The Problem of Induction and The Axiom of Order

Daniel:

Just to remind readers, you're presenting arguments which purport to invalidate inductive thinking, the so-called "problem of induction."

I will offer a proposed solution to that problem in this post.

Induction, to those less familiar with such subjects, is a kind of reasoning described as:

"... inference of a generalized conclusion from particular instances."

http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=induction

"It is to ascribe properties or relations to types based on limited observations ... or to formulate laws based on limited observations of recurring phenomenal patterns."

http://en.wikipedia.org/wiki/Induction_%28philosophy%29

It is contrasted with deductive reasoning.

Many philosophers have considered it problematic since David Hume. Daniel will present examples of how that problem is posed. There is more here:

http://en.wikipedia.org/wiki/Problem_of_induction

Daniel writes:
....

I'll give you a quick overview.

Problem No.1 - reasoning from past events to future ones, or from the *known* to the unknown.

Say I've got 6 white eggs in a carton. Am I justified in saying "all eggs are white"?

Say I've got 100 eggs in a carton. Am I justified yet?

Say I've got 20,000 eggs etc. How about that?

Obviously the answer is no to all three. (If you think "yes" at say 20,000, or 20,000, 000, you'd have to explain why this is a magic justifying number, and 6 or 100 is not).

Now the situation does not change by adding "probably white" to the formula either, because there has never been a method of reliably estimating the probability of such a thing (tho many have tried) If you think the probability increases with each observation, it's kind of like the gambler's fallacy (in reverse). Actually, there is no "necessary connexion" between each observation, just like ther is no necessary connection between dice throws. Nothing in logic that takes you from the particular to the general. You simply don't *know.*

This problem itself is enough to destroy the idea we can find certainty, or probability, using inductive means. But there are others.

Actually, and I hope you'll forgive me if I sound glib, this doesn't seem such an intractable problem as it might first appear.

First, interestingly, this argument uses induction to invalidate induction. It reasons from SOME circumstances for which induction cannot give us quantitative or relative information* to the inference that NO circumstances can do so.

NH:

*How could we possibly ascertain the probability of a finite number of circumstances as compared to an unknown, open-ended, or infinite number?

Second, it's a mistake to reason that because a precise mathematical probability cannot be calculated that NOTHING can be known about imprecise degrees of "more likely that not."

Third, that the value of induction is assessed using induction is not a fatal flaw. Actually, it is a necessary fact, as I'll discuss in a moment.

Daniel writes:

Yet the certain truth of this principle can be known only *inductively* itself, and thus requires reference to a higher principle to be certain, and so on into an infinite regress, which is a logical no-no.

The Axiom of Order

The way out of regress is axiomatic. I've previously posted two (one of Rand's and one of mine) which are apparently irreducible. (The so-called axiom(s) of identity are not.)

The Axiom of Existence (Existence exists.)
The Axiom of Values (Values vary).

To which we can add:

The Axiom of Order (Order exists.)

The need for this is clear, whether as a metaphysical or an epistemological axiom. Without order, regularity, we have a universe filled with floating, formless, orderless mush.

This has a number of implications, but I'll only discuss those which relate to reasoning:

Deductive reasoning would not be possible without order, regularity. Without order, nothing can be differentiated and nothing can be integrated. Classification would not occur.
Order implies some degree of efficacy for inductive thinking, so long as the thinker can perceive that order.

To the degree the universe is ordered, to that degree inductive thinking can be effective.

For example, we have apparently had sunrises at a given place on Earth for 4,000,000,000 years or so. All other knowledge aside, such as knowledge about WHY the sun rises, does the knowledge of four thousand million sunrises make the assertion "Tomorrow there will be a sunrise" more likely to be true?

Yes, to the degree that there is a regularity, order, underlying the inductive conclusion.

By axiomatizing order we justify induction, as induction is a thinking process based upon order, regularity. We are then free to use induction to further validate induction, i.e., inductive reasoning has worked well in the past for many circumstances, so it will probably continue to do so in the future.

The logical regress has been solved because induction has a logical basis in the Axiom of Order and the presumption of consciousness.

The numerical problem is solved because induction makes no certain or quantified predictions. This is the generalized deductive logic of induction:

Axiom: Order exists.
Premise: Order tends to occur more than once.
Premise: I am perceiving order.
Conclusion: This order is likely to occur again under these circumstances.

Daniel writes:

Problem 3. I;ll throw a third related problem in there just for good measure - that is, in induction we make observations which, as they build in number, we turn into theories.

However, this idea, commonsensical as it sounds, is illogical for the following reason: *we must already have a theory before we make an observation*

I believe that problem, such as it is, is solved by the Axiom of Order, which is both a metaphysical axiomatic fact and an epistemological axiom. (For example, babies have a built-in ability to sort out order. The "theory" is built into us epistemically.)

