# When the universe is empty

1. Feb 14, 2005

### honestrosewater

Do I understand correctly that the universe, x, is empty just in case $$[\exists x (Px)]$$ is false and $$[\forall x (Px)]$$ is true?
Is there anything interesting about empty universes? I don't have any problems with them yet, but I'm not sure how to think about them. That is, they seem to be lacking some meaningfulness that non-empty universes have. For instance, in an empty universe, $$[(\forall x (Px)) \wedge (\forall x (\neg Px))]$$ is always true. So what does Px mean in a non-empty universe? What does it mean when $$[(\forall x (Px)) \wedge (\forall x (\neg Px))]$$ is false? I think I am wrongly attributing some kind of meaning to Px and, perhaps, to propositions in general.
Edit: Eh, I guess when the universe is empty, the truth-values of all propositions are already determined. Or, rather, determining that the universe is empty is equivalent to determining the truth-values of all propositions.? I mean, doesn't determining the truth values of $$[\forall x (Px)]$$, $$[\neg (\forall x (Px))]$$, $$[\forall x (\neg Px)]$$, and $$[\neg (\forall x (\neg Px))]$$ tell you everything?

Last edited: Feb 14, 2005
2. Feb 14, 2005

### chound

What???????????????????????????????????

3. Feb 14, 2005

### honestrosewater

Ooh, I have another question.
$$\begin{array}{|c|c|c|c|c|c|}\hline \forall x (Px)&\neg (\exists x (\neg Px))&T&T&F&F\\\hline \neg (\forall x (Px))& \exists x (\neg Px)&F&F&T&T\\\hline \forall x (\neg Px)& \neg (\exists x (Px))&T&F&T&F\\\hline \neg(\forall x (\neg Px))&\exists x (Px)&F&T&F&T\\\hline \end{array}$$
(Eh, this is not a truth table and is conveniently redundant. Propositions in the same row are logically equivalent. Rows 1 & 2 and 3 & 4 are contradictory.)

Column 3 lists truth-values in the empty universe, IOW, when $$[(\forall x (Px)) \rightarrow (\exists x (Px))]$$ is false. Columns 4, 5, & 6, together, list possible truth-values in a non-empty universe, IOW, when $$[(\forall x (Px)) \rightarrow (\exists x (Px))]$$ is true.
What about columns 4, 5, and 6, considered separately? Are they inconsistent? I'm asking this mostly because I think it's an interesting question, not so much because I want a yes or no answer.

4. Feb 14, 2005

### honestrosewater

Sorry, it took me a while to make that table (my first).
Do you not understand what I said, or do you understand it but have a problem with it?
Eh, I don't mean the "real" universe- it's just a logical term.

Last edited: Feb 14, 2005
5. Feb 14, 2005

### chronon

I don't think that you can claim this if you assume that "For all x ..." implied that "There exist x ..." (as Aristotle seemed to do). For instance taking the universe of unicorns in my garden, (which seems to be empty at present), and P to be the property of having a horn, then the first expression in your second row says that "it is false that all unicorns in my garden have a horn", which is true with Aristotle's assumption, while the second expression says that there exists a hornless unicorn in my garden, which is false.

6. Feb 14, 2005

### chound

Dont bother ecksplainin it to me. I wouldn't understand! Not in a million years!

