on certain necessarily true conditions. And there you have
it: The concept of contingency preserved as well as the concept of
necessity. Hmmm .... Well, let's see ...
o_ho...@my-deja.com
slightly...@my-deja.com wrote:
>
>
> > No, []~p or ~<>p means 'it is necessarily false that p'.
>
> Oh, well ...
>
> > > For any shoe to exist it must necessarily be true that there is at
> > > least one object x that is a shoe.
>
> > No way! A proposition is necessary if we can prove it's truth by
some
theorem
> > (deductively proven) then it is necessary.
>
> Yes, we did. It states: If p is a theorem then it is necessarily
> *true*. Necessarily true for any theorem, then, means that within the
> body of axioms - constitutive for any particular logical system - that
> sentence cannot be false. It is true because of the constitution of
the
> system - henceforth: Necessarily true.
Correct.
> What might be wrong here is that any true theorem p entails
necessarily
> true p. However, any necessarily true sentence p does not entail a
true
> theorem p.
Wrong, it is an axiom, of propositional Modal Logic, that;
[]p->p, for all p theorem or not.
> Our particular necessarily true sentence []Ex(x=y) is
> clearly not a theorem.
Wrong again, Ex(x=y) is a theorem, see; *13.17 of Principia Mathematica.
Therefore []Ex(x=y) is a theorem by (Ax1).
>It is however not a faulty use of the
> term 'necessary' because of this:
>
> Imagine a logical system (like the one we move about now) consisting
of
> the axioms constitutive of the system + one additional sentence 'y
> exists' - not derived or deduced from the axioms, but simply
> stipulated. Given this state of logical affairs it would seem
> reasonable to state that []Ex(x=y) is valid.
Yes, it is valid. This is my objection to modern predicate logic and
the point of my post.
x=x<->E!x is valid, and x=x is an axiom therefore E!x is a theorem, ??
> > It is necessary that there is at least one object x that is a shoe,
> > cannot be proven.
>
> I do not claim that this sentence can be proven. Again, it seems that
> you forget the stipulation or assumption that 'y exists' - I claim
that
> the sentence 'For any object y to exist it must be necessarily true
> that there exists an object x that is y' can be proven. Indeed, I
claim
> that you have exactly proven this sentence. And cleverly so, I might
> add.
>
> But you have actually proven even more: You have proven the
> *theorem*: 'It is necessarily true that for any object y to exist it
> must be necessarily true that there exists an object x that is y' -
> this sentence: [](Ay[]Ex(x=y)) - following your own reasoning. And, I
> conjecture, this sentence might be said to be constitutive of our
> particular body of logic or merely derivative thereof (It is a
theorem,
> hence necessarily true).
Yes, []AyEx(x=y) and Ay[]Ex(x=y) are theorems.
> > Indeed it is contingent on the time frame in which
> > there are shoes, eg. it was not true 1000000 years ago, and not true
> > 1000000 years in the future.
> > No contingent statements are necessary. To say of a fact that it is
> > necessary is contradictory.
> > The same applies to Bill Clinton.
>
> I don't care about shoes or Bill Clinton. The sentence, you have
> proven, can be refrased as follow: For any POSSIBLE object y,
possesing
> a finite number of properties, to exist it must be necessarily true
(or
> simply necessary) that there exists at least one object x that
posseses
> the exact same finite number of properties.
>
> It would seem to me that you think this is the same as saying: 'it is
> necessarily true that there exists one object x'. It isn't. It is,
> however, the same as saying that the existence of an object y is
> CONTINGENT on certain necessarily true conditions. And there you have
> it: The concept of contingency preserved as well as the concept of
> necessity. Hmmm .... Well, let's see ...
It is an implicit axiom of predicate logic, that there exists at least
one value of the individual variable. Indeed, in virtue of the axiom
x=x, we can prove that every value of the individual variable
necessarily exists. Leading to my post. I deny that it can be true,
that is, we cannot accept the axiom x=x! Surely there are contingent
existences, eg. (my car)=(my car). Can we say that it's necessary?
I don't think so.
Logic needs correcting in this regard!
We need a 'free' logic to do the job, free from existential assumption.
