Knowledge and induction

Assume that in fact all ravens are black. Suppose you are sequentiallly observing ravens, and noting each one to be black. After observing n ravens, your evidence that the next raven is black will typically be significantly better than the evidence that all ravens are black. Now at some point, say after observing n_A ravens, your evidence that all ravens are black will rise to the level of knowledge. Thus, plausibly, at an earlier point in the sequence, call it n_N, your evidence that the next raven is black will have risen to the level of knowledge.

Suppose now you have observed n_A − 1 ravens, and you have been handed a raven in an opaque box, which you are certain you are about to open. Since n_N < n_A, at this point you have reached n_N. Hence:

1. You do not know that all ravens are black.

2. You do know that the next raven is black.

3. You know that when you observe the next raven, you will have sufficient evidence for knowledge that all ravens are black.

But note that while you know you will have sufficient evidence for knowledge that all ravens are black, you don’t know that you will know that all ravens are black. There is nothing deeply surprising about this distinction. We might well say about someone who has been subjected to misleading or Gettiered evidence that they have sufficient evidence to know something but nonetheless they don’t know, though the case at hand feels different.

One interesting thing about this case, as I read it, is that it contradicts the thesis that K = E, i.e., that knowledge is evidence. For if knowledge is evidence, and you know that the next raven is black, then you already have the evidence you will gain by observing the next raven, and hence you are already in the position to know that all ravens are black.

Another interesting thing is that it shows that you can know something and nonetheless it be rational for you to investigate it. For you know that the next raven is black, but it’s worth investigating further, since it is only upon observation that your knowledge of the next raven’s blackness turns into the kind of evidence that gives you knowledge that all ravens are black.

All this might make one think that I have misconstrued the epistemic facts, and it is false that there can be a point n_N prior to n_A at which you know that the next raven is black. Here is one way to back up my intuition that there can be such a point n_N < n_A. Suppose that we know for sure we live in a world where the color distributions of birds are always uncorrelated between the males and the females of the species, so that information about the color of members of one sex are irrelevant to the color of members of the other sex. Also assume that you know for sure that ravens have equal numbers of each sex, that you are observing ravens in an alternative female-male-female-male-… sequence, and that your priors for the color distributions of the two sexes of ravens are the same. Then if p_M and p_F are the probabilities that all male ravens are black and all female ravens are black, and p_A is the probability that all ravens are black, then at any given point in the observation sequence p_A = p_Mp_F. Let n_M and n_F be the points in the sequence where you know that all male and all female ravens are black, respectively. Then, n_M < n_A and n_F < n_A, since p_A = p_Mp_F is significantly smaller than either p_M and p_F at all points in the sequence except when we’ve observed all the ravens of one sex, and since p_M and p_F rise fairly gradually as we go through the sequence. Thus, at the point n_A − 1, we will have already reached knowledge that all the male ravens are black and the knowledge that all the female ravens are black. In particular, then, we know that the next raven is black, since if n_A is even, the n_Ath raven is male and we know all male ravens are black, and if n_A is odd, then the n_Ath raven is female, and we know that all female ravens are black.

An impartiality premise

In an argument that David Lewis’s account of possible worlds leads to inductive skepticism, I used this premise:

1. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) does not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

This is less clear to me now than it was then. Self-locating evidence might be a counterexample to this principle. I know that the tallest person in the world is the tallest person in the world. But suppose I now learn that I am the tallest person in the world. It doesn’t seem entirely implausible to think that at this point it becomes reasonable (or at least more reasonable) to infer that the number of people in the world is small. For on the hypothesis that the number of people is small, it seems more likely that I am the tallest than on the hypothesis that the number of people is large. (Compare: That I won some competition is evidence that the number of competitors was small.)

But I think I can fix my argument by using this premise:

2. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) and that a uniformly randomly chosen person (or other occupied location) is x would not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

There are multiple versions of (b) depending on how the random choice works, e.g., whether it is a random choice from among actual persons or from among possible persons (cf. self-sampling vs. self-indication).

It takes a bit of work to convince oneself that the rest of the argument still works.

