Thursday, March 31, 2011

On the differences between deduction and induction

I’m finally taking the time to comment on a December post from the Maverick Philosopher, this one concerning the nature of deduction and induction. Dr. Vallicella makes some good points, but seems to overlook a key distinction.

Dr. Vallicella begins in a dialectic form, explaining why the usual notions of deductive reasoning moving from the universal to the singular and inductive moving from the singular to the universal are inadequate. I generally agree with his comments on this. However, I would hope in a discussion designed to illustrate reasoning, a better choice would be used for the example of a deductive argument that goes from the singular to the general than the use of a contradiction of singulars to derive a universal general, a usage even Dr. Vallicella acknowledges is artificial. A better choice would have been an argument that is at least hypothetically sound, rather than valid yet vapid. For example, from the singular propositions John is the only chess player born on Feb. 29, 1992 and John is fat, you could come to universal conclusion that every chess player born on Feb. 29, 1992 is fat without having to rely on a contradiction.

However, it is with the last paragraph that I find the thinking clumsy. In saying "To be a bit more precise, a deductive argument is one that embodies the following claim: Necessarily, if all the premises are true, then the conclusion is true", Dr. Vallicella basically removes the meat from the sandwich. A deductive argument is a demonstration of that relationship between the premises and the conclusion by the use the accepted rules of argumentation. That is the separation from the inductive argument, which makes no offer of demonstration except example. In the example of "All As are Bs; All Bs are Cs; ergo, All As are Cs", this is a straight-forward syllogism, and we use the rules of inference to determine it. My example is a little more complex syllogistically, but is straightforward in symbolic logic.

Also, while "the universal to the singular" and "the singular to the universal" are not strictly true, they are simplifications of a more accurate understanding of the scopes of the conclusions. For example, if you interpret propositions as statements about sets, deductive arguments are basically arguments that the set of the conclusion is a subset of the intersection of the sets containing the premises. That is, A∧B ⇒ C is another way of saying C ⊆ A∧B. So, deductive arguments go can decrease their inclusivity from premises to conclusion, but never increase it. Meanwhile, inductive arguments increase inclusivity, by extending membership in a set to something when it was not previously considered a member of that set. So, this is another good way of looking at the difference between deductive and inductive arguments, even if the word choice Dr. Vallicella rejected is inferior.

20 comments:

aintnuthin said...

Vallicella said: "If the claim is correct, then the argument is valid, and invalid otherwise."

This is not the terminology I learned in logic, and I think it is misleading. Where I come from, the "validity" of a deductive argument is quite independent of the "correctness" of the premises.

A deductive argument can be both (1) valid and, at the same time, (2) contain "incorrect" premises. In such a case the logical argument is said to be "valid," but "unsound" (as opposed to "invalid"). As I think he said elsewhere, the validity of a deductive argument is determined by its form alone, irrespective of it's content.

I may be misreading his intended meaning, which seems somewhat ambiguous to me, though.

As I said in another thread, some have argued that all inference is ultimately deductive. I guess the idea here is that there are always tacit assumptions involved in any conclusion. Take, for example, this "inductive" argument:

1. Every member of Jazzfans I've ever seen is fat.
2. Eric is a member of Jazzfans.
3. Therefore Eric is fat.

Throw in one more (tacit) premise, as follows, and you have a deductive argument:

3. What is true of every member of Jazzfans that I've ever seen is true of all members of Jazzfanz.

Others have argued that all logical reasoning is ultimately inductive. I guess the argument for that position goes something like this:

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

At bottom, the first premise, though universal in form, is actually inductive in nature. Because every man I've ever heard of is mortal, I conclude that Socrates must be mortal. But, who knows, Socrates may be the exception and be immortal. Until I see him die, I can't confidently (i.e., absolutely) conclude that he is mortal.

I not sure that it's very important catagorize every argument as either inductive or deductive, but if it is, the distinction may not ultimately be that clearcut.

One Brow said...

aintnuthin said...
Vallicella said: "If the claim is correct, then the argument is valid, and invalid otherwise."

This is not the terminology I learned in logic, and I think it is misleading. Where I come from, the "validity" of a deductive argument is quite independent of the "correctness" of the premises.


The claim he was referring to at that point was the claim that the truth of the premises ensured the truth of the conclusion, i.e., the claim of the validity of an argument.

I not sure that it's very important catagorize every argument as either inductive or deductive, but if it is, the distinction may not ultimately be that clearcut.

When you take deductive reasoning out of it's domain (a formal system) and try to use it in another domain (empirical world), naturally the results will look murky.

aintnuthin said...

