Two Kinds of "Induction": Important similarities and trivial differences

[BaalChatzaf]

Here's the Table of Contents to The Art of Reasoning by David Kelley to show just how little he understands deduction and induction and how far removed from science he is.

...................snipped to save bandwidth..................................

(Each chapter is further broken down into technical sections, but I did not list those.)

This text is accompanied by an online supplementary tutorial program (see here) and another book called The Art of Reasoning: Readings for Logical Analysis by Hicks and Kelley (containing 44 essays by Bacon, Mencken, Doyle, Freud, Reagan, Rousseau, Machiavelli, Jefferson, and a host of others from a broad spectrum of authors and fields, from literature to case law to science and other areas, to illustrate the different types of reasoning). An Instructor's Manual and Test Bank with 500 problems are also available as supplements.

Michael

It is the section on modern deductive logic that makes the book worthwhile.

While categorical syllogisms are correct, they are so restrictive in their form that they are almost never used to do proofs in a mathematical context. My favorite challenge problem to those who think that logic begins and ends with the categorical syllogism is to ask them to put the proof of Euclid's Proposition 47 (the right triangle theorem attributed to Pythagoras) into a set of interlocking categorical syllogisms. It is a virtual impossibility. No mathematician I have read or spoken to, uses that style of proof. Rather, they use a mode of proof that is readily translated (symbolically) into first order logic, which Kelley refers to as modern deductive logic in the table of contents that you provided (and I snipped).

I have not read Kelley's book but if one can judge by the t.o.c. it looks comparable to Copi's textbook, which is widely used in colleges. Perhaps I will see if I can get it from my local library and scan it. Thanks for posting the t.o.c.

Ba'al Chatzaf

[tjohnson]

I think this whole "problem" with induction stems from not differentiating between mathematics and natural language. For example from Wikipedia;

For all natural numbers n, 2·n = n + n

is an example of a universally quantified statement which is also true only because we can semantically include ALL possible numbers, n. This is the important difference between mathematics and natural language. But it makes no sense to speak of a universally quantified statement in a natural language, like physics, for example, because we cannot observe ALL of "the existents" (to use objectivist terminology). This is why our "universal' statements in natural language must be considered as only relatively (or contextually, etc.) true.

This particular example is true, because you could put any natural number in for n and the statement "2·n = n + n" would be true. In contrast, "For all natural numbers n, 2·n > 2 + n" is false, because if you replace n with, say, 1 you get the false statement "2·1 > 2 + 1". It doesn't matter that "2·n > 2 + n" is true for most natural numbers n: even the existence of a single counterexample is enough to prove the universal quantification false.

This illustrates Popper's falsification principle. We should note that it applies in both mathematics and natural language but in different ways.Since there is no such thing as a TRUE universally quantified statement in natural language the falsification only works as a method of eliminating structural dissimilarity and not establishing absolute falseness.

Edited October 14, 2007 by general semanticist

[BaalChatzaf]

This illustrates Popper's falsification principle. We should note that it applies in both mathematics and natural language but in different ways.Since there is no such thing as a TRUE universally quantified statement in natural language the falsification only works as a method of eliminating structural dissimilarity and not establishing absolute falseness.

I see you are up to your usual standards. Try this one: all bachelors are unmarried. That is a universal proposition, it is true and it is in natural language.

Taking too many lessons from The Count may cause brain-rot.

Ba'al Chatzaf

[tjohnson]

I see you are up to your usual standards. Try this one: all bachelors are unmarried. That is a universal proposition, it is true and it is in natural language.

Taking too many lessons from The Count may cause brain-rot.

What do you mean by 'bachelor' and what do you mean by 'married'?

[tjohnson]

In relation to mathematics Bertrand Russell has said: "Logic is the youth of mathematics, mathematics is the manhood of logic." This brilliant mot of the eminent philosopher of mathematics is no doubt just and is profoundly significant; the least it can teach us is that it is useless to try to find a dividing line between logic and mathematics, for no such line exists; to seek for one serves merely to betray one's ignorance of mathematical philosophy.