Humeanism about causation and functionalism about mind

Suppose we combine a Humean account of causation on which causation is a function of the pattern of intrinsically acausal events in reality with a functionalist account of consciousness. (David Lewis, for instance, accepted both.)

Here is an interesting consequence. Whether you are now conscious depends on what will happen in the future. For if the world were to radically change 14 billion years from the Big Bang, i.e., 200 million years from now, in such a way that the regularities that held for the first 14 billion years would not be laws, then the causal connections that require these regularities to be laws would not obtain either, and hence (unless we got lucky and new regularities did the job) our brains would lack the kind of causal interconnections that are required for a functionalist theory of mind.

This dependence of whether we are now conscious on what will happen in the future is intuitively absurd.

But suppose we embrace it. Then if functionalism is the necessary truth about the nature of mind, the fact that we are now conscious necessarily implies that the future will not be such as to disturb the lawlike regularities on which our consciousness is founded. In other words, on the basis of the fact that there are now mental states, one can a priori conclude things about the arrangement of physical objects in the future.

Indeed, this opens up the way for specific reasoning of the following sort. Given what the constitution of humans brains is, and given functionalism, for these brains to exhibit mental states of the sort they do, such-and-such generalizations must be special cases of laws of nature. But for there to be such laws of nature, then the future must be such-and-such. So, we now have a room for substantive a priori predictions of the future.

This all sounds very un-Humean. Indeed, it sounds like a direct contradiction to the Humean idea that reasoning from present to future is merely probabilistic. But while it is very counterintuitive, it is not actually a contradiction to the Humean idea. For on functionalism plus Humeanism about causation, facts about present mental states are not facts about the present—they are facts about the universe as a whole!

(This was sparked by some related ideas by Harrison Jennings.)

Fundamentality and anthropocentrism

Say an object is grue if it is observed before the year 2100 and green, or it is blue but not observed before 2100. Then it is reasonable to do induction with “green” but not with “grue”: our observations of emerald color fit equally well with the hypotheses that emeralds are green and that they are grue, but it is the green hypothesis that is reasonable.

A plausible story about the relevant difference between “green” and “grue” is that “green” is significantly closer to being a “perfectly natural” or “fundamental” property than “grue” is. If we try to define “green” and “grue” in fundamental scientific vocabulary, the definition of “grue” will be about twice as long. Thus, “green” is projectible but “grue” is not, to use Goodman’s term.

But this story has an interesting problem. Say that an object is pogatively charged if it is observed before Planck time 2ⁿ and positively charged or it negatively charged but not observed before Planck time 2ⁿ. By the “Planck time”, I mean the proper time from the beginning of the universe measured in Planck times, and I stipulate that n is the smallest integer such that 2ⁿ is in our future. Now, while “pogatively charged” is further from the fundamental than “positively charged”, nonetheless “pogatively charged” seems much more fundamental than “green”. Just think how hard it is to define “green” in fundamental terms: objects are green provided that their emissive/refractive/reflective spectral profile peaks in a particular way in a particular part of the visible spectrum. Defining the “particular way” and “particular part” will be complex—it will make make reference to details tied to our visual sensitivities—and defining “emissive/refractive/reflective”, and handling the complex interplay of these, is tough.

One move would be to draw a strong anti-reductionist conclusion from this: “green” is not to be defined in terms of spectral profiles, but is about as close to fundamentality as “positively charged”.

Another move would be to say that projectibility is not about distance to fundamentality, but is legitimately anthropocentric. I think kind of anthropocentrism is only plausible on the hypothesis that the world is made for us humans.

Induction to the Causal Principle?

I’m curious whether one can infer the causal principle C that everything that comes into existence has a cause inductively on the basis of our observations of things with causes.

There are a couple of issues with such an inference. First, let’s think about the inductive evidence about causes globally. It seems to consist primarily in these two observations:

1. we have found causes for many things that come into existence, but

2. there are many things that come into existence for which we have yet to find causes.

It is worth noting that in terms of individuals, (b) vastly outnumbers (a). Consider insects. Of the myriad insects that we come into contact daily, we have found the causes of very few. Of course, we assume that the others have causes, causes that we suppose to be parent insects, but we haven’t found the parents.

