Generally speaking, there are many aspects to Buddhist logic, but the following is the main point; the continuation of material things and the continuation of consciousness are something separate.
Because the continuation of consciousness is understood through the experience and awareness that there is clarity in our mind, it is not something you can actually see, or experiment with from the outside.
Because in this century people are much more materialistic, it is very difficult to demonstrate to them this fact of the mind’s continuum, for them to understand the fact of the mind continuing.
Although there could be different methods, traditionally meditation is one way to understand it. Through meditation your basic or unrefined consciousness goes deeper. It becomes more and more clear and subtle. At a certain level you can actually remember your past lives. So therefore, when you go deeper into your consciousness, at a more subtle level of consciousness, real clarity can be experienced.
It also sometimes happens that your mind can be brought into a deeper consciousness through medical science, and then maybe sometimes you can also experience certain memories of the past. So there are also other ways.