Definitions of Induction and Deduction -- Help a Confused Beginner

[drkitten]

Thanks. Let me ask one more embarrassingly basic question:

When you say "which they did basically by denying that "all" statements had any such implications and allowing people to add in existential import via conjunctions," what, precisely, do you men by existential import? Do you mean that the composer of the argument can now stipulate a non-empty set?

Broader than that. Under Aristotelian logic, every concept you name must have an instance; the idea that I could put together a meaningful set of words that referred to nothing didn't exist.

Under modern logic, people can name and manipulate concepts without necessarily having instances of them in mind.

 

[drkitten]

drkitten, if it bemuses you, what do you think about having a specific quantifier for each of the statement types as in my "Mega Logic"? Due to my sz I need to think very specifically about things due to Voices and delusions and so need to be very conscious of fictions (empty sets) and facts (non-empty sets). So I needed a specific statement I could use for every possible case. That's how I came up with Mega Logic to help me do this.

I think it's notational overkill; as you point out, these 24 types arise in rather specific combinatorics. Why not specify the combinatoric primitives instead?

This is particularly true since some of your distinctions are synonymous in ordinary English reading, so you're making important philosophical distinctions that are completely opaque to the average reader. And more importantly, the average reader will THINK she understands, and therefore get the argument wrong. At least with the gibberish that is formal first order logic, a novice will not think they understand.

E.g.

So in Mega Logic I did the following:

1. Modern logic could keep all and no to make it easier:
+/- Universal Non-existential statements - All S are P == No S are not P

2. Mega Logic thus needed to add universal, existential statements:
+/- Universal Existential statements - Every S is P == None of the S are not P

To my eyes, "No S" and "None of the S" are synonymous. As, for that matter, are "every" and "all."

It's also not clear where the existential import comes in. If you say "Every S is P," does that imply that S is non-empty, that P is non-empty, or both?

Furthermore, some of the combinatoric possibilities simply don't exist, so you can't get all four forms.

E.g. if "every S is P," then there are three possibilities, but you can't have an empty P in conjunction with a non-empty S.

 

[Garrette]

Broader than that. Under Aristotelian logic, every concept you name must have an instance; the idea that I could put together a meaningful set of words that referred to nothing didn't exist.

Under modern logic, people can name and manipulate concepts without necessarily having instances of them in mind.

Ah. Then I think I see my two major problems:

1. I hadn't realized that Aristotelian logic had that limitation, and

2. I need to study up on the notation and terminology; the concepts don't seem to be beyond me.

Thanks again.

 

[case#46cw39]

Under modern logic, people can name and manipulate concepts without necessarily having instances of them in mind.

Funny you should say that because that's exactly what I needed to do to help me deal with my schizophrenia (hearing Voices, etc). I learned to be able to name and manipulate concepts without having any actual instance. It's like building a story about mind controlling demons being the Voices you hear, while consciously realizing that this demon set is empty. It doesn't stop the pressing need to name and manipulate the concepts without choice, but it helps me realize the concepts aren't real which reduces anxiety, panic attacks, and suicidal thoughts.

So I use modern logic's, universal, non-existential statements and the "certain" statement when thinking about the Voices to give me some sense of their being empty sets and nothing to really worry about. And this is one way logic can help a paranoid schizophrenic cope with Voices and delusions.

 

[ddt]

Broader than that. Under Aristotelian logic, every concept you name must have an instance; the idea that I could put together a meaningful set of words that referred to nothing didn't exist.

Under modern logic, people can name and manipulate concepts without necessarily having instances of them in mind.

Why would nowadays anyone even consider using Aristotelian logic? I admit to not having studied it in detail, so please bear with me if my impressions are mistaken.

First of all, Aristotelian logic can only deal with quantified statements, and the syllogisms are quite a bear to spell out in modern logical symbols.

Secondly, first-order predicate logic is sound and complete, i.e., every true logical predicate can be derived with a dozen-or-so inference rules. Aristotelian logic has no such set of syllogisms that make it complete, AFAIK.