For observations (a) and (b) to support C, these observations have to be more likely on C than on C’s negation. But now we have two problems. First, on the negation of C it doesn’t seem like we can make any sense of the probability that some item has or does not have a cause. Causeless events have no probabilities. Second, even if somehow assign such a probability, it is far from clear that the observations of (a) and (b) are more to be expected on C than on not C.

Second, I suspect that often when we claim to have found y to be the cause of x, our reason for belief that y is the cause of x depends on our assumption of C. Our best candidate for a cause of x is y, so we take y to be the cause. But I wonder how often this inference isn’t based on our dismissing the possibility that x just has no cause.

None of this is meant to impugn C. I certainly think C is true. But I think the reasons for believing C are metaphysical or philosophical rather than inductive observation.

Naturalness and induction

David Lewis’s notion of the naturalness of predicates may seem at first sight like just the thing to solve Goodman’s new puzzle of induction: unlike green, grue is too unnatural for induction with respect to grue to be secure.

But this fails.

Roughly speaking, an object is green provided its emissivity or reflectivity as restricted to the visible range has a sufficiently pronounced peak around 540 nm. But in reality, it’s more complicated than that. An object’s emissivity and reflectivity might well have significantly different spectral profiles (think of a red LED that is reflectively white, as can be seen by turning it off), and one needs to define some sort of “normal conditions” combination of the two features. Describing these normal conditions will be quite complex, thereby making the concept of green be quite far from natural.

Now, it is much easier to define the concepts of emissively black (eblack) and emissively white (ewhite) than of green (or black or white, for that matter) in terms of the fundamental concepts of physics. And emeralds, we think, are eblack (since they don’t emit visible light). Then, just as Goodman defined grue as being observed before a certain date and being green and or being observed after that date and being blue, we can define eblite as existing wholly before 2100 and being eblack or existing wholly after 2100 and being ewhite. And here is the crucial thing: the concept of eblite is actually way more natural, in the Lewis sense of “natural”, than the concept of green. For the definition of eblite does not require the complexities of the normal conditions combination of emissivity and reflectivity.

Thus, if what makes induction with green work better than induction with grue is that greenness is more natural than grueness, then induction with eblite (over short-lived entities like snowflakes, say) should work even better than induction with green, since ebliteness is much more natural than grueness. But we know that we shouldn’t do induction with eblite: even though all the snowflakes we have observed are eblite, we shouldn’t assume that in the next century the snowflaskes will still be eblite (i.e., that they will start to have a white glow). Or, contrapositively, if eblite is insufficiently natural for induction, green is much too unnatural for induction.

Moreover, this points to a better story. Lewisian unnaturalness measures the complexity of a property relative to the properties that are in themselves perfectly natural. But this is unsatisfactory for epistemological purposes, since the perfectly natural properties are ones that we are far from having discovered as yet. Rather, for epistemological purposes, what we want to do is measure the complexity of a property relative to the properties that are for us perfectly natural. (This, of course, is meant to recall Aristotle’s distinction between what is more understandable in itself and what is more understandable for us.) The properties that are for us perfectly natural are the directly observable ones. And now the in itself messy property of greenness beats not only grue and eblite, but even the much more in itself natural property of eblack.

This can’t be the whole story. In more scientifically developed cases, we will have an interplay of induction with respect to for us natural properties (including ones involved in reading data off lab equipment) and in themselves natural properties.

And there is the deep puzzle of why we should trust induction with respect to what is merely for us natural. My short answer is it that it is our nature to do so, and our nature sets our epistemic norms.

Humeans should be (Kenneth-)Pearceans

I have long thought that Humeanism leads to strong inductive scepticism about the future—the thesis that typical inductive generalizations about the future aren’t even more likely than not—roughly because there are a lot more induction-unfriendly worlds with our world’s history than induction-friendly ones.

But this argument assumes that there isn’t some extra-systemic explanation of why we have an induction-friendly physical reality. If there is, then the mere counting of worlds does nothing. Now, standard theism provides such an extra-systemic explanation. But standard theism is incompatible with Humeanism, because God-to-world causation is incompatible with the Humean understanding of causation.

