Definitions of Induction and Deduction -- Help a Confused Beginner

[drkitten]

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.

It is indeed. You're looking at the wrong part of the sentence, though. Don't focus on "certain." Focus on what it means for there to be an 80% chance that Thumper is white.

Let me give you an example. Based on demographic statistics, about 51% of the American population is female. From that, may I conclude that you have a 51% chance of being female?

What does it mean for you to "have a 51% chance of being female"? Would your mother also agree that you had a 51% chance of being female? She doesn't know whether or not you'll show up in at her house for Thanksgiving dinner wearing a bra or not, because there's a 51% chance that you're female?

Most statisticians would accept that you're either female or you aren't, but the idea of assigning a "probability" to that statement doesn't make much sense.

From a traditional (frequentist) perspective, "probability" only makes sense when applied to repeated or repeatable events. E.g. we can talk about the probability of drawing a white marble from the bag, but not the probability of the French Revolution. From a Bayesian perspective, "probability" is not a property of the object but of our knowledge of the object -- i.e. we can say that we are 80% confident that a statement is true given what we know about the world, but the world still is what it is and the statement is either true or false. (And two different people would assess the probability differently, because they know different things.)

 

[fls]

All rabbits are white.
Thumper is a rabbit.
Therefore Thumper is white.

Some rabbits are white.
Thumper is a rabbit.
Therefore Thumper is white or not white.

Some rabbits are white.
Thumper is a rabbit.
Therefore Thumper is white.

Using drkitten's example where we know 80% of rabbits are white and the rest are not...

The first two syllogisms are valid in form, while the last is not, so it can be disregarded. If 80% of rabbits are white, the initial premise is false in the first syllogism. If the premises are true in the second syllogism, the conclusion is true.

The fault appears to be in assigning probability to a universal statement (i.e. All bunnies are white (probability .8)), or using an invalid syllogism (i.e. Any (some) bunny is white (probability .8)...) not with forming deductive arguments when statements do not apply to all members of a set. If one wants to go about assigning probabilities, then you are adding a bunch of hidden assumptions about what probabilities mean, not changing the validity of the argument.

Linda

 

[case#46cw39]

-snip -
Some rabbits are white.
Thumper is a rabbit.
Therefore Thumper is white.

Using drkitten's example where we know 80% of rabbits are white and the rest are not...
-snip -
The fault appears to be in assigning probability to a universal statement (i.e. All bunnies are white (probability .8)), or using an invalid syllogism (i.e. Any (some) bunny is white (probability .8)...) not with forming deductive arguments when statements do not apply to all members of a set. If one wants to go about assigning probabilities, then you are adding a bunch of hidden assumptions about what probabilities mean, not changing the validity of the argument.

Linda

Thanks. I was just wondering about that. Still rusty on induction. I got that from the manual's statement:

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

I took it to be universal, but like you say, here a universal statement can only have a probability of 1.0. "Probabilities vary from 0 to 1. The value of 1 represents absolute confirmation -- a condition that applies only to deductive conclusions." from DHS Manual So really it's a particular statement with a numeric probability of .95 ...

"Some (95%) of the first-line supervisors in the Border Patrol, have taken the basic supervisory course."

So below is an invalid deduction like you say (could be one of the rabbits who aren't white):

Some rabbits are white.
Thumper is a rabbit.
Therefore Thumper is white.

However, this is valid inductively:

Some (95%) rabbits are white (in a universe of discourse).
Thumper is a rabbit (in the universe of discourse).
Therefore There is a 95% chance Thumper is white.

 

[case#46cw39]

- snip -
From a traditional (frequentist) perspective, "probability" only makes sense when applied to repeated or repeatable events. E.g. we can talk about the probability of drawing a white marble from the bag, but not the probability of the French Revolution. From a Bayesian perspective, "probability" is not a property of the object but of our knowledge of the object -- i.e. we can say that we are 80% confident that a statement is true given what we know about the world, but the world still is what it is and the statement is either true or false. (And two different people would assess the probability differently, because they know different things.)

Thanks, that was helpful. The public domain DHS manual I'm using as a resource says the following about induction and probability:

"Probability refers to a number between 0 and 1 that indicates the likelihood that an event will happen or that a conclusion is true. The value of 1 represents absolute confirmation -- a condition that applies only to deductive conclusions. See also degree of confirmation. Degree of confirmation refers to the interpretation of probability in inductive logic. The probability of a conclusion is the degree of confirmation provided for the conclusion by the premises.

Inductive reasoning, as you learned in Unit I, is reasoning that occurs in the absence of complete information. Induction leads to conclusions that are not necessarily true but only have some probability of being true. In inductive reasoning, the evidence does not guarantee the truth of the conclusion, but it does give us a good reason to believe in the truth of the conclusion. We can only say that the premises confirm or support the conclusion. This is in contrast to deductive reasoning, in which the truth of the evidence makes the truth of the conclusion certain."

 

[blobru]

...However, this is valid inductively:

Some (95%) rabbits are white (in a universe of discourse).
Thumper is a rabbit (in the universe of discourse).
Therefore There is a 95% chance Thumper is white.

