At 39a–42a in Plato's Apology, which fallacy does Socrates use in his premises to argue that death is not to be feared?

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Socrates's first logical fallacy is a form of post hoc ergo propter hoc—that is, after this, because of this—and is centered on Socrates's argument that his personal god or spirit, the daimonion, who normally opposes him when he does something likely to harm himself, has remained silent from...

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Socrates's first logical fallacy is a form of post hoc ergo propter hoc—that is, after this, because of this—and is centered on Socrates's argument that his personal god or spirit, the daimonion, who normally opposes him when he does something likely to harm himself, has remained silent from the time Socrates left for the trial to its conclusion:

Probably what has occurred to me has turned out to be good. ... For there is no way that the accustomed sign [from his personal spirit] would not have opposed me unless I were about to do something good.

Socrates assumes that there exists a direct cause and effect relationship between the spirit's silence and the rightness of his actions during the trial. In other words, because the spirit fails to speak to Socrates, Socrates takes no other actions or makes further arguments to save himself, his conclusion being that death cannot be bad if he was not warned to avoid it.

A second logical fallacy occurs when Socrates argues that being dead results in one of two things:

For either it is like being nothing and the dead man has no perception of anything, or else, in accordance with the things that are said, it happens to be a sort of change and migration of the soul from the place here to another place.

This fallacy is called begging the question (petitio principii), which does not mean "prompting the question" as used commonly by people who don't know its proper meaning. Begging the question means that one assumes a conclusion that is not proven. Socrates makes the argument that death is not bad because it is either nothingness, a never-ending sleep with no dreams, or one's soul migrates to Hades, the Greeks' underworld, where, after having been released by honest judges in the underworld, a good man would be free to

... associate with Orpheus, Musaeus and Hesiod and Homer, how much would any of you give? For I am willing to die many times if these things are true. ...

Here, Socrates assumes that an afterlife based on "things that are said" will be his destiny—certainly, he believes in this afterlife, but believing is not the same as knowing—and so his conclusion, based on assumption, may or may not reflect reality. Socrates's logic here is similar to someone arguing that the Bible must be true because it is the word of God—basing an argument on another argument that is a belief rather than a fact.

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In Plato's Apology, Socrates makes numerous statements containing logical fallacies about why one should not fear death.

For this fear of death is indeed the pretense of wisdom, and not real wisdom, being the appearance of knowing the unknown; since no one knows whether death, which they in their fear apprehend to be the greatest evil, may not be the greatest good.

Here, Socrates's argument contains Appeal to Closure. This fallacy relies on the listener's refusal to have an argument remain open. Therefore, the listener readily accepts the argument for one reason only: closure. Socrates states that because we do not possess "real" wisdom about death, we cannot make an announcement about death being good or evil (and thus, we cannot fear it).

Whether I am or am not afraid of death is another question.

Here, Socrates's argument contains the Red Herring logical fallacy. This is a deliberate diversion from the topic at hand. In this example, Socrates diverts the audience's attention away from the topic (not fearing death) to the topic of whether or not he, himself, fears death.

Either death is a state of nothingness and utter unconsciousness, or, as men say, there is a change and migration of the soul from this world to another.

This example is one of a False Dilemma or Either/Or fallacy. In a False Dilemma, only two options are given as possible. Here, Socrates states that death is only a "state of nothingness" OR a "migration of the soul." Socrates fails to acknowledge other religious and personal perspectives on death and what comes after one's death. This argument could also be fallacious based upon the logical fallacy called Ad Populum. This fallacy, also called bandwagoning, exists when one concludes that an argument must be true because the greater whole believes in it.

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At this point in Plato's Apology, Socrates has been convicted of the charges against him and finds himself subject to a sentence of death. Socrates observes that most people regard death as a bad thing and something to be feared. Socrates, however, reasons that death is not a bad thing because his personal daimon always warns him if he is about to make some sort of mistake or error. Accordingly, Socrates reasons that because his personal spirit did not oppose him in anything he had done or said on the day of the trial, that what happened to him on that day must be a good thing. The silence of the daimon leads Socrates to "regard this as a proof that what has happened to me is a good, and that those of us who think that death is an evil are in error" (Benjamin Jowett translation).

So, in this situation, Socrates reasons as follows: My daimon warns me when bad things happen. My daimon did not warn me, therefore getting sentenced to death is not a bad thing.

There are a number of possible fallacies that this might be classified as. It could fall under the heading of an "oversimplified cause." Just because Socrates' daimon didn't warn him does not mean that death is not a bad thing or to be feared.

In the Cum hoc fallacy, a person incorrectly assumes that because two events coincide they must be related. Again, just because Socrates daimon does not oppose him on the day that he gets sentenced to death does not mean that death is neither bad nor to be feared.

I suppose some might also categorize this as a "hasty generalization" since Socrates may be relying on an unrepresentative example to arrive at his conclusion.

In short, there are a number of logical fallacies that Socrates' reasoning might be classified as in this situation.

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