Against nihilism

Argument A:

1. Necessarily, if there is nothing, it is impossible that anything exists.

2. Something exists.

3. So, by Brouwer Axiom, necessarily possibly something exists.

4. So, the consequent of (1) is impossible.

5. So, it is impossible that there is nothing.

The most controversial premise in this argument is (1). Premise (1) follows from a picture of modality on which possibility is prior to necessity, and the possibility of non-actual things is grounded in possibilifiers. Absent possibilifiers, nothing is possible. But suppose that instead we like a picture of modality as grounded in necessitators. Then instead we have this argument.

Argument B:

6. Necessarily, if there is nothing, no proposition is necessary.

7. It’s necessary that it’s necessary that 2+2=4. (Obvious, or else a consequence of S4 and the fact that it’s necessary that 2+2=4.)

8. So the consequent of (6) is impossible.

9. So, it is impossible that there is nothing.

And finally we have:

Argument C:

10. Necessarily, if there is nothing, either it is impossible that anything exists or no proposition is necessary.

11. Necessarily possibly something exists. (Premise (3))

12. It’s neccessary that it’s necessary that 2+2=4. (Premise (7))

13. So, the consequent of (10) is impossible.

14. So, it is impossible that there is nothing.