This is "scientific" proof that the soul exists.
It's very complicated, I don't understand what they think they're saying.
It looks like circular reasoning and presuppositionalism well blended together.

They start with dark ages philosophy, note that this philosophy has no place for modern science
thus proving modern science supports dark ages philosophy. Or something.
The article is too long to post, go to the link for the full explanation.

Here are some examples. The full article is more complete but not any more coherent.

Nor has neuroscience helped “all but kill off” the concept of a soul. It could do so only if it showed how thought could be reduced to neuro-processes. But many have pointed out the insuperable difficulties for such a reduction. Any argument advanced to support such a feat would logically undermine itself. For the point of the reduction would be to show that one’s thoughts are fully explained by the interactions of electrochemical processes operating according to physical, not necessarily logical, laws. But if one’s thought—including the reductionist’s argument itself—rests on such non-rational causes, it is undermined, since beliefs that are determined by non-rational causes, rather than reasons, are thereby made suspect. If my thoughts are merely the result of the electrochemical processes in my brain, then they are non-rational.

The proposed reduction of thought to neurochemical processes could succeed, however, only if the actions of the neural components, operating according to physical laws, determine the reasoning processes—that is, determine which conclusions one draws in an argument. On a reductive view of mental events, the premises (or the acts of accepting the premises) have the causal powers they do only in virtue of their physical properties, and so the logical laws—the relations among contents of thought just as such—will be utterly irrelevant. Thus, if thoughts are just neuro-processes, governed by physical laws, then the laws of logic are dispensable, and the physical antecedents of a thought (such as a conclusion) determine it regardless of the contents of those antecedents. But this renders the argument by which one defends the attempted reduction unworthy of acceptance. Thus, thought cannot be adequately explained by neuroscience alone.

Thus, some properties and causal powers of organisms belong to them as wholes rather than merely resulting from the sum of the properties and causal powers of their components, and so organisms are substantial entities rather than mere aggregates. But as complex substances, each organism must have a principle of unity making its components a single whole. This principle cannot itself be a concrete component; the resulting unity would not be a single substantial entity composed of parts, but one entity acting on others—an accidental whole, a mere aggregate. Nor can the source of unity be merely a relation accruing to those components, which remain what they are but acquire ordered relations to others. What is required is a factor that unifies the materials in order to make them one being, one substance, and makes the parts be what they are because of their place within that whole. It must be a principle of organization that is logically prior to and not merely the result of the causal properties of the parts. Such a principle is precisely what the Aristotelian tradition called a “substantial form.” In a living being, such a form is a soul.

One can of course rightly affirm many things without affirming the existence of a soul, but some of these affirmations cannot be made sense of without affirming a soul. One can agree that human beings are both animals and persons without first appealing to the notion of the soul—and one could even be derisive of that concept at the same time. But one can give no intelligible account of those affirmations of our nature as personal animals without the concept of a soul—as that term has traditionally been used and understood.

Moreover, while organisms are irreducible to the laws and properties of the chemicals and particles composing them, likewise the human person (as Sir Roger rightly suggests) is irreducible to the laws and properties of organisms. Human thoughts and choices cannot be fully explained by biological laws and properties: the dimensions of logic and morality are distinct and irreducible types of reality.