A failed Deep Thought

I was going to post the following as Deep Thoughts XLIII, in a series of posts meant to be largely tautologous or at least trivial statements:

1. Everyone older than you was once your age.

And then I realized that this is not actually a tautology. It might not even be true.

Suppose time is discrete in an Aristotelian way, so that the intervals between successive times are not always the same. Basically, the idea is that times are aligned with the endpoints of change, and these can happen at all sorts of seemingly random times, rather than at multiples of some interval. But in that case, (1) is likely false. For it is unlikely that the random-length intervals of time in someone else’s life are so coordinated with yours that the exact length of time that you have lived equals the sum of the lengths of intervals from the beginning to some point in the life of a specific other person.

Of course, on any version of the Aristotelian theory that fits with our observations, the intervals between times are very short, and so everyone older than you was once approximately your age.

One might try to replace (1) by:

2. Everyone older than you was once younger than you are now.

But while (2) is nearly certainly true, it is still not a tautology. For if Alice has lived forever, then she’s older than you, but she was never younger than you are now! And while there probably are no individuals who are infinitely old (God is timelessly eternal), this fact is far from trivial.

Trivial universalizations

Students sometimes find trivial universalizations, like "All unicorns have three horns", confusing. I just overheard my teenage daughter explain this in a really elegant way: She said she has zero Australian friends and zero Australian friends came to her birthday party, so all her Australian friends came to her birthday party.

The principle that if there are n Fs, and n of the Fs are Gs, then all the Fs are Gs is highly intuitive. However, the principle does need to be qualified, which may confuse students: it only works for finite values of n. Still, it seems preferable to except only the infinite case rather than both zero and infinity.

Triviality

For almost three years, I’ve occasionally been thinking about a certain mathematical question about infinity and probability arising from my work in formal epistemology (more details below). I posted on mathoverflow, and got no answer. And then a couple of days ago, I saw that the answer is trivial, at least by the standards of research mathematics. :-)

It’s also not a very philosophically interesting answer. For a while, I’ve been collecting results that say that under certain conditions, there is no appropriate probability function. So I asked myself this: Is there a way of assigning a finitely additive probability function to all possible events (i.e., all subsets of the state space) defined by a countable infinity of independent fair coin tosses such that (a) facts about disjoint sets of coins are independent and (b) the probabilities are invariant under arbitrary heads-tails reversals? I suspected the answer was negative, which would have been rather philosophically interesting, suggesting a tension between the independence and symmmetry considerations in (a) and (b).

But it turns out that the answer is positive. This isn’t philosophically interesting. For the conditions (a) and (b) are too weak for the measure to match our intuitions about coin tosses. To really match these intuitions, we would also need a third condition, invariance under permutations of coins, and that we can’t have (that follows from the same method that is used to prove the Banach-Tarski paradox). It would, however, have been interesting if just (a) and (b) were too much.

Oh well.

Non-triviality of conditionals

Here's a rough start of a theory of non-triviality of conditionals.

A material conditional "if p then q" is trivially true provided that (a) the only reason that it is true is that p is false or (b) the only reason that it is true is that q is true or (c) the only reasons that it is true are that p and q are true.

A subjunctive conditional "p □→ q" is trivially true provided that (a) the only reason that it is true is that p and q are both true or (b) the only reason that it is true is that p is impossible or (c) the only reason that it is true is that q is necessary or (d) the only reasons that it is true are that p is impossible and q is necessary.

For instance, "If it is now snowing in Anchorage, then it is now snowing in the Sahara" understood as a material conditional is trivially true, because the falsity of the antecedent (I just checked!) is the only reason for the conditional to be true. The contrapositive "If it not now snowing in the Sahara, then it is not now snowing in Anchorage" is trivially true, since it is true only because of the truth of the consequent. On the other hand, "If I am going to meet the Queen for dinner tonight, I will wear a suit" is non-trivially true. It is true not just because its antecedent is false--there is another explanation.

Likewise, "Were horses reptiles, then Fermat's Last Theorem would be false" and "Were Fermat's Last Theorem false, horses would be mammals" are "Were I writing this, it would not be snowing in Anchorage" are trivially true, in virtue of impossibility of antecedent, necessity of consequent and truth of antecedent and consequent, respectively. But "Were horses reptiles, either donkeys would be reptiles or there would no mules" is non-trivally true--there is another explanation of its truth besides the impossibility of antecedent, namely that reptiles can't breed with mammals and mules are the offspring of horses and donkeys.