The Problem of Induction and The Axiom of Order

Laj writes
>So you could say that Popper is not so interested in how the belief arrived in your head or how you rank the importance of beliefs, but is way more interested in how you criticize or test the belief.

Hi Jordan (if you're still around)

This is the key point. Popper said he is completely uninterested in the *source* of our theories. He simply saw that the inductivist idea that theory follows observation was another illogical aspect of it, and criticised it accordingly. A theory might come from your imagination, from God's Voice, from a dream someone told you about, a book you read or misread, from the impact of a light wave impacting on your retina and triggering a electrochemical process inside your nervous system. It doesn't really matter, as sorting it all out is like trying to sort out a puddle back into its original raindrops.

What we really need is not debate over which original *authority* our knowledge should properly spring from. What is needed is means for *testing* our knowledge - and the better the means of testing it, the better our theories will be. Because the dogmatic assertions of God's Voice are unlikely to stand up to open debate and rational criticism. He saw that the logical problems of inductive testing were too many to overcome - and that these problems opened the door to irrationalist philosophies, in which logic and argument don't matter - and so found a more rationally sound solution.

As I've said before, there is little point in criticising his lack of positive certainty in his result, as his inductive critics are unable to offer anything better!

- Daniel

(Edited by Daniel Barnes
on 6/02, 6:34pm)

Post 41

Friday, June 3, 2005 - 2:31am

Daniel:

Even Hume himself, having impeccably demonstrated that induction was illogical...

And I've undemonstrated it.

You have only to refute my original post in the thread, and my post on the raven paradox to convince me otherwise.

This empirical method, Popper proposed, was *falsification* - the idea that while we cannot logically find via inductive inferences definitive confirmations, we can find, via the modus tollens, definitive *refutations*. No number of white swans observed will finally confirm a theory that "all swans are white", a single black swan will refute it.

I'm afraid that doesn't meet Hume's objections. Falsification only disposes. Induction PROPOSES. Unless there's a logical basis for using induction to PROPOSE, why use it? "Falsification" does not create a logical basis for induction, only for invalidating inductive beliefs.

Ultimately, some of the things induction proposes can NEVER be falsified. Moreover, many thing may have the theoretical potential for falsification, but may be completely beyond the realm of the practical.

So the answer is - and this is the simplest reply to Nathan's idea too: *why bother* with trying to justify the inductive method of empiricism in the first place, when we now have a logically sound alternative?

Daniel, you posed the questions about Hume's (et al) objections to induction. I have answered them--successfully, I believe. That's the answer to "why bother?" Perhaps there are other solutions, of course.

But you have not shown how Popper's method (to which I assume you're referring) is a "logically sound alternative." (Unless I overlooked something.) Would you care to elaborate?

Nathan Hawking

Post 42

Saturday, June 4, 2005 - 5:03pm

Hi Daniel,

Found a computer.

Popper said he is completely uninterested in the *source* of our theories....doesn't really matter, as sorting it all out is like trying to sort out a puddle back into its original raindrops.

I just disagree with Popper here in that I think it's a mistake to ignore the stuff Popper is ignoring. Again, when choosing between two as-of-yet unfalsified theories, Popper offers no satisfactory answer of which to choose. I'd say that making a theory falsifiable is probably necessary but not sufficient for making it good.

As I've said before, there is little point in criticising his lack of positive certainty in his result, as his inductive critics are unable to offer anything better!

It's a problem they all suffer from, but that doesn't make it any less a problem.

Jordan

Post 43

Saturday, June 4, 2005 - 8:50pm

Right, sorry about the slight delay, been tied up. Now where were we?

Nathan:
>You have only to refute my original post in the thread, and my post on the raven paradox to convince me otherwise.

I haven't read the raven paradox post as the header warned me that it contained scholastic arguments, which I try to avoid! But I would refute your original post as follows:

>Axiom: Order exists.

The first question is: as this axiom is intended as the logical successor of Rand's preceding fundamental axioms, how does this axiom stack up compared to its predecessors?

Obviously an axiom in the sense Rand uses it is a self evident proposition; or put another way, a proposition the negation of which would entail a contradiction, and is thus impossible.

For example, with "Existence exists", we would get a negation like "non-existence exists" - which it obviously cannot.