Daniel writes:

Anyway, it's a complex subject, this is just the raw overview, read up on Hume etc to get the full deal. But overall, it wasn't looking good for the basis of human rationality - until Popper came along.

Thanks for stimulating discussion on this, Daniel. I don't know what Popper's thinking on this was, but I'd be surprised if it resembled my solution.

I doubt that I'll get much more comment on this than on my previous axioms thread. I don't imagine Objectivists will know what to make of this for some time. I can live with that. LOL

Nathan Hawking

(Edited by Nathan Hawking on 5/30, 3:01am)

(Edited by Nathan Hawking on 5/30, 12:31pm)

Post 1

Monday, May 30, 2005 - 8:42am

Hi Nathan,

First, interestingly, this argument uses induction to invalidate induction. It reasons from SOME circumstances for which induction cannot give us quantitative or relative information* to the inference that NO circumstances can do so.

The skeptic is more cunning that this. She concedes that we might arrive at a universal truth after extrapolating from circumstances, but that we lack justification for why we should accept that truth as universal.

Second, it's a mistake to reason that because a precise mathematical probability cannot be calculated that NOTHING can be known about imprecise degrees of "more likely that not."

Why is that a mistake?

The Axiom of Order (Order exists.)

The need for this is clear, whether as a metaphysical or an epistemological axiom. Without order, regularity, we have a universe filled with floating, formless, orderless mush.

Not exactly. First, the skeptic will argue that just because we have order today, doesn't mean we'll have it tomorrow. Oh sure, it's awfully handy for us that the universe thus far has been pretty regular, but that's not stopping the universe from going haywire on us later. Second, the skeptic can question whether the regularity in our universe is the only type of regularity. Perhaps our universe could switch from one type of regularity to another overnight. Third, the skeptic can argue that we impose order, pursuant to our conceptual faculty, on an otherwise chaotic world. This would render your "axiom" as an epistemic rather than metaphysical. Fourth, the Objectivist will argue that an axiom is that which is undeniably and inescapably true. As we can see from the skeptic's argument, one can successfully reject order as a necessarily ever-present attribute of existence.

You (and plenty other philosophers) have identified a pretty important assumption required for inductive efficacy -- that the universe behaves regularly. But I don't see a solid justification for that assumption from you. And even if you did justify it, you'd then have to explain why our little sample size of that regular (and vast) universe is sufficient basis for successful generalization.

Anyway, I think there're interesting ways to reduce (though not overcome) the problem of induction.

(A) One way is to abandon "non-advancing" theories. I tried to explain this to Laj in the other thread. Consider the two theories: (1) all ravens are black and (2) some ravens are nonblack. If ravens exist, then one of these is true, and the other false.

Consider the evidence: I see one black raven, then another, then another. Every time, thus far, that I've seen a raven, it's been black. Which theory is advanced by this evidence? I say that (1) is advanced because, thus far, it still might be the case that all ravens are black. Our evidence might've converged on the truth, even if we can't be certain of it. I say that (2) is not advanced by this evidence because it doesn't matter whether I see 5 black ravens or 5,000, I will never get closer to (2) until I actually see one nonblack raven, at which point I should, of course, adopt (2). Utility suggests we abandon the dead-end theory in favor of the theory advanced by the evidence. We might be wrong about (1), but we'll get more mileage from it given our evidence.

You might ask: why not just adopt (3) some ravens are black? This is tricky, and I don't have a super great answer for this, but here's one of the better ones. The antithesis of (3) would be (4) all ravens are nonblack. Clearly, the evidence is against (4), which means that we could safely abandon it and safely accept (3). But the problem with (3) is that it's not advanced by the evidence because it doesn't matter whether I see 5 black ravens or 5,000, it will still just be the case that some ravens are black. So while (3) is safe, it is not advanced by our evidence, and therefore is less useful than (1).

(B) Another way to reduce (but not overcome) induction -- although I'm not sure how useful this is -- is to severely limit the universe. Pretend the universe is a 5 by 5 matrix (25 squares total). Let us accept that any entity, including a raven, can fill exactly and only one square in the matrix. How probable is (1) that all ravens are black? After I see one raven in the matrix, I think I can say with certainty that there's at least a 1 in 25 chance that all ravens are black. When I see a 2nd black raven, there's a 2 in 25 chance. But then something weird happens: I see a copy of Atlas Shrugged in the matrix as well. Does the existence of an Atlas Shrugged make (1) any more or less likely? In my limited universe, it does. It means we can say that there's a 2 in 24 chance of (1). Then I see a copy of The Fountainhead, which gives us a 2 in 23 chance of (1), then I see another black raven (a 3 in 23 chance of (1)), then a brown fox (a 3 in 22 chances of (1)), etc. This is weird because no matter what I observe, whether it's a raven or otherwise, so long as it's not a nonblack raven, the chances of (1) improve. This is weird (and somewhat counterintuitive, I think) because what the hell does the existence of a brown fox have to do with the proposition that all ravens are black? It's weird to say that the more brown foxes I see, the more likely it is that all ravens are black. But in a severely limited universe we gain much ground by pointing out instances of non-ravens.

