Another modal cosmological argument

[punkforchrist]

I have been working on this for some time, so I thought I would test it out. Feedback, as always, is welcome.

Standard versions of the Principle of Sufficient Reason (PSR) usually state something like this: every existing things has an explanation of its existence, either in the necessity of its own nature, or in an external cause.

The strongest version (S-PSR) states: every state of affairs has an explanation . . .

I think the PSR is true, but suppose we weaken it:

W-PSR: Every state of affairs is at least partially explicable

where “explicable” means “possibly explained.” More precisely, then, the W-PSR states: Every state of affairs is possibly at least partially explained. I prefer the former way of saying it because the latter is a mouth-full.

The question is: can we demonstrate the existence of a necessary entity with the W-PSR? I think we can:

1. Every existing entity is either contingent or necessary. (Definition)

contingent = possibly can-be, possibly can not-be
necessary = cannot not-be (e.g. must-be)

In short, a contingent entity is something that can exist but can also fail to exist. A necessary entity is something that, if it exists at all, cannot fail to exist. It’s also important to note that we’re not presupposing the existence of anything contingent or necessary in (1). Instead, we are stating our available options. The only other option is that an entity be impossible, in which case it cannot exist, anyway.

1. There is a possible state of affairs S in which nothing contingent exists. (Premise)

If one part of a house can fail to exist, then it is reasonable to conclude that the house as a whole can fail to exist. This is an instance, I think, in which the whole really is like its parts. In any case, there doesn’t appear to be any logical contradiction in the idea of a collective failure to exist of anything contingent.

1. Every state of affairs is at least partially explicable. (Premise, W-PSR)

Imagine that a brick pops into existence without any explanation whatsoever. I don’t think this is possible, but let’s assume for the sake of argument that it is. Even granting this as a possibility, it is also possible for the brick to be explained in some possible set of circumstances, e.g. the brick’s popping into existence is explicable, even if not actually explained.

1. A state of affairs is explicable only if something exists. (Premise)

If nothing exists, then nothing would be able to explain the state of affairs.

1. Hence, a necessary entity possibly exists. (From 1 - 4)

If nothing contingent exists in S, then the only thing capable of explaining (even if only partially) S is a necessary entity. Given that S is possible, it follows that a necessary entity is possible.

1. Therefore, a necessary entity exists. (Conclusion, from 5 and S5)

Whatever is possibly necessary is necessary, via the S5 axiom of modal logic.

Any thoughts?

 

[R_Daneel]

Well, first of all one must consider “necessary” and “contingent” existence. The usual way to talk about it is to talk about “possible worlds”. A “possible world” is one which is different from our world in some respect of another.

If one considers a causal chain like A → B → C, then B is contingent upon A and necessary in relation to C. But this is a different usage of “necessary” and “contingent”. I only mention it, because the two usages are sometimes mixed up, and lead to confusion.

So, let’s consider all the possible worlds. If the intersection of all these worlds is not a null-world then we have a “necessary” being. The question is, is there a non-null intersection? The answer depends on how specific the “being” is, how “being” is defined. If one wishes to be specific, then one must consider two possible worlds, one containing exactly one electron, the other containing exactly one positron (the same particle as the electorn with an opposite electrical charge). Obviously the intersection is a null-world. So there is no “necessarily” existent being.

However, if the “being” is defined loosely, then one can say that both of these possible worlds are made of STEM (space-time-energy-matter). In this case, a looser definiton of “being” will yield a “necessary” being, which is simply STEM.

It is a matter of personal choice, depending upon how “being” is defined.

 

[punkforchrist]

R Daneel:

Well, first of all one must consider “necessary” and “contingent” existence. The usual way to talk about it is to talk about “possible worlds”. A “possible world” is one which is different from our world in some respect of another.

That’s correct. More precisely, a possible world is a compossible set of propositions describing the totality of existence. If X cannot not-be, that would result in X’s existence in all possible worlds.

If one considers a causal chain like A → B → C, then B is contingent upon A and necessary in relation to C. But this is a different usage of “necessary” and “contingent”. I only mention it, because the two usages are sometimes mixed up, and lead to confusion.

Yes, it can be confusing, and I hope to avoid that in explaining the MCA. “Contingent,” as you use it here is also called “dependent.”

So, let’s consider all the possible worlds. If the intersection of all these worlds is not a null-world then we have a “necessary” being.

I suppose one could hypothesize that only some contingent entity A exists in w1 and only a different contingent entity B exists in w2. This would mean that there is no necessary entity, since only one (contingent) entity exists in each of these possible worlds. Of course, I don’t think these really are possible worlds, so I defer my own additional comment for a bit later.

The question is, is there a non-null intersection? The answer depends on how specific the “being” is, how “being” is defined. If one wishes to be specific, then one must consider two possible worlds, one containing exactly one electron, the other containing exactly one positron (the same particle as the electorn with an opposite electrical charge). Obviously the intersection is a null-world. So there is no “necessarily” existent being.

Well, an electron’s negative charge isn’t literally less than 0, or less than nothing. It’s really just a helpful description in distinguishing between its own charge and the charge of a positron.

However, if the “being” is defined loosely, then one can say that both of these possible worlds are made of STEM (space-time-energy-matter). In this case, a looser definiton of “being” will yield a “necessary” being, which is simply STEM.

It is a matter of personal choice, depending upon how “being” is defined.

I’m willing to use “being,” but I figured I would use “entity,” since the latter is often not immediately perceived to be personal.

If there is a possible world in which STEM does not exist, then of course STEM isn’t necessary. Perhaps there is a possible world in which only abstract objects exist, if they can be said to exist at all. What’s interesting here, though, is that abstract objects do not stand in causal relations, so they cannot be the necessary entity (being) that the MCA concludes exists.

 

[R_Daneel]

That’s correct. More precisely, a possible world is a compossible set of propositions describing the totality of existence. If X cannot not-be, that would result in X’s existence in all possible worlds.

Fine, though I would say the opposite. If something is present in all the possible worlds, then it is a necessary entity. The starting point is the “global” existence, the adjective comes later.

Yes, it can be confusing, and I hope to avoid that in explaining the MCA. “Contingent,” as you use it here is also called “dependent.”

Yes, you did. I just expressed my agreement.

I suppose one could hypothesize that only some contingent entity A exists in w1 and only a different contingent entity B exists in w2. This would mean that there is no necessary entity, since only one (contingent) entity exists in each of these possible worlds. Of course, I don’t think these really are possible worlds, so I defer my own additional comment for a bit later.