So, what's the use except for historians of science?

 

[Garrette]

Why would nowadays anyone even consider using Aristotelian logic? I admit to not having studied it in detail, so please bear with me if my impressions are mistaken.

First of all, Aristotelian logic can only deal with quantified statements, and the syllogisms are quite a bear to spell out in modern logical symbols.

Secondly, first-order predicate logic is sound and complete, i.e., every true logical predicate can be derived with a dozen-or-so inference rules. Aristotelian logic has no such set of syllogisms that make it complete, AFAIK.

So, what's the use except for historians of science?

Well, it does allow egotistical ignoramuses like me to make fools of ourselves when we challenge someone who has, you know, actually studied it.

 

[Robin]

No. He's saying that for something to be an inductive argument, the conclusion must be less certain than the premises.

But that was not what he said - he said "to conclude that there is a certain probability that it is true", which is why I asked. In most cases of inductive logic there is no way of concluding anything about the probability that the conclusion is true.

 

[drkitten]

But that was not what he said - he said "to conclude that there is a certain probability that it is true", which is why I asked.

Only if you egregiously misrepresent his statement. In context, he's merely pointing out the difference between certainty and reliable uncertainty.

Deduction leads to necessary conclusions based on "complete" evidence, meaning complete to guarantee the truth of the conclusion ... it MUST be true. The same can't be said for inductive conclusions which are at best probably true , and could well be false.


Induction leads to probable but not necessary conclusions based on "incomplete" evidence, meaning not enough to guarantee the conclusion, but just to conclude there is a certain probability it's true (but could be false).

Deductive conclusions are as reliable as the premises; if the premises are rock-solid, so are the conclusions. Inductive conclusions are less reliable than the premises, whether you choose to formalize that lack of reliability into a formal probability space or not.

 

[drkitten]

Why would nowadays anyone even consider using Aristotelian logic? I admit to not having studied it in detail, so please bear with me if my impressions are mistaken.

First of all, Aristotelian logic can only deal with quantified statements, and the syllogisms are quite a bear to spell out in modern logical symbols.

And that's one reason to use Aristotelian syllogisms; if the problem is naturally framed as a syllogism, the conversion process is tedious, error-prone, and unnecessary.

More generally, though,.... I think there are a lot of people around -- they would probably not object to being called "artistic" types, although they'd object to the scare quotes -- who dislike the tyranny of mathematics and prefer a more "humanistic" method of inquiry. The friesian.com site referenced earlier seems to be a good example; the author objects to the use of "Set Theory" as a base for logic because it is "question-begging" and because "It should now be obvious that logic is not just the neutral and unbiased description of forms of rational inference."

What it really boils down to, stripped of diplomatic niceties, is that formal math is hard and most modern philosophers who aren't logicians are terrible mathematicians. So rather than knuckle down and learn the actual math, they prefer to denigrate the difficult logic-that-works as being "problematic" and "trendy" and various other negatives, and to work with a system that's stripped down enough that they can understand.

 

[fls]

Garrette, I think the problem that you had with the example given in your OP was because the inclusion of the phrase "scientifically certainly true, .95%" (surely the poster means either 0.95 or 95%) is misleading. And the explanations have given the impression that it is the presence or absence of uncertainty, or the presence of universals (all, none) vs. particulars (some), which makes something deductive or inductive.

Certainty and uncertainty in deductive reasoning applies to whether or not we are certain as to the form of our premises - universal vs. particular - not uncertainty as to whether the premise is true or false. If only mortal men have been observed, then this statement is true:

1.All or some men are mortal.

From that we can deduce that:

2.No men are mortal is false.
3.All men are mortal is true or undetermined.
4.Some men are immortal is false or undetermined.

Adding premises or gathering additional information does not alter this, no matter what else may be true. So even though we do not know whether our observations are complete, that is, we are uncertain as to whether any men are immortal, the argument is still valid. It's just not particularly enlightening.

