It will come down to some kind of combinatorics fundamentally. Because all probabilities are frequencies. And all frequencies are combinatoric (even if only in ratios rather than absolute counts).

In terms of pedagogy, it’s certainly easier to just do probability arithmetic (adding a supposition not known to be more or less likely than not halves any probability because 0.5 x 0.5 = 0.25, and so on). But when someone goes one level down on you and asks “why” it is that that happens mathematically, you can’t sate them until you get to the point that that is because there are half as many ways to get that supposed result than without supposing it. So it’s always going to be some kind of combinatoric answer at the bottom of all epistemology.

Whether you discourse on that level (or at some higher level of abstraction) is a free parameter.

Another way of looking at it, and in general why I’m not sure combinatorics or listing possibilities works or is even fundamental, is that it is mainly useful for when we’re indifferent between possibilities. This maps to having all of those possibilities in a hat, and then we’re picking a random possibility out of it. But this of course is rarely the case, if ever, apart from certain designed contexts like games of chance. In order for all possibilities to be in a hat, they must already exist, and listing possibilities doesn’t really take into account the generative complexity of each hypothesis, which is what I feel matters more.

I realize my communication of these ideas might not be the best, so if you have any better words to summarize these ideas, or if they already exist, that would be great.

That’s a good question.

And yes, it comes down to “we have no reason to pick a possibility over another.” The fallacy in critiques of induction is to circularly assume any possibility is therefore probable, the very thing they are trying to claim induction does, but induction is actually adducing evidence for its probabilities, while the critics are literally just making their probabilities up. This is most obvious in the case of Cartesian Demons. Which are a good example here:

Yes, there are “infinitely many” conceivable Cartesian Demons. But that does not increase the probability that one exists. Possibility does not multiply probability. It divides it. When we say there are more ways to have what we observe without a CD than with, we are not simply counting the logical possibilities but the ratios. So, for example, a good CD is extremely complex and thus requires a huge number of assumptions (even assumptions against evidence require yet more assumptions to dismiss that evidence, and so on), and with no epistemic evidence for any of those assumptions being true, they are at best each 50/50 (“so far as we know,” hence this is always about warrant, not factual reality independent of that: we are, after all, supposed to be talking about epistemology here, not confusing it with ontology), and so by deductive necessity they split the probability space.

So, one assumption and your theory is p = 0.5; two, 0.25; three, 0.125; etc. By geometric progression, specified complex theories logically necessarily occupy almost zero probability space—within the field of warranted belief. That belief (“justified true knowledge”) can increase that probability for one theory over another only with evidence (there is no other logically possible way to do it) and that happens as well by the same kind of logical necessity (evidence is measured by P(e|h) and P(e|~h) so P(e) is splitting the probability space between h and ~h and again with no reason to favor either, it’s 50/50 which has no effect). So when there is no evidence, an assumption like “everything will change tomorrow” has an extremely low p by logic alone.

By contrast, observations can be explained in many more ways without a CD. So, for example, in the simplest sense, consider “all explanations that omit a CD,” those occupy nearly all the p-space (by law of converse probability). If you narrow it, like say, “natural laws of physics plus existential inertia,” that is vastly simpler than a CD and thus there are many more ways it can split the p-space than a CD can, and so the sum of them all (by disjunction) occupies far more p-space.

This is really just the exact same math by which we derive the laws of thermodynamics from statistical mechanics: there may be “infinitely many” ways to arrange a gas in a tank, but there are still relatively fewer ways to arrange it that are highly ordered. So the random distribution of a gas is complex, but it is unspecified, and thus is actually highly probable by raw permutation. That isn’t saying the specific arrangement is more probable (every one is equally probable), but that the pattern that state belongs to belongs to far more arrangements than can be claimed by more ordered patterns (like CDs and other “maybe it will all suddenly change” explanations, which always require more specificity and therefore always count fewer, relative to low-specificity explanations).

This relates to the Configuration of the Stars fallacy: the denier of the logical validity of inductive inference is committing some form of that fallacy.

I agree with your closing assessment (and I get the impression it better represents Armstrong than the SEP) that “the lower probability event occurring simply brings in more mystery (and thus seems to intrinsically need more explanation) than the higher probability event.”

I don’t know that I would choose the words “need for explanation” though (as if we were declaring a principle of sufficient reason). I don’t think the need is relevant. Even if we did not care what explained something, it would still be the case that one explanation is more likely than another given the information we have at that time. And that does come down to relative permutation counts (all the ways there are to have a highly specific explanation, given all the indifference-assumptions you need to build it, in ratio to all the ways there are to have a far less specific explanation, i.e. a simpler theory in regards components, which is not just a physical difference in probability, but a logical one in the absence of allowing “evidence” to change anything, which is the anti-inductionists POV, and thus their POV is self-refuting on those grounds). If one is concerned about the transfinite arithmetic stymying ratios, this can all be done with random sampling again (which operates on infinite sets the same as finite), only this time on possibility space (the way Archimedes did with his Mechanical Theorems).

