Definitions of Induction and Deduction -- Help a Confused Beginner

[Garrette]

Well, I haven't had the time to read the things I had intended to read about this, but since I started the thread, I figured I should weigh in.

First, let me say to case: Since you have defined what you mean by induction and deduction, it is okay to use them that way even if I disagree with the definitions. Caveated by what has been pointed out in the other thread about needing to accept the premises (and start from the same null hypothesis) if you're going to bring certainty into it. So even if your definition is consistent, your application appears to be lacking. But that isn't the point of this thread, so that's all I'll say on it.

Second, the definitional issue seems to hinge on whether or not it has been "updated." My understanding about specific-to-general and general-to-specific was correct at one time; we all agree. The question is whether it is still correct.

Now I'm writing while thinking without thinking it out completely, so I could be off, but it seems to me that it comes down to a matter of Sets.

For the mortal/man/Socrates scenario, they work like this if rephrased:

Set(Mortal Men) = Set(Men)
Socrates is in the Set(Men)
Therefore Socrates is in the Set(Mortal Men)

Certainty. Ergo, deductive.

===

Set(Mortal Men) < Set(Immortal Men)
Set(Mortal Men) + Set(Immortal Men) = Set(Men)
Socrates is in the Set(Men)
Therefore Chance-Socrates-is-in-Set(Mortal Men) > Chance-Socrates-is-in-Set(Immortal Men)

Not certain, therefore inductive.

ETA: Actually, the manner in which I wrote the conclusion about chance socrates in a certain set is Certain, therefore it is deductive. It is only if I change the conclusion to remove the chance and say Socrates is in Set(Mortal Men) that it becomes uncertain and therefore inductive. Correct?

At least, it's correct if we accept that the definitions have been updated.

 

[Robin]

The wiki quote had a [2] citation which referred to the Internet Encyclopedia of Philosophy which is "a Peer-Reviewed Academic Resource". So let's agree that this and the Stanford Encyclopedia of Philosophy are good resources.

SEP identifies Mach and Carnap as being neutral monists - which is a little like identifying Richard Dawkins and James Randi as prayerful Roman Catholics.

So let's not agree that.

 

[Robin]

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).

So are you saying that in order for something to be an inductive argument it must be possible to assign a probability to the conclusion?

 

[case#46cw39]

SEP identifies Mach and Carnap as being neutral monists - which is a little like identifying Richard Dawkins and James Randi as prayerful Roman Catholics.

So let's not agree that.

Well since you won't accept any of my sources, and you offer no other sources, I can't very well talk to you about this.

So I will leave you with John Vickers <John.Vickersatcgudotedu> at SEP. Contact him if you still can't believe that this contemporary shift in understanding due to modern logic has occurred; he's the expert. He wrote (2010):

Until about the middle of the previous century induction was treated as a quite specific method of inference: inference of a universal affirmative proposition (All swans are white) from its instances (a is a white swan, b is a white swan, etc.) ... It is no longer possible to think of induction in such a restricted way; much synthetic or contingent inference is now taken to be inductive; some authorities go so far as to count all contingent inference as inductive.
http://plato.stanford.edu/entries/induction-problem/

 

[case#46cw39]

First, let me say to case: Since you have defined what you mean by induction and deduction, it is okay to use them that way even if I disagree with the definitions.

Well I tried. That's about the best I can do here. I stand by it and my many resources who say the same thing. There has been a quantum leap in understanding of i and d due to modern logic. The old guard (Aristotelian logic of existential statements, etc.) has fallen and the new kid is in town (first order predicate logic) or modern logic, with its new understanding of inductive reasoning and deductive reasoning.

Today's logic students don't even learn about Aristotelian logic accept maybe a brief intro. One logic professor told me he had taken 10 logic classes and not one of them was on Aristotelian logic. All these modern logic students of today will tell you immediately that the Bramantip syllogism is invalid. Two universals can't draw to a particular in modern logic! But of course in the old guard way it is valid. But nobody studies it that way in logic anymore.

 

[Garrette]

Well I tried. That's about the best I can do here. I stand by it and my many resources who say the same thing. There has been a quantum leap in understanding of i and d due to modern logic. The old guard (Aristotelian logic of existential statements, etc.) has fallen and the new kid is in town (first order predicate logic) or modern logic, with its new understanding of inductive reasoning and deductive reasoning.

Today's logic students don't even learn about Aristotelian logic accept maybe a brief intro. One logic professor told me he had taken 10 logic classes and not one of them was on Aristotelian logic. All these modern logic students of today will tell you immediately that the Bramantip syllogism is invalid. Two universals can't draw to a particular in modern logic! But of course in the old guard way it is valid. But nobody studies it that way in logic anymore.

