It's not as much a logical problem as it is psychological.
(What follows are not my own thoughts, but what I came to understand was the Buddha's way of thinking as he came up with his Dharma. My specific presentation may stumble and limp but I believe Buddha's overall logic is correctly captured below.)
Imagine you found some perfect state or a way of being, that is fully 100% satisfactory. If this state was potentially finite or impermanent, or somehow dependent on a condition, this very quality would be its limit and flaw, so a finite or conditionally perfect state cannot be considered truly perfect.
And what is a "perfect" state? It must be a state, when, by definition, whatever is presently being experienced leaves no room for further craving for an even better state. Indeed, if there was some other state, "better" than the first state, could the first state be considered truly perfect? No, because then there is still room for craving.
So a perfect state must be 1) permanent/unconditional and 2) completely satisfactory without any craving for anything better. Such theoretical state is known under the name of "Nirvana".
Given the conditional nature of all aggregate phenomena and therefore the impermanence of any state, the only "state" that can possibly be permanently and unconditionally without craving is the (dynamic!) state of seeing things as they are, in their moment-by-moment transience, emptiness, and conditionality. This state of experiencing things as they are, without craving, is known as "Non-Abiding Nirvana" and the object of this experience is known as tathata or "suchness".
If we are looking for Ultimate Satisfaction we must understand what Satisfaction is and what Unsatisfaction is, psychologically, and then from this we can logically infer the only way in which the Ultimate Satisfaction can possibly exist.
So it's not about binary nor catuskoti logic nor the question of having or not having beginning and ending, but rather about the nature of experience. It is about fully understanding why any conditional state cannot be satisfactory. A conditional state cannot be satisfactory because any condition defines a limited area within which things are right, and outside of that area things are still wrong. A condition creates a potential for conflict, for disharmony, between "is" and "should". This very conflict between "is" and "should" is the exact nature of Dukkha or Unsatisfactorness. Conflict is the nature of im-perfection and the enemy of the Perfect state we're seeking.
This is why the Fourth Noble Truth defines a path that leads to progressive reduction and gradual elimination of conflict as the source of suffering, ending up in removing even the subtlest conflict - the conflict of deep subconscious attachment to conditionality, craving for "better". This last phase of removing attachment to conditionality is reflected in the progression of jhanas.
So at the end we're talking about complete transcendence of craving that comes from qualitative judgement of experience. This, of course, is the cherry on top of the giant pyramid of buddhist practice that leads to elimination of objective causes of conflict. Taken together, the "external" practice of removing the sources of conflict, plus the "internal" practice of removing all craving and any attachment to conditionality, make up the complete path to attainment of the perfect harmony of unconditional suchness.
Having no beginning and no end is a purely theoretical construct that has hardly anything to do with this ethical and psychological endeavor.