Definitions of Induction and Deduction -- Help a Confused Beginner

[Garrette]

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).

1: Don't know. It seems a pointless distinction. It should be basic, but obviously is not, with the obvious example being me.

2. Certain

3. Complete

4. Probabilistic

5. Incomplete

6. Please rephrase

 

[case#46cw39]

According to the manual induction and deduction are considered basic thinking skills. Also referred to as general thinking skills. And since statistical reasoning is figuring out probabilities it's an aid to induction. So I'll give you a pass for unit 1.(j/k) Unit 2 let's say involves existential import. Classical universal logic statements have existential import, meaning their subjects exist, and modern universal logic statements don't have existential import, meaning their subjects can be an empty set or a hypothesized entity not yet known to exist, for example tomorrow's weather.


Unit 2 Modern Logic (my try anyway)

This difference with classical logic comes from modern logic’s usage of existential import. Modern logic interprets all universal statements (e.g. All men are mortal and No men are mortal) as having no existential import. In other words, the subject terms in universal terms are assumed not to be existing things (empty sets), since, in modern logic:

*** the A statement has been reinterpreted to say, “If there are men, then they are mortal,” and
Formal Statement: ∀x(Sx → Px)


*** the E statement has been reinterpreted to say, “If there are men, then they are not mortal”).
Formal Statement: ∀x(Sx → ~Px)

Likewise particular statements (e.g. Some men are mortal and Some men are not mortal) are taken as having existential import. In other words, the subject terms in particular statements are said to assert the actual existence of their subjects.

*** the I statements has been interpreted to say, “There is an x such that x is a man and x is mortal,”
Formal Statement: ∃x(Sx & Px)


*** the O statement has been interpreted to say, “There is an x such that x is a man and x is not mortal.”)
Formal Statement: ∃x(Sx & ~Px)

 

[Garrette]

--snip--

And since statistical reasoning is figuring out probabilities it's an aid to induction.

--snip--

I suspected that was the answer. The reason I asked for a rephrasing is because of earlier discussion about how a conclusion with probabilities can still be certain, e.g.,

80% of bunnies are white
Ralphie is a bunny
Therefore, Ralphie is 80% likely to be white.

It uses probabilities but is still, unless I am mistaken, deductive.

So I'll give you a pass for unit 1.(j/k) Unit 2 let's say involves existential import. Classical universal logic statements have existential import, meaning their subjects exist, and modern universal logic statements don't have existential import, meaning their subjects can be an empty set or a hypothesized entity not yet known to exist, for example tomorrow's weather.


Unit 2 Modern Logic (my try anyway)

This difference with classical logic comes from modern logic’s usage of existential import. Modern logic interprets all universal statements (e.g. All men are mortal and No men are mortal) as having no existential import. In other words, the subject terms in universal terms are assumed not to be existing things (empty sets), since, in modern logic:

*** the A statement has been reinterpreted to say, “If there are men, then they are mortal,” and
Formal Statement: ∀x(Sx → Px)


*** the E statement has been reinterpreted to say, “If there are men, then they are not mortal”).
Formal Statement: ∀x(Sx → ~Px)

Likewise particular statements (e.g. Some men are mortal and Some men are not mortal) are taken as having existential import. In other words, the subject terms in particular statements are said to assert the actual existence of their subjects.

*** the I statements has been interpreted to say, “There is an x such that x is a man and x is mortal,”
Formal Statement: ∃x(Sx & Px)


*** the O statement has been interpreted to say, “There is an x such that x is a man and x is not mortal.”)
Formal Statement: ∃x(Sx & ~Px)

This is helpful, thanks, though I still have to learn the notation. (I can gather it from context here)

 

[case#46cw39]

I suspected that was the answer. The reason I asked for a rephrasing is because of earlier discussion about how a conclusion with probabilities can still be certain, e.g.,

80% of bunnies are white
Ralphie is a bunny
Therefore, Ralphie is 80% likely to be white.

It uses probabilities but is still, unless I am mistaken, deductive.