However, it’s occurred to me today that there is a non-standard theism that could furnish the Humean with an escape: Kenneth Pearce has advocated a theism on which God explains the contingent world in a non-causal way.

I don’t know of another option for the Humean in the literature. I know of three candidates for extra-systemic explanations of physical reality:

1. there isn’t one

2. there is one, and it’s theistic

3. there is one, and it’s necessitarian (e.g., Optimalism).

The Humean can’t take the necessitarian way out, because Humeanism is strongly opposed to such necessities. The first option leads to inductive scepticism. That leaves 2. But Humeans cannot accept causal theism. So that leaves them non-causal theism.

Yet another counterexample to Nicod's Principle

Nicod’s Principle says that the claim that all Fs are Gs is confirmed by each instance.

Here’s yet another counterexample. Consider the claim:

1. All unicorns are male.

We take this claim to be true, albeit vacuously so, since there are no unicorns.

But suppose an instance of (1), namely a male unicorn, were found. We would immediately conclude that (1) is probably false. For if there is a male unicorn, likely there is a female one as well.

The problem here is that when we learn of Sam that it is a male unicorn, we also learn that there are unicorns. And as soon as we learned that there are unicorns, that undercut the reason we had for believing (1), namely that we thought (1) was vacuously true.

Provability and numerical experiments

A tempting view of mathematics is that mathematicians are discovering not facts about what is true, but about what is provable from what.

But proof is not the only way mathematicians have of getting at truth. Numerical experiment is another. For instance, while we don’t have a proof of Goldbach’s Conjecture (each even number bigger than two is the sum of two primes), it has been checked to hold for numbers up to 4 ⋅ 10¹⁸. This seems to give significant inductive evidence that Goldbach’s Conjecture is true. But it does not seem to give significant evidence that Goldbach’s Conjecture can be proved.

Here’s why. Admittedly, when we learned that that the conjecture holds for some particular number n, say 13, we also learned that the conjecture can be proved for that specific number n (e.g., 13 = 11 + 2 and 11 and 2 are prime, etc.). Inductively, then, this gives us significant evidence that for each particular number n, Goldbach’s conjecture for n is provable (to simplify notation, stipulate Goldbach’s Conjecture to hold trivially for odd n or n < 4). But one cannot move from ∀n Provable(G(n)) to Provable(∀n G(n)) (to abuse notation a little).

The issue is that the inductive evidence we have gathered strongly supports the claim that Goldbach’s Conjecture is true, but gives much less evidence for the further claim that Goldbach’s Conjecture is provable.

The argument above is a parallel to the standard argument in the philosophy of science that the success of the practice of induction is best explained by scientific realism.

Inductive evidence of the existence of non-spatial things

Think about other plausibly fundamental qualities beyond location and extension: thought, charge, mass, etc. For each one of these, there are things that have it and things that don’t have it. So we have some inductive reason to think that there are things that have location and things that don’t, things that have extension and things that don’t. Admittedly, the evidence is probably pretty weak.

A problem for some Humeans

Suppose that a lot of otherwise ordinary coins come into existence ex nihilo for no cause at all. Then whether a given coin lies heads or tails up is independent of how all the other coins lie in the sense that no information about the other coins will give you any data about how this one lies.

It is crucial here that the coins came into existence causelessly. If the coins came off an assembly line, and a large sample were all heads-up, we would have good reason to think that the causal process favored that arrangement and hence that the next coin to be examined will also be heads-up.

But now suppose that I know that Humeanism about laws is true, and there is a very, very large number of coins lying in a pile, all of which I know for sure to have come to be there causelessly ex nihilo, and there are no other coins in the universe. Suppose, further, that in fact all the coins happen to lie heads-up. Then when the number of coins is sufficiently large (say, of the order of magnitude of the number of particles in the universe), on Humean grounds it will be a law of nature that coins begin their existence in the heads-up orientation. But if the independence thesis I started the post with is true, then no matter how many coins I examined, I would not have any more reason to think that the next unexamined coin is heads than that it is tails. Thus, in particular, I would not be justified in believing in the heads-up law.