This seems to be the crux of the matter. Why would you say that these are not “reallly” possible worlds? There are no other criteria, except that a possible world is different from our to some dgree. These possible worlds are quite different, for sure. But the definition does not talk about the difference in any descriptive manner. After all we are dealing here with a thought-experiment.

Well, an electron’s negative charge isn’t literally less than 0, or less than nothing. It’s really just a helpful description in distinguishing between its own charge and the charge of a positron.

The terms “positive” and “negative” are just a convenient way to describe the difference. We could talk about quarks of different flavors, and stipulate two possible worlds, in each there is just one quark, and they do not have have the same flavor.

I’m willing to use “being,” but I figured I would use “entity,” since the latter is often not immediately perceived to be personal.

I am more than happy to go with this terminology. Precisely because I also think that “being” has a “personal” overtone.

If there is a possible world in which STEM does not exist, then of course STEM isn’t necessary.

Correct. The null-world does not contain anything, and it is a possible world. Since it is empty, there can be no necessarily existing entities. The trouble is that the null-world is usually rejected as a possible world, though no one has made an argument why it should be rejected.

Perhaps there is a possible world in which only abstract objects exist, if they can be said to exist at all. What’s interesting here, though, is that abstract objects do not stand in causal relations, so they cannot be the necessary entity (being) that the MCA concludes exists.

I do not accept the concept of “abstract objects”. As you correctly noted the word “existence” cannot be applied to these concepts.

 

[punkforchrist]

R Daneel:

Fine, though I would say the opposite. If something is present in all the possible worlds, then it is a necessary entity. The starting point is the “global” existence, the adjective comes later.

When I say “result,” I just mean that we know something exists in all possible worlds if it cannot not-be. I don’t mean that one is causally prior to the other.

Yes, you did. I just expressed my agreement.

Okay, great.

This seems to be the crux of the matter. Why would you say that these are not “reallly” possible worlds? There are no other criteria, except that a possible world is different from our to some dgree. These possible worlds are quite different, for sure. But the definition does not talk about the difference in any descriptive manner. After all we are dealing here with a thought-experiment.

It seems to me that if one says that, “there is no necessary entity, but it is necessarily the case that at least one contingent entity exists,” then he is expressing a highly implausible statement. For one, the statement is ad hoc and doesn’t explicitly reject any of the premises of the MCA. In other words, in order for that proposition to be correct, one of the MCA’s premises must be wrong, and I just don’t know which one would possibly be mistaken.

Moreover, the non-existence of something is not plausibly an explanation of why some other thing exists. For example, if everything other than a unicorn failed to exist, that wouldn’t mean that a unicorn exists. Yet, “it is necessarily the case that something contingent exists,” seems to imply that.

The terms “positive” and “negative” are just a convenient way to describe the difference. We could talk about quarks of different flavors, and stipulate two possible worlds, in each there is just one quark, and they do not have have the same flavor.

We could do that, but then one of the MCA’s premises must be rejected. If it is even possible for S to obtain, and if S is even possibly explained, it follows that a necessary entity possibly exists and therefore exists in all possible worlds.

I am more than happy to go with this terminology. Precisely because I also think that “being” has a “personal” overtone.

Okay. It’s just if we call a quark a “being,” we don’t mean that the quark is a personal agent. I would also call a quark a being, but then I don’t normally ascribe personality to everything that has being.

Correct. The null-world does not contain anything, and it is a possible world. Since it is empty, there can be no necessarily existing entities. The trouble is that the null-world is usually rejected as a possible world, though no one has made an argument why it should be rejected.

It’s difficult to see how if a world is literally empty, how it could be a world at all. “World” seems to denote something, as opposed to nothing.

I do not accept the concept of “abstract objects”. As you correctly noted the word “existence” cannot be applied to these concepts.

I think they exist as concepts of the mind, but that’s another issue altogether.

 

[R_Daneel]

When I say “result,” I just mean that we know something exists in all possible worlds if it cannot not-be. I don’t mean that one is causally prior to the other.

I think we must clarify something. If something exists in one particular world “necessarily”, that is not “necessary existence”. Only if the same thing exists is all possible worlds, can we speak of “necessary existence”. Let’s use the quark example, since at the current time they are considered the “most” elemantary particles. ( en.wikipedia.org/wiki/Quark )

For example, in the hypothetical example of a possible world (W1) which contains exactly one quark of a some kind (let’s call it: Q1) and we remove that quark, we shall get a different possible world. That means that the existence of Q1 is necessary in W1. Let’s take another possible world (W2), which contains a different quark (Q2). For the same reason, the existence of Q2 in W2 is necessary, since its removal would lead to another possible world.

However, the intersection of W1 and W2 is the null-world (W0). It does not matter if the null-world is considered a “true” possible world, or not. The fact is that the two worlds of W1 and W2 have no common element, and thus there is no necessarily existing “thing”.

It seems to me that if one says that, “there is no necessary entity, but it is necessarily the case that at least one contingent entity exists,” then he is expressing a highly implausible statement.

For sure, but only if “necessary” is taken imprecisely. One can imagine two possible worlds (W1 and W2 above), where both have an existing particle, yet these particles are not the same.

For one, the statement is ad hoc and doesn’t explicitly reject any of the premises of the MCA. In other words, in order for that proposition to be correct, one of the MCA’s premises must be wrong, and I just don’t know which one would possibly be mistaken.

We are still very far from addressing the particulars. The concept of “necessary existence” must be clarified first.

It’s difficult to see how if a world is literally empty, how it could be a world at all. “World” seems to denote something, as opposed to nothing.

Well, it is an abstraction, all right. But so is the concept of “different possible worlds”. Yet these abstractions are cruical to the concept of “necessary” and “dependent” (or contingent) existence. Mathematicians routinely speak of an “empty set” (a set which contains no elements), and this usage is far from meaningless.

I think they exist as concepts of the mind, but that’s another issue altogether.

Strongly agreed!

 

[punkforchrist]

I think we must clarify something. If something exists in one particular world “necessarily”, that is not “necessary existence”. Only if the same thing exists is all possible worlds, can we speak of “necessary existence”.

It depends on what you mean. If X is the type of entity that cannot not-be, and X is instantiated in W, then X must be logically (and not just metaphysically) necessary in W. This means that X must also exist in all possible worlds, given that what is logically necessary in one possible world is logically necessary in all possible worlds (S5 axiom).

Let’s use the quark example, since at the current time they are considered the “most” elemantary particles. ( en.wikipedia.org/wiki/Quark )

Sure.