At the “basement” maybe something exists that literally has no explanation. But that itself would just be another proposed explanation. I say more about that in my articles on cosmology (and my discussion of the Argument from Uniformities).

And at the basement of motivation, I do agree one does need to want to care to ever appreciate any of this, but that’s a different question from, once you do want that, what would work (see my Epistemological End Game for a really old discussion of that, which actually touches on your question, too).

Thank you for your answer.

I’m a bit confused as to how you’re using combinatorics here. You said that the number of ways in which a pattern can suddenly change is fewer than the number of ways a pattern can continue.

But how is “number of ways” defined here? It seems that one can logically at least imagine an infinite number of ways for the pattern to change than for the pattern to continue.

Secondly, why would the number of ways here even matter? The principle of indifference seems to only make sense if we have no reason to pick a possibility over another. But in this case, we do have a reason. Namely, that if the pattern didn’t continue, it just adds more mystery: why is the pattern breaking? Whereas the pattern continuing simply does not seem to require as much explanation.

This is a bit like the concept of existential inertia for objects that Graham Oppy (and I believe yourself?) discussed against some of the arguments Edward Feser had made but for the physical dispositions of patterns instead of objects. If something exists, it doesn’t seem to be the case that we need further explanation for its existence if no forces change or destroy the object (even if the number of ways one can imagine this happening is infinite).

So I guess what I’m trying to wonder is whether all of this just reduces to “need for explanation”. Suppose one of two events occurred: a coin landed on heads or a dice rolled on 6. You are then asked what is more rational to believe in? You will clearly pick the coin. If someone asks, why, you will say “because one has a probability of 1/2 and the other has a probability of 1/6.” I am interested in the further question of an ardent questioner who then says, “so? Why does that matter? All you’re saying is that one number is smaller than another based upon your mentally constructed sample spaces.” Now, I can’t just recourse my answer back to probability, since that would be circular. It seems that the true reason, and maybe I’m wrong here, is that the lower probability event occurring simply brings in more mystery (and thus seems to intrinsically need more explanation) than the higher probability event.

I don’t know. I have not read its proponents in detail (like Armstrong or Foster—though I did read some Armstrong a long time ago, and that might have influenced me), but that section of SEP does not present it as similar to mine (so, either it is straw manning them, or my answer would have to be no).

I could only say yes if we changed the presentation in SEP (and that we have to do that is why I don’t like labels like this, because they commit the Baggage Fallacy, assuming I adopt all the baggage the SEP claims for the position, which is false). Contrary to the SEP we would have to say:

(1) The IBE is just a colloquialization of Bayes’ Theorem (Proving History, index).

(2) Bayesian hypothesis testing is an extension of the principles of frequentist statistics, whereby random samples have deductively fixed probabilities of matching the unobserved portion of the data, and the frequency of being right approaches the objective frequency of the phenomenon as information increases (Proving History, end of ch. 6).

(3) Such that the probability of a pattern continuing goes up with time because the number of ways it could exist and suddenly change are fewer than the number of ways it can continue (as the product of some causal system, i.e. the “nomoi” in “nomological”), subject to knowable exceptions (when we have intervening propositions, also well-tested, that nomologically entail secession, e.g. the sun will die out eventually, everyone eventually stops brushing their teeth every day because they die, etc.).

(4) All of this is only getting us an epistemic probability. With increased data the epistemic probability does approach the objective probability (with perfect knowledge they are identical).

That last point the SEP critic seems to be missing, and I think you are on to that point when you say this is all just “what we are warranted in believing given what we have” and not a statement of certainty (the definition of knowledge, “justified true belief,” entails knowledge only exists as to the probability a proposition is true, not that a proposition is true: PH, Ch. 6 and my article The Gettier Problem). The critic is mistaking “what is the case” with “what we are warranted in believing is the case” and thus appears to not even be understanding how epistemic probability reasoning even works.

I am not sure how to parse your worm/human brain analogy, since it is not clear what proposition we are supposed to be testing there. Maybe if you specified that I could give a more useful answer. But for an analogy I worked up using the classic “grue” thought experiment in the philosophy of induction, see Hypothesis: Only Those Who Don’t Really Understand Bayesianism Are Against It.

The bottom line is that a large or complete past sample is logically equivalent to a random sample, and a random sample has deductively certain implications for the probabilities in the unobserved sample space (e.g. future outcomes). In a straightforward random sample (colored beads in a jar—as there is no logical difference between extensions in space and extensions in time), if you “randomly” select 100 beads, you can mathematically work out to a certainty the probability that the remaining beads will match the frequency of colors drawn in the sample (to a confidence level and interval, which are adjustable: i.e. for every confidence level you want, there is a corresponding confidence interval). This is all deductively certain so no inductionist complaints can be raised against it.