7. Feb 14, 2005

### honestrosewater

They are equivalent in predicate logic. google DeMorgan laws for quantifers or quantifer negation.
You're talking about subalternation (and the reverse, superalterntation) in Aristotle's logic. From what I've read, he actually phrased O statements as "Not every S is P" instead of the familiar "Some S is not P". So he avoided the problems arising from subalternation from E to (the familiar) O. (Aristotle rocks.)
Anywho, he did assume subalternation from A to I statements. The thing is that (I don't know all the implications of this but) statements in categorical logic translate to predicate logic as follows:
A: $$[\forall x (\neg Sx \vee Px)]$$
E: $$[\forall x (\neg Sx \vee \neg Px)]$$
I: $$[\exists x (Sx \wedge Px)]$$
O: $$[\exists x ( Sx \wedge \neg Px)]$$
So, for instance, A does not mean $$[\forall x (Px)]$$, though they do seem to say the same thing at first. Your example would be $$[[\forall x (\neg Ux \vee Hx)] \rightarrow [\exists x (Ux \wedge Hx)]]$$, where Ux: (x is a unicorn in my garden) and Hx: (x is horned). Ux is false, so the conditional is false, but the universe is not necessarily empty.
In "All S are P", existential import applies to S, while, in $$[\forall x (\neg Sx \vee Px)]$$, existential import applies to Sx not to x. At least, that's how I understand things.

Eh, actually, maybe that wasn't your example, but the point still holds. It seems the universe (x) doesn't have any properties; Only elements of the universe (a, b, c, ...)have properties (P, Q, R, ...). This is confusing because the universe can apparently be empty and non-empty, which certainly sound like properties to me. Perhaps they are "metaproperties", for lack of a better word.

Last edited: Feb 14, 2005
8. Feb 14, 2005

### cen2y

[QUTOE]Do I understand correctly that the universe, x, is empty just in case $$[\forall x (Px)]$$ is false and ~ is true? [/QUTOE]
No, it's when there's nothing possessing P(x), i.e. $$[\forall \neg x(Px)]$$
Then it is true, given that P is the property which defines whether something is "something in the universe" or not.

$$\forall x(\forall y(x(y)) \leftrightarrow \neg \exists y(\neg x(y))[/itex] Is a common rule posing no problems in classical logics what so ever. That there's something that isn't both P and not P, i.e. there's something. When correctly configured, yhey are all equivalent with the above stated ones, knowing the others means knowing these.' Row 3 and 4 aren't contradictionary. Row 3 states: Everything's not P, second says, There's no thing that is p. (I don't have a clue what your T/Fs for). You need to separate things which are supposed to be in the universe from things that aren't. Such as the universe itself. No need for "metaproperties": 9. Feb 14, 2005 ### honestrosewater I have never seen [tex][\forall \neg x]$$ before (and I've seen a lot of different notations), and I don't understand what it means. Does $$[\neg (\exists x (Px))]$$ not say the same thing?
I don't understand this either- I don't understand the x(y) part or whether the ~ in "~x(y)" applies to x or to x(y).
I was thinking more of what it means for properties. Do you know how predicates defined?
How are rows 3 and 4 not straightforward negations of each other?
How does row 4, $$[\exists x (Px)]$$, say there's no thing that is P?
True and False. The propositions listed are either true or false- the table lists all the possible combinations.
That's what I meant "meta" to do.

10. Feb 15, 2005

### Philocrat

The best Logical form to answer your question is the very TITLE itself:

The universe is empty.......WHEN..... the universe is empty!

Or if you wanted to take the TL (Transitional Logic) route, you could formally declare:

The Universe is empty ....UNTIL...the universe is not empy!

Generalisation may be useful in demostrating knowledge of Formal Logic, which you have successfully done here, but rarely does it ever answer questions that are being asked about the Epistemological Status of the world or the universe. Yes, you may successfully demonstrat in general terms what logically applies given various logical possibilities as some of you have done, but you still have to epistemologically, let alone metaphysically, demonstrate how a given possibility is or may be actually the case. You need some criteria deduced from an argument as to under what actual conditions may we scientifically declare the universe completely empty. What does it really take for a universe to be empty? You could still use the same Predicate calculus as you have done and say something like.

Take anything, if it has the property of being a universe, is completely empty when conditions x1, x2, x3.......xn are true?

When you do this, you are now fundamnetally operating at the level metaphysics even though you are still formally bound by logic. My fear is that at this metaphysical level things may get fuzzier and your argument crumbles and ends in a non-logical verdict of INDETERMINACY OF THE EPISTEMOLOGICAL STATUS OF THE UNIVERSE. Under this result it becomes impossible to know when to declare the universe empty.