For example, in the hypothetical example of a possible world (W1) which contains exactly one quark of a some kind (let’s call it: Q1) and we remove that quark, we shall get a different possible world. That means that the existence of Q1 is necessary in W1. Let’s take another possible world (W2), which contains a different quark (Q2). For the same reason, the existence of Q2 in W2 is necessary, since its removal would lead to another possible world.

These quarks wouldn’t be necessary in the relevant sense, then. I think you’re adopting metaphysical necessity in describing Q1’s necessity in w1, for example. This wouldn’t mean that Q1 cannot not-be, but rather that if Q1 exists in w1, then it must exist at all times in w1.

However, the intersection of W1 and W2 is the null-world (W0). It does not matter if the null-world is considered a “true” possible world, or not. The fact is that the two worlds of W1 and W2 have no common element, and thus there is no necessarily existing “thing”.

I’m afraid I’m not sure what you mean by “intersection.” I agree that Q1 and Q2 are distinct quarks, but the proponent of the MCA will conclude that if w1 and w2 contain only logically contingent entities (Q1 and Q2, respectively), then neither are they possible worlds. Does this make sense? If every premise of the MCA is correct, it follows that a (logically) necessary entity exists; and that if any possible world is conceptualized that does not include this necessary entity, it follows that said possible world is not actually possible.

In order to reject the MCA, however, one must reject one of its premises. Do you disagree with any of the premises?

Well, it is an abstraction, all right. But so is the concept of “different possible worlds”. Yet these abstractions are cruical to the concept of “necessary” and “dependent” (or contingent) existence. Mathematicians routinely speak of an “empty set” (a set which contains no elements), and this usage is far from meaningless.

I’m not saying it’s meaningless, but that there is no concrete instantiation of an empty set or a null world. The existence of an abstract object is pertinent to the mind of the mathematician, etc., but it’s not something literally “out there,” so-to-speak.

 

[In_Spiration]

punkforchrist,

Read the modal argument to which I’ve linked, in response to R Daneel, in this post. I think you’d like it. I can’t address your argument at length, so for now I’ll just sorta unload what comes to me initially. Take it for what it’s worth.

I’m not exactly sure what you take the essence of full or “partial” explication to be, or what you think constitute its necessary and sufficient criteria. For instance, how could anything be part-explicable and, whatever the “rest” is, part-inexplicable? Would its complexity entail anything meaningful about its possible or impossible status as necessary? Wouldn’t it make more sense to think that the explicable “part” is actually a separate being and the inexplicable itself simple and/or necessary?

This claim seems dubious to me:

Even granting this as a possibility, it is also possible for the brick to be explained in some possible set of circumstances, e.g. the brick’s popping into existence is explicable, even if not actually explained.

If a “brute brick” really did just simply “pop” into existence, as either a necessary, brick entity (whatever that would be) or a contingent one, then I can’t conceive of any merely possible or partial explanation. Wouldn’t an “explanation” of the brick and its existence have to be only an attempted explanatory theory given by a mind, or perhaps a pseudo-explanation, at best? In reality, this brick would be beyond any explanation, right?, since its explanation would render it not a “brute brick” at all, i.e., any possible explication or reason “behind” or “grounding” it would make “it” an entirely different thing.

Do you distinguish between absolute necessity and relative necessity? That may be what’s at issue here. I think Daneel may have hit upon that distinction – I’ve only skimmed so far. You agree, I hope, that an absolutely necessary brick would be impossible under the conventional meaning of “brick,” as undeniably being a physical, materially differentiated object. Such a brick, existing without any cause, could only be classified a brick in only the loosest of senses and is surely only “conceivable” by some vague process of a confused imagination and uncritical intellect.

A relatively necessary brick holds more prima facie conceivability as a genuine possibility, and I’d have to think about it more re: whether one could then make the move from such a relatively necessary brick-existence to the absolute possibility of relative necessity being necessarily grounded in an actual, absolutely necessary entity.

 

[R_Daneel]

A few words are due as a starting point. A set “S” is collection of elements. It is described as S = {e1, e2, e3,…} a list of the elements in curly braces. If you have two sets, the intersection of them is the set of the elements, which appear in both sets. Take two sets: "S1 = {1, 2, 3} and S2 = {0, 2, 4}. The intersection is S1 * S2 = {2}, since 2 is the only common element in the two sets. The unity would S1 + S2 = {0, 1, 2, 3, 4}, the collection of the elements, which appear on at least one of the sets. If S1 = {1, 3, 5} and S2 = {2, 4, 6}, then the intersection is an empty set: S1 * S2 = {} - since there are no common elements.

A “world” is a collection of the entities it contains. So the set is the same as a world, and the constituting entities are the elements of the set. The empty set is the equivalent of the null world.

It depends on what you mean. If X is the type of entity that cannot not-be, and X is instantiated in W, then X must be logically (and not just metaphysically) necessary in W. This means that X must also exist in all possible worlds, given that what is logically necessary in one possible world is logically necessary in all possible worlds (S5 axiom).

What do you mean by “logically” and “metaphysically” necessary?

These quarks wouldn’t be necessary in the relevant sense, then. I think you’re adopting metaphysical necessity in describing Q1’s necessity in w1, for example. This wouldn’t mean that Q1 cannot not-be, but rather that if Q1 exists in w1, then it must exist at all times in w1.

Yes, it is the case. If that Q1 would not be there, or some other entity would be “added”, the resulting world would not be W1 any more - it would be a different Wn, instead. We have a set of all the possible worlds S = {W1, W2, W3,…} These are all distinct worlds. One possible world is defined as the set of the objects which are the elements of the world. In other words: “Wx = {e1, e2, e3, …}”. There is only one restriction, namely that the world Wx cannot contain a logical contradiction, for example it cannot contain a “married bachelor”, or a “square circle”. There are no other restrictions. You cannot “add” or “take away” any of these elements, since then you would get one of the other possible worlds. Remember, the set S is the collection of all the possible worlds.

There is another thing to mention, and at first glance it will seem to be outrageous. There is no “time” in these possible worlds, these worlds are static. I suspect you will feel that this assumption makes the whole model worthless. But it is not true. Suppose you have a dynamic world Wx, which changes. Maybe another entity is added, or maybe an entity is removed. What is the result? The result is just another possible world Wy - since the set S contains all the possible worlds. So we can disregard “time” from our analysis.

Necessary existence is defined as “If there is an entity X, which is part of all the possible worlds, then the existence of X is necessary”. The question is: “Is there an entity X which does appear in all the possible worlds?”. If there is, then X exists necessarily. If there is not, then there is no necessarily existence.

