### Necessary Existence

Okay since we're not doing necessary existence, I'm going to present what I take to be a fairly upsetting argument that is discussed at length in the readings for the 'necessary existence' section of the course syllabus.

First, note that often when people use 'everything' and related quantifiers, they don't mean it. So a used car salesman may say 'everything is on sale' and not mean to imply thereby that my sweater is on sale. Similarly you might say 'there's no beer in the fridge' even if there's a tiny puddle of beer at the bottom of the fridge.

But in order to do metaphysics, it must make sense to use 'everything' to mean absolutely everything, rather than some restricted domain. So for example if you make a metaphysical claim, like 'God does not exist', and you're only quantifying over some subset of everything (like the used car salesman does), then you have not succeeded in making an appropriately metaphysical claim. That's because if your domain of quantification is restricted, then it is compatible with what you said that God does exist--he just is outside the domain you are quantifying over. Similar considerations apply to any other putative metaphysical claim.

I will assume without argument that it's actually possible to do metaphysics. Since this seems to require that we can quantify over absolutely everything, I'll infer from the assumption that we can do that too. So we have (1):

1. We can quantify over absolutely everything.

But if (1) is true, we can ask the following question: Is it possible that there's something such that, possibly, absolutely everything is distinct from it?

The answer is 'no'. Take any candidate. If it's possible that that thing is such that, possibly, absolutely everything is distinct from it, then it is possibly distinct from itself. (Remember, we're quantifying over absolutely everything.) But nothing is possibly distinct from itself. (That is, if a = a, then necessarily, a = a.)

So we established the following:

2. If (1), then it's not possible that there's something such that, possibly, absolutely everything is distinct from it.

By modus ponens, we may infer (3):

3. Therefore, it's not possible that there's something such that, possibly, absolutely everything is distinct from it.

But (3) is logically equivalent to saying that necessarily, it's not the case that there's something such that, possibly, absolutely everything is distinct from it. So we have (4):

4. If (3), then necessarily, it's not the case that there's something such that, possibly, absolutely everything is distinct from it.

But the consequent of (4) is logically equivalent to saying that necessarily, absolutely everything is such that, it's not the case that, possibly, absolutely everything is distinct from it. So we have (5):

5. If necessarily, it's not the case that there's something such that, possibly, absolutely everything is distinct from it, then necessarily, absolutely everything is such that, it's not the case that, possibly, absolutely everything is distinct from it.

But the consequent of (5) is logically equivalent to saying that necessarily, absolutely everything is such that, necessarily, it's not the case that absolutely everything is distinct from it. So we have (6):

6. If necessarily, absolutely everything is such that, it's not the case that, possibly, absolutely everything is distinct from it, then necessarily, absolutely everything is such that, necessarily, it's not the case that absolutely everything is distinct from it.

But the consequent of (6) is logically equivalent to the claim that necessarily, absolutely everything is such that, necessarily, something is identical to it. That is, necessarily, absolutely everything necessarily exists. So we have (7):

7. If necessarily, absolutely everything is such that, necessarily, it's not the case that absolutely everything is distinct from it, then necessarily, everything necessarily exists.

By (2-7) and multiple modus ponens, we infer the conclusion:

8. Therefore, necessarily, everything necessarily exists.

!!!!!!!!

The ways out seem three-fold: deny metaphysics is possible (i.e., deny (1)), deny necessity of identity (i.e., deny (2)), or deny the logical equivalence between (2) and (8) (so deny one of (3-7)). Not an attractive set of options.

First, note that often when people use 'everything' and related quantifiers, they don't mean it. So a used car salesman may say 'everything is on sale' and not mean to imply thereby that my sweater is on sale. Similarly you might say 'there's no beer in the fridge' even if there's a tiny puddle of beer at the bottom of the fridge.

But in order to do metaphysics, it must make sense to use 'everything' to mean absolutely everything, rather than some restricted domain. So for example if you make a metaphysical claim, like 'God does not exist', and you're only quantifying over some subset of everything (like the used car salesman does), then you have not succeeded in making an appropriately metaphysical claim. That's because if your domain of quantification is restricted, then it is compatible with what you said that God does exist--he just is outside the domain you are quantifying over. Similar considerations apply to any other putative metaphysical claim.