This is important because, metaphysically, you would be confronted with so many problems such as determining what constitutes the actual things that can occupy a universe? What should count as things and what should not count as things that could occupy it? What about the actual physical laws that hold the universe together....would you include or exclude these? Even if our universe were structurally like an inflated baloon, would you count both its internal and external boundaries as things?

And even more a more problematic metaphysical question is this:

Supposing our universe were a SEMI-CLOSED SYSTEM such that it became partially dependent upon external things for self-sustainment, would you count these externally introduced dependencies as things that could occupy it?

The questions are just endless, whether metaphysically, formally or epistemologically. I am only suggesting that your question is not the type that necessarily requires formal logical proof. There is more to this question than taking the formal dimension.

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

### honestrosewater

Philocrat, Thanks for trying, but as I already said, I'm not talking about that kind of universe. I am talking about the "universe of discourse". I said "universe" because it's shorter, and I thought it would be clear that I meant it as a logical term. The universe, as I intend it, is a collection of individual constants, like unicorns, men, sentences, numbers, etc., and, in my examples, is denoted by the individual variable "x", as I said. My question is about logic not physics or metaphysics or ontology or epistemology.

12. Feb 17, 2005

### Chronos

I agree with Philocrat. Since observation suggests the universe permits you to ask this question, it clearly implies it is not empty. That renders the question unphysical and logically meaningless. If it is any consolation, there are many mathematical solutions to science problems that are also regarded as meaningless, because they too are unphysical [unsupported by observation].

Last edited: Feb 17, 2005
13. Feb 17, 2005

### honestrosewater

What universe? The physical universe? Because I am not talking about the physical universe! Why don't you believe me?
And if it's unphysical, it's logically meaningless? Since when is logic an observational or physical science?
Edit: Sorry, I don't mean to be rude, if it sounded that way. Could you explain what you think the relationship is between being physical and being logically meaningful?
BTW, if that conditional is true, it's contrapositive is true. So if something is not logically meaningless, it's not unphysical. This may be the same as saying if something is logically meaningful, it's physical, depending on how some other things are defined.

Last edited: Feb 17, 2005
14. Feb 18, 2005

### Philocrat

The "universe of discourse" is a well understood quantificational device in formal logic. In Predicate Calculus we know that you use variables to formally denote and pick out the indvidual members of it. But the question that I keep on asking is:

Is that sufficient?

For the reality of things is that, long after you have successfully determined how to formally track the truth values of a given universe of dsicourse, you still have to demonstrate how all this epistemologically translate into scientifically valuable world view. You still have to say something about what it takes to declare that the real universe is empty. Formal Logic does formally model reality but says nothing about the actual facts of the world.

One other fundamental mistakes that is often made by the formalists is to assume that

1) When they are METAPHYSICALLY talking about the world, that there is no formal logic involved or that Logic as a whole is completely abandoned in the process

2) When they are making demands about the EPISTEMOLOGICAL STATUS of the world, that no logic of some sort is involved.

None of these two assumptions can be so straightforwardly true. Once you have formally modelled the world as we often do, no logic is abandoned when we are making direct metaphysical and epistemological demands about the actual state of the world. To say that the Universe is empty, or may be empty, implies that we have good and carefully deduced actual reasons for saying so. Would you say that this statement is formally redundant?

The universe is, or may be, empty because n numbers of actual physical conditions are, or may be, true.

There is nothing formally far removed from this direct statement about the universe. The formal content of this proposition is still 100% intact. From this point onward, it is therefore up to us or anyone to say what those actual n conditions are. Hence, epistemologically and metaphysically, the actual universe that we describe in this way is formally equivalent to your 'universe of discourse', except that here we are formally grounding things in actuality. Logic formally talks, but it is even stronger and more relevant when it says something concrete about the actual state of the world. On this, you are (or anyone esle is) free to blame me for trying to blur the line between Logic, Metaphysics and Epistemology.