We must examine all the possible worlds, and see if there is such an element. Since there are infinitely many possible worlds, it looks like a daunting (or impossible) task. Indeed, to prove that there is a necessarily existing entity X, truly all the worlds must be examined. However to prove that there is NO necessarily existing X, we can examine a subset of the possible worlds, and that is exactly what I have done, by postulating W1 = {Q1} and W2 = {Q2}. Neither of these worlds contains a contradiction. Both are physically possible, therefore they are also logically possible. Yet, their intersection W1 * W2 = {}, or the empty (null) world. Therefore there is no necessarily existing entity X.

I’m afraid I’m not sure what you mean by “intersection.”

Explained above.

I agree that Q1 and Q2 are distinct quarks, but the proponent of the MCA will conclude that if w1 and w2 contain only logically contingent entities (Q1 and Q2, respectively), then neither are they possible worlds. Does this make sense? If every premise of the MCA is correct, it follows that a (logically) necessary entity exists; and that if any possible world is conceptualized that does not include this necessary entity, it follows that said possible world is not actually possible.

No, it does not. In the world W1 = {Q1} Q1 is “just there”. It is not logically dependent on anything, since there is nothing else to be dependent upon. Remember, I am not even considering the MCA or its premises. I am concentrating upon the concept of a necessarily existing entity.

I’m not saying it’s meaningless, but that there is no concrete instantiation of an empty set or a null world. The existence of an abstract object is pertinent to the mind of the mathematician, etc., but it’s not something literally “out there,” so-to-speak.

Yes, I agree with this.

Now, I understand that the above analysis might be hard to follow. If you have any questions, feel free to ask.

 

[punkforchrist]

R Daneel:

. . . A “world” is a collection of the entities it contains. So the set is the same as a world, and the constituting entities are the elements of the set. The empty set is the equivalent of the null world.

Okay, that makes sense.

What do you mean by “logically” and “metaphysically” necessary?

Something logically necessary is true in all possible worlds. On the other hand, a metaphysical necessity doesn’t have to exist in all possible worlds, but if it exists in some world W, then it must exist at all times in W.

Yes, it is the case. If that Q1 would not be there, or some other entity would be “added”, the resulting world would not be W1 any more - it would be a different Wn, instead. We have a set of all the possible worlds S = {W1, W2, W3,…} These are all distinct worlds. One possible world is defined as the set of the objects which are the elements of the world. In other words: “Wx = {e1, e2, e3, …}”. There is only one restriction, namely that the world Wx cannot contain a logical contradiction, for example it cannot contain a “married bachelor”, or a “square circle”. There are no other restrictions. You cannot “add” or “take away” any of these elements, since then you would get one of the other possible worlds. Remember, the set S is the collection of all the possible worlds.

I would prefer to call the set of all possible worlds something else, since I already defined S as a state of affairs in which nothing contingent exists.

Q1, as you have described it, is only metaphysically necessary. What is logically necessary doesn’t vary from world to world. The example you give of a married bachelor is helpful, since “it is impossible for there to be a married bachelor” is true in all possible worlds if it is true in at least one possible world. The conclusion of the MCA is that we can know a logically necessary entity exists so long as we can show that it exists in one possible world.

There is another thing to mention, and at first glance it will seem to be outrageous. There is no “time” in these possible worlds, these worlds are static. I suspect you will feel that this assumption makes the whole model worthless. . . .

No, that’s legitimate.

Necessary existence is defined as “If there is an entity X, which is part of all the possible worlds, then the existence of X is necessary”. The question is: “Is there an entity X which does appear in all the possible worlds?”. If there is, then X exists necessarily. If there is not, then there is no necessarily existence.

Right.

We must examine all the possible worlds, and see if there is such an element. Since there are infinitely many possible worlds, it looks like a daunting (or impossible) task. Indeed, to prove that there is a necessarily existing entity X, truly all the worlds must be examined.

The task is actually much easier once S5 is introduced. For, all that needs to be shown is that there is one possible world in which a logically necessary entity exists. Given that what is logically necessary does not vary from world to world, it follows that a necessary entity’s possible existence implies its necessary (and therefore, actual) existence.

However to prove that there is NO necessarily existing X, we can examine a subset of the possible worlds, and that is exactly what I have done, by postulating W1 = {Q1} and W2 = {Q2}. Neither of these worlds contains a contradiction. Both are physically possible, therefore they are also logically possible. Yet, their intersection W1 * W2 = {}, or the empty (null) world. Therefore there is no necessarily existing entity X.

Okay, now I understand your inclusion of the null world. As mentioned, I would simply deny that w1 and w2 really are possible worlds. The mere conceptualization of a possible world does not suffice for us to infer that it really is a possible world. For example, you can conceptualize a possible world W in which there is no logically necessary entity. If this is all that were needed to prove that W really is a possible world, then we come across a contradiction. For, I can just as easily conceive of a possible world W in which a necessary entity does exist. Given S5, this would lead us to the conclusion that a necessary entity both exists and does not exist, which is absurd.

No, it does not. In the world W1 = {Q1} Q1 is “just there”. It is not logically dependent on anything, since there is nothing else to be dependent upon.

This begs the question, though, in favor of the conclusion that a necessary entity does not exist. I realize you can conceive of a possible world in which only Q1 exists, but conceivability does not necessarily entail actual possibility.

Remember, I am not even considering the MCA or its premises. I am concentrating upon the concept of a necessarily existing entity.

I think you would need to reject one of the MCA’s premises in order to conclude consistently that there is a possible world without a necessary entity.

Now, I understand that the above analysis might be hard to follow. If you have any questions, feel free to ask.

I think I understand what you’re saying now. What confused me was the whole discussion of the null world and intersections, but I follow it now. Are my distinctions between conceivability and possibility clear? Moreover, do you accept S5?

 

[R_Daneel]

Something logically necessary is true in all possible worlds. On the other hand, a metaphysical necessity doesn’t have to exist in all possible worlds, but if it exists in some world W, then it must exist at all times in W.

I think we are getting somewhere.

So you use the “logically necessary” and “necessary entity” interchangably. And the actual question is: “is there something that exists in all the possible worlds?”. This is where we have the contention.

I would prefer to call the set of all possible worlds something else, since I already defined S as a state of affairs in which nothing contingent exists.

Sorry, if I misused your notation. It was not intentional. We can use a different symbol if you so choose. Let then Z be the set of all possible worlds. Everything else stays the same.