The usual armchair objection is to say “maybe” the sample isn’t random, but that places the burden on the claimant: if something was making it nonrandom we should have observed that (e.g. as we can for the sun given our knowledge now of nuclear physics, so our epistemic probability the sun will come up every day has gone down, but we were still justified a thousand years ago in believing it would obey Laplace’s Rule of Succession—just because we were wrong does not relate to what we were justified in believing at the time, a distinction I am getting the sense that the SEP author doesn’t understand).

In terms of “making things up” (the fallacy of possibiliter ergo probabiliter: see index to PH) to change this: you cannot logically declare the sample nonrandom. Without evidence that it is not, it is at best 50/50, and typically much less likely (as with grue; your points about Armstrong etc. hold here). The overall point is that it cannot be the case that evidence never increases the probability of anything, because we have no reason to believe that—to the contrary, we know for an effective certainty that had we ever started believing that, we’d have been dead within a day. So it is clearly not a justifiable belief. So it cannot be interjected as a premise by any logically valid mechanism. Obviously evidence increases probabilities. The reason it does is random sampling mathematics. Which is deductive. And therefore cannot be toppled by an induction objection.

It’s similar to saying “well, if our universe is in a false vacuum, it could instantly collapse in one minute and everything would disintegrate, so no patterns would continue.” And that’s true (it’s been worked out in physics). It’s just that with no evidence warranting belief that “we are in a false vacuum” or even “we are and it will collapse in a minute and not a billion years from now” you cannot be justified in believing it, and therefore it cannot be raised as an objection.

Everything after that is a straightforward fallacy of logic. For example, “well, it’s been hanging on for 14 billion years, so surely odds are high it will collapse in a minute” is a gambler’s fallacy (it is deductively false: the probability it will collapse next minute is the same as the probability it would have collapsed any prior minute, and after 14 billion years, it is deductively certain that that probability is not likely to be even remotely high, just like if we did this with billions of beads in a jar and were talking about drawing the single black bead).

The only difference seems to be that instead of saying that, for example, a law continuing to be true is more probable than breaking for example, philosophers like Armstrong and Foster argue that a law that wouldn’t be uniform would be intrinsically more mysterious and require more explanation.

I wonder thus if probability is ultimately just an estimate for “need for explanation”, and something deeper than frequencies, for even if probability approximates to frequencies, we can ask why we should think that the past frequencies should be projected into the future. And even in cases where the idea of a frequency may not apply (imagine comparing the complexity of a human brain vs a worm brain but while knowing nothing else about the world), there seems to be something apriori about the human brain being more complex, and that would be the idea that more complex things just have more stuff posited in reality which is inherently more mysterious and thus needs more explanation.

I suppose some of these points are just semantic but I wonder what you think about your approach connecting with Armstrong’s.

That it is impossible to be wrong about the three basic laws of thought (particularly the law of non-contradiction, which really grounds and entails the other two) I detail in Sense and Goodness without God (check “contradiction, nature of” in the index).

That it is impossible to be wrong about immediate experience existing is that it is self-referential and thus requires no additional information (so there is no other condition allowing it to be false).

For something to be “wrong” there has to be a possible state of being in which the claimed fact doesn’t exist. But there is no possible state of being in which “I am experiencing this right now” and at the same time “I am not experiencing this right now.” The latter refers to the absence of the former—so can never be true in its presence.

For more on why small closed loops like that are undeniables, see my article Epistemological End Game.

Sorry for the typo, I did mean uninterpreted.

You used the law of non-contradiction in your reasoning, isn’t that something which we could be getting wrong. More generally, since we could be wrong about anything else, why is raw uninterpreted present experience an exception.

I am not sure what you are referring to.

When I speak of “raw [UN-]interpreted present experience” I mean Cartesian facts (that you are experiencing x now), not inferences (that x indicates something objectively happening outside your mind, or even inside your mind apart from the mere experience itself).

It is logically impossible to experience x and not experience x at the same time (a fact and its negation cannot be simultaneously true: the Law of Non-Contradiction), so Cartesian knowledge is always undeniable. So you can be “100% sure” of it. It’s just that that isn’t very helpful. You generally are not concerned whether you are experiencing having fallen into the sea, but with whether you have actually fallen into a sea (and thus had better start swimming or treading or looking for floatation).

But interpreted experience can indeed always be false—even if the probability is so low you needn’t concern yourself with it. Hence, it is possible that when you experience yourself falling into the sea that you are hallucinating or dreaming or something; but it is so unlikely that you are, that you had better assume you really did fall into a real sea, and act accordingly.

This references the Cartesian Demon problem. On which see We Are Probably Not in a Simulation.