Proof by reductio ad absurdum

1. Assume there is no necessary thing. (for reductio)

D1. A necessary thing is a thing which cannot fail to exist. (Def.)

D2. A contingent thing is a thing which can fail to exist. (Def.)

2. Then all things are contingent. (from 1 by definitions D1 and D2)

I.e., Each thing which exists is a contingent thing.

Therefore, each thing which exists can fail to exist. (from 2 and D2)

3. "Anything which is capable of not existing at some time or other does not exist." (P 50b)

I.e., If a thing can fail to exist, then at some time it does not exist.

Therefore, each thing which exists at some time does not exist. (from 3 with subconclusion from 2)

4. "If therefore all things are capable of not existing, [then] there was a time when nothing existed in the Universe." (P 50b)

I.e., If each thing which exists at some time does not exist, then at some time no thing existed. (Quantifier change fallacy, plus tense assumption)

Therefore, at some time no thing existed. (Modus ponens from 4 and subconclusion from 3)

5. "But if this is true, [then] there would also be nothing in existence now . . ." (P 50b) ("ex nihil nihil fit")

I.e., If at some time no thing existed, then nothing exists now.

Therefore, nothing exists now. (Modus ponens from 5 and subconclusion from 4)

6. There is something in existence now. (obvious)

I.e., something exists now.

7. So, Contradiction: there is nothing in existence but something is in existence.

8. So, there is a necessary thing. (by reductio)