I guess this make you ignorant of mathematical philosophy Baal, no wonder you don't like Korzybski.

[tjohnson]

In a previous post I said "But it makes no sense to speak of a universally quantified statement in a natural language, like physics, for example, because we cannot observe ALL of "the existents" (to use objectivist terminology)." Well it's worse than that - we can't even include all the characteristics in our definitions of these 'existents'. So in Baal's example of "all bachelors are unmarried" it is not possible to define absolutely what we mean by 'bachelor', for example - we went through this with the discussion about Popper. Our classifications of things in natural language do not have "sharp edges" like the Venn diagrams we use in set theory. There are always cases where there is some question whether or not a particular object should be regarded as a member of the set or not. For example, if you define 'bachelor' as a man who has never been married what if the 'man' used to be a woman and had a sex change operation? What if 'she' had been married before?

[John Dailey]

Ellen:

~ In your post #442, you assert...

It is logically inconsistent...to claim certainty for one's empirical generalization.

Well, speaking in terms ONLY of 'deduction', I guess so; if that's all one sees 'logic' as consisting of. I do not; but, that's 'the rub' in all this no? Begging the question establishes nothing. --- I suspect that you would not agree that THAT quote of yours is a 'generalization.' I suspect that you would NOT logically 'argue', er, soritely show the deduction, (though you clearly innuend) that you are 'certain' of it (generalized or specific-only-to-moi.)

~ In that post, you start off with "I recommend thinking about it until you see it." --- HOW does one 'think' about that which they yet do not 'see'? I recommend that *you* 'think' about what you...advise.

~ Let's get a bit more cogent.

LLAP

J:D

[John Dailey]

Ellen:

~ In that same post, you end with how you see 'the problem of induction'; to wit...

When do you know your empirical generalizations are correct? (That's how AR states the problem as well, btw.) The answer is, You don't, ever.

~ Fine. Ok. Guess what my next question is, re your 'answer'? Nm; I'll spell it out...

~ My question is: "Is the 'You' which you specify therein, one you find applicable to others OTHER THAN *you* (aka: not ONLY *you*)? If so...uh...why and how, without any Humeanly-worried 'generalizing' or Mill'sian procedures, should such be considered as...'empirically'...applicable to 'others'? --- We're not sliding into solipsism here, I hope, because, if so, (aka: ONLY *you*), that's where this subject is headed, and, well, that obviously leaves me out of it.

LLAP

J:D

Edited October 15, 2007 by John Dailey

[John Dailey]

Ellen:

~ Again, in that same post, you point out *your* view of 'knowledge' (er, 'certainty'? Not sure which, nor even if it's their lack you mean), to wit, re

the conservation laws, the bedrock bedrock of modern physics aren't known for certain to be true.

I think we need to distinguish, since clearly distinction need be made, re the use of the terms 'certain' and 'knowledge'; I mean, that statement is really confusing, as put. Does 'uncertain' KNOWLEDGE really have any meaning? If not, you're being confusingly redundant here. If so, c'mon: 'tentative' knowledge (aka: 'suspicions') considerations merely confuse this subject unnecessarily.

~ I gather you don't mind the word 'context'...unless the term 'certain' or 'knowledge' is added; correct? If not, is 'knowledge' therefore...non-'contextual'? Also, if not, is 'knowledge' therefore not 'certain'? (This is apart from the question about 'scientific "laws"' being more than mere postulates.)

LLAP

J:D

[John Dailey]

Ellen, ADDENDUM:

~ Just trying to get a definitional clarification of *your* use of these terms...in this subject's context.

LLAP

J:D

[John Dailey]

Ellen:

~ You specify...

being used as a part of a thinking process is not the issue; the issue is validation...

