A puzzle about actions

Here is a philosophical puzzle with respect to actions:
(1) If A is an action, then there is an event, E, such that A is identical to E.
(2) S is the agent of action A only if S causes E (A = E).
(3) An event can only be caused by another event.
(4) Persons are not events.

Therefore:
(5) A person can never be the agent of an action.

Also:
(6) S is the agent of an action only if S is a person.

Therefore:
(7) Actions do not have agents.
But:
(8) Actions do have agents.

[Also, one could write down “living being” instead of “person” in (4) and (6) and still infer (7)]

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