Q1, as you have described it, is only metaphysically necessary. What is logically necessary doesn’t vary from world to world. The example you give of a married bachelor is helpful, since “it is impossible for there to be a married bachelor” is true in all possible worlds if it is true in at least one possible world. The conclusion of the MCA is that we can know a logically necessary entity exists so long as we can show that it exists in one possible world.

We seemed to agree that “propositions” do not “exist” in and by themselves. So the proposition “there are no married bachelors” can only exist and is only meaningful if that world contains sentient beings, if those sentient beings use a conceptual “marriage” which is distinct from “not married”. In a world of intelligent bacteria or plants the concept of “marriage” does not exist so the concept of bachelor is meaningless.

The task is actually much easier once S5 is introduced. For, all that needs to be shown is that there is one possible world in which a logically necessary entity exists. Given that what is logically necessary does not vary from world to world, it follows that a necessary entity’s possible existence implies its necessary (and therefore, actual) existence.

Ah, but I reject the axiom of S5. It is just an elaborate hand-waving act.

S5 says: “If there is an entity wich is possibly necessary, then it is necessary”. Makes no sense at all. It is redundant, in the first place. If something is necessary than it is possible, too. The reverse is not true, however, and no mental gymnastics will make it happen. Leave out the word “possibly” and you get a nice, tautological proposition: “If there is an entity wich is necessary, then it is necessary” - which earns a resounding DUH.

Okay, now I understand your inclusion of the null world. As mentioned, I would simply deny that w1 and w2 really are possible worlds. The mere conceptualization of a possible world does not suffice for us to infer that it really is a possible world.

Except that these worlds are physically possible. There are no limits on the number of particles must be present in a possible world.

This begs the question, though, in favor of the conclusion that a necessary entity does not exist. I realize you can conceive of a possible world in which only Q1 exists, but conceivability does not necessarily entail actual possibility.

See right above. These worlds are physically possible.

The null-world cannot be “physically actualized”, that is for sure. But the possible world does not include the criterion that it must be possible to have it physically actualized. As long as it contains no physical contradiction, it is a possible world, and as such the null-world qualifies. But I am not adamant on this. If you so choose we can include the possibility of physical actualization in the criteria of a possible wolrd, and then we exclude the null-world. However, this does not apply to the “one”-worlds, which only contain one physical partice, or the “two”-worlds, which contain exactly 2 physical particles, or the “n”-worlds, which contain exactly “n” physical particles. All of these can be physically actualized, so they are “physically possible” worlds. And their intersection is still the null-world.

I think I understand what you’re saying now. What confused me was the whole discussion of the null world and intersections, but I follow it now. Are my distinctions between conceivability and possibility clear? Moreover, do you accept S5?

I am only concerned with physical possibilities. If something is physically possible, then it is metaphysically possible and therefore logically possible, too.

And I vehemently reject S5, as I said above.

 

[punkforchrist]

I think we are getting somewhere.

So you use the “logically necessary” and “necessary entity” interchangably. And the actual question is: “is there something that exists in all the possible worlds?”. This is where we have the contention.

Yes, that’s correct.

Sorry, if I misused your notation. It was not intentional. We can use a different symbol if you so choose. Let then Z be the set of all possible worlds. Everything else stays the same.

It’s not a big deal. I just want to make sure if anyone is following this that they won’t get confused.

We seemed to agree that “propositions” do not “exist” in and by themselves. So the proposition “there are no married bachelors” can only exist and is only meaningful if that world contains sentient beings, if those sentient beings use a conceptual “marriage” which is distinct from “not married”. In a world of intelligent bacteria or plants the concept of “marriage” does not exist so the concept of bachelor is meaningless.

A proposition can be meaningful even if there is no application in a given circumstance. For example, in the actual world, we can say that there are no married bachelors in any possible world regardless of whether there are bachelors in a possible world or not.

Ah, but I reject the axiom of S5. It is just an elaborate hand-waving act.

S5 says: “If there is an entity wich is possibly necessary, then it is necessary”. Makes no sense at all. It is redundant, in the first place. If something is necessary than it is possible, too. The reverse is not true, however, and no mental gymnastics will make it happen. Leave out the word “possibly” and you get a nice, tautological proposition: “If there is an entity wich is necessary, then it is necessary” - which earns a resounding DUH** **

S5 is a foundational axiom of modal logic. Of course, if one doesn’t accept S5, then that would explain why one rejects the MCA. Both necessary and contingent entities/propositions are possible, as you allude to. You’re also correct in saying that if something is necessary, then it is possible, but that the reverse is not true. It’s true that the reverse is not true, but that’s not what S5 states. S5 states if something is possibly necessary, then it is necessary. The reason this is true is that what is logically necessary doesn’t vary from world to world, as I explained.

In fact, S5 isn’t even a controversial axiom among logicians.

Except that these worlds are physically

possible. There are no limits on the number of particles must be present in a possible world.

Physical possibility is a subset of metaphysical and logical possibility. I might be willing to grant that these worlds are physically possible if I rejected one or more premises of the MCA, but I think each of the premises is highly plausible, to say the least.

The null-world cannot be “physically actualized”, that is for sure. But the possible world does not include the criterion that it must be possible to have it physically actualized. As long as it contains no physical contradiction, it is a possible world, and as such the null-world qualifies. But I am not adamant on this. If you so choose we can include the possibility of physical actualization in the criteria of a possible wolrd, and then we exclude the null-world. However, this does not apply to the “one”-worlds, which only contain one physical partice, or the “two”-worlds, which contain exactly 2 physical particles, or the “n”-worlds, which contain exactly “n” physical particles. All of these can be physically actualized, so they are “physically possible” worlds. And their intersection is still the null-world.

I just don’t see any reason to think these are possible worlds independent of a rejection of S5 and the MCA’s premises. Keep in mind that the mere conceptualization of a possibility does not mean it is actually possible.

I am only concerned with physical possibilities. If something is physically possible, then it is metaphysically possible and therefore logically possible, too.

Sure, but physical possibility cannot be determined by mere conceptualization.

 

[R_Daneel]

A proposition can be meaningful even if there is no application in a given circumstance. For example, in the actual world, we can say that there are no married bachelors in any possible world regardless of whether there are bachelors in a possible world or not.

I agree. But this proposition itself exist in this world, though its referent is a hypothetical world. The proposition does not exist in a hypothetical world, where there are no beings who “marry”. Suppose that there is a hypothetical world, where there is a proposition: “there are no wirgal porzous”. In that world the word “wirgal” and “porzous” refer to something real (just like “married” and “bachelor” do in our world). This proposition would be meaningless in our world, because there are no referents to those words.