This reminds me of when I read Skinner's BEYOND FREEDOM AND DIGNITY where he, 2/3rds through the book asserted that free-will 'was not the issue.' It was, and he made clear there and then that he wasn't going to deal with it. I see the same here, Ellen. INDUCTION IS a thinking process which IS 'the [original!] issue.' It's just not the one you wish to deal with...Here. You wish to change it to...

the issue is validation, ie., proof (that's the sense in which "validation" was being used) that one's generalization is correct, that it holds, always has held, always will hold without exception.

~ I like your tacking on of that last, (can we say begging the whole 'induction' subject-question?) part. Your Popper preference is causing you to do a 'switch-and-bait' re this thread's subject matter.

LLAP

J:D

[John Dailey]

Ellen:

~ "Validation" has everything to do with repetitively confirming a pattern of consistency...of what's happened IN THE PAST. Indeed, 'validation' is merely a synonym for 'confirming'. It has nothing to do with discovering anything new in anything, much less 'science;' nothing to do with putting 2 and 3 together and getting more than '23.' It has nothing to do with 'fitting' a new item with all old ones. --- Such must be left to 'trial-and-error' ("let's see what happens if we randomly try this..."), serendipity ("WHOA-H! Lookit THAT!"), very rare 'intuitive' insight ("EUREKA! I STUMBLEY FOUND IT!")...or...Induction (think Al's 'theory' wasn't? Methinks one should read him [his own book] again.)

LLAP

J:D

[John Dailey]

~ I gather that many just don't (or is it won't?) see Feynman's 'thinking process' followed by his 'show-and-tell' re the O-Rings as 'validating' (not to mention deducedly 'showing') his...Induction about the cause of the Challenger fiasco. --- So be it.

~ Yes, I imply that 'INDUCTION' is validatable...just not sensorially (aka: 'empirically', in this term's myopic useage.) I've implicitly argued such throughout re how others have argued AGAINST its useability, and, its useage.

~ Clearly, for some, 'validation' need, as per Hume, be unconcerned with conceptual validifying/validating, but, need only sensorially 'empirical' observings. Why is Mill (who was as 'empirical' in this context as one could get?) so consistently ignored? Pardon my 'induction', but, probably because he accepts 'causes', whereas Hume didn't (nor his followers do.)

LLAP

J:D

[John Dailey]

~ Sheesh! I'm just getting into the 'Gardner-vs-Popperians' seguing and have a need to comment about the gist of the commentings.

~ 'Falsification' is a great PART of scientific methodology! It's better even than mere 'confirmation' (verification, if you will.) It affects a hypothesis (and even that over-used term, 'theory'.) --- But, it ADDS nothing to any explanatory hypothesis, 'theory', 'LAW', whatsoever. It thereby lacks in usefulness re 'explanatoriness.' Without the latter, there are no DIRECTIONS to pick out as preferable to expend effort in attempting to acquire new 'knowledge.'

~ Only a 'hypothesis' which shows a 'confirmed' prediction (making it a 'theory') does. The prediction is 'induced', falsely or truely, mathematically (Dirac's/Einstein's/Hawking's) or conceptually (Aristotle's/Smith's/Rand's), and confirmed (by some's criteria [!])...or not.

LLAP

J:D

[John Dailey]

gs:

~ If you think Martin Gardner was a 'bottom feeder'...you obviously haven't read much by him. Think twice 'fore you speculate castigations. He was a 'thinker'/analyst par excellence as Dragonfly and Baal can tell you. He was a routine article writer in SCIENTIFIC AMERICAN 'fore it went to 'the masses', and after my 1st-yr subscription became the predominant reason I suscribed for the next decade +. He knew how to analyze a prob in science perplexities, and not just myopically parse how the prob should be worded for proper terminological-analysis.

LLAP

J:D

[John Dailey]

DB:

~ Re your post #480 about MG's belief in God: he was pushed to admitting that, and clearly he didn't want to discuss it. Re an analogy on your comment therein on his admission...

