Stories can continue

 The lowest level assumption of transitivity is that 

A story can continue from where it left off. 

To illustrate the meaning of this, consider the statement: 

If you can get from A to B and from B to C, then you can get from A to C. 

This is functionally false if you forget everything that just happened when you get to B. Who remembers A at that point? Or, by contrast, if you do remember getting to B from A, then you can remember that as you get to C and remember the whole sequence. It must play as a story that is being continued.

Another example is: if a<b and b<c then a<c. Mathematically we say b-a is positive and c-b is positive, so c-a is positive. But that is because c-a = (c-b) + (b-a) and the sum of two positives is a positive. Adding two numbers together is a narrative that is continued from a, to a+, to a+b.

In these cases, there is a preserving of information through the intermediate step, embodied in the idea of a story continuing. Am I overcomplicating this? The idea is that information from an earlier part of the story is still available if the story is continued. In my linguistics, this is built into the persistence of a ledger.