Did Socrates believe that animals possessed souls? I come to a logical contradiction when I apply his teachings to the question.

I would be curious to hear what contradiction you arrive at. It's hard to guess from what you say just which aspects of his teaching imply that animals do have souls and which aspects imply that they don't. But I can summarize some of the things he says in Plato's dialogues and suggest one path to a contradiction.

By the way, I say "in Plato's dialogues" by way of making clear that I will talk about the Socrates we encounter in those works, not about a version of Socrates we get from other authors and not about the hypothetical, hard to pin down, "Socrates who really existed." The Socrates in Plato's dialogues says a lot more of a philosophical nature about souls than any other Socrates, so we should begin with him.

Although there are many dialogues in which Socrates does not discuss the soul, at some points Plato has him argue at length that it survives a person's death. Plato's PHAEDO is devoted to this question, but we also find arguments for immortality of the soul in MENO, PHAEDRUS, and REPUBLIC. And in addition to those arguments there are stories or myths about the afterlife in the GORGIAS, PHAEDO, PHAEDRUS, and REPUBLIC.

Finally Plato's TIMAEUS describes the passage of souls into and out of bodies; you may or may not want to call this a myth, but it probably shouldn't enter the discussion given that Plato makes Timaeus talk about souls there and not Socrates. Socrates is just present in that dialogue and listening.

The arguments for immortality presuppose or assert different conceptions of the soul. In one argument, the essential quality of the soul seems to be that it knows the Forms. In another one, it is the cause of life (alternatively, the cause of motion) in an organism that has a soul. Or the soul may be that which acts virtuously when healthy, viciously otherwise. In each case the conclusion that Socrates reaches is that the soul survives the death of the body and has some type of existence apart from the body it is now in.

And depending on which conception of the soul is being assumed, animals may have souls or may not. If a soul is the cause of life or cause of motion, then horses and fish and even trees and blades of grass ought to have souls. But if the essential property of the soul is that it knows the Forms (beauty as such, justice as such), souls in non-human animals would seem to be impossible. Nothing in any Platonic dialogue suggests that a squirrel has a conception of true justice, or that a spider knows what largeness as such is.

That is what we end up with when we come to the question through arguments about the soul. Stories about the afterlife are less ambiguous. Socrates generally presents some version of a theory of reincarnation such as the reincarnation associated with Pythagorean philosophers. The Pythagoreans apparently believed that non-human animals not only had souls, but could specifically have souls that had previously belonged to human beings. You are human now but you might come back as a dog, and then (it seems) come back from being a dog to a new human life.

Plato's REPUBLIC associates your next life with a life your soul would choose in the period between lives; and that myth (the myth of Er, told in the second half of Book 10 of the REPUBLIC) has some people choosing the lives of animals: an eagle, an ape. If this story is true, then the animals we encounter have formerly human souls -- although the myth does not specify that they all do. In other words, some eagles might have the souls of former humans while other eagles just have souls that had previously belonged to eagles. It's not clear whether you can tell from looking at an eagle which kind of soul it possesses.

We come back to the question of whether these human souls in animal bodies would know philosophical generalities as souls in human bodies do.

Some passages in Plato try to draw a distinction between the types of human souls that can wind up in animals. In the PHAEDRUS we are told that souls go into humans, in their first incarnation, if they have experienced the Forms during their otherworldly existence. So there's some qualitative difference between the soul to be found in a human and the soul found in another animal. The TIMAEUS (which again is not attributed to Socrates) makes reincarnation as a non-human the punishment for not being virtuous. Like the PHAEDRUS, this account at least makes human and non-human souls different. If you are flighty as a human you come back as a bird; and so on.

You asked for one answer and I gave several. That's because there are several answers available in the dialogues. Sometimes it has to follow from an argument about the soul that all living things possess souls, sometimes that only humans do. That's one kind of contradiction. If we go by what the myths say in the dialogues, we get the closest thing to a mixed answer, namely that humans have human souls and non-humans have a different kind.

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