You've still got the odd conclusion though (because objectively, Thumper is either white [100% chance] or not-white [100%] -- as fls points out, there seems a problem assigning probabilities to universal statements; and "Thumper is white", although it doesn't look like it, can be read as a universal statement, with universe = Thumper: "All rabbits which are this rabbit named 'Thumper' are white [95%]"). To meet drkitten's objections, you might have to rephrase it:

{Bayesian}

x knows 95% of rabbits are white (in a sample / 'universe of discourse')
x knows Thumper is a rabbit (in the sample)
x can be 95% confident Thumper is white

{frequentist}

95% of rabbits (in a sample) are white
a rabbit (from the sample) will be randomly selected and called "Thumper"
there is a 95% chance Thumper will be white

It's a bit nit-picky, maybe; but without a more precise formulation, valid premises lead to ambiguous, unintended, and potentially nonsensical conclusions (similar to the existential fallacy earlier discussed).

 

[case#46cw39]

You've still got the odd conclusion though (because objectively, Thumper is either white [100% chance] or not-white [100%] -- as fls points out, there seems a problem assigning probabilities to universal statements; and "Thumper is white", although it doesn't look like it, can be read as a universal statement, with universe = Thumper: "All rabbits which are this rabbit named 'Thumper' are white [95%]"). To meet drkitten's objections, you might have to rephrase it:

{Bayesian}

x knows 95% of rabbits are white (in a sample / 'universe of discourse')
x knows Thumper is a rabbit (in the sample)
x can be 95% confident Thumper is white

{frequentist}

95% of rabbits (in a sample) are white
a rabbit (from the sample) will be randomly selected and called "Thumper"
there is a 95% chance Thumper will be white

It's a bit nit-picky, maybe; but without a more precise formulation, valid premises lead to ambiguous, unintended, and potentially nonsensical conclusions (similar to the existential fallacy earlier discussed).

wow thanks, I think I'm starting to get these Frequentist/Bayesian views of probability.

What you said seems to line up with the DHS manual I have in the following way.
Am I reading this right?

From the DHS Manual:
------------------------------------------------------------------------
{Bayesian} ...
"We have been using the concept of probability to represent the likelihood that a conclusion is true, given certain evidence. Another way of expressing this is to say that probability represents the degree of confirmation for the conclusion provided by the premises. This is the common interpretation of probability in inductive logic. Probabilities vary from 0 to 1. The value of 1 represents absolute confirmation -- a condition that applies only to deductive conclusions. A value of 0 represents no confirmation. For the range between 0 and 1, numerical values may or may not be assigned.

Various words are used to refer to the degrees of certainty within this range. For example, a typical meaning for the word “probably” is “more likely than not” or “with a probability greater than .5.”
-------------------------------------------------------------------------
{frequentist} ...
"The word probability can also be used to represent relative frequency, in other words, how often an event of interest occurs relative to other possible events. For example, in the case of the first-level supervisors in the agency, our premise told us that 95 out of 100 of these supervisors had taken the basic course in supervision. The relative frequency of 95/100 provided our best estimate of the probability that any one first-level supervisor had attended the training.

We will use the concept of probability throughout this unit on induction."

 

[drkitten]

wow thanks, I think I'm starting to get these Frequentist/Bayesian views of probability.

What you said seems to line up with the DHS manual I have in the following way.
Am I reading this right?

Pretty much. A frequentist uses probability to represent the frequency which which something happens -- obviously, this doesn't make sense when we're talking about a one-off event.

A Bayesian uses probability to represent belief or confidence -- but just because you believe strongly in something doesn't make it true, as millions of astrologers have learned to their dismay.

Unless you're drawing Thumper from a pool of similarly-situated rabbits, Thumper himself is a one-off event and his color is what it is. Therefore, you can't talk meaningfully about the probability of his color from a frequentist point of view. You can be strongly confident that he's white (from a Bayesian point of view), but again, this just means that you think you know what you're doing, not that you actually do. Either way, the statement that "Thumper is 80% likely to be white" doesn't actually mean what a logician would like it to mean.

 

[case#46cw39]

Pretty much. A frequentist uses probability to represent the frequency which which something happens -- obviously, this doesn't make sense when we're talking about a one-off event.

A Bayesian uses probability to represent belief or confidence -- but just because you believe strongly in something doesn't make it true, as millions of astrologers have learned to their dismay.

Unless you're drawing Thumper from a pool of similarly-situated rabbits, Thumper himself is a one-off event and his color is what it is. Therefore, you can't talk meaningfully about the probability of his color from a frequentist point of view. You can be strongly confident that he's white (from a Bayesian point of view), but again, this just means that you think you know what you're doing, not that you actually do. Either way, the statement that "Thumper is 80% likely to be white" doesn't actually mean what a logician would like it to mean.

That was great thanks drkitten. This took me down the rabbit's hole into Baysian Inference Methods in the deep end. I say this because this public domain DHS MAnual I'm using for inductive logic (employing probability, both frequentist and Bayesian) says:

"We have been using the concept of probability to represent the likelihood that a conclusion is true, given certain evidence. Another way of expressing this is to say that probability represents the degree of confirmation for the conclusion provided by the premises. This is the common interpretation of probability in inductive logic."

So this interpretation of probability in induction is referring to the Bayesian (subjective) Inference Methods. Now as an old programmer this immediately reminded me of certainty factors for expert system inference engines which were Bayesian. Certainty factors could be combined to calculate the certainty factor for the conclusion. I forget the equation.

As i was looking this up I found the following Microsoft article which shows the shortcomings of certainty factors and overcoming them with the modern usage of Belief Networks.

"Each variable in a belief network is associated with a set of probability distributions. (A probability distribution is an assignment of a probability to each instance of a variable.) In the Bayesian tradition, these distributions encode the knowledge provider’s beliefs about the relationships among the variables"
http://research.microsoft.com/en-us/um/people/heckerman/h92encyclopedia.pdf