One might worry that I couldn’t know, much less know for sure, that the coins are there causelessly ex nihilo. A reasonable inference from the fact that lots of examined coins are all heads-up would seem to be that they were thus arranged by something or someone. And if I made that inference, then I could reasonably conclude that the coins are all heads-up. But my conclusion, while true and justified, would not be knowledge. I would be in a Gettier situation. My justification depends essentially on the false claim that the coins were arranged by something or someone. So even if one drops the assumption that I know that the coins are there causelessly ex nihilo, I still don’t know that the heads-up law holds. Moreover, my reason for not knowing this has nothing to do with dubious theses about the infallibility of knowledge. I don’t know that the heads-up law holds, whether fallibly or infallibly.

There is no problem for the Humean as yet. After all, there is nothing absurd about there being hypothetical situations where there is a law but we can’t know that it obtains. But for any Humean who additionally thinks that our universe came into existence causelessly, there is a real challenge to explain why the laws of our world are not like the heads-up law—laws that we cannot know from a mere sample of data.

This problem is fatal, I think, to the Humean who thinks that our universe started its existence with a large number of particles. For the properties of the particles would be like the heads-up and tails-up orientations of the coins, and we would not be in a position to know all particles fall into some small number of types (as the standard model in particle physics does). But a Humean scientist who doesn’t think the universe has a cause could also think that our universe started its existence with a fairly simple state, say a single super-particle, and this simple state caused all the multitude of particles we observe. In that case, the order-in-multiplicity that we observe would not be causeless, and the above argument would not apply.

The problem of induction in mathematics

Let's say I have some algorithm that generates the sequence of numbers, and I run ten iterations on the computer and get

• 1
• 1.5
• 1.41666666667
• 1.41421568627
• 1.41421356237
• 1.41421356237
• 1.41421356237
• 1.41421356237
• 1.41421356237
• 1.41421356237
• 1.41421356237

I will now be very confident that the sequence of numbers converges, and indeed that it converges to the square root of two. But why? Convergence is a property that the sequence has at infinity. The first 10 items in the sequence are an infinitely short proportion of infinity. Moreover, why do I assume that the sequence converges to the square root of two. Maybe it converges to the square root of two plus e⁻¹⁰⁰. Such possibilities ensure that my credence that the limiting value is the square root of two is strictly less than one. But the credence stays high.

In other words, the standard problems of induction come up not in just in science, but in mathematics. We should, thus, hope that whatever solutions we adopt to the problems as they come up in science will apply in the mathematical cases as well.

My favorite story about induction, the theistic story that God would have good reason--and hence be not unlikely to--create a well-ordered universe does not apply to mathematics, since pace Descartes, God doesn't choose the truths of mathematics. A relative of the story does apply, however. The truths of mathematics are grounded in the necessary nature of the mind of God, as Augustine held, and it is to be expected that the necessary nature of the mind of God will exhibit beauty and elegance. And where there is beauty and elegance, there is pattern and some purchase for induction.

Clumps and continuity

Our backyard had been free of black cats for as long as we've lived in this house, well over 400 days, except that over the last two nights, a black cat has visited our yard, meowing at the doors and windows. It's reasonable to think that it will visit again tonight. Yet 99.5% of evenings have been free of black cats. So how can it be inductively reasonable to think a black cat will visit tonight?

Presumably, it is because the data from the last two days is more relevant than the data from the earlier days, even though there are two orders of magnitude more black-cat-free days. But why is that data more relevant?

Granted, yesterday and the day before are more temporally similar to today than the other days. But why should temporal similarity override other kinds of similarity? No doubt there are many features (say, temperature, lunar phase, etc.) in respect of which today is more like some other day in the past 400 than like yesterday or the day before—after all, the earlier 398 days have a wide diversity of properties. But temporal similarity seems particularly important.

Maybe it is because we expect clumping, both in time and in space. Two black-cat evenings suggest the beginning of a clump.

I am curious: Is our expectation of clumping a priori justified or only a posteriori? Clumping seems to be a kind of
continuity. Is an expectation of continuity a priori justified or only a posteriori?