The reason I'm not sure whether this is very useful is because our universe would be a friggin huge matrix and entities don't fit nicely into each square. Still, in our universe, pointing out brown foxes does get us closer to (1), but only minisculely and negligably so.

(Aside: I should not that I'm importing at least one inductive inference into my ontology: that solids can't overlap. While this inference renders (B) even less effective, I think (A) can successfully survive it.)

-Jordan

Post 2

Monday, May 30, 2005 - 9:45am

Nathan,

"I don't know what Popper's thinking on this was,.."

This is pretty obvious.

"... but I'd be surprised if it resembled my solution."

How can you consider yourself a first rate philosopher if you don't bother to read the writings of one of the most respected philosophers of the twentieth century?

You begin with:

"Just to remind readers, you're presenting arguments which purport to invalidate inductive thinking,"

It is not my sense that this is what Popper did. He pointed out the landmine in induction and a way to recognize when induction was likely to fail. Perhaps you are trying to do the same thing. You yourself have given "problems" of induction but are not invalidating induction itself. Are you saying if inductive conclusions are not "testable" they are not valid?

Note: I would replace your "Axiom of Order" with causation.

Let me state that I DON'T want to get into a long winded discussion about this, I'm just interested in your reply to this one question.

Post 3

Monday, May 30, 2005 - 12:28pm

Mike E:

Nathan,

"I don't know what Popper's thinking on this was,.."

This is pretty obvious.

Really?

"... but I'd be surprised if it resembled my solution."

How can you consider yourself a first rate philosopher if you don't bother to read the writings of one of the most respected philosophers of the twentieth century?

I'm not "rating" myself, first or otherwise. Nor do I care to.

My first interest is in novel solutions to unsolved problems, and devising novel paths is easier in fresh snow. I often begin by thinking first, and reading much of what others have written afterward. Then modifying my views if necessary.

You begin with:

"Just to remind readers, you're presenting arguments which purport to invalidate inductive thinking,"

It is not my sense that this is what Popper did.

That was referring to Daniel's presentation of the "problem of induction," not Popper's.

He pointed out the landmine in induction and a way to recognize when induction was likely to fail. Perhaps you are trying to do the same thing. You yourself have given "problems" of induction but are not invalidating induction itself.

I did not pose those problems. They were Daniel's version of what others have said.

Are you saying if inductive conclusions are not "testable" they are not valid?

No, I didn't say that. There is no way to decisively test our inductive conviction that there are "laws" of nature. We can only corroborate, never certify.

Let me state that I DON'T want to get into a long winded discussion about this, I'm just interested in your reply to this one question.

Note: I would replace your "Axiom of Order" with causation.

Reply to which question, about replacing the Axiom of Order with the notion of causation?

If so, then I would not do so because the notion of causation is insufficiently primitive to qualify as an axiom.

Causality is reducable, at a minimum, to a combination of ALL THREE of my listed axiomatic metaphysical circumstances: existence, values, and order.

Nathan Hawking

Post 4

Monday, May 30, 2005 - 1:52pm

Nathan,

"There is no way to decisively test our inductive conviction that there are "laws" of nature. We can only corroborate, never certify."

I think Popper would agree with this statement. As do I.

"the notion of causation is insufficiently primitive to qualify as an axiom."

To my philosophically unsophisticated eye, causation seems more primitive than your "axiom of order". I believe causation has been an integral part of mankind's reasoning about nature since the beginning of human thought. Order needs to come from something, have a "cause"? Not quibbling, just not used to thinking of things in this way.

I have to go out and make a gate for my yard. I won't be back until this evening.

Post 5

Monday, May 30, 2005 - 2:27pm

Jordan:

Hi Nathan,

First, interestingly, this argument uses induction to invalidate induction. It reasons from SOME circumstances for which induction cannot give us quantitative or relative information* to the inference that NO circumstances can do so.
The skeptic is more cunning that this. She concedes that we might arrive at a universal truth after extrapolating from circumstances, but that we lack justification for why we should accept that truth as universal.

That's just the point. The Axiom of Order posits that there IS order in the universe, not that EVERYTHING IS ORDERLY.