Yeah, I see what you mean and this is pretty deep into inductive reasoning, not intro. But it's real interesting. As I understand it (please correct me if I'm wrong) but according to my resources this is an inductive argument. Their answer I believe would be that this argument does not have "complete" information about the set of all bunnies (just 80% of the set), therefore the conclusion is probabilistic and so the reasoning is inductive.

This is because the conclusion of these premises is not certain 100% but possible (80%). They would write this argument formally as:

1. There is an 80% chance that a bunny is white. (80% chance)
2. Ralph is a Bunny. (100% chance)
------------------------------------------------------------------
C. Therefore, the probability is no higher then 80% that Ralph is white.
In other words: Ralph is white. (80% chance)

So the real issue of this argument is "Ralph, white or not?". And as you see we would guess white, but we could be wrong, Ralph could be black. We can't be certain in our conclusion (guess), ergo inductive argument.

Here's an example of an inductive argument in the manual that has the same general form as above:

All of the new trainees (in a specific occupation) are college graduates.
There is a 50% chance that your office’s new employee this year will be a college
graduate.
-----------------------------------------------------------------------------
Therefore, the probability is no higher than 50% that your office’s new employee
will be one of the new trainees.

 

[Malerin]

Yeah, I see what you mean and this is pretty deep into inductive reasoning, not intro. But it's real interesting. As I understand it (please correct me if I'm wrong) but according to my resources this is an inductive argument. Their answer I believe would be that this argument does not have "complete" information about the set of all bunnies (just 80% of the set), therefore the conclusion is probabilistic and so the reasoning is inductive.
This is because the conclusion of these premises is not certain 100% but possible (80%). They would write this argument formally as:

1. There is an 80% chance that a bunny is white. (80% chance)
2. Ralph is a Bunny. (100% chance)
------------------------------------------------------------------
C. Therefore, the probability is no higher then 80% that Ralph is white.
In other words: Ralph is white. (80% chance)

So the real issue of this argument is "Ralph, white or not?". And as you see we would guess white, but we could be wrong, Ralph could be black. We can't be certain in our conclusion (guess), ergo inductive argument.

Here's an example of an inductive argument in the manual that has the same general form as above:

All of the new trainees (in a specific occupation) are college graduates.
There is a 50% chance that your office’s new employee this year will be a college
graduate.
-----------------------------------------------------------------------------
Therefore, the probability is no higher than 50% that your office’s new employee
will be one of the new trainees.

That's not necessarily true. If the entire set of rabbits was 80 white, 20 non-white, then there would be "complete" information.

But anyway, the first premise defines the set:
80% of bunnies are white
All men are mortal.

No difference, really.

 

[case#46cw39]

That's not necessarily true. If the entire set of rabbits was 80 white, 20 non-white, then there would be "complete" information.

But anyway, the first premise defines the set:
80% of bunnies are white
All men are mortal.

No difference, really.

I think here by "complete" information they mean a piece of information about the entirety of the set, i.e., that can be said about the entirety of the set. Could be 'all the members of the bunny set are this', or 'no members of the set are that', in other words a universal statement.

For this statement:

"Of all first-line supervisors in the Border Patrol, 95% have taken the basic supervisory course."

Manual says: "Our premise does not give us complete information about the entire set of people who have taken the course. Therefore, if the only information we have is that a particular person has taken the course, we can conclude that there is some chance that the person is a first-level supervisor in the Border Patrol..."

 

[case#46cw39]

In deductive reasoning there is not just a chance that the conclusion is true; it's guaranteed that the conclusion is true (impossible to be false, 100% certainly true).

"The power of deductive arguments comes from this compelling rationality. That is, as you may know, one of the great attractions of mathematics, especially geometry, which is entirely deductive in nature. Hence, someone who can frame an argument in a deductive structure has the most powerful rational means of persuasion available.

Now, what is important to notice about such a deductive argument is that the truth of the conclusion is compelling. If we are rational, then we have to agree. To accept the truth of the first and second statements and to agree that they have been combined reasonably, and then to decline to accept the truth of the third would be to violate a basic principle of reason. I am free in a modern liberal society to reject that conclusion, but I cannot do so and claim that I am acting rationally, unless I can prove that there is something wrong with either of the first two statements or with the way they have been put together."