Induction over brute facts, and the initial state of the universe

Suppose that we've observed a dozen randomly chosen ravens and they're all black. We (cautiously) make the obvious inference that all ravens are black. But then we find out that regardless of parental color, newly conceived raven embryos have a 50% chance of being black and a 50% chance of being white, and that they have equal life expectancy in the two cases. When we find this out, we thereby also find out that it was just a fluke that our dozen ravens were all black. Thus, finding out that it's random with probability 1/2 that a given raven will be black defeats the obvious inference that all ravens are black, and even defeats the inference that the next raven we will see will be black. The probability that the next raven we observe will be black is 1/2.

Next, suppose that instead of finding out about probabilities, we find out that there is no propensity either way of a conception resulting in a black raven or its resulting in a white raven. Perhaps an alien uniformly randomly tosses a perfectly sharp dart at a target, and makes a new raven be black whenever the dart lands in a maximally nonmeasurable subset S of the target and makes the raven be white if it lands outside S. (A subset S of a probability space Ω is maximally nonmeasurable provided that every measurable subset of S has probability zero and every measurable superset of S has probability one.) This is just as much a defeater as finding out that the event was random with probability 1/2. It's still just a fluke that the dozen ravens we observed were all black. We still have a defeater for the claim that all ravens are black, or even that the next raven is black.

Finally, suppose instead that we find out that ravens come into existence with no cause, for no reason, stochastic or otherwise, and their colors are likewise brute and unexplained. This surely is just as good a defeater for inferences about the colors of ravens. It's just a fluke that all the ones we saw so far were black.

Now suppose that the initial state of the universe is a brute fact, something with no explanation, stochastic or otherwise. We have (indirect) observations of a portion of that initial state: for instance, we find the parts of the state that have evolved into the observed parts of the universe to have had very low entropy. And science appropriately makes inferences from the parts of the initial state that have been observed by us to the parts that have not been observed, and even to the parts that are not observable. Thus, it is widely accepted that the whole of the initial state had very low entropy, not just the part of it that has formed the basis of our observations. But if the initial state and all of its features are brute facts, then this bruteness is a defeater for inductive inferences from the observed to the unobserved portions of the initial state.

So some cosmological inductive inferences require that the initial state of the universe not be entirely brute.

Induction, naturalness and physicalism

Something is grue provided that it is now before the year 3000 and it is green or it's the year 3000 or later and it's blue. From:

1. All observed emeralds were grue

we should not infer that all emeralds will be grue. But from

2. All observed emeralds were green

we should infer that all emeralds will be green. A standard thought (e.g., Sider in his Book book) is that the relevant difference between (1) and (2) is that "green" carves reality more at the joints, is more natural, than "grue".

Suppose that we understand naturalness in a Lewisian way: a concept is more unnatural the longer its expression in a language whose bits refer to perfectly natural stuff. And suppose we think that among the sciences only the terms of fundamental physics refer to perfectly natural stuff. Now consider:

3. All observed electrons were nesitively charged

where an object is nesitively charged provided it's negatively charged and it's before the year 3000 or it's positively charged and it's 3000 or later. We had better not infer that all electrons will be nesitively charged. But "nesitively charged" is an order of magnitude more natural than "green". Consider this beginning of an account of "green":

4. in electromagnetic radiation of the 484-789 THz range, reflecting or transmitting primarily that in the 526-606 THz range.

And this account is not finished. To make this be in terms of the perfectly natural stuff, we'd need to specify the units (terahertz) in microphysical terms, presumably in terms of Planck times or something like that, and we'll get quite messy numbers. Moreover, we need an account of reflection and transmission. I suspect that we can more easily give an account of nesitive charge: "positive" and "negative charge" seem to already be perfectly natural or close to it; the year 3000 is a bit tricky, but we can count it (or maybe just some other "neater" date) in Planck times from the Big Bang.

If naturalness then correlates with brevity of microphysical expression, "green" is not more natural, and probably is less natural, than "nesitive charge". And so we had better not base induction on naturalness.