Regardless of whether your premises are particular or universal, and regardless of whether you are certain or uncertain about the truth of a premise/conclusion, the argument may be deductive. If uncertainty can alter the validity of the argument (i.e. the argument is not necessarily valid, so the validity depends upon the quality of your information), then it is inductive.

Linda

 

[Robin]

Only if you egregiously misrepresent his statement.

I quoted his exact words and asked for a clarification. How exactly is that misrepresentation?

It is am In context, he's merely pointing out the difference between certainty and reliable uncertainty.

In which case it is important to point out the ambiguity in the terms "probable" and "probability".

Some of the things of which we can be reliably certain cannot even be assigned a rough probability.

 

[drkitten]

I quoted his exact words and asked for a clarification. How exactly is that misrepresentation?

Cherry-picked and out of context.

The same misrepresentation technique that most creationists use when they point out the "problems" that Darwin is supposed to have with the eye. The actual quote is "To suppose that the eye, with all its inimitable contrivances for adjusting the focus to different distances, for admitting different amounts of light, and for the correction of spherical and chromatic aberration, could have been formed by natural selection, seems, I freely confess, absurd in the highest possible degree." Those are his actual words. "How exactly is that misrepresentation?"

In which case it is important to point out the ambiguity in the terms "probable" and "probability".

Some of the things of which we can be reliably certain cannot even be assigned a rough probability.

Yes, but since everyone except you who has participated in the thread appears to already have known that already knew that, that doesn't seem to be an important clarification.

In the same way that it's probably not important for me to point out that "things" don't have probabilities; only "events" do. If I were to blatantly misread your statement above to apply the word "things" to "objects" and not to "events," it would be horrifically wrong. But the wrongness would have been introduced by my (deliberate?) misreading, not by your writing.

More worrisome is that if we can be "reliably certain" of an event, can't we assign that event probability 1, by definition?

 

[Robin]

... creationists...

 

[Robin]

In the same way that it's probably not important for me to point out that "things" don't have probabilities; only "events" do.

For a start it would not be true - I wouldn't call the link between parental smoking and cot death an event for example.

But if the ambiguity between "thing" as object or "thing" as event, as in "things to do", were relevant to the discussion you would be right to point it out.

More worrisome is that if we can be "reliably certain" of an event, can't we assign that event probability 1, by definition?

I don't see why it is worrisome, but it is more relevant at least. It is a good question.

I would say that if we were reliably certain that some event would happen then it would, by definition have a probability of one.

Just as if we were to say that some event would probably happen then by definition there is a probability of greater than 0.5 of it happening.

But whether we can assign that probability would depend upon whether or not the phrase "we can assign a probability" implied that we had a basis for assigning that probability.

 

[case#46cw39]

So to help the "confused beginner" mentioned in the OP I think by now we can all agree on the following skill level 1 principles when it comes to induction and deduction (from an official US Government technical manual):

Self-Test: Unit I
For each of the statements below, decide which of the two words in parentheses makes the
statement a true statement.
1. Deduction and induction are (basic, advanced) types of thinking skills.
2. Deduction is the type of reasoning that leads to a (certain, probabilistic) conclusion.
3. Deductive conclusions are based on (incomplete, complete) evidence.
4. Induction is the type of reasoning that leads to a (certain, probabilistic) conclusion.
5. Inductive conclusions are based on (incomplete, complete) evidence.
6. Statistical reasoning is an aid to (deduction, induction).

 

[blutoski]

For a start it would not be true - I wouldn't call the link between parental smoking and cot death an event for example.

Yeah, I agree.

Events have probabilities for sure, but my impression is that objects can as well. For example, if I am shown a photo of an Eastern Cougar shredding today's newspaper in his claws, it increases the probability that this species still exists and is not extinct after all.

Skeptics work in this subject matter that surrounds debating what does or does not exist.

 

[blutoski]

Yes, but since everyone except you who has participated in the thread appears to already have known that already knew that, that doesn't seem to be an important clarification.