Now, when we negate "Order exists" the same way we get something like "non-order exists" or more commonly "disorder exists"

Now is this impossible, thus guaranteeing the self evidence of the proposition? Hardly. In fact, only the strictest ultra determinist woud disagree that there is some amount of disorder in the universe - and indeed, you describe yourself as a physical indeterminist. So even if it *were* axiomatic, you yourself would have to disagree with it!

Or you could put the negation another way: "Order does not exist". Once again, as this must be true in at least *some* parts of time and space if you assume any degree of indeterminism this likewise undermines the axiomatic status of your proposition. So I do not even need to resort to a Hericlatean view of the universe - although I quite reasonably could *as well* - in order to demonstrate the non-self evident, thus non-axiomatic, nature of your proposition.

>Premise: Order tends to occur more than once.

Now, this premise is closer to being axiomatic than your axiom...;-) I would suggest you drop the tentative "tend"!

>Premise: I am perceiving order.

To which the first answer is: so what if you are...? The fact that you *perceive* order does not mean that 1) the universe is ordered, or less trivially 2) the universe is ordered *in the way you perceive it is*. This is very simple to demonstrate, in that humans often *perceive orders that don't exist*. Examples of falsely perceived orders include everything from conspiracy theories to the old idea that the sun goes round the earth, or any modern scientific theory that was disproved yesterday. Further, as well as false orders, there is the related problem of cognitive bias, which means we tend to view events *in a way that suits our existing theories*. Further, as well as "perceiving order" humans also perceive *disorder*! And perceive disorder where order is later discovered!

All this is damaging enough for your premise. But even if we assumed it was all untrue, and you had perceived a particular order correctly, your premise is *entirely misses* Hume's argument, the very nub of which is that there is no logical reason *the order you perceived in past experiences* can be used to *predict future events*!

So this premise, even if we grant you it in its strongest form - and there are plenty of excellent reasons not to - is *irrelevant*. I strongly suggest you take time to absorb Hume's arguments before refuting them....;-)

>Conclusion: This order is likely to occur again under these circumstances.

You appear to have neglected your second premise in your conclusion, which itself seems little more than a rephrasing of first premise, which is basically the idea that order by definition consists of more than one event! So other than defining "order" we do not seem to have got very far in addressing any of the various problems induction faces. And even if you did manage to come up with a better basic argument, your real challenge still lies well in front of you - the task of *formulating a strict principle of induction*.

Simply, this would read something like:
"We require ? number of observations to establish a universal law with certainty"
or
"We require ? number of observations to establish a universal law with ?% probability"

There is no-one in the history of science who has been able to successfully formulate the above principle (as an aside. on the basis of this past experience, the sincere inductivist might feel some duty to admit his own principle will never be formulated in future...;-))

Whereas as I can formulate the principle of falsification as follows:
"We require 1 observation to negate a universal law with 100% certainty"

(and yes the same problems of "certainty" in the sense of *accepting* an observation - precision of instruments, reliability of observations etc - applies to induction equally as to falsification)

You are of course welcome to keep searching for justifications for the inductive method. But I can't help but ask again why a sharp fellow like yourself feels it necessary to strain nerve and sinew in stapling inductive custard to the ceiling, when an alternative empirical method that *is* logically justified already exists?

In fact, as a final blow, Popper argues that induction is even *inductively invalid*!

"...Any strong positive reply to Hume's logical problem (say, the thesis that induction is valid) would be paradoxical. For, on the one hand, if induction is the method of science, then modern cosmology is at least roughly correct (I do not dispute this); and on the other, modern cosmology teaches us that to generalise from observations taken, for the most part, in our incredibly idiosyncratic region of the universe would almost always be invalid. Thus if induction is 'inductively valid' it will almost always lead to false conclusions; and therefore it is inductively invalid." (Section 14, "Replies To My Critics", Popper, cited in "Popper Selections")

;-)

- Daniel

(Edited by Daniel Barnes
on 6/04, 8:59pm)

Post 44

Sunday, June 5, 2005 - 2:50am

Jordan:
>Again, when choosing between two as-of-yet unfalsified theories, Popper offers no satisfactory answer of which to choose.

Hi Jordan

Can I ask what part of Popper's work gave you this impression? He actually goes into the issue of exercising a "critical preference" for one theory over another - including unfalsified ones - in some depth in his work.