I'll have to take your word on how it's being taught or not taught, but I found this site which indicates that the new form of teaching does not really say the old form is invalid but "Sometimes it is simply said that the old theory was wrong. But what has happened instead is that a new interpretation of meaning has been imposed on the propositions, and this interpretation comes from Set Theory."

(On a side, egotistical note, I crafted my Set post above before I found this site.)

While I have not fully studied the site I linked to, it seems complete and coherent.

 

[drkitten]

So are you saying that in order for something to be an inductive argument it must be possible to assign a probability to the conclusion?

No. He's saying that for something to be an inductive argument, the conclusion must be less certain than the premises. Whether or not this is formalizable as a probability is irrelevant. E.g. if I can show a possible world where the premises are true but the conclusion isn't, then it's an inductive argument, even if we can't assess probability across distributions of possible words.

 

[drkitten]

Today's logic students don't even learn about Aristotelian logic accept maybe a brief intro. One logic professor told me he had taken 10 logic classes and not one of them was on Aristotelian logic. All these modern logic students of today will tell you immediately that the Bramantip syllogism is invalid. Two universals can't draw to a particular in modern logic! But of course in the old guard way it is valid. But nobody studies it that way in logic anymore.

Well, there's a very good reason for that; the old way is error-ridden, because we have a better understanding of existential import now.

Just like we now know that the Black Prince's Ruby is really a spinel, because we understand that there are different types of red gems. (The difference, chemically, is that some red stones contain magnesium; these we term spinels.)

You can certainly make a version of Bramantip that is valid; you simply need to make explicit the fact that you're talking about non-empty sets. But without that assumption, explicit or implicit, Bramantip is indeed invalid. The ancients didn't know that you needed that assumption. That's not a mere difference of opinion; that's actual ignorance on their part.

 

[drkitten]

I'll have to take your word on how it's being taught or not taught, but I found this site which indicates that the new form of teaching does not really say the old form is invalid but "Sometimes it is simply said that the old theory was wrong. But what has happened instead is that a new interpretation of meaning has been imposed on the propositions, and this interpretation comes from Set Theory."

Shrug; I think the site is wrong.

From that site: "So there is nothing wrong with the four "invalid" syllogisms except when a new interpretation of meaning introduces the possibility of the Empty Set. If it is postulated that the circle representing an entire term cannot be blanked out, because it cannot be the Empty Set, then Darapti, Felapton, Bramantip, and Fesapo are all quite valid inferences."

The problem is that a "new interpretation of meaning" isn't what introduces the possibility of the Empty Set. God, or if you prefer, the Real World, is what introduces that possibility. Unless you've got the sets you're discussing sitting in front of you right now you have no idea what their status is.

Or to put it another way, if you postulate that no sets are empty, then you're a fool, and modern logic avoids this particular foolishness.

This point was brought forcibly home to me a few years ago, when I was delivering a lecture that included some set-theoretic formalism, and I was using the Beatles as an illustrative example. As it happened, it was the very night that Harrison's death was reported, which meant that my example of "living right-handed Beatles" was the empty set without my knowing it..

 

[case#46cw39]

E.g. if I can show a possible world where the premises are true but the conclusion isn't, then it's an inductive argument, even if we can't assess probability across distributions of possible words.

(*underlining is mine*)


That's the hallmark of inductive reasoning, no guarantee the conclusion is true, could be false. So the inductive evidence is considered "incomplete" for a guaranteed conclusion. The hallmark of deductive reasoning is that the conclusion MUST be true (if premises are true). It's guaranteed so by the deductive argument structure. So the deductive evidence is considered "complete" for a guaranteed conclusion i.e. it entails the conclusion.


As for the Bramantip syllogism controversy that link Garrette gave is famous in logic, and very informative as to this break between classical logic and modern logic. It's a long story though.

 

[Garrette]

Shrug; I think the site is wrong.

From that site: "So there is nothing wrong with the four "invalid" syllogisms except when a new interpretation of meaning introduces the possibility of the Empty Set. If it is postulated that the circle representing an entire term cannot be blanked out, because it cannot be the Empty Set, then Darapti, Felapton, Bramantip, and Fesapo are all quite valid inferences."

The problem is that a "new interpretation of meaning" isn't what introduces the possibility of the Empty Set. God, or if you prefer, the Real World, is what introduces that possibility. Unless you've got the sets you're discussing sitting in front of you right now you have no idea what their status is.

Or to put it another way, if you postulate that no sets are empty, then you're a fool, and modern logic avoids this particular foolishness.

This point was brought forcibly home to me a few years ago, when I was delivering a lecture that included some set-theoretic formalism, and I was using the Beatles as an illustrative example. As it happened, it was the very night that Harrison's death was reported, which meant that my example of "living right-handed Beatles" was the empty set without my knowing it..

