For one thing, a mental continuum (or continuum of consciousness), separate and permanently identifiable from other continuums.
This includes a unique isolated effect of actions - an individual "future life".Mind you, the things that cannot possibly exist in order to entail the LOGICAL impossibility of rebirth must themselves be logically impossible, such as round squares
Yes, indeed. (noting that I only say that literal rebirth is logically impossible, and not rebirth)
The logical impossibility of literal rebirth is the same as the logical impossibility of self existent things.
This logical impossibility applies to "mental continuums" or "continuums of consciousness" or "karmic streams" or "karmic seeds", or an "atma" or "self" which transmigrates from one life to the next, etc.
Such things are logically impossible.
I'm not going to go into detail here about why self-existent things are impossible. I will assume you know the logical reasons.
I believe that we have been thinking about the mental continuum differently then. It is not self existent and it is indeed the result of quite a few non-mental continuums which give rise to one mental continuum. You cannot say that we cannot isolate specific effects of actions, which are by definition unique. For example, if I tip a glass of water off the table I can isolate one unique and specific effect of that glass of water tipping over, such as the effect that water spilled out of the glass. I can isolate that and speak about it without referring to the fact that the glass also fell off the table and shattered.
First, let us agree that individual mental continuums do in fact exist, i.e. I have a mental continuum that I do not share with you (except indirectly via communication) and you have a mental continuum that you do not share with me. Second, let us concede that these mental continuums arise in dependence on causes and conditions. Third, in order for a mental continuum to, well, continue, these causes and conditions must be ongoing, obviously. Now the question arises: What continuums are the ones responsible for producing this mental continuum? Well, I'm inclined to say having atoms arranged in certain ways is probably a necessary condition but putting that aside one could conceive of mental continuums being caused by continuums not comprised of atoms, say perhaps higg's bosons or something of the sort. Certain quantum particles seemingly pop in and out of existence in space all the time according to some scientists and according to many other scientists/mathematicians we most likely do not live in a purely 4 dimensional world, which means that non-elemental (as in the periodic table) non-mental processes could conceivably converge to sustain/cause a mental continuum while at the same time not being bound to have to travel through space in some linear fashion. Now I say conceivably, because I am referring to logical possibility not physical. Anyway, put simply as I'm rather tired and my brain is beginning to slow down, it is conceivable that a multitude of causal processes are continuously converging to produce a single mental continuum of which the mental events comprising the mental continuum play a part in the conditioning of the next mental event along with other non-mental processes. Basically, a mental continuum is not self-sustaining but is the result of many other processes and thus literal rebirth is logically possible when conceived of in this light. All you need is a series of ongoing conditions to keep individuality.