Criteria include: 1) how well do they solve the problem posed? 2) how bold are they? ie: which would have the greater *informative content* if it survived testing? 3) which would produce the most interesting subsequent problems etc etc

I suppose it also might be worth asking a) if you consider that induction does provide such an answer, or if not b) what you consider a satisfactory answer to such a problem would look like?

regards
Daniel

Post 45

Sunday, June 5, 2005 - 3:53am

Daniel:

>Axiom: Order exists.

The first question is: as this axiom is intended as the logical successor of Rand's preceding fundamental axioms, how does this axiom stack up compared to its predecessors?

Obviously an axiom in the sense Rand uses it is a self evident proposition; or put another way, a proposition the negation of which would entail a contradiction, and is thus impossible.

Rand's definition of an axiom is self-defeating, and leads to all manner of semantic gymnastics with "existence."

Axioms (or axiomatic facts) do not have to be nonnegatable. They only have to be true for the system in question and irreducible.

For example, with "Existence exists", we would get a negation like "non-existence exists" - which it obviously cannot.

That merely attempts to turn the fatal flaw into a virtue.

For everything in the entire universe, saying that something exists has a logical and necessary contrary, that it may (logically) NOT exist under some circumstances. For EVERYTHING in the universe, to say that something "exists" IS ONLY MEANINGFUL if there is some sense in which it CANNOT exist. Otherwise, "exists" as opposed to what?

Rand completely ignores that fact by proclaiming "existence" an "existent." In what meaningful sense is it an existent? None.

What she should have said is: Existence is a fact.

Or even better: Existents exist.

For our universe, "existents exist" is an irreducible (and thus axiomatic) fact. It also uses "exist" in a meaningful way, one which HAS a possible logical negation.

But it is just as resistant to contrary argument, for it is not possible to argue that "existents DO NOT exist" without assuming the very thing one is repudiating.

Now, when we negate "Order exists" the same way we get something like "non-order exists" or more commonly "disorder exists."

That's fine. Disorder does exist.

But "order" is axiomatizable for exactly the same reasons as "existents." It is apparently TRUE, and it is an irreducible fact--it has no components.

Now is this impossible, thus guaranteeing the self evidence of the proposition? Hardly.

Your logic is severely flawed.

Either order exists in our universe, or it doesn't. It is either an irreducible fact, or order is not present. The presence of disorder does not negate the presence of order, does it?

If order is an irreducible fact, then it is a metaphysical axiom. Semantics and flawed logic won't change that.

In fact, only the strictest ultra determinist woud disagree that there is some amount of disorder in the universe - and indeed, you describe yourself as a physical indeterminist.

But the Axiom of Order does not preclude some absence of order, does it? Just as the Axiom of Existents (you'll note my adoption of the modification) does not preclude some things from going out of existence. You see the parallel?

>Premise: Order tends to occur more than once.

Now, this premise is closer to being axiomatic than your axiom...;-) I would suggest you drop the tentative "tend"!

Not at all. Recurrance is a form of order, and rests on an axiom of order. "Tend" is necessary because order can exist and recurr, then cease.

>Premise: I am perceiving order.

To which the first answer is: so what if you are...? The fact that you *perceive* order does not mean that 1) the universe is ordered, or less trivially 2) the universe is ordered *in the way you perceive it is*.

You miss the point. Perception is a necessary link in the chain of induction. What makes induction work is that our perceptions OFTEN match an underlying objective order.

If NO perceptions matched the universe, they wouldn't be perceptions.

This is very simple to demonstrate, in that humans often *perceive orders that don't exist*.

I have no quarrel with this. It's what makes induction less than certain. But obviously perception must have objective referents in reality SOME of the time or it's not perception and then it wouldn't be induction.

All this is damaging enough for your premise.

I'm sorry, Daniel, but you haven't even scratched it. The argument was fatally flawed in these ways:

Flawed definition of "axiom."
Meaningless axiom of existence and use of "exists."
Mistaken belief that disorder negates an axiom of order.
Failure to demonstrate that "order" is reducible into components.
Misunderstanding of the role of perception in induction.

But even if we assumed it was all untrue, and you had perceived a particular order correctly, your premise is *entirely misses* Hume's argument, the very nub of which is that there is no logical reason *the order you perceived in past experiences* can be used to *predict future events*!

Nope. The Axiom of Order SUBSUMES Hume's argument, for recurrance is a form of order.