(*underlining is mine*)


That's the hallmark of inductive reasoning, no guarantee the conclusion is true, could be false. So the inductive evidence is considered "incomplete" for a guaranteed conclusion. The hallmark of deductive reasoning is that the conclusion MUST be true (if premises are true). It's guaranteed so by the deductive argument structure. So the deductive evidence is considered "complete" for a guaranteed conclusion i.e. it entails the conclusion.


As for the Bramantip syllogism controversy that link Garrette gave is famous in logic, and very informative as to this break between classical logic and modern logic. It's a long story though.

Well, as I said, I haven't studied it yet, but I will concede my position for two reasons:

1. I know from long interaction that drkitten is usually right about these things (therefore beginning my inductive process...) and

2. It is becoming more apparent that case knows far more about it than I do.

So, case, my apologies. And my thanks to you both for letting me learn something new. Even if I don't fully understand it. Yet.

 

[case#46cw39]

Well, there's a very good reason for that; the old way is error-ridden, because we have a better understanding of existential import now.

You can certainly make a version of Bramantip that is valid; you simply need to make explicit the fact that you're talking about non-empty sets. But without that assumption, explicit or implicit, Bramantip is indeed invalid. The ancients didn't know that you needed that assumption. That's not a mere difference of opinion; that's actual ignorance on their part.

That was interesting. So you are saying empty sets are a natural part of circumstances and so automatically ruling them out, as Aristotelian logic did, doesn't fit circumstances. So it's not a matter of opinion on this but awareness of circumstances as they are.

I added to this break in logic, in my book, that we do need statements of existential import for certain discourse. In other words non-empty sets are also useful in discourse and so not providing for them, as modern logic does, doesn't fit circumstances either.

I want statements for all possibilities. I even want a statement for particular non-existential statements "Certain unicorns are female." (t) So I built a "Mega Logic" that includes all the statement types possible, 24 in total.

 

[drkitten]

That was interesting. So you are saying empty sets are a natural part of circumstances and so automatically ruling them out, as Aristotelian logic did, doesn't fit circumstances.

Yes.

I added to this in my book that we do need statements of existential import for certain discourse. In other words non-empty sets are also useful in discourse and so not providing for them, as modern logic does, doesn't fit circumstances either.

I don't understand. Modern logic does provide for non-empty sets; that's the purpose of the "there exists" predicate that I don't have a font for on my machine but that looks like a backwards E. I've even seen some notation used to provide for explicitly singleton sets ("there uniquely exists"), although of course you need an identity predicate to do this properly. And if you have notation for set cardinality -- and who doesn't, really? -- you can provide for sets of any definable size.

I want statements for all possibilities. I even want a statement for particular non-existential statements "Certain unicorns are female." (t) So I built a "Mega Logic" that includes all the statement types possible, 24 in total.

Again, I don't know what this is supposed to mean, so I can't comment on it. But if what you're suggesting is that if unicorns existed, some of them would be female, that's easy enough to express in first order logic. And the nice thing about that is that in that notation, it meshes well with everything else we know about the real world, so your possible world with unicorns introduces no absurdities into the real one or vice versa.

That's the ultimate problem with Aristotelian logic; it doesn't describe the real world, and if you try to merge the two, you get discontinuities. That's similarly a problem with higher-order logics (thank Godel for that). But modern first order logic really does seem to work well with physics.

 

[case#46cw39]

I should add quickly that modern logic is talking about and making doctrinal the modern logic way to rectify this perceived shortcoming of it by some. I thought I had invented this technique and told a professor at Niagara University, and then found it in the SEP. LOL

Every modern logic text must address the apparent implausibility of letting ‘Every S is P’ be true when there are no Ss. The common defense of this is usually that this is a logical notation devised for purposes of logic, and it does not claim to capture every nuance of the natural language forms that the symbols resemble. So perhaps ‘∀x(Sx → Px)’ does fail to do complete justice to ordinary usage of ‘Every S is P’, but this is not a problem with the logic. If you think that ‘Every S is P’ requires for its truth that there be Ss, then you can have that result simply and easily: just represent the recalcitrant uses of ‘Every S is P’ in symbolic notation by adding an extra conjunct to the symbolization, like this: ∀x(Sx → Px) & ∃xSx. This defense leaves logic intact and also meets the objection, which is not a logical objection, but merely a reservation about the representation of natural language.
http://plato.stanford.edu/entries/square/

ETA: I mean there are no (formal) universal existentials (just all/no) or non-existential particulars (I use "certain") in modern logic proper.

 

[drkitten]

Well, as I said, I haven't studied it yet,

I'm sorry. I was a little too terse perhaps.

Bramantip is a classical syllogism of the following form:

1. All P is M
2. All M is S
3. Therefore, some S is P

For example : all cats are mammals, all mammals are animals, therefore some animals are cats. Which seems reasonable until you consider the possibility that P is an empty set.