~ Einstein did believe in some 'Logos' which he called "the Old Man." Guess that makes even him a 'non-starter' in his scientific theories, huh? Like, we should scratch "E=MC2" and start over since his 'reasoning' was false and he just 'lucked out' into this nuclear-energy stuff?

~ Man, you guys really don't know who you're talking about! (But, considering most Rand-criticizers, what's new?)

~ I just hate it when someone gets 'criticized' for stuff irrelevent to what they've produced. Shall we talk about Popper's 'love-life' or shall we skip to Martha Stewart's immorality? Or can we just skip 'gossip' irrelevencies altogether...at least in THIS thread?

LLAP

J:D

Edited October 15, 2007 by John Dailey

[John Dailey]

MSK:

~ How you 'deal' with all this...stuff, I can only wonder. And I'm only talking about THIS thread!

~ At this point of my reading herein, were I too have to monitor all, I don't know if it would put me TO sleep, or keep me FROM sleeping.

LLAP

J:D

Edited October 15, 2007 by John Dailey

[Daniel Barnes]

~ Einstein did believe in some 'Logos' which he called "the Old Man." Guess that makes even him a 'non-starter' in his scientific theories, huh? Like, we should scratch "E=MC2" and start over since his 'reasoning' was false and he just 'lucked out' into this nuclear-energy stuff?

JD,

Unfortunately it's you who isn't following the debate. You've got the wrong guy. It was MSK who appealed to Martin Gardner's reputation as a science writer as if it somehow validated Gardner's logical and philosophical criticisms of Popper.

But Gardner is neither a logician nor a philosophical writer, and his talents in science do not validate his opinions here, any more than they validate his belief in God. While he's done good work in popular science writing he stone-cold crapped out on this one. His errors are quite clearly articulated by Jan Lester in the reply cited.

So you've mistakenly put forward as my argument the very type of argument I'm criticising; that is, because someone is right about one thing, they're automatically right about another! That's MSK's line of argument, mate, not mine. You've got it backwards. Sound off at him.

~ Man, you guys really don't know who you're talking about! (But, considering most Rand-criticizers, what's new?)

Looks like in fact the boot's on the other foot. Read in haste, repent at your leisure.

Edited October 15, 2007 by Daniel Barnes

[Michael Stuart Kelly]

Unfortunately it's you who isn't following the debate. You've got the wrong guy. It was MSK who appealed to Martin Gardner's reputation as a science writer as if it somehow validated Gardner's logical and philosophical criticisms of Popper.

But Gardner is neither a logician nor a philosophical writer, and his talents in science do not validate his opinions here, any more than they validate his belief in God. While he's done good work in popular science writing he stone-cold crapped out on this one. His errors are quite clearly articulated by Jan Lester in the reply cited.

So you've mistakenly put forward as my argument the very type of argument I'm criticising; that is, because someone is right about one thing, they're automatically right about another! That's MSK's line of argument, mate, not mine. You've got it backwards. Sound off at him.

Daniel,

This is a perfectly stupid post. I merely checked Gardner's credentials when he was being denigrated as a bottom feeder and said he didn't sound like one. I dare you to cite one post I have ever made anywhere claiming that if "someone is right about one thing, they're automatically right about another." You won't find it because I never said it. Ever.

Like I said, you used to be good. Now you are making things up for some damn reason.

Michael

[Daniel Barnes]

This is a perfectly stupid post. I merely checked Gardner's credentials when he was being denigrated as a bottom feeder and said he didn't sound like one. I dare you to cite one post I have ever made anywhere claiming that if "someone is right about one thing, they're automatically right about another." You won't find it because I never said it. Ever.

Well, obviously you came out with this very argument right here!

Specifically your:

MSK:

Hey... I didn't really bring it up. It was published in a science vehicle with broad circulation by a respected writer. I merely reported it since I found it suddenly pertinent to something I have noticed in what little I have read of Popper's works. Gardner is some non-starter, huh?

Let's follow your argument here:

Gardner is a respected science writer...therefore ...his claim about Popper that you cite (and that I referred to as a "non-starter") must have merit.