Instead of the axiom x=x, we have E!x->x=x, ie. E!x<->x=x is then a
theorem. we also need x=y->(E!x &. E!y &. Fx<->Fy) as axiom instead of
x=y->(Fx<->Fy).
In second order logic we need a new definition of identity.
instead of the Leibnitz-Russell definition:D1. x=y = AF(Fx<->Fy) Df.
we have D2. x=y = (E!x & E!y & AF(Fx<->Fy)) Df. IMHO.
Regards,
Owen
slightly...@my-deja.com
> Logic needs correcting in this regard!
>
> We need a 'free' logic to do the job, free from existential
assumption.
> Instead of the axiom x=x, we have E!x->x=x, ie. E!x<->x=x is then a
> theorem. we also need x=y->(E!x &. E!y &. Fx<->Fy) as axiom instead of
> x=y->(Fx<->Fy).
>
> In second order logic we need a new definition of identity.
> instead of the Leibnitz-Russell definition:D1. x=y = AF(Fx<->Fy) Df.
> we have D2. x=y = (E!x & E!y & AF(Fx<->Fy)) Df. IMHO.
A strong claim: Logic needs correcting. In the sense that some logical
systems needs rebuilding axiomatically - which I think is your point -
I agree. Indeed, such an endeavor has been undertaken exactly
concerning the notions of contingency, necessity and possibility (the
modalities). Kripke's work is to some extent such a rebuilding.
Before I go into any specific details: Principia Mathemathica is
probably not the best choice for a canon regarding logic. This is not
related to the consistency of the work itself but rather to Gödel's
claim of the relationship between consistency and completeness of any
logical system. It clearly showed that Russell and Whitehead's quest
for the axioms of mathematics was hopeless. It could never be complete
or consistent.
Now, it would seem to me that our troubles arises from what you have
established is a bi-implication between necessarily true sentence and
theorem. This means of course that the phrases necessarily true
sentence and theorem become interchangeable. Given the existence of y
it then becomes a theorem that there must exist at least one y.
My claim is this: 'There exists y' is axiomatic. Not as an object, an
individual - not even refering to one. But as the axiomatic condition
for any body of logic: Namely the notion of 'substance' to which
properties are ascribed. I think this is similar to your point here (we
must in some sense axiomatically assume existence or spell it out in
some of our definitions):
> Instead of the axiom x=x, we have E!x->x=x, ie. E!x<->x=x is then a
> theorem. we also need x=y->(E!x &. E!y &. Fx<->Fy) as axiom instead of
> x=y->(Fx<->Fy).
Does this notion say that any individual exists with necessity? No, it
doesn't because an individual embodies a distinct set of properties -
of which we have said nothing. Does it say that given any individual
with a distinct set of properties that it exists necessarily? Yes, it
does. But this is no problem: The necessary existence of any object
would be conditioned on this object possessing certain distinct
properties. Clinton, for instance: For any object Bill Clinton to exist
with necessity there must be at least one object x that embodies every
distinct property of Bill Clinton. These distinct properties are not
only the concrete bodily proportions of Bill Clinton, his thoughts and
wishes but these properties also encompass something like the entire
history of the earth, of mankind. Would it be fair to say that if these
conditions obtain then Bill Clinton exists with necessity? I think so.
It has to do with our conception of necessity: In all thinkable worlds
with properties exactly like ours Bill Clinton's existence would be
necessary. In all other worlds with properties not quite like or not at
all like ours Bill Clinton would exist contingently - dependant on our
conjuring up properties similar to our world. This spells out a lot
like Kripke's view of necessity and contingency. And I believe strongly
that a broader discussion of these terms is the only way one can treat
these issues of necessity.
Regards,
o_ho...@my-deja.com
slightly...@my-deja.com wrote:
>
>
> > Logic needs correcting in this regard!
> >
> > We need a 'free' logic to do the job, free from existential
> assumption.
> > Instead of the axiom x=x, we have E!x->x=x, ie. E!x<->x=x is then a
> > theorem. we also need x=y->(E!x &. E!y &. Fx<->Fy) as axiom instead
of
> > x=y->(Fx<->Fy).
> >
> > In second order logic we need a new definition of identity.
> > instead of the Leibnitz-Russell definition:D1. x=y = AF(Fx<->Fy) Df.
