First Parmenidean Argument

1. There are no non-existent objects.
2. If (1) is true, then there are no non-existent objects that have properties.
3. Therefore, there are no non-existent objects that have properties.
4. If (3) is true, then whatever has any property exists.
5. Therefore, whatever has any property exists.
6. If (5) is true, then if A has the property of being spoken or thought of, then A exists.
7. Therefore, if A has the property of being spoken or thought of, then A exists.
8. If (7) is true, then if A can be spoken or thought of, then A exists.
9. Therefore, if A can be spoken or thought of, then A exists.

Here is the proper symbolization of (1-9):
1. A
2. A --> B
3. Therefore, B
4. B --> C
5. Therefore, C
6. C --> (D --> E)
7. Therefore, D --> E
8. (D --> E) --> (F --> E)
9. Therefore, F --> E

Here are explanations for each of the premises:

Explanation of (1): First a word about existence and identity. If A exists, then something is identical with A. In addition, if x = y, then x is F if, and only if, y is F. Finally, 'there are Fs' means that there exist Fs. Now for the reasoning in favor of (1). Suppose (1) is false. Then there exist objects that are non-existent. But that is impossible, since it is contradictory for there to exist objects that are non-existent. So (1) is true.

Explanation of (2): A property is any feature, attribute, or way that something is. Now note that (2) is a conditional ("if-then") premise. So, as with explaining all conditional premises, we must assume the "if"-part and give a reason for accepting the "then"-part. So assume that (1) is true. Then there exist no objects that are non-existent. But if that's so, then there exist no objects that are non-existent that have properties. (Compare: if there are no dogs, then it follows that there are no dogs that have fleas. If I have no dinner, then I have no steak dinner. Etc.) So (2) is true.

Explanation of (3): (3) follows from (1) and (2) by Modus Ponens.

Explanation of (4): (4) is another conditional premise. So assume that (3) is true. Suppose now that A has properties. Then either A exists or A does not exist. If A does not exist, then there is a non-existent object that has properties. But this contradicts our assumption, (3). So A exists. But there is nothing special about A. The same reasoning applies to any other thing. So (4) is true.

Explanation of (5): (5) follows from (4) and (3) by Modus Ponens.

Explanation of (6): Assume (5) is true. Assume A has the property of being spoken or thought of. Then A must exist; otherwise, we contradict our assumptions. So (6) is true.

Explanation of (7): (7) follows from (5) and (6) by Modus Ponens.

Explanation of (8): Assume (7) is true. Assume A can be spoken or thought of. Then A has the following property: being something that can be spoken or thought of. But then, by (7), it follows that A exists. So (8) is true.

Explanation of (9): (9) follows from (7) and (8) by Modus Ponens.