I think the lesson of this is that we either shouldn't think of degrees of unnaturalness as distance from the perfectly natural, or we shouldn't limit the perfectly natural (even in the concrete realm) to the microphysical. The latter gives us reason to accept some kind of antireductionism about the special sciences and ordinary language.

Simple and full induction

A followup on the previous post.

Simple induction: F₁ is G, F₂ is G, ..., F_n is G, so probably: F_n+1 is G.

Full induction: F₁ is G, F₂ is G, ..., F_n is G, so probably: F_k is G for all k.

Intuitively, simple induction seems to be always the better inference than full induction. Indeed, in cases where there are rare exceptions that didn't occur for F_k where k≤n, simple induction typically gives the right answer but full induction gives the wrong answer. Moreover, the conclusion of the full induction is logically stronger (modulo the existence of F_n+1), so it seems clear that simple induction is the better inference.

But no! Let's say that I, Jon and Trent (and a number of others!) entered a raffle held for a charity where there is only one prize. That Jon and Trent didn't win is some weak evidence that nobody won the raffle—namely, that the charity raffle was crooked. So we do have some evidence for the full inductive conclusion. But that Jon and Trent didn't win is also some evidence that I won. This is true even if we admit the possibility that nobody won, as long as we insist that it is certain that there is only one prize, and hence at most one person won. For P(Jon and Trent didn't win | I won) = 1, but P(Jon and Trent didn't win | I didn't win) < 1, and so that they didn't win supports that I won.

On Bayesian grounds, if the existence of all the F_k is in the background, that F₁ is G, F₂ is G, ..., F_n is G will never be evidence against that all the F_k are G, and in contingent regular cases will be evidence for the universal claim. But it could well be evidence against that F_n+1 is G.

Lightbulbs and induction

My colleague Trent Dougherty brought to me the very interesting question of how we inductively confirm that the sun will rise tomorrow given background knowledge that the sun one day won't rise.

This makes me think of an oddity. If I know that a lightbulb worked yesterday, that gives me reason to think it will work today. But if I know that it worked for the hundred preceding days, that gives me less reason to think it will work today, because it also gives me evidence that a burnout is due.

So given appropriate background knowledge—in this case, that lightbulbs burn out—more inductive cases do not necessarily raise the probability of the outcome, but can even lower it.

Burnout cases aren't the only ones like this. If I bought a lottery ticket, the more people I learn did not win, the more likely it is that I won.

Nothing greatly exciting here, except that we need to be careful to avoid flatfooted statements of how induction works.

Metric temporal logics and induction

Prior's metric temporal logic comes with two sentential operators, Pn and Fn, which respectively mean something like "n ago in the past" and "in n in the future", where n is a duration. On Prior's metric logic, omnitemporal universal quantification becomes a triple conjunction. Thus,

1. Always, all ravens are black

becomes:

2. (x)(Rx→Bx)&(n)Pn(x)(Rx→Bx)&(n)Fn(x)(Rx→Bx),

i.e., all ravens are black, all ravens have always been black and all ravens will always be black. But this is unsatisfactory. Suppose my data is that all ravens have always been observed to be black. This data does not significantly support the first and third conjunct of (2), but only the second. To conclude to (2) is like trying to draw inferences about jade, which is in fact a disjunction of jadeite and nephrite, on the basis of observations of jadeite alone. Consequently, if Prior's metric temporal logic captures fundament temporal structure, induction from past to future is dubious indeed.

Suppose now that time is in fact discrete. Then one could use a temporal logic with only one sentential operator @n, where n is an integer, which means "in n moments". Since n could be negative, zero or positive, this can be used to talk about the past, present and future. And (1) becomes:

3. (n)@n(x)(Rx→Bx),

an elegant and simple hypothesis that it is reasonable to take to be supported by the fact that all pastly observed ravens were black.

But logic shouldn't be tied to how things actually are. So this temporal logic is only plausible if time not only is (which is controversial enough) but must be discrete. And that would be very controversial.