I will assume without argument that it's actually possible to do metaphysics. Since this seems to require that we can quantify over absolutely everything, I'll infer from the assumption that we can do that too. So we have (1):

1. We can quantify over absolutely everything.

But if (1) is true, we can ask the following question: Is it possible that there's something such that, possibly, absolutely everything is distinct from it?

The answer is 'no'. Take any candidate. If it's possible that that thing is such that, possibly, absolutely everything is distinct from it, then it is possibly distinct from itself. (Remember, we're quantifying over absolutely everything.) But nothing is possibly distinct from itself. (That is, if a = a, then necessarily, a = a.)

So we established the following:

2. If (1), then it's not possible that there's something such that, possibly, absolutely everything is distinct from it.

By modus ponens, we may infer (3):

3. Therefore, it's not possible that there's something such that, possibly, absolutely everything is distinct from it.

But (3) is logically equivalent to saying that necessarily, it's not the case that there's something such that, possibly, absolutely everything is distinct from it. So we have (4):

4. If (3), then necessarily, it's not the case that there's something such that, possibly, absolutely everything is distinct from it.

But the consequent of (4) is logically equivalent to saying that necessarily, absolutely everything is such that, it's not the case that, possibly, absolutely everything is distinct from it. So we have (5):

5. If necessarily, it's not the case that there's something such that, possibly, absolutely everything is distinct from it, then necessarily, absolutely everything is such that, it's not the case that, possibly, absolutely everything is distinct from it.

But the consequent of (5) is logically equivalent to saying that necessarily, absolutely everything is such that, necessarily, it's not the case that absolutely everything is distinct from it. So we have (6):

6. If necessarily, absolutely everything is such that, it's not the case that, possibly, absolutely everything is distinct from it, then necessarily, absolutely everything is such that, necessarily, it's not the case that absolutely everything is distinct from it.

But the consequent of (6) is logically equivalent to the claim that necessarily, absolutely everything is such that, necessarily, something is identical to it. That is, necessarily, absolutely everything necessarily exists. So we have (7):

7. If necessarily, absolutely everything is such that, necessarily, it's not the case that absolutely everything is distinct from it, then necessarily, everything necessarily exists.

By (2-7) and multiple modus ponens, we infer the conclusion:

8. Therefore, necessarily, everything necessarily exists.

!!!!!!!!

The ways out seem three-fold: deny metaphysics is possible (i.e., deny (1)), deny necessity of identity (i.e., deny (2)), or deny the logical equivalence between (2) and (8) (so deny one of (3-7)). Not an attractive set of options.

## 2 Comments:

so the last line of this proof would read:

(box)(upside-down A)x(box)(backwards E)y(x=y).

This could mean as little as "necessarily for everything in the domain, it is necessarily in the domain". Even if the domain is everything, does this prove everything necessarily exists?

Case 1, we can quantify over non-existent objects:

In this case, when the negation sign passes through the last "for all", wouldn't it be more appropriate to change that to crazy G, rather than backwards E? After all, to say "it's not the case that for all y... blah", all it takes is a non-existent object to not be blah to make the statement true (if you can quantify over non-existents). So the statement would be changed to "(crazy G) not blah". So the end result of this long line of proof would read "necessarily, for all things necessarily there is something which is that thing". This doesn't look quite as bad.

Case 2, we can't quantify over non-existent objects:

I think having this view in the first place requires one to be "serious". The proof seems to go through ok. That means whatever you've got in your ontology, it's necessary that all those individual things are necessarily in your ontology. Maybe I'm wrong about this, but I don't think this would worry a serious metaphysicist much. They seem like the type who would (likely) be modal skeptics anyway.

I take it back, this strategy won't work even if you can quantify over non-existent objects. If you restrict the domain to existent objects, then the proof goes through as normal. That is, it is not possible that there exists an object such that possibly all (existent) objects are distinct from it. The proof follows through and says that necessarily for every existent object, necessarily there exists an object equal to it, so necessarily every object necessarily exists.

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