S5 is a foundational axiom of modal logic. Of course, if one doesn’t accept S5, then that would explain why one rejects the MCA.

Actually, I am baffled that anyone would accept the S5 as a correct proposition much less an axiom. You correctly pointed out that just because we can imagine something, it may not have an actual existence. This valid statement will not change, just because we insert “necessary” into it.

Both necessary and contingent entities/propositions are possible, as you allude to.

In the examples I brought up, there is no “necessarily” existing entity.

You’re also correct in saying that if something is necessary, then it is possible, but that the reverse is not true. It’s true that the reverse is not true, but that’s not what S5 states. S5 states if something is possibly necessary, then it is necessary. The reason this is true is that what is logically necessary doesn’t vary from world to world, as I explained.

But that is circular reasoning. What is “logically necessary” is something that is supposed to exist in all possible worlds. You cannot say that it is possible to have some entity existing in all possible worlds, and then say: “this is possible, therefore it exists”. And that is precisely what S5 says.

Physical possibility is a subset of metaphysical and logical possibility. I might be willing to grant that these worlds are physically possible if I rejected one or more premises of the MCA, but I think each of the premises is highly plausible, to say the least.

The concept that a “simple” world is physically possible does not hinge on anything, except physics.

I just don’t see any reason to think these are possible worlds independent of a rejection of S5 and the MCA’s premises. Keep in mind that the mere conceptualization of a possibility does not mean it is actually possible.

Well, then apply this definitely true proposition to the “necessary” existence. Just because we can imagine, or conceptualize a “necessary entity”, it does not mean that it is actually possible - contrary to what S5 says.

Sure, but physical possibility cannot be determined by mere conceptualization.

The hypothesized simpe worlds are a “subset” of our current, existing world. A physically existing world cannot be made of physically impossible entities. Therefore the hypothesized “simple” worlds are physically possible.

 

[R_Daneel]

Let’s play a simplified thought experiment. Let’s assume that there are only 2 possible basic entites, “A” and “B”. Even in this case there are infinitely many possible worlds.

W1 = {A}, W2 = {B}, W3 = {A, B}, W4 = {A, A}, W5 = {B, B}, W6 = {A, A, B}, W7 = {A, B, B} etc… and there can be complex entities, too, like: W8 = {AA}, W9 = {AB}, …

Observe, W4 is not the same as W8 and W3 is different from W9… But no matter how long we continue this sequence, there will always be W1 and W2, and W8 and W9, (and many similar ones) which are all different worlds… and none of which contains the same basic or even complex entities. Therefore, even in this very simplified example there are no entities, which appear in all the possible worlds - therefore there is no “necessary” entity. One cannot single out, for example W1 and say: that is not a possible world. On what grounds could anyone assert that?

Isn’t set theory beautiful?

 

[punkforchrist]

R Daneel:

I agree. But this proposition itself exist in this world, though its referent is a hypothetical world. The proposition does not exist in a hypothetical world, where there are no beings who “marry”. Suppose that there is a hypothetical world, where there is a proposition: “there are no wirgal porzous”. In that world the word “wirgal” and “porzous” refer to something real (just like “married” and “bachelor” do in our world). This proposition would be meaningless in our world, because there are no referents to those words.

I don’t think that’s the case. If it had a meaning in some other world, then it would be true in our world that the other world has no “wirgal porzous.”

Actually, I am baffled that anyone would accept the S5 as a correct proposition much less an axiom. You correctly pointed out that just because we can imagine something, it may not have an actual existence. This valid statement will not change, just because we insert “necessary” into it.

S5 really isn’t even a controversial axiom of modal logic. It’s not that we just add “necessary” to a statement, but that there really are only two types of (possible) things: necessary and contingent. What I think the skeptic ought to say in response to the MCA is not that a necessary entity is possible and therefore actual, but that a necessary entity is literally impossible. Of course, that would again mean that in order to be consistent, the skeptic would have to reject one of the MCA’s (apparently benign) premises. Other than S5 (which, as I said, is relatively uncontroversial), I just don’t know which premise the skeptic would want to reject or undermine.

In the examples I brought up, there is no “necessarily” existing entity.

Yes, but whether these really are possible worlds is in contention.

But that is circular reasoning. What is “logically necessary” is something that is supposed to exist in all possible worlds. You cannot say that it is possible to have some entity existing in all possible worlds, and then say: “this is possible, therefore it exists”. And that is precisely what S5 says.

It would be circular reasoning if no additional reasons were given for supposing that a necessary entity exists in a possible world. However, the premises of the MCA do lead us to that very conclusion. For example, I would rightly be accused of question-begging if I merely stated:

A. A necessary entity exists in some possible world W.

However, that is not a premise of the MCA, but a conclusion. Circular arguments use the conclusion as one of the premises, so the MCA isn’t circular.

The hypothesized simpe worlds are a “subset” of our current, existing world. A physically existing world cannot be made of physically impossible entities. Therefore the hypothesized “simple” worlds are physically possible.

Notice, however, that you haven’t established that a necessary entity does not exist in the actual world. If a necessary entity does exist in the actual world, then the subset of our current, existing world will include maybe one quark (as you say), but it will also have to include a necessary entity.

See what I mean? What is physically possible isn’t the only consideration for what constitutes a possible world.

One cannot single out, for example W1 and say: that is not a possible world. On what grounds could anyone assert that?

Well, if these contingent entities (or their absence) are even possibly explained, it follows that a necessary entity is possible. Are you willing to commit yourself to a universal negative in which contingent entities are literally inexplicable (not even possibly explained)?

Isn’t set theory beautiful?

Yes, and its beauty points to an Intelligent Designer.

 

[R_Daneel]

I don’t think that’s the case. If it had a meaning in some other world, then it would be true in our world that the other world has no “wirgal porzous.”

How could a proposition be “true” if its constituent parts are meaningless? Not all propositions can be decided whether they are true or false. Here is one: “God is omnipresent, therefore God exists to the north from the North Pole”.

S5 really isn’t even a controversial axiom of modal logic. It’s not that we just add “necessary” to a statement, but that there really are only two types of (possible) things: necessary and contingent.

I really wish that the words “necessary” and “contingent” would not be used. They carry all sorts of “overtones”. Something that is called “contingent” does not imply that it needs some “precursor”, on which it depends on.

The idea of “necessary existence” is simply an unfortunate shortcut for “something that exists in all the possible worlds”. There is nothing “necessary” about it.

The idea of “contingent existence” is just another shortcut for “something that exists in some of the possible worlds, but not in all of them”. There is nothing “contingent” about it.

