General, non-contextual, direct (one-step) Connections -

Considering the nature of and the role played by connections, connections are at the heart and soul of any intellectual or mental activity. So we need to define connections.

As a real, cognitive phenomenon, what is a connection? It relates to immediacy of thought-occurrence, in the mind, of something, from something. If this is present, there is a connection. Also, we are talking about direct one-step connections (hence the immediacy). ('John - surgeon' is a two-step connection since John's mother is a surgeon. John is connected to his mother (one-step) and his mother to surgeon (one-step)).   

So we need to study the game of - if one thing is said, what other thing immediately comes to your mind? (Like the rapid-fire rounds in TV shows).

So we need to know about storage of knowledge in the mind, where these immediacies stem form. Lets look into this. 

We think we “know” a lot. So I would like to segregate all that we apparently “know” into 2 parts. Before coming to those 2 parts, lets us first see an example.

Suppose someone is teaching you about and to drive a car. The first thing he tells you is - A car cannot run without petrol. If there is no petrol, the car cannot move ahead. Later, on going to a petrol -pump, he tells you - here you fill petrol in the car. So I learn 2 things about cars - ‘a car needs petrol to run’ AND ‘you fill petrol in the car i.e. there is something like “filling petrol” into the car’. These are 2 things you have learned, and now you know them.

Suppose on some occasion later you happen to tell someone else, say a friend of yours - Hey, if you don't fill petrol, how will the car run? Now, it appears from this sentence that you also “know” - if you don't fill petrol into a car, the car cant run. But no one told you this, as it is; you have combined the 2 pieces of knowledge learned, mentioned above. This is actually thinking - (but) you have COMMONSENSICALLY, and hence, swiftly and easily, combined the “need of petrol for running” with the “filling (the concept of filling)” from the 2 pieces respectively to create this new statement which appears to be something you directly know. 

So, coming to the 2 kinds of knowledge,  the first kind of knowledge is what is impressed upon memory, as it is, in a certain given format, while learning about something (like the 2 pieces of knowledge learned, mentioned before). Hence, it is that which was perceived by us, as it is, while learning about something. And the second kind of knowledge is what happens to be easily and swiftly combined (since it is commonsensically so), from those of the 1st kind, via “thinking”. The latter is so easy and rapid that it creates the impression that we know it (and for all practical purposes, can also be said to be something we know.)

Lets call the first type as type-A knowledge-piece and the second type as type-B knowledge-piece.

Thus, the immediacy of thought, that we talked about in the very beginning, is the commonsensical combination of the literal memory-impressed data while learning.

So, coming back to the attempt of defining connections, 2 entities are connected if there exists a type-B knowledge-piece combining the 2 entities, stemming from the type-A knowledge-pieces about the 2 entities, (or, obviously, if the type-A knowledge pieces stemming from them directly coincide). 

(Otherwise, anything can be connected to anything. The above explains why a rat is not connected to the International space station (ISS), though some distant far-fetched connection can be drawn between the 2, just for the technical heck of it.)