Last edited: Feb 18, 2005
15. Feb 18, 2005

### Philocrat

CLEARING UP A FEW MISUNDERSTANDINGS

As I have mentioned before in some of my postings on this PF, ALL PROPOSITIONS ARE CONCLUSIONS OF FULLY DEDUCED FACTS ABOUT LOGICAL SPACE OR THE REAL WORLD. A Logical Space and the Physical World are formally equivalent. Now, let me say something about TRUTH and FALSITY of propositions.

1 LOGICAL TRUTH AND FALSITY

A proposition is LOGICALLY TRUE or FALSE by vitue of its LOGICAL FORM alone. For example:

a) If P then Q
b) If P then Q | P | therefore, Q
c) There is One and only one Universe and it is empty (Russell)

and so on. These are just a few of all the well-known logical forms that if true or false do so in virtue of their logical forms alone.

2 EMPIRICAL TRUTH AND FALSITY

A Proposition is Empirically true or false in virtue of the verifiable physical facts that it conveys about reality or the world. The mistake that is often made by the formalists is to assume that emperical facts automatically exclude logical facts from the causal and relational structure of the world. Of course, statements of fact about the world also contain logical facts. We visually detect the facts and logically order and validate them to combinantly secure their truth or falsity. Would you say that the following statements are quantificationally and formally redundant?

a) If am standing, then it is necessarily true that I am not sitting.

b) If I play any game, then it is necessarily true that there will be only one of three possible outcomes: Win, Lose or draw.

c) If Play any game, then it is possibly true that I will win.

d) The sun is shining

e) John is tall

f) Possibly, it will rain tomorrow

g) Come here now!

h) Have you stopped beating your wife? (Peter Strawon's Presuppositions)

i) Goodness me, what a waste!

j) The FAT sound filled the room

and so on. Well, I personally don't think so. All these types of statements have factual information about the world. Yet, their logical contents or forms are so subtle in some of them such that over the years many philosophers, especially the so-called 'analytical philosophers', have naively declared them epistemologically and logically truth-valueless. This is typical of questions, metaphors, exclamations and commands. That is, they are neither true no false. They refuse to accept that you can make a true or false statement even when you are only asking a question or commanding someone to do something. For example, if you command someone to get you a red book from an empty table, can your cammand statement be construed as containing or being devoid of truth values (true or false)? Or if you make an exclamation 'What a beautiful picture!' while looking at an empty wall, would you construe this as being true or false?

Last edited: Feb 18, 2005
16. Feb 20, 2005

### honestrosewater

Yes, since I am only talking about logic.

No, I don't. I'm not even sure what to say to that. I think your argument is doomed, by definition, but I'll see if I can get some clarity from someone.

Well, lucky me- grime et al have already made my point in "Is mathematics empirical?"

Last edited: Feb 20, 2005
17. Feb 20, 2005

### Philocrat

Aquamarine, is right! If you read the second part of my posting #14 and the whole of posting #15 you should see and realise that Aquamarine and I are saying almost the same thing. Logic, regardless of its type, rigour and formal structure, must reflect completely the causal and relational structure of the physical reality or world.

REDUCTION OF MATHEMATICS TO LOGIC?

I think we have been wasting a great deal of time on this ...... and we still do! For both are one of the same thing. Homework for the formalists: try and extract the LOGICAL and QUANTITATIVE contents of the following statements:

1) John is tall

2) Smith has given all Cambridge telephones (Wittgenstein's Foundation of Mathematics)

3) At least there is one Universe and at most there is one universe, and whatsoever has the property of being a universe is empty (Russell)

Which is equivalent to simply saying:

'There is one and only one universe and it is empy'

4) Have you stopped beating your wife (Peter Strawson's Presuppositions)

5) The voluptuous innocense of the artworks at the Tate Gallery screemed at the visitors (metaphors).