FROM DEDUCTION AND INDUCTION[This text, which has been prepared by Ian Johnston of Malaspina University-College, Nanaimo, BC, is in the public domain and may be used, in whole or in part, without permission and without charge, released May 2000]


So, the deductive conclusion must be true (not just a chance of being true), ... "unless I can prove

(lacks 'premise acceptability' -) that there is something wrong with either of the first two statements

(lacks 'premise sufficiency' -) or with the way they have been put together."

 

[Garrette]

That's not necessarily true. If the entire set of rabbits was 80 white, 20 non-white, then there would be "complete" information.

But anyway, the first premise defines the set:
80% of bunnies are white
All men are mortal.

No difference, really.

I think here by "complete" information they mean a piece of information about the entirety of the set, i.e., that can be said about the entirety of the set. Could be 'all the members of the bunny set are this', or 'no members of the set are that', in other words a universal statement.

For this statement:

"Of all first-line supervisors in the Border Patrol, 95% have taken the basic supervisory course."

Manual says: "Our premise does not give us complete information about the entire set of people who have taken the course. Therefore, if the only information we have is that a particular person has taken the course, we can conclude that there is some chance that the person is a first-level supervisor in the Border Patrol..."

In deductive reasoning there is not just a chance that the conclusion is true; it's guaranteed that the conclusion is true (impossible to be false, 100% certainly true).

"The power of deductive arguments comes from this compelling rationality. That is, as you may know, one of the great attractions of mathematics, especially geometry, which is entirely deductive in nature. Hence, someone who can frame an argument in a deductive structure has the most powerful rational means of persuasion available.

Now, what is important to notice about such a deductive argument is that the truth of the conclusion is compelling. If we are rational, then we have to agree. To accept the truth of the first and second statements and to agree that they have been combined reasonably, and then to decline to accept the truth of the third would be to violate a basic principle of reason. I am free in a modern liberal society to reject that conclusion, but I cannot do so and claim that I am acting rationally, unless I can prove that there is something wrong with either of the first two statements or with the way they have been put together."

FROM DEDUCTION AND INDUCTION[This text, which has been prepared by Ian Johnston of Malaspina University-College, Nanaimo, BC, is in the public domain and may be used, in whole or in part, without permission and without charge, released May 2000]


So, the deductive conclusion must be true (not just a chance of being true), ... "unless I can prove

(lacks 'premise acceptability' -) that there is something wrong with either of the first two statements

(lacks 'premise sufficiency' -) or with the way they have been put together."

I think I'm with Malerin on this (but I'm still learning).

Let me try it as a slightly different example:

1. 80 of the 100 marbles in this bag are white
2. The remaining 20 (of 100) marbles in this bag are black
3. John is randomly choosing a marble from the bag, noting the color, replacing it in the bag, and repeating forever
-------------------
C: The proportion of white marbles randomly picked to total marbles picked will approach 80/100

 

[Robin]

That's not necessarily true. If the entire set of rabbits was 80 white, 20 non-white, then there would be "complete" information.

But anyway, the first premise defines the set:
80% of bunnies are white
All men are mortal.

No difference, really.

Even without complete information the argument is still deductive:

P1 There is an 80% chance that a bunny is white
P2 Thumper is a bunny
Conclusion: There is an 80% chance that Thumper is white

It is deductive in that if the premisses are true then so is the conclusion.

 

[fls]

I think I'm with Malerin on this (but I'm still learning).

Let me try it as a slightly different example:

1. 80 of the 100 marbles in this bag are white
2. The remaining 20 (of 100) marbles in this bag are black
3. John is randomly choosing a marble from the bag, noting the color, replacing it in the bag, and repeating forever
-------------------
C: The proportion of white marbles randomly picked to total marbles picked will approach 80/100

A marble picked from the bag will be black or white.