A consequence is that NO inductive belief can ever be "justified" and certified truth or universal. At best there will be a preponderance of apparent likelihood.

Second, it's a mistake to reason that because a precise mathematical probability cannot be calculated that NOTHING can be known about imprecise degrees of "more likely that not."
Why is that a mistake?

Because the Axiom of Order assumes that order exists. Consciousness, to be conscious, MUST perceive order in at least SOME degree.

Without assuming these things, we have no basis for ANY mental activity. We have not just repudiated the notion that "induction provides NO degree of certitude," as Daniel's examples purported to demonstrate, but the whole basis for thought itself.

The Axiom of Order (Order exists.)

The need for this is clear, whether as a metaphysical or an epistemological axiom. Without order, regularity, we have a universe filled with floating, formless, orderless mush.

Not exactly. First, the skeptic will argue that just because we have order today, doesn't mean we'll have it tomorrow.

Oh sure, it's awfully handy for us that the universe thus far has been pretty regular, but that's not stopping the universe from going haywire on us later.

And I would agree with the second part. But that is no less true for existence itself. We ASSUME that things will persist in the realm of existence, but due to unknown factors everything could blink out of existence tomorrow.

Second, the skeptic can question whether the regularity in our universe is the only type of regularity. Perhaps our universe could switch from one type of regularity to another overnight.

That's fine. The Axiom of Order simply assumes order. That is a fitting metaphysical and epistemological assumption because both the universe and our thinking appear to exhibit order. We cannot even deny order without using order in the thoughts and sentences of denial.

Our axioms afford no guarantee. They're AXIOMS, not certificates of Eternal Guarantee with doiley-fringe patterns printed on gilded edges.

Third, the skeptic can argue that we impose order, pursuant to our conceptual faculty, on an otherwise chaotic world. This would render your "axiom" as an epistemic rather than metaphysical.

The skeptic would be foolish to so argue.

Imposed order would have no predictive power in a chaotic world. The nature of our consciousness and of the universe is predictive, which leads us to conclude that the order is an objective reality.

Fourth, the Objectivist will argue that an axiom is that which is undeniably and inescapably true. As we can see from the skeptic's argument, one can successfully reject order as a necessarily ever-present attribute of existence.

The Objectivist would be foolish to so argue.

First, axioms are things we ASSUME. We assume them because they APPEAR true and irreducible to components. Axioms do not come with guarantees.

Second, no rule says axiomatic facts need apply to all possible universes.

You (and plenty other philosophers) have identified a pretty important assumption required for inductive efficacy -- that the universe behaves regularly. But I don't see a solid justification for that assumption from you.

I'll buy that last statement when you can demonstrate that the Axiom of Order is any less "justified" than the Axiom of Existence.

And even if you did justify it, you'd then have to explain why our little sample size of that regular (and vast) universe is sufficient basis for successful generalization.

The success itself proves sufficiency (i.e., it allows conscious entities to predict circumstances with sufficient success to ensure survival).

Anyway, I think there're interesting ways to reduce (though not overcome) the problem of induction.

(A) One way is to abandon "non-advancing" theories. I tried to explain this to Laj in the other thread. Consider the two theories: (1) all ravens are black and (2) some ravens are nonblack. If ravens exist, then one of these is true, and the other false.

Consider the evidence: I see one black raven, then another, then another. Every time, thus far, that I've seen a raven, it's been black. Which theory is advanced by this evidence? I say that (1) is advanced because, thus far, it still might be the case that all ravens are black. Our evidence might've converged on the truth, even if we can't be certain of it. I say that (2) is not advanced by this evidence because it doesn't matter whether I see 5 black ravens or 5,000, I will never get closer to (2) until I actually see one nonblack raven, at which point I should, of course, adopt ...

You have restated, in different words, the numerical problem Daniel cited. The problem lies in three words you use: true, false, and certain. These simply do not apply, in the way you're using them, to inductive knowledge.

Metaphysically, it is either true or not true that all ravens are black (assuming no 'shades' of black).

Epistemologically, we cannot EVER have logical certainty of the truth without a full knowledge of the raven population. (Knowledge of raven genetics and the black-producing mechanism is no help, unless we are willing to classify a white raven, defective only in pigment-producing genes, a "nonraven.")

(2). Utility suggests we abandon the dead-end theory in favor of the theory advanced by the evidence. We might be wrong about (1), but we'll get more mileage from it given our evidence.

I'm not sure what you mean by this.