So this premise, even if we grant you it in its strongest form - and there are plenty of excellent reasons not to - is *irrelevant*. I strongly suggest you take time to absorb Hume's arguments before refuting them....;-)

That's nice of you, but I could say the same of my arguments and your attempted refutation. LOL

>Conclusion: This order is likely to occur again under these circumstances.

You appear to have neglected your second premise in your conclusion, which itself seems little more than a rephrasing of first premise, which is basically the idea that order by definition consists of more than one event!

There is a distinct difference you have overlooked. The operative word is THIS. To review my formula:

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.

First an axiom, then a premise about recurrance derived from that axiom, then a perception, then a concusion about THIS order, reasoning from the general to the specific.

So other than defining "order" we do not seem to have got very far in addressing any of the various problems induction faces. And even if you did manage to come up with a better basic argument, your real challenge still lies well in front of you - the task of *formulating a strict principle of induction*.

Define what you mean by that, if you would.

Simply, this would read something like:
"We require ? number of observations to establish a universal law with certainty"
or
"We require ? number of observations to establish a universal law with ?% probability"

You've gone full circle and forgotten that neither are appropriate demands of induction. Induction is induction without claims of certitude or recourse to percentages. They are not in the nature of induction.

There is no-one in the history of science who has been able to successfully formulate the above principle (as an aside. on the basis of this past experience, the sincere inductivist might feel some duty to admit his own principle will never be formulated in future...;-))

For the same reason that no one has formulated a square circle. So long as you see "the problem" of induction, you are not understanding its nature, I'm afraid.

That nature is: CORROBORATION but not certainty, and RELATIVE LIKELIHOOD but not measurable probabilities. It is the numerically undefined classes which give rise to both these aspects of the nature of indiction.

In effect, the critics of induction are saying: "Induction is that form of reasoning from some members of a numerically undefined class to all members. It is invalid because no percentage of certainty can be computed."

That is like invalidating triangles because they are not circles. Or plane geometry because it doesn't work on spheres.

One may as well be saying that "We require the color of induction."

Whereas as I can formulate the principle of falsification as follows:

"We require 1 observation to negate a universal law with 100% certainty"

So what? You still required induction to ascertain what to falsify, and you still require induction to assess the falsifier.

Point to one falsifying event whose observation does not rest in part upon induction. Even if falsification somehow magically solved some imagined problem with induction, you cannot falsify without using it.

Did Popper recognize that fact?

You are of course welcome to keep searching for justifications for the inductive method. But I can't help but ask again why a sharp fellow like yourself feels it necessary to strain nerve and sinew in stapling inductive custard to the ceiling, when an alternative empirical method that *is* logically justified already exists?

What do you imagine this "alternative method" to be? Is "falsification" it?

If so, then you're in trouble. For falsification as a process also depends upon induction.

Nathan Hawking

(Edited by Nathan Hawking on 6/05, 1:58pm)

Post 46

Sunday, June 5, 2005 - 5:34am

You guys are using induction to argue about induction. Would those who think it is not a valid form of forming conclusions please stop using induction from this point forward. Thank you.

Post 47

Sunday, June 5, 2005 - 2:21pm

David:

You guys are using induction to argue about induction. Would those who think it is not a valid form of forming conclusions please stop using induction from this point forward.

I haven't said anything about that until my preceding post, wondering if someone else would notice it. If they did, I missed their post.

A minor point: The only thing I would add is similar to my comment about using "corroborate" for induction rather that "proof" in philosophical writing. Perhaps "hypothesis" would technically be a better word than "conclusion." Conclusion has a connotatation of certitude which doesn't match the nature of induction.

'All men are mortal and Socrates is a man therefore Socrates is mortal' takes the form of a "proven conclusion."

'The sun has come up daily for 4,000 million years and therefore will likely come up tomorrow' is a "corroborated hypotheses".

Unless someone can suggest an alternative for formal use.

Nathan Hawking

Post 48

Sunday, June 5, 2005 - 3:32pm

Nathan (Post 45)
>Rand's definition of an axiom is self-defeating, and leads to all manner of semantic gymnastics with "existence."

Actually I am critical of Rand's axioms for similar reasons. But the fact that 1) you are an Objectivist (I assume), and Objectivists surely have to accept her fundamental definitions and 2) you invoked Rand's fundamental axiom *without* such criticism in your original post lead me to believe you accepted her definition.

But if you have rejected Rand's definition of axioms then that is an interesting position for an Objectivist!

