1. Feb 25, 2005

### Owen Holden

#1
Nothing exists, means, It is not the case that something exists.

(Nothing exists) <-> ~(Something exists)
Something exists <-> ~(Nothing exists)

Something exists, means, there is an x such that: x exists.
Something exists, means, Ex(x exists). (ExE!x)

x exists, is defined, there is some y such that: x is equal to y.
E!x =df Ey(x=y).

Something exists, means, Ex[Ey(x=y)].

Nothing exists, means, ~Ex[Ey(x=y)].

Because, (nothing exists) <-> ~(something exists).

But, ExEy(x=y) is a theorem.

1. Ax[x=x] and 2. AxAy[x=y -> (Fx <-> Fy)] are the axioms of identity theory, within first order predicate logic.

ExEy(x=y)

Proof:

1. Ax(x=x) -> a=a
2. a=a -> Ey(a=y).
3. Ey(a=y) -> Ex[Ey(x=y)].
4. Ax(x=x) -> ExEy(x=y).
5. ExEy(x=y).
By axiom 1, Ax(x=x).

If we use the second order Leibnitz-Russell definition of identity,
x=y =df AF(Fx <-> Fy), then we can prove that Ax(x=x) is a theorem.

x=x means AF(Fx <-> Fx), which is clearly tautologous for all x.

i.e. Nothing exists is a contradiction.

~(Nothing exists), is a theorem of classical logic.

2. Feb 25, 2005

### nnnnnnnn

nothing is nothing is also a contradiction...

If nothing were nothing then something (specifically nothing) would be nothing.

so is nothing not nothing?

also something is nothing is a contradiction...

Last edited: Feb 25, 2005
3. Feb 25, 2005

### Hurkyl

Staff Emeritus
This is a flaw. You've assumed that there is a constant "a". Thus, you've already assumed something exists.

4. Feb 25, 2005

### Owen Holden

1. Ax(x=x) -> a=a

Not so. Variables refer to names, and names refer to objects.
I only assume that there are names such as 'a'.

5. Feb 25, 2005

### Hurkyl

Staff Emeritus
If there are no objects, there are no constant symbols, names, or whatever you want to call it, and the quoted step is not even wrong -- it's not even a logical statement.

6. Feb 25, 2005

### honestrosewater

"For a sound and complete proof system for logic with an empty domain, see (Tennant 1990)." - Logic and Ontology, Thomas Hofweber.
There are several other people who write about the empty domain and "free logics", ex. Lambert, Williamson, Goe, Hodges. I haven't read all of them yet (I do intend to), and the articles I tried reading assumed more knowledge than I have, so I can't really add any ideas of my own. But if the assumption that the universe of discourse is non-empty is unnecessary- as it seems to be-, I'd rather not assume it. (I gather that "domain" and "universe of discourse" are interchangeable.) Why do you have a problem with letting the universe be empty?

As for names v. objects, doesn't a use-mention distinction, where, for example, "Socrates is a man" and "'Socrates' is a word" are both true propositions, allow a language to talk about itself? Or is that not the distinction you meant?
Edit: Okay, I think you meant the distinction between intension (or connotation) and extension (or denotation). I'm always getting sidetracked in my reading.

Last edited: Feb 25, 2005
7. Feb 25, 2005

### Doctordick

Hurkyl, I used to think you were a rational educated individual but your inability to pick up on rather simplistic things is beginning to bother me. Just exactly what is your background anyway.

Have fun -- Dick

8. Feb 26, 2005

### Owen Holden

Me too. I Have not read much about 'free' logics either and it does sound interesting.

I dont have a problem with admitting the emty domain, but classical logic does.

If we admit language and a sytem of logic then we can say:
1. Ax(Fx) -> Fy, 2. Fy -> Ex(Fx), 3. Ax(Fx) -> Ex(Fx) ..are true for all F, including when F={}. {} =df {the x's: ~(x=x)}

That is, (x e {}) or ({}x) are contradictory for all x.

1a. Ax({}x) -> {}y, 2a. {}y -> Ex({}x), 3a. Ax({}x) -> Ex({}x) ..are true.

{}y is false, Ex({}x) is false, and Ax({}x) is false.
~({}y) is true, Ex[~({}x)] is true, and Ax[~({}x)] is true, ..in all domains.

In the empty domain Fx is contradictory for all x, for all F.
There is no x such that x exists, ~Ex(E!x) is true there.

i.e. 'F' behaves in the same way as {} does for every F, in the empty domain.

The empty domain does exist but, nothing does not exist.
The null set/property exists by axiom, and it is something. EF(F={}).

In Fa, (a) refers to the object that "a" names if such there be.
If (a) does not refer to any object, then Fa is contradictory for all F.

e.g. Vulcan rotates, is contradictory because Vulcan (that planet which accounts for the unusual orbit of Mercury within Newtonian physics) does not exist.

Owen

9. Feb 26, 2005

### Owen Holden

I agree, it is absurd (contradictory) to assume that no-thing exists.

Nothing is not a thing at all, rather, it is a way of talking about things.

I exist, entails the existence of: I, existence, truth, language, etc., etc..

I exist, cannot be denied by anyone who understands it, because, the process of denying it shows that I do exist.

Descartes Dictum, Fa -> E!a, follows from the definition of existence.

E!x =df EF(Fx).

Last edited: Feb 26, 2005
10. Feb 26, 2005

### Hurkyl

Staff Emeritus
Do you agree or disagree that your proof fails when using a language that does not contain name symbols?