You might ask: why not just adopt (3) some ravens are black? This is tricky, and I don't have a super great answer for this, but here's one of the better ones. The antithesis of (3) would be (4) all ravens are nonblack. Clearly, the evidence is against (4), which means that we could safely abandon it and safely accept (3). But the problem with (3) is that it's not advanced by the evidence because it doesn't matter whether I see 5 black ravens or 5,000, it will still just be the case that some ravens are black. So while (3) is safe, it is not advanced by our evidence, and therefore is less useful than (1).

This is getting a bit convoluted. I think I'd like to work on the "raven paradox" another time.

(B) Another way to reduce (but not overcome) induction -- although I'm not sure how useful this is -- is to severely limit the universe.

I think the Axiom of Order fully overcomes the "problem of induction." I think you just don't recognize that yet.

Pretend the universe is a 5 by 5 matrix (25 squares total). Let us accept that any entity, including a raven, can fill exactly and only one square in the matrix. How probable is (1) that all ravens are black? After I see one raven in the matrix, I think I can say with certainty that there's at least a 1 in 25 chance that all ravens are black. When I see a 2nd black raven, there's a 2 in 25 chance. But then something weird happens: I see a copy of Atlas Shrugged in the matrix as well. Does the existence of an Atlas Shrugged make (1) any more or less likely? In my limited universe, it does. It means we can say that there's a 2 in 24 chance of (1). Then I see a copy of The Fountainhead, which gives us a 2 in 23 chance of (1), then I see another black raven (a 3 in 23 chance of (1)), then a brown fox (a 3 in 22 chances of (1)), etc. This is weird because no matter what I observe, whether it's a raven or otherwise, so long as it's not a nonblack raven, the chances of (1) improve. This is weird (and somewhat counterintuitive, I think) because what the hell does the existence of a brown fox have to do with the proposition that all ravens are black? It's weird to say that the more brown foxes I see, the more likely it is that all ravens are black. But in a severely limited universe we gain much ground by pointing out instances of non-ravens.

The reason I'm not sure whether this is very useful is because our universe would be a friggin huge matrix and entities don't fit nicely into each square. Still, in our universe, pointing out brown foxes does get us closer to (1), but only minisculely and negligably so.

(Aside: I should note that I'm importing at least one inductive inference into my ontology: that solids can't overlap. While this inference renders (B) even less effective, I think (A) can successfully survive it.)

I think it's a mistake to apply notions of mathematical probability to the problem of induction. That kind of thinking is what creates the appearance of a dilemma where none exists. I can see several problems with what you just wrote, but don't have the time at present to unwind all that.

If you wish to simplify an objection or problem you find with my solution, one I haven't dealt with, that would be a little closer to my current agenda. I'd like to consider the raven thing another time, though.

I enjoyed your post.

Nathan Hawking

Post 6

Monday, May 30, 2005 - 3:22pm

Mike E:

NH wrote:

"the notion of causation is insufficiently primitive to qualify as an axiom."

To my philosophically unsophisticated eye, causation seems more primitive than your "axiom of order". I believe causation has been an integral part of mankind's reasoning about nature since the beginning of human thought. Order needs to come from something, have a "cause"?

Order can conceivably exist without causation. Imagine an eternal, static universe filled with a single kind of noninteracting "particle" spaced in a cubical lattice arrangement.

We cannot assume causation for either the existence of such particles or for the order, as both could be uncaused, ultimate metaphysical facts of existence.

I agree with you on the importance of causality, and consider it fundamental to consciousness, but since not all order need imply causality, but all causality implies at the very least temporal order, I think order the more fundamental fact.

I'm not sure how much value there is in rigidly dividing existence and values and order from each other, or if it's even possible. But positing order as an axiom seems to solve what's commonly called the problem of induction, so there may be value in that.

It remains to been seen whether someone will actually vitiate the idea.

Nathan Hawking

Post 7

Monday, May 30, 2005 - 5:57pm

Hi Nathan,

That's just the point. The Axiom of Order posits that there IS order in the universe, not that EVERYTHING IS ORDERLY.

Then how do you know that the stuff we're observing and basing our predictions on is the orderly stuff? Saying that there's some order somewhere is hardly helpful.

We ASSUME that things will persist in the realm of existence, but due to unknown factors everything could blink out of existence tomorrow.

Existents can and do blink out of existence, but existence as a whole can't stop existing, for the alternative would be that "nothing exists," which is contradictory.

Third, the skeptic can argue that we impose order, pursuant to our conceptual faculty, on an otherwise chaotic world. This would render your "axiom" as an epistemic rather than metaphysical.
The skeptic would be foolish to so argue.

Imposed order would have no predictive power in a chaotic world. The nature of our consciousness and of the universe is predictive, which leads us to conclude that the order is an objective reality.

I would discourage you from calling people foolish when you disagree with their arguments. The skeptic can support her "imposed order" idea by pointing to all the false causes and false theories we have used throughout the eons, yet managed to survive with just the same.