> > we have D2. x=y = (E!x & E!y & AF(Fx<->Fy)) Df. IMHO.
>
> A strong claim: Logic needs correcting. In the sense that some logical
> systems needs rebuilding axiomatically - which I think is your point -
> I agree. Indeed, such an endeavor has been undertaken exactly
> concerning the notions of contingency, necessity and possibility (the
> modalities). Kripke's work is to some extent such a rebuilding.
>
> Before I go into any specific details: Principia Mathemathica is
> probably not the best choice for a canon regarding logic. This is not
> related to the consistency of the work itself but rather to Gödel's
> claim of the relationship between consistency and completeness of any
> logical system. It clearly showed that Russell and Whitehead's quest
> for the axioms of mathematics was hopeless. It could never be complete
> or consistent.
Godel's theorems do not alter the approach to axiomatic systems. That
there are truths within any deductive system that cannot be proven
within the system is self-evident, the axioms themselves are examples.
We still develop systems to-day, in the same way that Russell did.
It's the only way that it can be done. We move frome better systems to
better again, without ever dreaming of an absolute best.
> Now, it would seem to me that our troubles arises from what you have
> established is a bi-implication between necessarily true sentence and
> theorem. This means of course that the phrases necessarily true
> sentence and theorem become interchangeable. Given the existence of y
> it then becomes a theorem that there must exist at least one y.
>
> My claim is this: 'There exists y' is axiomatic. Not as an object, an
> individual - not even refering to one. But as the axiomatic condition
> for any body of logic: Namely the notion of 'substance' to which
> properties are ascribed. I think this is similar to your point here
(we
> must in some sense axiomatically assume existence or spell it out in
> some of our definitions):
>
> > Instead of the axiom x=x, we have E!x->x=x, ie. E!x<->x=x is then a
> > theorem. we also need x=y->(E!x &. E!y &. Fx<->Fy) as axiom instead
of
> > x=y->(Fx<->Fy).
>
> Does this notion say that any individual exists with necessity? No, it
> doesn't because an individual embodies a distinct set of properties -
> of which we have said nothing. Does it say that given any individual
> with a distinct set of properties that it exists necessarily?
No, no physical object can have the property of necessary existence.
Only abstract objects, eg. numbers necesarily exist.
[](3>2) implies [](3 exists)
slightly...@my-deja.com
> Godel's theorems do not alter the approach to axiomatic systems. That
> there are truths within any deductive system that cannot be proven
> within the system is self-evident, the axioms themselves are examples.
> We still develop systems to-day, in the same way that Russell did.
> It's the only way that it can be done. We move frome better systems to
> better again, without ever dreaming of an absolute best.
I agree of course. I do not make the claim that axiomatic systems are
in anyway invalid as an approach to our reasoning in logic or that we
can do any better in other ways (whatever these ways might be).
However, I do claim, that any axiomatic system cannot stand alone if we
want to import notions from this system into our everyday language.
This is what you have done with the Bill Clinton example: You have
taken a notion of necessity - defined in terms of a theorem, within a
body of axioms - and applied it to our everyday language. Now, this
clearly cries out for further clarification of the notion of necessity
itself - because as it stands in relation to your logical findings it
has no bearing on our everyday language, it is (stricly speaking) an
empty logical qualification. That is what I have been trying to do (on
my second try - I agree that one cannot attack your findings with
regards to the notion of existence, as I tried when starting out).
> > Does this notion say that any individual exists with necessity? No,
it
> > doesn't because an individual embodies a distinct set of
properties -
> > of which we have said nothing. Does it say that given any individual
> > with a distinct set of properties that it exists necessarily?
>
> No, no physical object can have the property of necessary existence.
>
> Only abstract objects, eg. numbers necesarily exist.
>
> [](3>2) implies [](3 exists)
Again, I must stress that I do not find that any individual exists with
necessity - instead I find that for any individual to exist certain
necessary conditions must apply. This is a completely different
approach to interpreting your findings. And I believe that this is what
you have proven.
We need a certain interpretation of the notion of necessity in everyday
language use (for instance the many worlds-interpretation) to actually
spell out your findings. It is not enough to simply say that the
sentence Bill Clinton necessarily exists is in violation with our
notion of contingent and necessary existence if we haven't established
these notions in a broader sense than the mere logical.