Could one get this temporal logic working if time weren't discrete? One would need some way of measuring temporal distance that would allow distances to be negative (past), zero (present) and positive (future). Fixing a unit system, say seconds, makes it easy to do that. But there are three problems with such an approach. First, logic should not be dependent on a unit system. Second, temporal logic should apply in all worlds with time, while all the units of time (Planck times, seconds, etc.) that we have in our world are dependent on the particular laws of nature (just look at how a Planck time or a second is defined). Finally, a third problem is that if times ranged over, say, the real numbers, then the system wouldn't work in worlds with a non-Archimedean timeline.

One could, however, try the following move. The basic tense logic operator is @(n,u) where n is a real number and u is a duration, and we can read it as: in n us. Then (1) becomes something like:

4. (n)(u)@(n,u)(x)(Rx→Bx).

This even works in non-Archimedean timelines—u could be an infinitesimal or infinite duration. But I think there is a difficulty. All our observations of past black ravens can be explained by:

5. (u)@(−1,u)(x)(Rx→Bx),

i.e., the claim that all ravens have been black, which is just as simple as (4), but less committive. Why not go for (5) instead of (4)? Induction is still in danger. Besides, all of this metric time stuff is going to be a failure in worlds where there is no metric for time.

So if presentism is true, our best bet for an induction-friendly metric temporal logic is if time is discrete.

What if eternalism is true? Then our most plausible temporal logic is a non-metric one with an operator like #t, which says that something is true at t, where t is a possible time. Our omnitemporal quantification (1) becomes:

6. (t)(Time(t)→#t(x)(Rx→Bx)),

where Time(t) says that t is a time of the actual world. Could a presentist endorse (6)? I think so. But I also think there would be a pressure on the presentist to explain Time(t) as something like a triple disjunction: t is a past time or t is the present time or t is a future time. Certainly, that's how it will have to look for Crisp-style ersatz times. And now we have the problem of induction again, because plausibly our observations should be taken instead to justify belief that:

7. (t)(Past(t)→#t(x)(Rx→Bx)).

But perhaps the presentist could explain Time(t) tenselessly. Maybe: Time(t) if and only if #t(0=0). (This won't work on Crisp-style ersatz times. But it might work for times that are durations from a beginning--assuming there has to be a beginning.) I.e., t is a time of the actual world if and only if something (and what better candidate than a tautology?) is true at t. Without the triple disjunction in the definition of an actual time, we now have some hope.

All that said, I doubt that a non-metric #t operator with a tenseless and changeless definition of Time(t) will be attractive to the presentist. For I think the typical presentist wants truths about what happens at different times to be grounded in tensed truths, something like in Prior's metric temporal logic.

Note: The metric logic renderings also need something that says that a given n isn't out-of-range (before a beginning of time or after an end of time).

Induction, growing block theory and open future

According to induction, the future is like the past and present. But the past and present contain real events. Hence, probably, so does the future. Hence, probably, Growing Block theory, on which only the past and present contain real events, is false. Likewise, excluded middle is true for claims about the past and for claims about the present. So, probably, it's true for claims about the future. So, probably, Open Future views are false.

A principle about induction and explanation

Here's an intuition I have. Suppose that I somehow knew that a dozen of boxes have appeared ex nihilo for no cause (not even a stochastic one) in my office. I open half of them and each one was purple inside. Do I have good reason to think that the others are also purple inside? As long as I hold on to my knowledge that there is no explanation of the boxes' presence and character, I think not. It is rather like when I get heads six times in a row when tossing a fair coin—as long as I get to hold on to my knowledge that the coin is fair, I have no reason to think subsequent tosses will be heads.

This suggests to me that induction requires that the cases we do induction over be non-brute, that they have explanations. But not just any explanations will do. The cases need to have a common type of explanation. If one box was materialized in my office by aliens, and another was delivered by my best friend, and another coalesced from the drippings in a leaky ceiling, and so on, then I don't get to do induction across the cases.

Thus:

1. That all observed Fs are Gs gives me knowledge by induction that all Fs are Gs only if there is a common type of explanation as to why each F is a G.

Normally, when all observed Fs are Gs, that gives us reason to think that there is a common explanation, say that Fness is a natural kind that includes Gness. But when there is no such common explanation, then even if all Fs are Gs, we don't know it—we have Gettiered knowledge.