Incompatibilism

The following is a standard form of argument, but I've been trying to get to what I think is the most easily defensible, and least open to counterexamples, principle to base it on:

1. If x is responsible for an event E and E is causally necessitated without overdetermination, then x is responsible for at least one event causally upstream of E.

(Here, read "x is responsible for E" as "x is to some degree responsible for E".) Suppose now x is responsible for an event. Then unless x has an infinite history, there must be a first event x is responsible for. Let E be that first event. It follows from (1) that either E is not causally necessitated or E is overdetermined. Say that an agent is "like us" if the agent has a finite history that does not begin at the beginning of the universe's existence. Then:

2. Responsibility in an agent like us requires either (a) that the first event the agent is responsible for is overdetermined; or (b) causal determinism is false.

I think it is also plausible that:

3. Compatibilisms do not require overdetermination.

Given this, we can conclude that compatibilism (with respect to causal determinism) about agents like us is false.

It would be nice if we could handle the overdetermination option more neatly. Here is one way to do it. Say that an event E is strongly causally necessitated provided that it is causally necessitated, and if it is overdetermined, then each of the sets of overdeterming causes causally necessitates it. For instance, E's being strongly causally necessitated rules out the hypothesis that E is overdetermined by A and B, where A causally necessitates E while B causes E without causally necessitating E. If causal determinism holds, then every non-initial event is strongly causally necessitated. Then, we can replace (1) with:

5. If x is responsible for an event E and E is strongly causally necessitated, then x is responsible for at least one event causally upstream of E.

It follows from (5) (with some work) that if an agent like us is responsible for E, then causal determinism is false.