Well...surprise, surprise, it is a total non-starter! Ba'al pointed this error - which was logical - out in a heartbeat. So much for your point. And as for your "I didn't really bring it up...I merely reported it..." - I'm still chuckling. I'm not really bringing up the fact you're not kidding anyone, merely reporting it...

Like I said, you used to be good. Now you are making things up for some damn reason.

Didn't make up a thing, as it turns out. Can I help it if you don't recall your own arguments?

Edited October 15, 2007 by Daniel Barnes

[Michael Stuart Kelly]

Daniel,

You're stretching. Now you come with this "non-starter" crap to change the subject? My remark didn't even mean that the guy was right. Do I need to remind you of your words?

So you've mistakenly put forward as my argument the very type of argument I'm criticising; that is, because someone is right about one thing, they're automatically right about another! That's MSK's line of argument, mate, not mine.

Gimme a break.

You're wrong. Get over it.

Michael

[Ellen Stuttle]

It would be interesting to see you quote where I said I hadn't read David's book on logic (or any of his other books).

And answer, thus far, came there none explaining your understanding of where you "see the logical problem with universally quantified statements."

Ellen,

You mentioned you had not seen Kelley's beta manuscript of the Logical Structure of Objectivism, not The Art of Reasoning. The fact is you made no reference at all to the second book until I did. You just now say you read it, so I'll take you at your word, but from what I have seen skimming Kelley's book, it makes hash out of your smarmy comment about science not being his thing (made in agreement with the idea that he did not know what he was talking about as regards induction/deduction validating each other).

I'm glad for the acknowledgment that I'd made no reference to The Art of Reasoning. Technically, however -- and here I see a misunderstanding occurred -- I wasn't referring to the "beta manuscipt of The Logical Structure of Objectivism" either, since I didn't realize that that's what you were talking about. I haven't seen that, but I thought you were talking about some other work-in-progress of David's. You had made the comment (here):

I vastly prefer the approach David Kelley has used so far, validating induction with deduction and validating deduction with induction in return. The two are not standalone for scientific validation. They work together.

Daniel replied (here):

Kelley's monograph is not complete, so it is hard to judge, but from what I have seen so far his argument is no less feeble than Peikoff's.

I thought from Daniel's wording that the work being talked about was something David was writing attempting to address the historic "problem of induction" (i.e., how could you ever demonstrate the truth of a universal -- for all time, past, present, future -- empirical generalization?). A successful attempt to answer this question (could it be answered successfully, which I hold we know on grounds of logical contradiction it couldn't be) would need some significant expertise in science. AR herself thought the same, I'll remind you, so on this point I'm not in disagreement with her:

ITOE, 2nd Expanded, pp. 303-04

Prof. M [Larry]: The question is: when does one stop? When does one decide that enough confirming evidence exists? Is that in the province of the issue of induction?

AR: Yes. That's the big question of induction. Which I couldn't begin to discuss--because (a) I haven't worked on that subject enough to even begin to formulate it, and (b ) it would take an accomplished scientist in a given field to illustrate the whole process in that field.

There isn't anything "smarmy" in my comment about David's not having the scientific background. He doesn't have that background. I regret, however, that I said anything at all about David. He was part of my circle of O'ism-connected friends in the old days; I remain on friendly terms with him, though I haven't seen him since summer 2000 (Larry's seen him a few times more recently); I have a lot of respect for his intellect, and I don't want to debate the details of his background with you.

[...] the concept of quantification in any degree in Objectivist language is a measurement. (Ta daaa!!! Here we go again!!!) The way Rand would put it, the unit must be present in some quantity, but may be present in any quantity. In a certain manner of speaking, a universally quantified statement is measurement omission incarnate, and very much used (but not acknowledged) in Popper's idea of vagueness, going from his examples.

That shows by how wide a degree you aren't understanding the lingo.

