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The Phaedo proposes four arguments for the immortality of the soul. The third of these is the Affinity Argument, and goes something like this:

  1. There are two kinds of existences: (a) the visible world that we perceive with our senses, which is human, mortal, composite, unintelligible, and always changing, and (b) the invisible world of Forms that we can access solely with our minds, which is divine, deathless, intelligible, non-composite, and always the same (78c-79a, 80b).

  2. The soul is more like world (b), whereas the body is more like world (a) (79b-e).

  3. Therefore, supposing it has been freed of bodily influence through philosophical training, the soul is most likely to make its way to world (b) when the body dies (80d-81a). (If, however, the soul is polluted by bodily influence, it likely will stay bound to world (a) upon death (81b-82b).)

How can I evaluate this in an essay without going into the objections Simmias and Cebes pose (these objections are just other theories which contrast this one).

To me it seems as if Socrates if just issuing a probabilisitc argument, which as a result of being probabilistic proves nothing.

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The affinity argument of Socrates is to support the immortality thesis of the soul. The argument is achieved by an analogical reasoning. The world of the Form and the soul share similar characteristics. The world of the Form is divine, intelligible, non-composite, unchanging and immortal. The soul is also divine, intelligible, non-composite and unchanging. Thus, the soul must be also immortal.

To evaluate the argument, you need to examine whether the analogical reasoning holds. If you can show that the analogy fails, then you show the unsoundness of the argument. There are at least three ways to attack the argument. (Some ideas borrowed from https://plato.stanford.edu/entries/ancient-soul/)

  1. Non-exhaustiveness

The argument assumes that there are only two kinds of worlds relating to existence (your first premise): the world composed of mutable things, and the world composed of immutable things (Forms). But it is possible to think of a world that shares some properties of the world of the Form, but not all. Some philosophers of mathematicians, for example, believe that mathematical entities are intelligible and immutable, but their mode of existence is different from that of the Form (e.g., structuralism and Quine-Putnam indispensability argument for mathematical realism). That is, it is possible that there are more than two worlds relating to existence, and that the soul belongs to a third world. In this case, the fact that the soul shares some properties of the Form does not warrant the conclusion that the soul shares all the properties of the Form, specifically, the property of immortality.

  1. Psychophysical reductionism

The argument also assumes that a person is composed of the body and the soul (your premise 2). But many materialists believe that all mental states, including the soul whatever that is made of or does, are reduced to the brain states. To them, the soul dies when the body dies, just as a smartphone dies when it is drowned.

  1. Underdetermination of probabilistic reasoning

This objection utilizes your complaint that a probabilistic argument does not prove much. The affinity argument only asserts the akiness in properties between the soul and the Form. Since all Socrates showed is that the soul is more like objects in the world of the Form, it is still possible that the soul after all does not belong to the world of the Form. Thanks to the aforementioned properties, the soul can be placed higher than the body, but lower than the Form. In other words, it can be placed inbetween mortal existence and immortal existence, possessing a demigod-like existence. So, the soul might have a longer life than the body, but would die eventually. This objection is essentially Cebes' weaver argument (87a-88b). According to the weaver argument, the soul in the body is like a man (weaver) wearing a cloak (weaver:cloak::soul:body). The weaver outlasts many cloaks, but he doesn't outlast all cloaks. Similarly, even if the soul survives the deaths of some bodies, this doesn't mean that it survives the deaths of all bodies.

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