11. Feb 26, 2005

### Owen Holden

Neither 'Fa' nor 'a=a' have sense if there are no name symbols, imo.

12. Feb 26, 2005

### loseyourname

Staff Emeritus
By making the statement "nothing exists," you've proven the falsehood of the statement by virtue of the fact that something must exist in order to make the statement. But if nothing actually existed, nothing could make that statement and there would be no contradiction. You must consider that there would also be no systems of logic, which all presuppose their own existence.

13. Feb 26, 2005

### honestrosewater

That's interesting. Why must a logic presuppose its own existence? Seriously. The logic itself is a subclass of other classes, for example, a metalanguage which talks about the logic (or perhaps I should call it something else, like an interpretation or structure or some such). My logic book remarks repeatedly- and I find this quite amusing- that we're only talking about the formal language L we're studying, using English- we never actually say anything using L. I don't know what would be left if we didn't stipulate that L existed- what do you guys think? Would that restrict what we could prove in L? Could we "use" L but stay in the metalanguage, or is that cheating?

14. Feb 27, 2005

### Crackpot

What does nothing mean ?

Does nothing exist?
Does nothing exist as an entity?

The idea that nothing can exist as an entity or otherwise is not a part of western civilization, (the Judeao-Christian tradition of thought); as such there are no good words or reference points that describe the concept.
See URL: http://7777777s.com/modules.php?name=Forums&file=viewtopic&t=12

The idea of nothing in existence is one of the foundations of Mahayana Buddhist thought, i.e.
The idea of nothing was used by vonNeumann
What do you mean when you say nothing exists?

Last edited by a moderator: May 1, 2017
15. Feb 27, 2005

### Philocrat

Logically, it implies:

I deliberately omitted the metaphysically vexing term 'Exist', as there is currently an ongoing dispute in philosophy as to whether it is a Predicate. I am only being generous here. However, you still have not resolved the fundamental question of a metaphysical scale as to who makes the proposition at what point in or outside time and space. The standard metaphysical assumption is that if anything makes the proposition:

Then this is a contradiction, or simply, the maker is contradicting himself or herself or itself. The argument is that you cannot say "I am" and not exist, even if you were a Brain in the vat or a toy in the factory. This is the formal component of Decartes' Corgito Argument before it got caught up in the 'EXISTENCE-PREDICATE' controversy. I am just sick and tired of people who try time and time again to twist this fact around in a self-serving manner as if though they were born to be die-hard controversialists.

---------------------
Think Nature......Stay Green! And above all, Never harm or destroy that which you cannot create! May the 'Book of Nature' serve you well and bring you all that is Good!

Last edited: Feb 27, 2005
16. Feb 27, 2005

### Owen Holden

#1
"Nothing exists, means, It is not the case that something exists."

imo, you have misunderstood von Neumann. He does not say that the null set is nothing.
The null set, {}, exists by axiom. {} e {{}}, proves that the null set is something.

17. Feb 28, 2005

### Crackpot

Perspective - Point of view - Attitude - they count

Another very popular quote from Heisenberg
"In the beginning was the symmetry"

Consider the idea that absolutely nothing and absolutely everything are coexisting simultaneously in the same space, place, and time. This is the symmetry that Heisenberg was referring to. This basic paradoxical duality can be viewed as the start of real time. Two dimensional view of Absolutely Everything & Absolutely Nothing Coexisting simultaneously in the same place space & time.

Stepping away from the Platonic idea of absolutes to the same paradoxical duality, nothing and everything coexisting simultaneously in the same space, place, and time; we come to the empty set. The empty set cam be described as the nothing part of this idea. The shared link between them causes them not to be absolutes.Two dimensional view of Everything & Nothing Coexisting simultaneously in the same place space & time.

A description of nothing and/or absolutely nothing that can be understood usually includes the part of the duality that we normally perceive.

My point, the idea you presented is incomplete at a basic level.

18. Mar 6, 2005

### Thor

The problem with 'Nothing' is purely semantical

'Nothing' has two connotations :

'Nothing(L)' - in logical terms - is the null set (represented by the symbol 'Ø').

'Nothing(A)' - in the abstract - is 'that which does not exist'.

But, 'that which does not exist' does not exist. It is not the empty set. It is not a set at all. It has no properties or attributes.

To consider 'Nothing(A)' would be not to consider.

To perceive 'Nothing(A)' would be not to perceive.

To understand 'Nothing(A)' would be not to understand.

Imagine an inert, infinitesimal point in space - and then try to imagine that same point NOT in space. Logic requires definition. 'Nothing' - in the abstract context - is undefined, it does not exist, it is a fiction which has no physical manifestation in the Universe.

The only definition of 'Nothing' which applies to logic and reality is:
'Nothing(L)' - the null set (represented by the symbol 'Ø').

19. Mar 6, 2005

### Owen Holden

I don't agree.
The null set exists by axiom, therefore it cannot be 'nothing' in any sense.

{} e {{}}, is a theorem which proves that {} is something and that it is not nothing.

20. Mar 6, 2005

### Hurkyl

Staff Emeritus
Several things: {1, 2, 3}

One thing: {42}

Nothing: {}

While I'm not willing to say I agree with Thor's post, the empty set is certainly relevant to the concept of nothing.

One could say that {} describes "nothing" just as the set {..., -4, 2, 0, 2, 4, ...} describes "even integer".