>Your logic is severely flawed. Either order exists in our universe, or it doesn't.

No, your reading of my post is incorrect. I didn't deny order exists. I was simply pointing out that your "axiom" is not axiomatic in Rand's sense, and it's evident - now - you didn't mean it to be anyway.

>Perception is a necessary link in the chain of induction.

Perception is necessary for falsification too. So what?

>I'm sorry, Daniel, but you haven't even scratched it. The argument was fatally flawed in these ways:
>Flawed definition of "axiom."
>Meaningless axiom of existence and use of "exists."

All of which, as you say, are Rand's problem not mine. If you're going to place yourself in her intellectual tradition and invoke her fundamentals as you did - and I do not - you should probably note in advance that you disagree with them!

>Mistaken belief that disorder negates an axiom of order.

See the above confusion. It does, in Rand's sense of "axiom".

>Failure to demonstrate that "order" is reducible into components.

I don't recall trying to, or why I might need to. That "order" consists of "components" seems uncontroversial...;-)

>Misunderstanding of the role of perception in induction.

I am not sure you understand what I mean by 'induction'. See below:

>You've gone full circle and forgotten that neither are appropriate demands of induction. Induction is induction without claims of certitude or recourse to percentages. They are not in the nature of induction.

By induction I mean: the idea that theories develop from repeated observations, and that the more repetitions that have occurred in the past, the more likely they are to occur in the future. But you don't seem to mean the same thing by it, as statements like this show:

>Point to one falsifying event whose observation does not rest in part upon induction.

So perhaps we are talking at cross purposes. Can you explain what *do* you mean by "induction"?

>For the same reason that no one has formulated a square circle. So long as you see "the problem" of induction, you are not understanding its nature, I'm afraid.

The problem of induction is that it is *logically unjustified*. Here you are claiming that this is ok - that, because of its "nature" induction does not need to *be* logically justified.

If you want to argue this, go ahead! But it has certain consequences. For if induction does not need logical justification, but can be appealed to as an authoritative source of knowledge due to *its inherent nature* - *why can't any other illogical source* be appealed to in the exact same way?

- Daniel

Post 49

Sunday, June 5, 2005 - 3:36pm

David wrote:
>You guys are using induction to argue about induction.

As this is not the case as far as I can see, perhaps we are talking at cross purposes. I am arguing that induction as an emprical method is *logically* unjustified.

As I said to Nathan, by induction I mean: the idea that theories develop from repeated observations, and that the more repetitions that have occurred in the past, the more likely they are to occur in the future.

Likewise, can I ask what *you* mean by it?

- Daniel

Post 50

Sunday, June 5, 2005 - 5:18pm

Daniel:

>Rand's definition of an axiom is self-defeating, and leads to all manner of semantic gymnastics with "existence."

Actually I am critical of Rand's axioms for similar reasons. But the fact that 1) you are an Objectivist (I assume), and

That's an understandable conclusion.

I've always been dissatisifed with it, for the reasons already mentioned.

Objectivists surely have to accept her fundamental definitions and

I think it's possible to be an Objectivist and reject her theory of axioms and concepts. What counts is the belief that there IS an objective reality, and that humans can KNOW that reality.

That Rand and I differ on the nature and degree of that knowing doesn't bother me.

2) you invoked Rand's fundamental axiom *without* such criticism in your original post lead me to believe you accepted her definition.

Sure. No problem. As I say, I've always found the formulation of "existence exists" as clumsy and problematic. The point of my thread, however, was something besides existence, so I allowed her formulation to stand for now.

As it turns out, it has become an issue I couldn't avoid in the context of additional axioms--so I won't do that again.

But if you have rejected Rand's definition of axioms then that is an interesting position for an Objectivist!

I'm an unusual Objectivist.

>Your logic is severely flawed. Either order exists in our universe, or it doesn't.

No, your reading of my post is incorrect. I didn't deny order exists. I was simply pointing out that your "axiom" is not axiomatic in Rand's sense, and it's evident - now - you didn't mean it to be anyway.

LOL Your reading of my reading is incorrect. I did not say you claimed order does not exist. What I said addressed your implication (based upon your reasoning on 'existence exists') that disorder somehow 'negates' an order as an axiom.

>Perception is a necessary link in the chain of induction.

Perception is necessary for falsification too. So what?

Look, that's what you said before: so what? I'll tell you what--perception is an integral part of induction.