One Brow said: "The claim he was referring to at that point was the claim that the truth of the premises ensured the truth of the conclusion, i.e., the claim of the validity of an argument."

Yeah, that's what I took him to mean too. That's why I said that it was misleading and that his terminology was wrong. Valid deductive logic does not depend on, and it fact has nothing to do with, the "truth" of the premises. So he is stating this in a confused (and confusing) way.

Even if one premise is not true, the reasoning could still be valid and the conclusion could still "necessarily" follow. It is a mistake to even try to explain why deductive logic is valid (or invalid) by reference to the truth/falsity of the premises.

aintnuthin said...

One Brow said: "The claim he was referring to at that point was the claim that the truth of the premises ensured the truth of the conclusion, i.e., the claim of the validity of an argument."

His claim should have been stated as: "...the claim that the truth of the premises ensured the truth of the conclusion, i.e., the claim of the SOUNDNESS [not "validity"] of an argument."

He is confusing soundness with validity.

aintnuthin said...

Sometimes a concrete example helps (then, again, sometimes it doesn't):

The moon is made of green cheese.
Mice can eat cheese.
Therefore mice can eat the moon.

This argument is logically valid. It is not "invalid" just because the first premise (and therefore the conclusion) is false.

Although it is "valid," it is unsound, precisely because it contains a false premise.

The argument is valid, but unsound.

One Brow said...

aintnuthin said...
Even if one premise is not true, the reasoning could still be valid and the conclusion could still "necessarily" follow. It is a mistake to even try to explain why deductive logic is valid (or invalid) by reference to the truth/falsity of the premises.

I agree, and I think Dr. Vallicella would as well. He is attemting contrast, not justification.

michael- said...

What about fuzzy logic? Isn't each word hooked up to so many other entailing concepts? Logical structure could thus go wrong at the very root of each concept?

One Brow said...

Welcome, michael-.

When you are working in a fuzzy logic, you set up the rules of deduction before you begin, just like in classical or predicate logic. It doesn't call for fuzzy definitions, just fuzzy truth values attached to precise defintions. It's not as simple as classical logic, but once you have a solid set of rules laid out, it can't really go wrong.

On the other hand, fuzzy terminology could exhibit that sort of connectedness where everything goes wrong together.

aintnuthin said...

One Brow said: " He is attemting contrast, not justification."

Well, whatever he was attempting, what he said was: ""If the claim is correct, then the argument is valid, and invalid otherwise."

Nice try, but....

One Brow said...

The claim under discussion was that the truth of the premises ensured the truth of the conclusion. A valid argument is exactly that. He referred to it as a claim because, once in a while, someone says that A => B but doesn't do the proof correctly.

aintnuthin said...

One Brow said: "The claim under discussion was that the truth of the premises ensured the truth of the conclusion. A valid argument is exactly that."

Hmmmm, I don't know any other way to say what I've already said....but I guess I'll try.

For one thing, he went on to say that the argument would be INVALID if one of the premises was not true. This is simply wrong, as a general proposition, and again, it is a mistake, from the outset, to attempt to explain validity by reference to the truth/falsity of the premises. It's like trying to explain what a grass hut is by saying "it has a roof." All enclosures have roofs, but that doesn't make them grass huts. When you say: "A valid argument is exactly that," it's kinda like saying that a enclosure with a roof is "exactly a grass hut."

Granted, it will always to be true, that if you have a valid deductive argument, and if the premises are true, then the conclusion will be true. It is also true that a grass hut will always have a "roof." But this observation is essentially what distinguishes a sound argument from an unsound argument (because the argument would be "unsound"--not necessarily invalid--if one of the premises were false).

In short, I can't agree with your claim that "A valid argument is exactly that." Having a roof may be one aspect of a grass hut, but a roof is not "exactly" a grass hut.

I'm not just quibbling. I think it's important to understand that proving a conclusion to be false does NOT prove that the reasoning which led to the conclusion was deductively invalid. Something is not "illogical" just because it may turn out to be untrue. Some people actually do think that's what "logic" means, but they are confused, and don't understand logic.

aintnuthin said...

One Brow said: "I don't deny SR's conclusion, because SR's conclusion does not rely on absolute motion. I just your interpretation of that conclusion, which does."

Eric, it is quite typical of you to ignore all content of something someone says and for you to reject, without further thought, everything a person says, based on the way YOU define a word. You don't give a rat's ass about what the other person means by a word, the word (as ONLY you define it) decides the issue for you. That's why I asked this question:

What do you think "absolute" means, I wonder?

You responded: "That there is some inertial state that represent "really" not moving, and all other inertial states are really moving."