First, axioms are things we ASSUME. We assume them because they APPEAR true and irreducible to components. Axioms do not come with guarantees.

An undeniable and inescapable truth is what Objectivists refer to as an axiom. For Objectivists, axioms aren't just assumptions that appear true. They are assumptions that one must necessarily commit to, even if one tries not to.

The problem lies in three words you use: true, false, and certain. These simply do not apply, in the way you're using them, to inductive knowledge.

Says you. :P

I think I'd like to work on the "raven paradox" another time.

Alright, but the raven paradox nicely outlines what induction is and what the problem with it is. That said, I'll leave other "aids" to the problem of induction out of this thread for now.

I think it's a mistake to apply notions of mathematical probability to the problem of induction.

Don't know why you think that, but okay.

I think the Axiom of Order fully overcomes the "problem of induction." I think you just don't recognize that yet.

I read your confidence loud and clear, and you're right that I don't yet recognize how your "Axiom of Order" fixes the problem of induction. You're welcome to try and persuade me, but my interesting with this topic is waning.

-Jordan

Post 8

Monday, May 30, 2005 - 4:33pm

Nathan:
>The way out of regress is axiomatic.

If you say induction is the only certain or probable way of attaining truth, why do you suddenly need *deduction* to ascertain induction's truth?

;-)

-Daniel

Post 9

Monday, May 30, 2005 - 4:41pm

Nathan:
>First, interestingly, this argument uses induction to invalidate induction. It reasons from SOME circumstances for which induction cannot give us quantitative or relative information* to the inference that NO circumstances can do so.

Actually, as far as I can see, this argument uses a hypothetical situation from which logical difficulties emerge, not a series of inductive observations to form a conclusion.

- Daniel

Post 10

Monday, May 30, 2005 - 7:27pm

Jordan:

That's just the point. The Axiom of Order posits that there IS order in the universe, not that EVERYTHING IS ORDERLY.
Then how do you know that the stuff we're observing and basing our predictions on is the orderly stuff? Saying that there's some order somewhere is hardly helpful.

As I see it, for much of human experience, the inductive helps us form hypotheses and corroborate them, and the deductive enables us to falsify those hypotheses.

Suppose a person drinks milk and gets an upset stomach. They might think, I wonder if it's the milk? After a second and a third and a fourth experience, they become more certain it was the milk.

Is there enough apparent order there to warrant a hypothesis? Yes.

Does the perception of order match an objective physical order? We don't know, without further knowledge. And much of that further knowledge will be based upon this same process--induction, hypotheses and testing.

If the universe were only slightly ordered, this process would not work well for our survival (assuming we could even exist). But the fact that we're alive shows that some of the universe is well-ordered.

To your question: If it looks orderly, we rightly proceed on SOME DEGREE of belief that there is or may be an underlying order, depending upon the circumstance.

We ASSUME that things will persist in the realm of existence, but due to unknown factors everything could blink out of existence tomorrow.
Existents can and do blink out of existence, but existence as a whole can't stop existing, for the alternative would be that "nothing exists," which is contradictory.

What would justify your faith that some things can vanish but not all things? Inductive reasoning based upon it not happening in your experience?

Third, the skeptic can argue that we impose order, pursuant to our conceptual faculty, on an otherwise chaotic world. This would render your "axiom" as an epistemic rather than metaphysical.
The skeptic would be foolish to so argue.

Imposed order would have no predictive power in a chaotic world. The nature of our consciousness and of the universe is predictive, which leads us to conclude that the order is an objective reality.
I would discourage you from calling people foolish when you disagree with their arguments.

Fair enough. My exuberance got the better of me.

The skeptic can support her "imposed order" idea by pointing to all the false causes and false theories we have used throughout the eons, yet managed to survive with just the same.

So the skeptic would employ inductive reasoning to invalidate inductive reasoning?

Besides, a few false beliefs do not negate the vast number of times we're right. Sentient beings are right FAR more often that they're wrong (taking ALL of experience and it's regularity, not just intellectual beliefs). If we weren't we'd be dead.

First, axioms are things we ASSUME. We assume them because they APPEAR true and irreducible to components. Axioms do not come with guarantees.
An undeniable and inescapable truth is what Objectivists refer to as an axiom. For Objectivists, axioms aren't just assumptions that appear true. They are assumptions that one must necessarily commit to, even if one tries not to.

That's true for both the Axioms of Value and Order. Try to deny them without using them. For THIS universe they apply.

Beyond that, I don't see much use in splitting hairs over the meaning of "appear" vs. "necessarily commit." I think that's a whole discussion in itself.