I have no idea why you keep saying "So what?" I made it part of my logical formulation.

>I'm sorry, Daniel, but you haven't even scratched it. The argument was fatally flawed in these ways:
>Flawed definition of "axiom."
>Meaningless axiom of existence and use of "exists."

All of which, as you say, are Rand's problem not mine. If you're going to place yourself in her intellectual tradition and invoke her fundamentals as you did - and I do not - you should probably note in advance that you disagree with them!

One can't note everything in advance. As you did not note YOUR disagreement with the formulation of "axioms" and "existence exists." Yet you used it.

>You've gone full circle and forgotten that neither are appropriate demands of induction. Induction is induction without claims of certitude or recourse to percentages. They are not in the nature of induction.

By induction I mean: the idea that theories develop from repeated observations, and that the more repetitions that have occurred in the past, the more likely they are to occur in the future. But you don't seem to mean the same thing by it, as statements like this show:

>Point to one falsifying event whose observation does not rest in part upon induction.

I'm not sure whether you're missing the point, Daniel, or avoiding it. I'll check below.

So perhaps we are talking at cross purposes. Can you explain what *do* you mean by "induction"?

I've covered that in other posts. I expect we mean the same thing.

>For the same reason that no one has formulated a square circle. So long as you see "the problem" of induction, you are not understanding its nature, I'm afraid.

The problem of induction is that it is *logically unjustified*. Here you are claiming that this is ok - that, because of its "nature" induction does not need to *be* logically justified.

No, it's not "logically unjustified." That's just the point. By proceeding from a metaphysical axiom one CAN logically justify induction.

The so-called "problem of induction," as you've stated it, consists of illogical demands that induction do something which is contrary to its definition and nature. I have logically justified induction--you are persisting in applying ILLOGICAL (self-contradictory) demands and calling them "logical."

You are simply insisting, as did Hume and everyone I know of since who posits a problem, that "induction be justified" in a way which is contrary to its nature.

If you want to argue this, go ahead! But it has certain consequences. For if induction does not need logical justification, but can be appealed to as an authoritative source of knowledge due to *its inherent nature* - *why can't any other illogical source* be appealed to in the exact same way?

Your premise is incorrect. Induction is logically justified, if one dismisses ILLOGICAL demands like certitude and probabilites.

Besides, a very significant question you (and others) did not answer: Assuming you WERE right about logic invalidating induction , what is your logical justification for logic and deductive reasoning?

What's sauce for the goose...

Care to answer that question this time?

Also, you noticably failed to answer the most important question I asked about the purported "alternative method" to induction:

What do you imagine this Popperian "alternative method" to be? Is "falsification" it? Doesn't falsification as a process also depends upon induction?

Nathan Hawking

Post 51

Sunday, June 5, 2005 - 5:20pm

Daniel,
This will be woefully incomplete, but here goes.

Branden (or perhaps Piekoff) pointed out in lectures long ago (I say this not as argument from authority but simply to avoid taking credit for having originated it) that:

Yes, it's true that induction can not be justified by deduction, since deductive arguments use premises formed by induction, and any attempt to justify induction inductively begs the question. It does not follow that induction is therefore unjustified. That's what Foundationalism is all about. But a more extensive argument will have, unfortunately, to wait.

And perhaps I am misreading you to be asserting that induction is unjustified when you say 'logically unjustified'.

(Actually, if memory serves it was first pointed out by Aristotle. I'll have to refresh my memory on that.)

Post 52

Sunday, June 5, 2005 - 5:21pm

Nathan,

As Daniel points out, the problem of induction is one of logical justification. In your original post, you made a claim regarding something called an Axiom of Order: �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.�

I don�t think there�s anyone here disputing that we regard the universe as exhibiting order, in greater or lesser degree. The question is: what is our justification for this belief? In calling order an axiom, you seem to be arguing that it is in some sense an �irreducible� fact of the universe.

But how do you know this? Since you agree that disorder exists, the so-called axiom can hardly be undeniable, or universal in its application, surely two prerequisites for an axiom. If by irreducible, you mean it depends on no other facts, you would need to demonstrate this without begging the question.

What about order as perceivable? I can perceive various objects, but none of them do I label �order�. The perception of order involves the grouping of perceptions that we judge to be similar in some important way, but in order to do that we need to have in mind the notion that there is an underlying order or regularity in the universe. And that just takes us back to the original question: why order?