E.g., all invisible pink unicorns are mammals, all mammals are animals, therefore some animals are invisible pink unicorns.

In classical (Aristotelian) logic, statements of the form "All A are B" were interpreted as having existential import; that is to say, "All A are B" is only true if some A (that are B) actually existed.

This caused all sorts of problems with indirect proofs or proofs by case analysis, though. For example, if it's not the case that "all unicorns are mammals," does that mean that "not all unicorns are mammals"? That "some unicorns are not mammals," which does indeed seem to imply that unicorns exist.

It wasn't until the late 1800s that logicians like Dodson, De Morgan, and Boole came up with a reasonable framework for handling the existential implications of syllogistic reasoning, which they did basically by denying that "all" statements had any such implications and allowing people to add in existential import via conjunctions.

E.g. "All unicorns are mammals" is true under the modern interpretation, but "All unicorns are mammals and some unicorns exist," is false for a reason that we can explicitly manipulate. In fact, from those two statements, we can infer that "no unicorns exist."

 

[Garrette]

I'm sorry. I was a little too terse perhaps.

Bramantip is a classical syllogism of the following form:

1. All P is M
2. All M is S
3. Therefore, some S is P

For example : all cats are mammals, all mammals are animals, therefore some animals are cats. Which seems reasonable until you consider the possibility that P is an empty set.

E.g., all invisible pink unicorns are mammals, all mammals are animals, therefore some animals are invisible pink unicorns.

In classical (Aristotelian) logic, statements of the form "All A are B" were interpreted as having existential import; that is to say, "All A are B" is only true if some A (that are B) actually existed.

This caused all sorts of problems with indirect proofs or proofs by case analysis, though. For example, if it's not the case that "all unicorns are mammals," does that mean that "not all unicorns are mammals"? That "some unicorns are not mammals," which does indeed seem to imply that unicorns exist.

It wasn't until the late 1800s that logicians like Dodson, De Morgan, and Boole came up with a reasonable framework for handling the existential implications of syllogistic reasoning, which they did basically by denying that "all" statements had any such implications and allowing people to add in existential import via conjunctions.

E.g. "All unicorns are mammals" is true under the modern interpretation, but "All unicorns are mammals and some unicorns exist," is false for a reason that we can explicitly manipulate. In fact, from those two statements, we can infer that "no unicorns exist."

For the highlighted part, to which two statements do you refer? [ETA: I'm not arguing. I'm just having difficulty following.]

Now speaking more generally, I saw a problem with your example syllogism even before you explained it, but I didn't see it based on empty sets.

1. All P is M (but Set(P) may be smaller than Set(M))
2. All M is S (but Set(M) may be smaller than Set(S))
3. Therefore, some S is P (does not follow because the overlap of P and M need not indicate an overlap with S)

I can't think of an actual example off hand, but the problem seemed obvious to me.

 

[Garrette]

Scratch my previous post. Further thinking on it leads me to conclude that my objection is misplaced and that the only flaw is in reference to empty sets.

This is embarrassing. I know I'm not in the league of many here in regard to logic and rational discourse, but I really had not considered myself a beginner (despite the thread title).

 

[drkitten]

For the highlighted part, to which two statements do you refer? [ETA: I'm not arguing. I'm just having difficulty following.]

1. All unicorns are mammals (is true).
2. All unicorns are mammals and some unicorns exist (is false)

Now speaking more generally, I saw a problem with your example syllogism even before you explained it, but I didn't see it based on empty sets.

1. All P is M (but Set(P) may be smaller than Set(M))
2. All M is S (but Set(M) may be smaller than Set(S))
3. Therefore, some S is P (does not follow because the overlap of P and M need not indicate an overlap with S)

I can't think of an actual example off hand, but the problem seemed obvious to me.

Actually, that's not an issue. If set P is non-empty, then we know that there exists an x such that Px.

Since all P are M, Mx.

Since all M are S, Sx.

Therefore, there is an element x of S that is also P.

Hence, some S are P.

Or in your terminology, the "overlap" is not just overlap but inclusion. P is a subset of M, which in turn is a subset of S, therefore P is a subset of S. Hence if P is nonempty, then P is a non-empty subset of S.

ETA: I know you caught your own error, but I'll leave it here for the education of the Ghost of JREF Yet to Come.

 

[Garrette]

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?

 

[case#46cw39]

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.

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

3. Modern and classical logic share some as a particular, existential statement:
Particular, Existential Statements - Some S are P (Some ghosts are hostile = F)

4. Mega Logic thus needs to add particular, non-existential statements:
Particular, Non-Existential Statements - Certain S are P (Certain ghosts are hostile = T)

There are four forms of these six statement types for a total of 24 set statements in my Mega Logic.