Now for an argument less strong than the one of the previous post, primarily because of premise 8 (for which I offer no argument here), but it is perhaps nevertheless interesting:

Argument:

1. Let ‘A’ refer to every thing that exists or has existed.
2. For any x, x is not caused to exist by itself.
3. For any x, if x is caused to exist, x is only caused by at least one thing that exists or has existed.
4. Either all of A is caused to exist, or all of A is not caused to exist, or some of A is caused and some of A is not caused to exist.
5. Some of A is caused to exist.
6. If all of A is caused to exist and only finitely many things exist or have existed, then some x is caused to exist by itself.
7. If some of A is not caused to exist, then something exists or has existed which is not caused to exist.
8. Only finitely many things exist or have existed.
9. Therefore, something exists or has existed which is not caused to exist.

Argument for Premise 2:

1. For any a and any b, if b is caused to exist by a, then a is not identical to b.
2. If (1), then for any x, if x is caused to exist by itself, then x is not identical to itself.
3. For any x, x is identical to itself.
4. Therefore, for any x, x is not caused to exist by itself.

Argument for Premise 3:

1. Only something that exists or has existed can cause something to exist.
2. If (1), then for any x, if x is caused to exist, x is only caused by at least one thing that exists or has existed.
3. Therefore, for any x, if x is caused to exist, x is only caused by at least one thing that exists or has existed.

Argument for Premise 4:

1. It is impossible that all of A is caused to exist and all of A is not caused to exist or some of A is not caused to exist.
2. It is impossible that all of A is not caused to exist and all of A is caused to exist or some of A is caused to exist.
3. It is impossible that some of A is caused and some of A is not caused to exist and all of A is caused to exist or all of A is not caused to exist.
4. Therefore, either all of A is caused to exist or all of A is not caused to exist or some of A is caused and some of A is not caused to exist.

Argument for Premise 5:

1. Let ‘A’ refer to every thing that exists or has existed.
2. If there is something that is caused to exist, some of A is caused to exist.
3. There is something that is caused to exist.
4. Therefore, some of A is caused to exist.

Argument for Premise 6:

1. Let ‘A’ refer to every thing that exists or has existed.
2. If all of A is caused to exist, then for any a that exists or has existed, a has a causal series.
3. The causal series of a is either finite or infinite.
4. If there are only finitely many things that exist or have existed, then a has a finite causal series.
5. If a has a finite causal series and all of A is caused to exist, then a is caused to exist by a.
6. Therefore, if all of A is caused to exist and only finitely many things exist or have existed, then some x is caused to exist by itself.

Argument for Premise 7:

1. Let ‘A’ refer to every thing that exists or has existed.
2. If it is false that something exists or has existed which is not caused to exist, then it is false that some of A is not caused to exist.
3. If it is true that something exists or has existed which is not caused to exist, then it is true that some of A is not caused to exist.
4. It cannot be true and false that some of A is not caused to exist.
5. It cannot be true and false that something exists or has existed which is not caused to exist.
6. Therefore, it is impossible that it is true that some of A is not caused to exist and not true that something exists or has existed which is not caused to exist.
7. Therefore, if some of A is not caused to exist, then something exists or has existed which is not caused to exist.