In this post I shall examine the intuitions that lie behind the notion of the soul (as opposed to the notion of mind). This is a concept that has been around for millenia and in some form or another has been believed by a large number of people for much of that time. I should qualify this post by pointing out that it is not meant to be an argument against the existence of the soul (because otherwise I could be accused of knocking down straw men).
Before beginning I'd like to make it clear what my definition of soul will be: Firstly a soul cannot be understood as a computation that the brain carries out. Secondly the soul exists before conception and after death. Thirdly the soul is not physical/material.
Now for four intuitions that lie behind our conception of the soul:
1) The concepts of mind and soul are often viewed as the rational and the irrational components of our mentalities. There is an intuition that an entity dependent on the physical must necessarily behave in a rational manner. Here you might think of a ball rolling around on a hilly surface. The ball will move towards the bottom of the valley closest to it. The analogy would be that the ball is rationally trying to find the lowest point from its limited perspective. Another analogy would be the way light travels. It was noticed many centuries ago that the path a beam of light takes on its journey from point A to point B is often just the quickest path between those points. Light rays bend in water and glass because light travels at different speeds in those mediums and so the light is better off spending less time within the water/glass.
There are several potential problems with this intuition: Firstly the notion of rationality is difficult to pin down. In can be rational for instance to take away ones own ability to make future rational decisions (A protester does this when she chains herself to the railings outside the house of commons). Secondly what is rational depends on your aims (An act of self sacrifice is often described falsely as being irrational on the basis that it doesn't help you). The ball is behaving irrationally if we regard it as wanting to find the highest point. Computer programs that are designed to behave intelligently can and do often exhibit apparently irrational behaviour.
2) During the life of a loved one you develop habits of thought associated with that person. You wonder about what they might think about X or if they would like Y. When that person dies these habits of thought are still there. It is still possible to imagine in detail the persons likely response to their room being repainted or that politician being elected. The existence of these habits of thought and emotional connections after the death of a friend or a lover makes the world seem more confusing. Imagining that the person is still around somewhere else (just not here) makes it easier to deal with those parts of your mind still devoted to that person and the world seems to make more sense again.
It is clear that the existence of a mental space within your mind devoted to understanding something does not make that thing exist.
3) People often think in a mode of thought termed the "intentional stance". We reason about the way people, animals and even inanimate objects behave in terms of goals, desires, emotions, knowledge, belief and other mental states. Daniel Dennett argues we have evolved to do this as it is a very effective strategy of understanding a world largely made up of intentional beings. He also points out that in many situations that our ancestors probably had to deal with it is better to assume that an entity is intentional than not to. Imagine the situation of a lion hidden amongst bushes. A bush shakes in a strange way. Your ancestor is more likely to be the sort that would be scared and assume the shaking of the bush reflected a malevolent intent. It seems to me that the predisposition to see intent where there is none may explain part of the tendency towards the belief in souls.
4) In the early twentieth century it was proved that purely mechanistic processes are not sufficient to deduce all mathematical truth (this was called the incompleteness theorem). The idea has permeated out into Western culture (and probably Eastern culture as well). The idea shows that there are inherent limitations to algorithmic entities (I should note the theorem can be generalised to apply to various sorts of non algorithmic processes). We all like to think that in some, at least potential way, that we are not limited and that entails the existence something beyond the computational.
The main problem with this way of thinking about things (which Roger Penrose tends to favour) is that the incompleteness theorem can be generalised to every deterministic or non deterministic decision making system we can conceive of. The incompleteness theorem does not just place a limit on what computers can do it places a limit on what anything can do.
You can almost certainly tell that I am sceptical as to the existence of the soul (as defined above). I have addressed the above intuitions as I have found they are the apparent basis for belief in the soul amongst friends and acquaintances of mine. It is very easy to argue that souls are silly and so don't exist but unless you address the basic intuitions that lead people to such a belief your argument is unlikely to be convincing.