A “possible world” is “some hypothetical world, which differs from our physically existing world is some respect and which does not contain a physical impossibility or physical contradiction”. There are no other criteria to meet.

Of course these technical terms are widely used, which is very unfortunate. Just like in mathematics, where we speak of “rational” and “irrational” numbers, or “real” and “imaginary” numbers. Horribly misguided choice of words.

Yes, but whether these really are possible worlds is in contention.

By the definition of “possible world” they are. If you disagree with the definition of “possible world”, then we are talking past each other.

See what I mean? What is physically possible isn’t the only consideration for what constitutes a possible world.

You mean that there is a “physically possible” world, which is not “possible”? That does not make sense.

Well, if these contingent entities (or their absence) are even possibly explained, it follows that a necessary entity is possible. Are you willing to commit yourself to a universal negative in which contingent entities are literally inexplicable (not even possibly explained)?

Again, we are in a linguistic quagmire. Explanation means that one reduces something to an even more basic entity. This process cannot lead to an infinite regress, so there is some starting point, which is inexplicable, theoretically inexplicable - because it forms the basis of an explanation.

Yes, and its beauty points to an Intelligent Designer.

Pulling my leg, eh?

 

[punkforchrist]

R Daneel:

How could a proposition be “true” if its constituent parts are meaningless? Not all propositions can be decided whether they are true or false. Here is one: “God is omnipresent, therefore God exists to the north from the North Pole”.

What I’m saying is that it’s not meaningless to say, “there are no married bachelors,” in a world without bachelors, because it’s still possible for bachelors to exist.

I really wish that the words “necessary” and “contingent” would not be used. They carry all sorts of “overtones”. Something that is called “contingent” does not imply that it needs some “precursor”, on which it depends on.

Oh, I know. I’m not using “contingent” in the sense that it is interchangeable with “dependent.” I’m just saying that contingent entities are possibly caused, even if not actually caused. (I think the latter is true, as well, but I’m assuming that it’s not for the sake of argument.)

The idea of “necessary existence” is simply an unfortunate shortcut for “something that exists in all the possible worlds”. There is nothing “necessary” about it.

The idea of “contingent existence” is just another shortcut for “something that exists in some of the possible worlds, but not in all of them”. There is nothing “contingent” about it.

In other words, there is no necessarily actual causal implication involved. I agree.

A “possible world” is “some hypothetical world, which differs from our physically existing world is some respect and which does not contain a physical impossibility or physical contradiction”. There are no other criteria to meet.

Logical contradictions are relevant, too.

Of course these technical terms are widely used, which is very unfortunate. Just like in mathematics, where we speak of “rational” and “irrational” numbers, or “real” and “imaginary” numbers. Horribly misguided choice of words.

With that, I definitely agree. Still, I’m not committing myself to any equivocations in my defense of the MCA.

You mean that there is a “physically possible” world, which is not “possible”? That does not make sense.

No, I just mean that physicalities aren’t the only considerations we need to make. This is why I’ve been saying that, “there are no married bachelors” is meaningful even in a possible world without the physical presence of bachelors.

Again, we are in a linguistic quagmire. Explanation means that one reduces something to an even more basic entity. This process cannot lead to an infinite regress, so there is some starting point, which is inexplicable, theoretically inexplicable - because it forms the basis of an explanation.

That’s not quite what I was getting at. If S obtains, then there is a state of affairs in which nothing contingent exists (by definition). Now, if there is no necessary entity, that would imply that S isn’t even possibly caused (in any possible world); for if it were possibly caused, then the only entity that could cause it would be a necessary entity (since nothing contingent exists in S). This, in turn, would imply that a necessary entity is possible, and therefore exists.

To reiterate, in order to consistently maintain that a necessary entity does not exist, one must say that S is not even possibly caused (explained).

Pulling my leg, eh?

Yes, I thought you’d like that.

 

[R_Daneel]

What I’m saying is that it’s not meaningless to say, “there are no married bachelors,” in a world without bachelors, because it’s still possible for bachelors to exist.

First of all, who would say that in a world where there are no sentient beings? Second, we agreed that propositions do not exist as ontological entities, and it is meaningless to talk about propositions without sentient beings who say them. Third, in a world with sentient plants or bacteria (for example) the concept of marriage would not exist. So, for those plants the phrase “married bachelor” would not carry any meaning, just like “wjailu qopasi rthelow” has no meaning for us, even if in some hypothetical world it would be a meaningful phrase.

Logical contradictions are relevant, too.

Well, what makes a logical contradiction? How is it different from a physically impossible state of affairs? When we say that “married bachelors cannot exist”, what do we deny? The physical existence of a being who belongs to two, distinct groups, one: the group af bachelors, and two: the group of married people.

But have it your way. What kind of a logical contradiction can “be” in a world, which only contains one elementary particle? Obviously none. So, on what grounds do you say that such a world “may not be possible?”. If you can point to some actual contradiction, I will fold my cards, and grant you your point. But a mere "there may be… " simply does not cut it.

No, I just mean that physicalities aren’t the only considerations we need to make. This is why I’ve been saying that, “there are no married bachelors” is meaningful even in a possible world without the physical presence of bachelors.

I point to the first paragraph.

That’s not quite what I was getting at. If S obtains, then there is a state of affairs in which nothing contingent exists (by definition). Now, if there is no necessary entity, that would imply that S isn’t even possibly caused (in any possible world); for if it were possibly caused, then the only entity that could cause it would be a necessary entity (since nothing contingent exists in S). This, in turn, would imply that a necessary entity is possible, and therefore exists.

To reiterate, in order to consistently maintain that a necessary entity does not exist, one must say that S is not even possibly caused (explained).

So what? Some object is not “caused” - it simply exists as a brute fact. In the mini-world I presented, W1 = {A} and W2 = {B} have no cause for them, being elementary particles, but they are not “necessary” since neither of them exists in some possible worlds. This little example is a refutation of S5. Let’s rephrase S5, while retaining exactly what it says:

“If it is possible that every possible world contains the same element, then it is true that every possible world actually contains the same element”. This is a complete non-sequitur. And also it is false, as the mini-world proves.

But you again used a confusing terminology. It seems that “causation” and “explanation” are being used interchangably. I don’t think it is a good idea.

Let’s get back to the basic disagreement. On what grounds do you say that “W1” and/or “W2” are not possible worlds?

Yes, I thought you’d like that.

I sure did.

 

[punkforchrist]

R Daneel:

First of all, who would say that in a world where there are no sentient beings?

