The argument from silence (also called argumentum e(x) silentio in Latin) is that the silence of a speaker or writer about X proves or suggests that the speaker or writer is ignorant of X. Here is an example of a legitimate argument from silence:

John: Do you know any Spanish?

Jack: Of course. I speak it like a native.

John: That's good, because I need to know the Spanish phrase for "Happy Birthday".

Jack: Sorry, I don't have time for that right now. Maybe tomorrow. 'Bye.

If Jack continually refuses to give John the Spanish translation without any good explanation, John can reasonably conclude that Jack does not in fact know Spanish or does not know it well. In other words, his ignorance is the most plausible explanation for his silence. Here is an example of an illegitimate argument from silence:

John: Do you know your wife's email password?

Jack: Yes, I do as a matter of fact.

John: What is it?

Jack: Hey, that's none of your business.

If Jack continues to refuse to give John the password, John cannot reasonably conclude that he does not in fact know it. In other words, his ignorance is not the most plausible explanation for his silence.

Whether reasonable or not, it would be a logical fallacy to say that you have proven the premise to be false solely on the basis of argument from silence.

Scholarly uses of the argument

The argument from silence has also famously been used by skeptics against the virgin birth of Christ. St Paul, for example, does not mention the virgin birth, and skeptics therefore argue from his silence that he did not know of it. If this argument is used as an attempted proof of Paul's ignorance, it is a logical fallacy, because ignorance is only one possible reason for Paul's silence: it's also possible that he did not think the virgin birth was important or relevant to his reasoning, or that he referred to it in texts that have now been lost or mutilated. However, the argument from silence is not fallacious if it is used to prove that Paul may have been ignorant. From the fact that Paul refers to the resurrection of Jesus, it is certain that he knew of it; from the fact that Paul does not refer to the virgin birth, it is not certain that he knew of it, therefore he may have been ignorant of it.

Legal aspects

In some legislative systems juries are explicitly instructed not to infer anything because of an accused silence, the right to silence. For example if an accused said that he was giving a lecture in Spanish at the time a crime was committed, but refused to speak any Spanish, the jury should not conclude that he couldn't speak Spanish and the alibi is false. This in effect bars the use of argument from silence.

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