Linda

 

[Robin]

Yeah, I see what you mean and this is pretty deep into inductive reasoning, not intro. But it's real interesting. As I understand it (please correct me if I'm wrong) but according to my resources this is an inductive argument. Their answer I believe would be that this argument does not have "complete" information about the set of all bunnies (just 80% of the set), therefore the conclusion is probabilistic and so the reasoning is inductive.

This is because the conclusion of these premises is not certain 100% but possible (80%). They would write this argument formally as:

1. There is an 80% chance that a bunny is white. (80% chance)
2. Ralph is a Bunny. (100% chance)
------------------------------------------------------------------
C. Therefore, the probability is no higher then 80% that Ralph is white.
In other words: Ralph is white. (80% chance)

So the real issue of this argument is "Ralph, white or not?". And as you see we would guess white, but we could be wrong, Ralph could be black. We can't be certain in our conclusion (guess), ergo inductive argument.

Here's an example of an inductive argument in the manual that has the same general form as above:

All of the new trainees (in a specific occupation) are college graduates.
There is a 50% chance that your office’s new employee this year will be a college
graduate.
-----------------------------------------------------------------------------
Therefore, the probability is no higher than 50% that your office’s new employee
will be one of the new trainees.

The second is not of the same form as the first. The second is invalid whether considered as an inductive or a deductive argument.

There might be new employee's who were not trainees but nevertheless graduated college.

With the first argument - is there a possibility that the premisses are true and the conclusion false?

 

[drkitten]

The second is not of the same form as the first. The second is invalid whether considered as an inductive or a deductive argument.

There might be new employee's who were not trainees but nevertheless graduated college.

With the first argument - is there a possibility that the premisses are true and the conclusion false?

Certainly. The statement "Ralph is white" is either true (probability 1.0) or false (probability 0.0). The idea that "Ralph is white" has an 80% chance of being true isn't supportable either from a frequentist or Bayesian perspective; a frequentist would point out that probabilities apply only to multiple trials, and a Bayesian would point out that probability applies to strength of evidence, not to individual statements.

That's one of the reasons that the inductive/deductive distinction is not generally useful in probability theory.

 

[case#46cw39]

The Logical Million Dollar Challenge

If there were a million dollar contest to predict the color of a bunny behind a door, white or not, the player with the deductive conclusion is guaranteed to be right (probability 1.0), no doubt, compelling evidence, has complete evidence, 100% certain, bet the house on it.

If you were the other player who had the argument that drew to the conclusion that there was a 80% chance the bunny was white (probability .8), would you bet the house? Do you have a compelling argument, 100% certain to be true, to tell the spouse pack up I bet the house? No. What you would have is a suggestive argument, but not a compelling argument (deductive).

The reason is because it's an inductive argument where even though the premises are true, the conclusion can be false. You can lose in other words.

In a deductive argument if the premises are true and the structure sufficient, then the conclusion is guaranteed to be true (probability 1.0), it can't be false. It MUST be true. So bet the house.

 

[fls]

The Logical Million Dollar Challenge

If there were a million dollar contest to predict the color of a bunny behind a door, white or not, the player with the deductive conclusion is guaranteed to be right (probability 1.0), no doubt, compelling evidence, has complete evidence, 100% certain, bet the house on it.

If you were the other player who had the argument that drew to the conclusion that there was a 80% chance the bunny was white (probability .8), would you bet the house? Do you have a compelling argument, 100% certain to be true, to tell the spouse pack up I bet the house? No. What you would have is a suggestive argument, but not a compelling argument (deductive).

The reason is because it's an inductive argument where even though the premises are true, the conclusion can be false. You can lose in other words.

In a deductive argument if the premises are true and the structure sufficient, then the conclusion is guaranteed to be true (probability 1.0), it can't be false. It MUST be true. So bet the house.

I bet the bunny is white or non-white.

Linda

 

[Robin]

Certainly. The statement "Ralph is white" is either true (probability 1.0) or false (probability 0.0). The idea that "Ralph is white" has an 80% chance of being true isn't supportable either from a frequentist or Bayesian perspective; a frequentist would point out that probabilities apply only to multiple trials, and a Bayesian would point out that probability applies to strength of evidence, not to individual statements.