In NL (Natural Language), however vague things may sometimes appear, people do to a certain extent find their ways around these sorts of statements, in terms of trying to clearly express both their quantitative and logical contents. But from my own observation, it is the formalist who cause more chaos and problems than even the naive native speakers of NL. For it is the formalists who claim to know how to take Logic out of NL to purify by formalising it. While they claim to do so, some philosophers in the disciplines of Metaphysics and Philosophical Logic are beginning to:

(a) Examine the Fundamental relations between mathematics and logic

(b) Catigorise Logic. How many forms or types of logic are there?

(c) Question the formal structure of logic itself. How formal is Logic itself? Or how formal is formal?

(d) Question the relationship between Logic and Reality, which is equivalent to asking 'How much logic or formal structure of logic is contained in, or is captured by NL?

(d) Question how much mathematical content or structure of the world is captured by NL?

And so on. And worst still, the study of Philosophy of Language on its own is wrecking havoc in most of the logics, or logical forms and terms that were initially thought to be wholly conclusive. A deeper reflection on all this has revealed so many inconclusive terms and forms in logic and even mathematics itself.

NOTE: By this, I am not in any shape or form trying to discourage anyone from working with any of the logical forms and symbolisms that are currently in use. All that I am saying is that while more studies are being done in all these areas, we must as of now formally think and act in a manner that directly addresses the human reality. There is no room here for fantasy, let alone for trying to construct a separate formal language to enliven such fantasy, and, besides, even if this were possible at all in the first place, in the end such fantasy and its language must clearly reflect reality.

Last edited: Feb 20, 2005
18. Feb 21, 2005

### honestrosewater

I said proposition, and I meant proposition. I don't know what "true" is supposed to mean (what's with the scare quotes?).
My book says the assumption that the universe is non-empty "has some techinical advantages and is adopted by most modern authors. However, it is not essential and some authors (for example, Wilfrid Hodges, Logic, Penguin 1977) do allow structures with empty domain. [sic]" So maybe your claim is "true". :tongue2:
I don't see how you can say the case never arises anyway- don't non-emptiness and negation define emptiness? And if $$[\forall x (Px) \Rightarrow Ex(Px)]$$ in a non-empty universe, doesn't $$\neg [\forall x (Px) \rightarrow Ex(Px)]$$ in an empty universe? (That was my initial question.)
The empty universe is what I'm interested in and what leads to my other questions. If you aren't interested in it, fine. But saying there's no point in taking it any further because you can assume it's never the case doesn't satisfy me.

19. Feb 21, 2005

### CrankFan

The existential and universal quantifiers allow you to talk about various collections of things in the universe. You might possibly talk about an empty collection, but just because you can talk about empty collections doesn't mean that the domain of the quantifiers (the universe itself) is empty, and that's what empty universe or empty domain means.

(Ex)(Fx v ~Fx) is a theorem of FOPL. If you allow an empty universe it can't be a theorem.

I'm not sure if this is going to work but have a look at:

Going on about how empty domains work out in FOPL is sort of like someone asking, what happens in a field if we multiply (2/0) by 2. The only reasonable answer is: that doesn't happen.

I can't answer your question about what $$\neg [\forall x (Px) \rightarrow Ex(Px)]$$ means in an empty universe because I don't know enough about other logics that might permit that sort of thing. I'd suspect that there is more than one approach.

I don't have any problem with you perusing whatever interests you have. However if you're interested in something like empty domains of logic, division by zero, or non-well founded sets, then when you start to talk about those things with other people you shouldn't give the false impression that you're talking about FOPL, fields, or ZF set theory.

Also, remember that you jumped into a thread I was active in and posted something of questionable relevance. It's not like I came looking to disrupt your thread...

20. Feb 22, 2005

### honestrosewater

Yeah, sorry, I could have handled that better.
On the rest of your post, I give up.