As to your expression of interest in "[my] understanding of measurement in Objectivist concept formation," I've already said that I have no desire to get into it here. I think her theory of concept formation is mostly wrong. I hope to have time to write a piece on the subject.

As of this time, I have no more time for even trying to keep up with reading the posts here. Serious scientific meeting this weekend, and much still to do after that before Thanksgiving.

Ellen

___

Edited October 16, 2007 by Ellen Stuttle

[Brant Gaede]

If it's not a logical fallacy, who cares if it's induction and deduction validating each other? This is tail chasing at best.

--Brant

[Michael Stuart Kelly]

I'm glad for the acknowledgment that I'd made no reference to The Art of Reasoning. Technically, however -- and here I see a misunderstanding occurred -- I wasn't referring to the "beta manuscipt of The Logical Structure of Objectivism" either, since I didn't realize that that's what you were talking about. I haven't seen that, but I thought you were talking about some other work-in-progress of David's. You had made the comment (here):

I vastly prefer the approach David Kelley has used so far, validating induction with deduction and validating deduction with induction in return. The two are not standalone for scientific validation. They work together.

Daniel replied (here):

Kelley's monograph is not complete, so it is hard to judge, but from what I have seen so far his argument is no less feeble than Peikoff's.

I thought from Daniel's wording that the work being talked about was something David was writing attempting to address the historic "problem of induction" (i.e., how could you ever demonstrate the truth of a universal -- for all time, past, present, future -- empirical generalization?).

Ellen,

I have little doubt Daniel was referring to a post I made on September 25 where I quoted from the beta version. Incidentally, you can download the entire beta version of The Logical Structure of Objectivism for free here.

[...] the concept of quantification in any degree in Objectivist language is a measurement. (Ta daaa!!! Here we go again!!!) The way Rand would put it, the unit must be present in some quantity, but may be present in any quantity. In a certain manner of speaking, a universally quantified statement is measurement omission incarnate, and very much used (but not acknowledged) in Popper's idea of vagueness, going from his examples.

That shows by how wide a degree you aren't understanding the lingo.

Heh.

I notice that after giving a challenge about universally quantified statements based on the presumption that I would get it wrong (and being VERY CLEAR in your insinuations of that presumption in more than one post), you did not mention my answer in the classical jargon, i.e., "In the formal language of predicate logic, the logical problem is that they do not allow the domain of discourse to be conditioned enough to make absolutely true or false statements." That answer is given in my words (although the technical terms came from study) and I actually do understand the lingo. I simply see what Rand saw and it is as clear as daylight.

When a person as intelligent as you finds difficulty with the idea of quantification being measurement, we really are on different pages.

As to your expression of interest in "[my] understanding of measurement in Objectivist concept formation," I've already said that I have no desire to get into it here. I think her theory of concept formation is mostly wrong. I hope to have time to write a piece on the subject.

That is precisely the page I think you are on (proving Rand wrong) and I have stated this several times. I look forward to your piece.

Michael

[BaalChatzaf]

That is precisely the page I think you are on (proving Rand wrong) and I have stated this several times. I look forward to your piece.

Michael

Did Rand deal with concept formation by means of metaphor or analogy? That is where most of the mathematical ideas come from. They very rarely arise by comparison and contrast. How, using Rand's concept formation ideas does one come up with spaces with uncountably many dimensions, for example? Or bottles with no inside or outside (Klein Bottle). Or surfaces with only one side (Moebius Strip)? Or the Banach-Tarski conundrum of taking apart a pea and re-assembling it as a sphere bigger than the sun? I would say Rand had very little experience or grasp of how concepts are formed in mathematics. Dismissing these concepts curtly as (mere) concepts of method does not really deal with them. The science of physics is built -largely- on concepts of method. Very little concrete concept formation by comparison and contrast is done in theoretical physics.

The major technique is idealization which is a form of analogy. That is how Newton came up with his first law of motion. Where in the cosmos do you find bodies on which there are no forces acting? Answer; you don't. But Newton used this notion quite profitably.

Ba'al Chatzaf