Everybody: thanks for the emails. I do see them. I’ll be answering all on the weekend.
Would it surprise you to learn that to graduate with a degree in statistics—BS, MS, or PhD—you are nowhere required to take courses in epistemology, logic, mathematical logic, and so on? Did you know that this is also true for graduates of physics, chemistry, biology, and for most other sciences?
Oh, there are minor exceptions, but usually in the form of ethics classes, and only required of those who deal in subjects like medicine. Too, some voluntarily take a class or two in these areas. But the philosophy the majority of scientists assimilate is by osmosis; and that consisting of generally misunderstood or misremembered fragments of Popperian falsifiability, the ability to identify (usually) common fallacies, and recognizing that bit about Socrates.
Which is this:
(P1) All men are mortal
(P2) Socrates is a man
————————————————————————————
(C) Socrates is mortal
Yesterday, we said that given (P1) and (P2), the probability of (C) is 1. The conclusion is only true with respect to the information provided in the premises. Thus, logic is, as was suggested in comments, a measure of information.
We are not free to dispense with either of the given premises, nor can we implicitly use others that are not given. Think of the argument like a mathematical theorem (which, in a real sense, it is): we would never dispense with or add to the conditions of a theorem, and we cannot do so here. Logic is, of course, the study of why arguments like this are valid; and why arguments like “Given (P2) alone, (C) is true” are valid.
We can put this old argument in a more abstract form, which (somewhat against intuition) might help.
(P1) All F are G
(P2) X is an F
————————————————————————————
(C) X is G
Looks a lot more like math now, doesn’t it? (C) follows in a now familiar, analytical sense from (P1) and (P2). Further, we can see that we cannot dispense with either premise. And we are free to substitute other F, G, and X! It turns out (this is not of main interest here) that we have to put restrictions on what’s allowable for F, G, and X—but this kind of thing is also true in mathematics, where we often start theorems restricting the universe to certain kinds of numbers.
Stating the argument in this form allows us to remove distractions: it is drier and less emotional.
But let’s look at that argument a little more closely; specifically, examine (P1), which is “All men are mortal”. Is that true? Well, nothing can be true or false without reference to something. Given our experience, (P1) has been true so far. But will it always be true? Saying so is to make an inductive argument; and based on our knowledge of logic, we know that all inductive arguments give us conclusions that are not certain; which is, that they do not have conclusions with probabilities of 1 or 0, but somewhere in between.
(P1) is contingent: it might be that sometime in the future, the universe might be such that (P1) is false. But regardless whether (P1) is true, false, or something in between, accepting it is true (“for the sake of argument”) our conclusion (C) follows. Get it? It doesn’t matter whether (P1) is true or false or merely probable with respect to other information: all that matters is that we accept is as true in our argument.
Now, if we change (P1) to align with our experience, then we have
(P1) So far, all men have been mortal
(P2) Socrates is a man
————————————————————————————
(C) Socrates is mortal.
The conclusion no longer follows deductively; it is now an inductive argument, and the conclusion is only probably true. This also happens if we change our “analytic” form:
(P1) So far, all F have been G
(P2) X is an F
————————————————————————————
(C) X is G.
Once more, (C) does not follow deductively; it is only probably true.
Logic is a measure of information, a special case where probabilities are 0 or 1. Probability, then, is also a measure of information. We can modify our argument again:
(P1) Most F are G
(P2) X is an F
————————————————————————————
(C) X is G
We can now see that the probability (C) is true given (P1) and (P2) is not 0, nor is it 1; it is somewhere in between. Further, because of our knowledge of the word “most”, we can say that the probability (C) is true given (P1) and (P2) is greater than 0.5 but less than 1.
As before, we can’t let every conceivable X, F, or G, into the picture; but the schema is still useful for its illustrative purposes.
Quick update: Try substituting for X, F, and G, “Bob”, “Winged Horses”, and “Clever”.
IF “We are not free to dispense with either of the given premises … ”
THEN
” … arguments like “Given (P2) alone, (C) is true†are valid.”
is FALSE.
We are not free to dispense with either of the given premises, nor can we implicitly use others that are not given.
Right .
However if using the same premises one finds a second conclusion contradicting the first one then there is a problem .
Either the system is inconsistent (system here = formal logics) , the premises are contradictory or one of the conclusions is incorrect .
That’s what I just did in the previous thread about the “6 faced object” 🙂
TomVonk:
Please give a simple example.
While I cannot get inside your head I will speculate that you mean not, ” … if using the same premises one finds a second conclusion contradicting the first … ” but “If one finds an example that satisfies the premises but contradicts the conclusion … ”
(P1) All men are mortal
(P2) Socrates is a man
And then one day, while working in the emergency room you come upon a man named Socrates who has multiple stab wounds, blunt trauma and gunshots to the head but is sitting up drinking a cup of coffee, smoking a cigarette and discussing football you would be justified in saying that one or both of the premises are wrong.
You are right. When arguing, it’s OK to suppress a premise that’s commonly known and would be accepted by everyone present. However, we need to make all necessary premises explicit for the conclusion to be logically sound or valid when evaluating arguments.
So, educate me, and help me see why P(C1|P1,P2) is 1/6 without using the additional premise that the six-sided subject is fair by symmetry? (I did take a class on mathematical logic in Chinese during high school though; and the high school I attended was top-ranked. Does this count? ^_^)
I do prefer the abstraction, because it helps to avoid discussion about the physics of rolling a die. I think for the die example you could abstract this as:
(P1) f:{1,2,3,4,5,6} –> {0,1}
(P2) f(1)+f(2)+f(3)+f(4)+f(5)+f(6) = 1
——
(C) f(6)=1
In English, (P1) states that f is a function defined on the set {1,2,3,4,5,6} such that f(k)=0 or f(k)=1 for each k in that set.
