All people are virgins (proof by induction)

Question

I recently got inspired by the famous proof- All Horses are the Same Color©. I soon learned that the proof has flaws- namely, the Induction hypothesis fails at n=2. So, after much thought, I came up with my own foolproof version.

Theorem: All people are virgins

Proof:

1. Base case (n=1) Suppose only a single person (P1) exists. We know only one person cannot have sex with himself/herself, therefore P1 is a virgin.

2. n=2. Now suppose we add another person, P2. Now, P2 may or may not be a virgin. However, if P2 is not a virgin, the only person he/she could have had sex with is P1. Hence P1 is also not a virgin. We know this is false (see step 1). Therefore, P2 is also a virgin.

3. n=3 Another person, P3 is added. Same as above, if P3 is not virgin, atleast one of P1 or P2 must not be virgin. We know this is false (step 2). Therefore P3 is virgin. ...

Induction hypothesis: Suppose a group of n people are virgins.

Inductive step:
We want to show that n+1 people are all virgins Again, if we take a group of n people and add one person, the only way the (n+1)th person would be a virgin is if one of the n people already in the group is not a virgin. We know from this is false since the IH states all n people are virgins. Therefore, the (n+1)th person must be a virgin.

Therefore, all people are virgins. Q.E.D.

Can anyone find the error in my proof?

Answer 1

I’d say: The proof actually has no flaw, however it doesn’t quite prove the statement it seems to prove.

What it seems to prove is the following statement:

In every isolated community of $n$ humans, everyone is a virgin.

You need to restrict to isolated communities, lest your base clause fails. If a “community” of a single human isn’t isolated, that single person doesn’t need to be a virgin.

However what you really have proven is something like:

In every freshly created community of $n$ live-long eremites, everyone is a virgin.

And that’s certainly true. You have proven only this because in your induction step you assume that

• every added human could not have had sex with other people from outside the community, but only with people from within the already existing community, to which they are just introduced, so you basically assume they’re all eremites, and
• between adding new humans to the community, there hasn’t been enough time for the people within the already existing community to have had sex with each other. So the community is freshly created.

This is the general flaw with natural language statements about everyday world phenomena: They are ambiguous in interpretation.

Answer 2

The logic in the induction step doesn't connect, in the way that the "all horses are the same colour" proof does for all but one case.

If you have a collection of $n$ horses, remove one, you get a collection of $n - 1$ horses. By the induction hypothesis, these are all the same colour. Now, removing another horse will also obtain a collection of same-coloured horses, by assumption. If $n > 2$, then these two collections of $n - 1$ horses must have some non-empty intersection, so all the horses are the same colour. If $n = 2$, then we just get two individual horses, which form singleton collections each trivially all the same colour, but the induction hypothesis does not imply that the two horses are the same colour.

An important point in this false proof is that, when removing or adding horses into a collection, the colour of the horses doesn't change. This is fundamental to the logic of the horses argument: removing a single horse makes the remaining horses all the same colour, so they are and always were the same colour with the possible exception of the one you just removed.

In your virgin argument, adding or taking away people can very well change the possibility of losing their virginity. If you take two genuine virgins, leave them alone in a room, there is a non-zero chance that they will no longer be virgins afterwards. If you take two differently coloured horses, and leave them alone in a barn, they will still be the same colour.