I would say your claimed axiom is an assumption, that we have an inbuilt expectation that the universe exhibits order. That�s a very necessary assumption, but it�s not a logical justification, which is what you need to advance in order to establish induction as a logically valid procedure.

Brendan

Post 53

Sunday, June 5, 2005 - 6:30pm

Nathan:
>Can you explain what *do* you mean by "induction"?
I've covered that in other posts. I expect we mean the same thing.

Well, I'm beginning to doubt that. Not that I want to argue over words, but induction is often given a very broad general meaning - equated with empiricism, for example. I am criticising a more specific idea - the one I have stated. So if you could just quickly review what you mean by it, we can ensure we're all on the same page, and this isn't a case of the "narcissism of minor difference"...;-)

>what is your logical justification for logic and deductive reasoning?

Deduction is deductively justified (do you not accept this? If not, let us know!). Induction is *not* deductively justified. In contrast, falsification *is* deductively justified. Further, Popper points out that induction is not even *inductively* justified.

>Your premise is incorrect. Induction is logically justified, if one dismisses ILLOGICAL demands like certitude and probabilites.

Think this through, Nathan, and you will see you just destroyed your own case, and made mine. Because induction is the idea that that the future can be predicted from past events with 1) certainty or 2) probability. Yet you now say that to demand such a thing from induction is "illogical"!

You tell me then: what on earth is the point of the method you are defending so stoutly if it can't tell you anything?

-Daniel

Post 54

Sunday, June 5, 2005 - 6:38pm

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

Brendan:

As Daniel points out, the problem of induction is one of logical justification. In your original post, you made a claim regarding something called an Axiom of Order: �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.�

I don�t think there�s anyone here disputing that we regard the universe as exhibiting order, in greater or lesser degree. The question is: what is our justification for this belief? In calling order an axiom, you seem to be arguing that it is in some sense an �irreducible� fact of the universe.

There is a difference between "necessary" and "irreducible."

The Axiom of Order only posits that order DOES exist, not that it MUST exist.

"Irreducible," as I'm using it, does not mean that order MUST exist, but simply that it has no components, that order is a metaphysical fact which is not comprised of other metaphysical facts.

But how do you know this? Since you agree that disorder exists, the so-called axiom can hardly be undeniable, or universal in its application, surely two prerequisites for an axiom.

First, the axiom of order, as an axiom for THIS universe, IS undeniable--you cannot deny it without using it. The very nature of your sentences is orderly.

Second, axioms are not necessarily universal. The various forms of geometry, for example, use contrary forms of axioms. As a metaphysical axiom, order would not be true in a universe filled with amorphous, formless mush.

If by irreducible, you mean it depends on no other facts, you would need to demonstrate this without begging the question.

But that's the whole point of axiomatization, Brendan. They are posited and assumed, not proven. They cannot be refuted without self-contradiction, nor can they be demonstrated except by ostensive definition.

What about order as perceivable? I can perceive various objects, but none of them do I label �order�. The perception of order involves the grouping of perceptions that we judge to be similar in some important way ..

The first part is false--if you are actually perceiving "objects," then you perceiving order. The boundaries of an object are a form of order.

The second part is true.

, but in order to do that we need to have in mind the notion that there is an underlying order or regularity in the universe.

That's why the axiom of order is an epistemological axiom as well as a metaphysical axiom.

It must be BOTH is we are to perceive order based in objective reality.

And that just takes us back to the original question: why order? I would say your claimed axiom is an assumption, that we have an inbuilt expectation that the universe exhibits order.

Both are true. It is an assumption, and we DO have an inbuilt expectation.

On the first:

Axiom: "2 : a statement accepted as true as the basis for argument or inference."

It is in the NATURE of an axiom to be an assumption.

On the second:

We must have an innate expectation of order if we are even to begin perceiving it. One cannot bootstrap a consciousness or intelligence without that innate expectation. That's why it's also an epistemological axiom. One is the counterpart of the other.

That�s a very necessary assumption, but it�s not a logical justification, which is what you need to advance in order to establish induction as a logically valid procedure.

ALL logic, Brendan, proceeds from assumptions. Even logic and deduction itself.

I have simply identified the elementary fact which makes induction possible and axiomatized it. Induction follows as a LOGICAL consequence of order in the universe and the nature of consciousness. THAT IS it's primary logical justification.