The problem lies in three words you use: true, false, and certain. These simply do not apply, in the way you're using them, to inductive knowledge.
Says you. :P

Lord, what's the proper reply? A neener-neener? LOL

I think I'd like to work on the "raven paradox" another time.
Alright, but the raven paradox nicely outlines what induction is and what the problem with it is. That said, I'll leave other "aids" to the problem of induction out of this thread for now.
I think it's a mistake to apply notions of mathematical probability to the problem of induction.
Don't know why you think that, but okay.
I think the Axiom of Order fully overcomes the "problem of induction." I think you just don't recognize that yet.
I read your confidence loud and clear, and you're right that I don't yet recognize how your "Axiom of Order" fixes the problem of induction. You're welcome to try and persuade me, but my interesting with this topic is waning.

-Jordan

Maybe later this week I'll feel more like working on the raven paradox. Right now I have a writing project which requires some attention.

I appreciate the discussion!

Nathan Hawking

Post 11

Monday, May 30, 2005 - 7:34pm

Daniel:

Nathan:
>The way out of regress is axiomatic.

If you say induction is the only certain or probable way of attaining truth, why do you suddenly need *deduction* to ascertain induction's truth?

I made no such claim about induction being the "only" anything. I simply said that the Axiom of Order and consciousness (as an identifier of order) enable induction to work.

I do not disallow the efficacy of deduction. I think they're both necessary.

I own both a wrench and screwdriver, and use each where appropriate.

NH

Post 12

Monday, May 30, 2005 - 7:42pm

Daniel:

Nathan:
>First, interestingly, this argument uses induction to invalidate induction. It reasons from SOME circumstances for which induction cannot give us quantitative or relative information* to the inference that NO circumstances can do so.

Actually, as far as I can see, this argument uses a hypothetical situation from which logical difficulties emerge, not a series of inductive observations to form a conclusion.

If it only applies to PART of inductive thinking, then it is vitiated as a problem which invalidates induction.

So, either it applies to all and is an inductive conclusion, or it applies to part and presents no serious challenge.

Or do you think I'm overstating the case?

Either way, it's not terribly important because that was just an amusing tangent. The important thing is that the axiomatization seems to work.

Nathan Hawking

Post 13

Monday, May 30, 2005 - 11:49pm

Induction is not about gaining omniscience (ie. seeing the 'future'; seeing into 'possible' worlds where 'possible' truths may hold)--it is about this world, it is about here and now; it is about a noncontradictory integration of previous and current knowledge. This "constraint" really seems to bother some of you.

$64,000 question: How come it doesn't bother me?

Hmm.

Ed

Post 14

Tuesday, May 31, 2005 - 2:33am

Ed:
>Induction is not about gaining omniscience (ie. seeing the 'future'; seeing into 'possible' worlds where 'possible' truths may hold)--it is about this world, it is about here and now; it is about a noncontradictory integration of previous and current knowledge. This "constraint" really seems to bother some of you.

Well, you're right that it's not about "omniscience", which means knowing everything. The problem that bothers us is that induction is deductively unjustified as a source of truth. So if we put our faith in it as a reliable "integration of previous and current knowledge" that is exactly what we are doing - *putting our faith in it*, even though we know better. Because we know there is *no logical justification for induction*.

>$64,000 question: How come it doesn't bother me?

Have you thought about it from that angle? That you hold a strong belief that it turns out there is actually no logical justification for? Put like that, does it bother you now? It would me.

- Daniel

Post 15

Tuesday, May 31, 2005 - 4:11am

I want to make a plea again to those of you who find value in posts that utter such unconscionable drivel as "there is no logical justification for induction"�would you please give Daniel some sanctions so I don't have to endanger my health by reading this stuff in the moderator's queue? PLEASE!!!!!!

Linz

Post 16

Tuesday, May 31, 2005 - 4:15am

Daniel:

Ed:
>Induction is not about gaining omniscience (ie. seeing the 'future'; seeing into 'possible' worlds where 'possible' truths may hold)--it is about this world, it is about here and now; it is about a noncontradictory integration of previous and current knowledge. This "constraint" really seems to bother some of you.

Well, you're right that it's not about "omniscience", which means knowing everything.

True.

The problem that bothers us is that induction is deductively unjustified as a source of truth.

I don't think "deductively unjustified" is any longer true.

I presented logic which reasoned from several assumptions, including the order of the universe and the discernment of order by consciousness, and went from there. So far nobody has refuted my reasoning.

Perhaps you're stalled on the notion that induction, logically justified or not, is not a "source of truth." Is that it?

But that's not quite correct. It is a source of truth (knowledge) but not with the possibility of certitude we attribute to something like:

All men are mortal.
Socrates is a man.
Therefore Socrates is mortal.

