My synopsis contains a consequent objective (Objective 2) that can only be studied or examined if I satisfy/prove the hypothesis I present in (say) Objective 1. How do I justify this?
I am under the impression I need to justify the contingent nature of Objective 2, conveying that it could be studied if and only if the hypothesis in Objective 1 is proved. Is a plain statement enough?
I'm assuming your question is equivalent to this scenario: "I have to submit a thesis plan to my PhD defense committee saying what I will work on in the next few years. I have one goal, Objective 1, which I will definitely work on. Objective 1 is to prove a theorem. If I prove this theorem, then I will work on Objective 2, which is a problem that assumes the statement in Objective 1 is true and provable. How do I describe this strategy in my thesis plan, given that I will only work on Objective 2 if I accomplish Objective 1, and I don't yet know if Objective 1 is possible?" (I think the meat of this answer also applies if this is a "grant proposal" instead of a "thesis plan", except "being able to articulate branch points in the plan" moves from being a strength to being something that's expected)
I think there's basically two main points.
So in your case, I think you could draft a plan like:
"First, I will work toward Objective 1, which is to prove a particular statement. Even though this statement is not yet proven, I expect it to be true because X, Y, Z. Assuming I accomplish Objective 1, I will move onto Objective 2, which is a natural next step and very important for reasons A, B, C. If we find the theorem in Objective 1 is false, and therefore cannot accomplish Objective 1, we will instead pursue Alternative Objective. I rate the risk that we fail to achieve Objective 1 to be [very low] / [low] / [somewhat likely] / [50-50] / [whatever you think and can back up with the evidence you have]."
Just to give an example, "Alternative Objective" could be something like, "attempt to develop a Modified Objective 1, and use these results to study Modified Objective 2. If this fails, I will attempt to prove that no easy modification of Objective 1 is possible, thereby ruling out a line of attack toward Objective 2."
Obviously I can't tell you in detail whether my "Alternative Objective" makes any sense in your case and what any of these sentences really should be in detail, but thinking through different "large scale" contingencies in your plan, like what happens if you can't achieve Objective 1, is a good exercise. Now, you don't want to go overboard and try to think of every single possible branching possibility that you may find in your thesis, but if you can foresee a few places where major parts of your plan may change, it is helpful to point those out and be prepared for them.
I think most committees will view it very positively that you point out this dependency of Objective 2 on Object 1 and have thought through (a) how likely it is that you can't achieve Objective 1 and (b) what is your plan if that happens. I also think, as I said before, the committee will not fault you if your thesis deviates from your plan based on discoveries you make in the process of research that you could not have foreseen at the outset.