All the alleged "problems" I've seen to date that purport to invalidate induction are really logically flawed--they entail a contradiction of the nature of induction, a demand that something behave contrary to its nature.

Nathan Hawking

Post 55

Sunday, June 5, 2005 - 6:57pm

Daniel:

Nathan:
>Can you explain what *do* you mean by "induction"?

I've covered that in other posts. I expect we mean the same thing.
Well, I'm beginning to doubt that. Not that I want to argue over words, but induction is often given a very broad general meaning - equated with empiricism, for example. I am criticising a more specific idea - the one I have stated. So if you could just quickly review what you mean by it, we can ensure we're all on the same page, and this isn't a case of the "narcissism of minor difference"...;-)

My original post, post 0, covers that.

>what is your logical justification for logic and deductive reasoning?
Deduction is deductively justified (do you not accept this? If not, let us know!).

Can you demonstrate that without recourse to axioms? No.

All logic and deduction is based upon axioms, things we assume.

So your "justification" really represents no more than an internal consistency if one accepts axioms and certain rules for their use.

Induction is *not* deductively justified. In contrast, falsification *is* deductively justified. Further, Popper points out that induction is not even *inductively* justified.

I'd say we've come to the Is-too-Is-not stage of this discussion.

>Your premise is incorrect. Induction is logically justified, if one dismisses ILLOGICAL demands like certitude and probabilites.
Think this through, Nathan, and you will see you just destroyed your own case, and made mine.

Ya think?

Because induction is the idea that that the future can be predicted from past events with 1) certainty or 2) probability.

Who the hell said this? This is a complete inversion of my position.

Yet you now say that to demand such a thing from induction is "illogical"!

It is. I defy you to point to where I said otherwise.

You tell me then: what on earth is the point of the method you are defending so stoutly if it can't tell you anything?

I've said induction can't tell us EVERYTHING, not that it cannot tell us ANYTHING.

I note that once again you are avoiding questions I put to you in the last two posts. This, and the fact that you're beginning to repeatedly invert my position, suggests to me that we're probably past productive discussion.

Nathan Hawking

Post 56

Sunday, June 5, 2005 - 7:17pm

Nathan:
>Induction is logically justified, if one dismisses ILLOGICAL demands like certitude and probabilites.

Let's just revisit this remark for a moment. Formulated into a principle, this would read something like:

"No number of repeated observations can establish a universal law with any certainty or probability, and it is illogical to demand otherwise".

So - debate's over, and you're absolutely right!

- Daniel

Post 57

Sunday, June 5, 2005 - 7:27pm

Nathan:
>I note that once again you are avoiding questions I put to you in the last two posts. This, and the fact that you're beginning to repeatedly invert my position, suggests to me that we're probably past productive discussion.

No, you just agree with me (and Hume) far more than you think!

- Daniel

Post 58

Sunday, June 5, 2005 - 8:06pm

Daniel:

Nathan:
>Induction is logically justified, if one dismisses ILLOGICAL demands like certitude and probabilites.

Let's just revisit this remark for a moment. Formulated into a principle, this would read something like:

"No number of repeated observations can establish a universal law with any certainty or probability, and it is illogical to demand otherwise".

So - debate's over, and you're absolutely right!

- Daniel

Take out the word "any" and I'll agree, if by "probability" you mean computed probability.

Certitude (in the strong sense of that last term) and an exact measure of numerical probability are INHERENTLY NOT in the nature of induction--by definition!

By definition, induction also entails the unknown members of a class. One cannot attain certitude or compute percentages while the size and nature of the unknown portion of a class remain unknown. To demand either is a self-contradiction.

"David Hume addressed this problem in the 18th century in a particularly influential way, and no analysis since has managed to evade Hume's critique. Hume looked at ways to justify inductive thinking. He pointed out that justifying induction on the grounds that it has worked in the past begs the question. That is, it is using inductive reasoning to justify induction."

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

Assuming that order exists and that consciousness can perceive that order (complementary metaphysical and epistemological axioms) completely justifies induction in a logical way which does not beg the question.

One can LOGICALLY reason from the nature of order and the nature of consciousness and arrive at the possibility of induction, then predict that it would both 1) deny certitude or computability with regard to probabilities, and 2) permit corroborated belief in the likely nature of many objective realities.

For my money, it doesn't get any better than that.

Nathan Hawking

Post 59

Sunday, June 5, 2005 - 8:13pm