But this is close:

The sun has come up 4,000,000,000 x 365 times in a row.
Therefore the sun will probably come up tomorrow.

This belief has a general justification: the order in the universe;
This belief has an OBJECTIVE causal fact (perhaps unknown) to which it corresponds: the orderly rotation of the planet.

So if we put our faith in it as a reliable "integration of previous and current knowledge" that is exactly what we are doing - *putting our faith in it*, even though we know better. Because we know there is *no logical justification for induction*.

>$64,000 question: How come it doesn't bother me?

Have you thought about it from that angle? That you hold a strong belief that it turns out there is actually no logical justification for? Put like that, does it bother you now? It would me.

So you say.
What's wrong with this logic:

Axiom: Order exists.
Premise: Order tends to occur more than once.
Premise: I am perceiving order.
Conclusion: This order is likely to occur again under these circumstances.

If that's not "logical justification," then can you give me any idea why, and what a logical justification for a thinking process which entails partial knowledge would look like?

By the way, have you given any thought to the logical justification for deduction?

One of your points was the criticism of induction for using induction to "justify" it. But what justifies deduction? Are you going to, likewise, justify deduction with deduction? If not that, then is it faith?

Nathan Hawking

Post 17

Tuesday, May 31, 2005 - 10:22am

Hi Nathan,

What would justify your faith that some things can vanish but not all things? Inductive reasoning based upon it not happening in your experience?

Like I said, the alternative to "existence exists" is "nothing exists," which is contradictory, and contradictions cannot exist. Therefore, existence must always exist.

So the skeptic would employ inductive reasoning to invalidate inductive reasoning?

Not necessarily. The skeptic just shows how your view might be logically inconsistent. That doesn't require induction.

Sentient beings are right FAR more often that they're wrong (taking ALL of experience and it's regularity, not just intellectual beliefs). If we weren't we'd be dead.

Or perhaps just damn lucky.

This belief has a general justification: the order in the universe;

But what justifies the belief that there's order in the universe? You don't think that belief requires justification. Okay, but that's pretty much a show-stopper. If you want to justify it using induction, well, that's circular.

Cheers,
Jordan

Post 18

Tuesday, May 31, 2005 - 5:03pm

Jordan:

Hi Nathan,

What would justify your faith that some things can vanish but not all things? Inductive reasoning based upon it not happening in your experience?
Like I said, the alternative to "existence exists" is "nothing exists," which is contradictory, and contradictions cannot exist. Therefore, existence must always exist.

But what makes "nothing exists" contradictory?

Let us not confuse "existence" as an existential fact with "existence exists" as a philosophical statement about existence.

We cannot point to even one existential REASON why existence "must" exist, can we? The way we can know this is that Rand was forced to axiomatize it. (Assuming her thinking on this was comprehensive.)

As a philosophical statement about existence, "existence exists" will always be logically true at the moment of utterance because the act of utterance is self-evident proof of existence.

But as a statement for all time, implying that 'something always existed and something will always exist,' that is not a demonstrable truth. It would have to be taken on faith.

So the skeptic would employ inductive reasoning to invalidate inductive reasoning?
Not necessarily. The skeptic just shows how your view might be logically inconsistent. That doesn't require induction.

You said: The skeptic can support her "imposed order" idea by pointing to all the false causes and false theories we have used throughout the eons, yet managed to survive with just the same.

The only way such a skeptic could support an "imposed order" idea by pointing to "false causes and false theories" is to reason from the particulars to the general, induction.

That some instances of induction are wrong fits my view precisely.
Not all perceptions of order have a basis in objective fact. Some apparent order is accidental, not derived from specific order-causing events or circumstances. Tossing a coin heads three times in a row is an accidental form of order, with no predictive value.

Sentient beings are right FAR more often that they're wrong (taking ALL of experience and it's regularity, not just intellectual beliefs). If we weren't we'd be dead.
Or perhaps just damn lucky.

If that's not tongue-in-cheek, then I think that would underestimate the pervasiveness of our use of induction.

This belief has a general justification: the order in the universe;
But what justifies the belief that there's order in the universe? You don't think that belief requires justification. Okay, but that's pretty much a show-stopper. If you want to justify it using induction, well, that's circular.

We should not overlook the very point of this thread: Like "existence," order is axiomatized. Am I required to justify belief in existence? If not, why would I be required to justify the EQUALLY irreducible notion of ORDER?

Like existence, I define order ostensively. Like existence, I accept it as an axiomatic fact of this universe. Like existence, it cannot be denied without resorting to its use.

Nathan Hawking

Post 19

Tuesday, May 31, 2005 - 2:13pm