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Chains of logic or reasoning can have two types of problems. they can be materially incorrect (have the facts wrong) or they can be logically invalid (the conclusions do not follow from the premisses). In this case, there is a major flow in the logical structure. The reason teachers use symbols to explain this is that they want you to see the logical structure of the argument rather than the particular facts (the particular facts being irrelevant).
The first problem with this is the unexamined connection in premise 1: `If the soul is immortal, then thinking doesn't depend on brain activity.` This depends on an unstated premise that an immortal soul separated from the body thinks in the same way as a living person. But that assumption is not proven in the case you have stated.
Second, there is an incorrectly distributed middle term. Third, the conclusion should be that the soul is NOT immortal.
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