Of course, prior to that response, I had already told you that is NOT what I mean. Does that in any way lead you to reconsider or re-evalute what I have said? Naw.

Not that it would matter, when you ask questions like this:

One Brow asked: "If you can't put a number on the speed of Pioneer or on our speed, how do you know Pioneer is faster?"

Are you serious? You ARE serious, aren't you? I dunno, how can I tell which horse is winning the Kentucky Derby if I haven't make a precise calculation of the speed at which they are running?

You have some very confused thinking going on. I will give you the benefit of the doubt and assume that these questions are based on some assumptions (and not just arbitrarily asked for the sense of disruption). But what those assumptions could possibly be are beyond my ability to speculate. It does all kinda suggest some underlying platonic metaphysical views, though. Like it's either absolute or else it's pure illusion, ya know?

aintnuthin said...

Oops, wrong thread. Lemme repost, eh?

J said...

hey OB.

However, it is with the last paragraph that I find the thinking clumsy. In saying "To be a bit more precise, a deductive argument is one that embodies the following claim: Necessarily, if all the premises are true, then the conclusion is true", Dr. Vallicella basically removes the meat from the sandwich

Actually Mav-P. is not only clumsy here, he's WRONG. A deductive argument is VALID, because the correct form is used. Truth concerns the premises, or rather the premises are either true (tautologically), or a matter of verification. Validity of the argument form--which is to say conclusion-- is, thus, separate from the truth of the premises (ie, that's what proofs are about--showing the conclusion follows from premises. Not proving the premises are "true"). We can have valid arguments with dubious, or whimsical, or even ..false premises, as anyone who ever read and understood a bit of Lewis Carroll knows. Modus ponens is a valid form, even with whimsical premises:

If X's are glubs, then X's are Tworped.

X's ARE Glubs!

X's are Tworped!

valid, however ridiculous, but..unsound (ie the premises are meaningless..or is it false). A bit obvious, but philosophasters--even the elite sort like Billy V--continually conflate the validity issue with the truth issue. The Truth issue is really the ..inductive/empirical issue.

Actually I think Vallicelli's got mental issues, OB. He needs help.

One Brow said...

J,

Thanks for stopping by. I agree mostly with what you say about the logic, but I do want to add that sometimes, proofs are actually about proving a particular conjuntion of premises false, to dervie the negation of one premise as being the reslt of the others (in a classical two-valued system).

Also, while I certainly disagree with Dr. Vallicella on many issues, I think it is unfair to say he conflates truth and validity. He's quite careful to distinguish them.

J said...
This comment has been removed by the author.
J said...

A Reductio ad Absurdum? AS you probably are aware, with a RaA one starts by negating the conclusion and showing that a contradiction ensues (ergo, the original conclusion was valid). But the truth status of the premises is not established...merely that a particular conclusion follows from a set of premises, or not--tho perhaps inconsistency ..or bad form may be shown .


One reason I sort of admire Quine's Logic is that he nearly always used premises from natural sciences, or mathematics (however basic) in his deductive arguments . No whimsy, theology, or ghost metaphysics (ie, the Vallicelli MO).

It's also important to note that inductive reasoning is not necessary in the sense that formal logic is. (IM not sure MAv P gets that really--many catholics or crypto-catholics don't). It's a safe bet the freezing point of H20 will be 32 F for the next few dozen centuries and that the Sun will continue to exist-- but ...not necessarily true in the sense of the Law of the excluded middle, or...pythagorean theorem, etc.

Aquinas's arguments are themselves empirical-inductive IMHE--based on the older physics, mostly--, (apart from the "ontological"--even that crypto-inductive in a sense) --another point papists tend to overlook

One Brow said...

J,

I think what you said about raa is largely what I said.

There are reasons to prefer using empirical premises as axioms (as well as reaons to prefer using personal truths). Using mathematical theorems as starting points is really just starting in the middle, though, as mathematics is every bit as artificial a construct as as Aristotle's metaphysics.

I don't know about other Scholastics, but certainly Dr. Feser acknowledges an empirical basis for his metaphysics, albeit he feels the axioms selected are so obvious that they can be held in certainty. I disagree with ther last part, naturally.

Anonymous said...

Off topic.

Hello Hopper. I miss you. Please reinvent yourself. They think Millsapa is you and this should not stand.

Hello One Brow. I always enjoy your blog.

One Brow said...

Anonymous,

Thanks for stopping by. I agree anyone who really thinks aintnuthin is Millsapa is not thinking clearly, but many of the posters are merely treating it as a standing joke (just like the one where I am supposed to aintnuthin and/or JAZZFAN_2814 occasionally gets dragged out).