And so that means that the premiss is false.

It does not mean that the conclusion could be false if the premiss was true.

Which is what I asked.

It is clearly a deductive argument with a false premiss.

 

[drkitten]

If there were a million dollar contest to predict the color of a bunny behind a door, white or not, the player with the deductive conclusion is guaranteed to be right (probability 1.0), no doubt, compelling evidence, has complete evidence, 100% certain, bet the house on it.

The problem with this is that there is uncertainty about what exactly the "conclusion" of the argument is supposed to be.

There are at least two potential conclusions.

1. R: Ralph the Bunny is white
2. S: "R is true" is 80% likely

We could be in a situation where I could and would bet the house on statement S (with perfect confidence of winning) while still being unwilling to bet the house on statement R.

 

[Robin]

The Logical Million Dollar Challenge

If there were a million dollar contest to predict the color of a bunny behind a door, white or not, the player with the deductive conclusion is guaranteed to be right (probability 1.0), no doubt, compelling evidence, has complete evidence, 100% certain, bet the house on it.

If you were the other player who had the argument that drew to the conclusion that there was a 80% chance the bunny was white (probability .8), would you bet the house? Do you have a compelling argument, 100% certain to be true, to tell the spouse pack up I bet the house?

I bet the house that the conclusion could not be false if the premisses are true.

Look:

A(x) means that x is a bunny
B(x) means that x has an 80% chance of being white

A(x)->B(x)
A(Ralph)
--------
B(Ralph)

Clearly a deductive argument.

The first premiss is or course false. Meaningless even. But that is besides the point.

 

[Robin]

I bet the bunny is white or non-white.

Linda

I bet that Ralph has an 80% chance of being white if the premisses are true.

 

[drkitten]

And so that means that the premiss is false.

No, it doesn't. In fact, that's exactly what it doesn't mean.

The two premises are, respectively,

1. There is an 80% chance that a bunny is white.
2. Ralph is a bunny

The first is a frequentist statement; if we're looking at a cage full of sixteen white bunnies and four black ones, then there's an 80% chance that a randomly chosen bunny is white; this premise is true by observation. In this version, of course, the Bayesian interpretation is the same as the frequentist one because Bayesian statistics reduce to frequentist when we're looking at a specific set.

And the second one can be confirmed to be true by any biologist. Just pull Ralph out of the cage and check him out.

But when you do pull Ralph out of the cage, you'll notice that it's not 80% true that he's white, nor is he white 80% of the time, nor is he white over 80% of his body. He's either one of the white rabbits or one of the black rabbits.

It does not mean that the conclusion could be false if the premiss was true.

Er,.... wrong. The premises are true, but the conclusion false.

 

[case#46cw39]

Even without complete information the argument is still deductive:

P1 There is an 80% chance that a bunny is white
P2 Thumper is a bunny
Conclusion: There is an 80% chance that Thumper is white

It is deductive in that if the premisses are true then so is the conclusion.

I see what you mean here and agree with your principle:

Deduction: If the premises are true and of sufficient structure, then conclusion is 100% certain (probability 1.0).

Let's say that's true here. Then this conclusion has to have a probability of 1.0. So the 80% would have to here, now be a part of the statement itself.

Then we do indeed get a deductive structure:

P1 There is an 80% chance that a bunny is white. (probability 1.0)
P2 Thumper is a bunny (probability 1.0)
Conclusion: There is an 80% chance that Thumper is white. (probability 1.0)


But this conclusion makes for an odd sentence: "I'm certain that there is an 80% chance that Thumper is white." This seems unusual but I don't know.

Is this certainty the same certainty of above that says bet the house? Or is this just another way to say there is an 80% chance Thumper is white? I'm still on the side of inductive as its implied meaning in general. But I see you are certain it's 80%! But is that the real criterion here?

Sometimes formalizing makes things clearer:
P1 Any bunny is white (probability .8)
P2 Thumper is a bunny (probability 1.0)
Conclusion: Thumper is white (probability .8)