In the same way that it's probably not important for me to point out that "things" don't have probabilities; only "events" do. If I were to blatantly misread your statement above to apply the word "things" to "objects" and not to "events," it would be horrifically wrong. But the wrongness would have been introduced by my (deliberate?) misreading, not by your writing.

More worrisome is that if we can be "reliably certain" of an event, can't we assign that event probability 1, by definition?

Not necessarily, but that's just a language thing.

A lot of inductive reasoning is just that - reasoning. It seems that if we've seen nothing but white swans up to this point that it's probable - but still not guaranteed - that the next swan will be white, given no further information about swan colouration than this isolated past experience.

This isn't a 'finding' - it's a reasoning approach that seems to be supported by ongoing observation, and so described as still justified.

Naturalistic (sometimes called 'scientific') reasoning is inductive. Not just the innovative part about creatively inventing models to explain observations, but also the testing of these models. Testing increases our confidence in the epistemology, but never ultimately guarantees anything.

In contrast, a science like math or compsci or an arts discipline like logic can decree that there are no odd numbers evenly divisible by two and this become a solid fact with total confidence.

 

[case#46cw39]

1. I hadn't realized that Aristotelian logic had that limitation,

Well it's true one can't write statements about fictions or empty sets e.g. unicorns. Aristotelian logic's statements imply that the subject of the statement exists. This is "existential import". Thus in Aristotelian logic "All unicorns are single-horned." is saying there ARE unicorns and all of them are single-horned, which is obviously false because there are no unicorns (it's an empty set).

In modern logic's universal statements there is no existential import, no existence is implied and the statement can refer to an empty set. Thus in modern logic "All unicorns are single-horned" is saying IF there are unicorns, then all of them are single-horned which can be true.

So that's the basic distinction between classical logic and modern logic. Well you can see the confusion this can lead to: "All" means one thing in classical logic and the other in modern logic thus of course leading to different conclusions! But this is all due to definitions of the statements (the existential import of the statements) only. In other words, within Aristotelian logic's usage, the four controversial syllogisms are perfectly valid and useful. One just has to remember the "missed assumption" that drkitten pointed out when using classical logic: that the sets are not empty.

 

[drkitten]

Well it's true one can't write statements about fictions or empty sets e.g. unicorns.

Of course this isn't true.

"In the hole in a ground lived a hobbit." Hobbits are both fictions and empty sets, but a better writer than me wrote lots of statements about them. Here are some more categorical statements about hobbits, from the same source: "They are (or were) a little people, about half our height, and smaller than the bearded dwarves. Hobbits have no beards. There is little or no magic about them except the ordinary everyday sort which helps them to disappear quietly and quickly when large stupid folk like you and me come blundering along, making a noise like elephants which they can hear a mile off."

From this we can conclude, via Aristotelian reasoning, that nothing with a beard is a hobbit. Or we could, if Aristotelian logic allowed us to discuss hobbits at all..

 

[case#46cw39]

Originally Posted by Garrette
1. I hadn't realized that Aristotelian logic had that limitation,


Originally Posted by case#46cw39 in response to above quote:
Well it's true one can't write statements about fictions or empty sets e.g. unicorns.

Of course this isn't true.

"In the hole in a ground lived a hobbit." Hobbits are both fictions and empty sets, but a better writer than me wrote lots of statements about them. Here are some more categorical statements about hobbits, from the same source: "They are (or were) a little people, about half our height, and smaller than the bearded dwarves. Hobbits have no beards. There is little or no magic about them except the ordinary everyday sort which helps them to disappear quietly and quickly when large stupid folk like you and me come blundering along, making a noise like elephants which they can hear a mile off."

From this we can conclude, via Aristotelian reasoning, that nothing with a beard is a hobbit. Or we could, if Aristotelian logic allowed us to discuss hobbits at all..

I was referring to Aristotelian logic. My answer was an answer to a question about Aristotelian logic, and as I said one can't write statements about fictions (empty sets) using Aristotelian statements.

or as you put it: "Or we could, if Aristotelian logic allowed us to discuss hobbits at all."