We exist in the actual world, and we can make claims that have truth-value of other possible worlds.

Second, we agreed that propositions do not exist as ontological entities, and it is meaningless to talk about propositions without sentient beings who say them.

Same as above.

Third, in a world with sentient plants or bacteria (for example) the concept of marriage would not exist. So, for those plants the phrase “married bachelor” would not carry any meaning, just like “wjailu qopasi rthelow” has no meaning for us, even if in some hypothetical world it would be a meaningful phrase.

I’m not sure this is a different point from the previous two. Either way, in a possible world without the concept of marriage, we are still able to relate that there is no marriage in that world (and thus, no married bachelors). Moreover, it is still true in that possible world that if there were such a thing as marriage (even if there isn’t) that bachelors could not logically be married.

Well, what makes a logical contradiction? How is it different from a physically impossible state of affairs? When we say that “married bachelors cannot exist”, what do we deny? The physical existence of a being who belongs to two, distinct groups, one: the group af bachelors, and two: the group of married people.

It depends on whether non-physical entities exist, or possibly exist. Unless we limit the totality of reality to what is physical, then it doesn’t make sense to say that all logical possibility are physical possibilities.

But have it your way. What kind of a logical contradiction can “be” in a world, which only contains one elementary particle? Obviously none. So, on what grounds do you say that such a world “may not be possible?”. If you can point to some actual contradiction, I will fold my cards, and grant you your point. But a mere "there may be… " simply does not cut it.

The contradiction becomes evident when the premises of the MCA are accepted and a necessary entity is shown to exist. I realize you don’t accept this argument, though.

So what? Some object is not “caused” - it simply exists as a brute fact. In the mini-world I presented, W1 = {A} and W2 = {B} have no cause for them, being elementary particles, but they are not “necessary” since neither of them exists in some possible worlds.

Then that commits you to a universal negative: it is not even possible for these contingent entities to be explained.

This little example is a refutation of S5. Let’s rephrase S5, while retaining exactly what it says:

Technically, it’s not a refutation of S5, but a refutation of the possible existence of a necessary entity. Even if a necessary entity is not possible, it could still be true that if a necessary entity were possible, then it would necessarily exist.

“If it is possible that every possible world contains the same element, then it is true that every possible world actually contains the same element”. This is a complete non-sequitur. And also it is false, as the mini-world proves.

This part of your argument does reject S5, but it’s question-begging to postulate a possible world with no necessary entity unless you have premises that don’t entail the conclusion but lead to the conclusion.

But you again used a confusing terminology. It seems that “causation” and “explanation” are being used interchangably. I don’t think it is a good idea.

Maybe not. “Explanation” is a broad term in which causation is included.

Let’s get back to the basic disagreement. On what grounds do you say that “W1” and/or “W2” are not possible worlds?

Well, I already stated that the MCA itself gives us a good reason for thinking these are not possible worlds. This isn’t question-begging, though, since the conclusion of the MCA (that a necessary entity exists) is not part of any of the individual premises of the argument. However, an additional argument might make use of mathematical sets.

In w1, where only Q1 is said to exist, it is true that Q1 exists and that the set containing Q1 exists. This means that two things exist: Q1 and the set that contains Q1. However, now it is also true that Q1, the set containing Q1, and the set containing Q1 and the set of Q1 exists. This implies that three things exist. This process can be continued ad infinitum. This all implies that there is no possible world in which only one entity exists.

Of course, I don’t expect anyone to accept this latter argument unless they also accept some form of realism or theological conceptualism with respect to abstract objects, such as sets.

 

[R_Daneel]

We exist in the actual world, and we can make claims that have truth-value of other possible worlds.

We are going in circles. It is not the point that we, in this actual world can make correct propositions about a hypothetical world. Certainly we can. But in that hypothetical world there are NO propositions. And that is what we did agree upon at the beginning. Are you repudiating that agreement?

It depends on whether non-physical entities exist, or possibly exist. Unless we limit the totality of reality to what is physical, then it doesn’t make sense to say that all logical possibility are physical possibilities.

If those non-physical entities are concepts then they do not exist if there are no beings who are able to conceptualize. if those non-physical entities are no concepts, then what are they?

The contradiction becomes evident when the premises of the MCA are accepted and a necessary entity is shown to exist. I realize you don’t accept this argument, though.

Of course I don’t. It would be meaningless to assume the final outcome of the whole argument and also use it as a premise.

Then that commits you to a universal negative: it is not even possible for these contingent entities to be explained.

They are not contingent. They simply exist as a brute fact.

Technically, it’s not a refutation of S5, but a refutation of the possible existence of a necessary entity. Even if a necessary entity is not possible, it could still be true that if a necessary entity were possible, then it would necessarily exist.

This part of your argument does reject S5, but it’s question-begging to postulate a possible world with no necessary entity unless you have premises that don’t entail the conclusion but lead to the conclusion.

Sorry. I simply restated S5 while retaining it meaning. What S5 is, it is a non-sequitur.

Well, I already stated that the MCA itself gives us a good reason for thinking these are not possible worlds. This isn’t question-begging, though, since the conclusion of the MCA (that a necessary entity exists) is not part of any of the individual premises of the argument. However, an additional argument might make use of mathematical sets.

In w1, where only Q1 is said to exist, it is true that Q1 exists and that the set containing Q1 exists. This means that two things exist: Q1 and the set that contains Q1. However, now it is also true that Q1, the set containing Q1, and the set containing Q1 and the set of Q1 exists. This implies that three things exist. This process can be continued ad infinitum. This all implies that there is no possible world in which only one entity exists.

Of course, I don’t expect anyone to accept this latter argument unless they also accept some form of realism or theological conceptualism with respect to abstract objects, such as sets.

Unfortunately what you say is even formally (mathematically) incorrect. And it also presumes the acceptance of “abstract objects”, and we both agreed that this is a nonsensical proposition.

To say that the postulated W1 = {Q1} also contains a “set” is not acceptable. A set is a logical construct, not an ontological entity.

But let’s examine what happens if we accept this idea.

Take the orininal W1 = {Q1}. You say that W1 is not “really” W1, rather it is W1’ = {Q1, {Q1}}. But these two worlds are not identical W1 <> W1’ (W1 prime). And further W1’’ = {Q1, {Q1}, {Q1, {Q1}}… etc. Now it is either true that W1 is a possible world, and W1’ and W1’’ are also possible worlds, not identical to W1, or none of them are possible worlds - and as such there are no possible worlds at all, including our current actual world. And that is absurd.