It’s clear from (P1) and (P2) that f(k)=1 for exactly one k in {1,2,…,6}. If I understand Briggs correctly, he contends that Prob(C|P1&P2)=1/6 is an assumption that comes from a rational system unifying logic and probability.
JH,
I thought I had when I proved that the inclusion of “fair” made the argument circular. And how about those winged horses examples?
I think the difficulty is that most people cannot leave real dice out of their minds when they think about probability as logic. Yet I never mentioned a die, nor anything like one. I certainly didn’t when talking of Johnstons and Freihaufs and winged horses.
You might think you have won the day by (mistaken) arguments against the die example, but you’d also have to prove why the other examples are invalid, too.
Tom,
I’m really following Jaynes here.
Steve Brookline,
Your premises are much more informative than mine were. I had no information about the other sides.
All,
Just quick notes. Have to run.
I haven’t won any day, in fact, I often lose several days trying to figure out what I don’t understand. Not understanding bugs the hell out of me.
No need to prove why other classic examples are invalid, nothing wrong with them and I understand them just fine; but they have only convinced me the premises you provided are not sufficient to warrant the conclusion that P(C1|P1,P2)=1/6.
I have tried to explain why I don’t understand, and provided some possible situation for the six-sided object under which, I think, that the conclusion P(C1|P1,P2)=1/6 is logically sound. I have also carefully considered other readers’ explanations. Nope, I still haven’t found the missing link or convincing arguments.
Of course, I’d asked Mr. JH who did a double-major in philosophy and mathematics during undergraduate, nope, not helping, because he said you were wrong and he is currently busy with a music project.
I apparently just don’t get it. I’ll stop thinking about this and wait until school starts.
What kind of idiot would not realize that the conclusion is only true with respect to the information provided in the premises?
(P1) Most F are G
(P2) X is an F
————————————————————————————
(C) X is G
“We can now see that the probability (C) is true given (P1) and (P2) is not 0, nor is it 1; it is somewhere in between. Further, because of our knowledge of the word “mostâ€, we can say that the probability (C) is true given (P1) and (P2) is greater than 0.5 but less than 1.” – Briggs
In the case above, as with the Freihauf problem, I say that the probability is very near to 0.75, on the grounds that all G fractions of F greater than 1/2 are equiprobable, based on P1.
Of course, this depends on the definition of “most.” For example, someone who says that “most” does not include fractions very slightly greater than 1/2 will calculate some other probability. Also, a person can take into account that there are various definitions, and in this case he may be uncertain about which probability he should calculate, or which probability someone else will calculate, but this is not uncertainty about some objective thing called “the probability.”
I really don’t understand the idea that a probability can be uncertain. Probability is about uncertainty. It does not make sense to say that a statement of uncertainty is itself uncertain.
Mr. Briggs,
Oh… Game 7 of the NBA Finals is on… what mistaken arguments have I made against the die example? Copy and paste, please. It would be greatly appreciated if you could also explain why they are incorrect.
“I have tried to explain why I don’t understand, and provided some possible situation for the six-sided object under which, I think, that the conclusion P(C1|P1,P2)=1/6 is logically sound. I have also carefully considered other readers’ explanations. Nope, I still haven’t found the missing link or convincing arguments.” – JH
From what I can see, you’re arguing that, because there are 6-sided objects for which P(6) would not be 1/6, P(6)=1/6 is not supported. The trouble with that idea is that probability is a best bet, based on what one knows; without knowing the type of object, one’s best bet is P(6)=1/6. For example, it would be wrong to say that P(6)=1/5; there’s nothing to support it.
I think Mr. Briggs agrees with this, but I don’t see that his agreeing would gibe with statements he’s made that a probability is between something and something, i.e., that a probability may be uncertain.
To understand disputes about this type of thing, you have to know about interpretations of probability. The Subjectivist interpretation supports P(6)=1/6. I’m not sure whether the Frequentist interpretation would, since, to support it, it might require that all possible types of 6-sided objects form a distribution in which P(6)=1/6 corresponds to the mean, and, for all I know, there may be a mathematical reason why it’s not so. The Subjectivist interpretation has no problem with this; it says that since I don’t know whether the types form a distribution in conflict with P(6)=1/6, I reasonably assume that they do not, and this makes 1/6 my best bet. If someone else knows that they do, his best bet is something other than 1/6. If someone else believes, based on (say) Chinese superstition, that 6 appears more often or less often, his best bet is yet something else.
Smoking Frog,
Six is a lucky number according to Chinese superstition! In life, we are often forced to make a “best bet†conclusion or decision with insufficient premises/information. What is the best bet? It probably depends on the situation, who you are and what you want.
However, the explanation of “best bet†implies that you are not certain about the conclusion. The point is that one has to be certain that the conclusion is true (w.r.t. the premises), regardless of the interpretation of probability.
If I understand correctly, Tom V’s comment is trying to address this point.
It’s interesting to read what people think that Briggs might have in mind. And we all are reading the same thing! It sure would be nice if we were all in a comment room where we are all physically present. Much easier to clear things up.
And, gentlemen, thank you all for your explanations.
The die problem, the Freihauf problem and in fact all others necessarily carries the assumption that all possible values are equally likely unless there is evidence to the contrary. That, I think, is what most people have in mind when imagining “fair” dice. Briggs’s claim of circularity is itself circular as you can only reach the conclusion of P(any side | lack of any other knowledge than it is a 6-sided object) = 1/6 by assuming that all values are equally likely. I tried to point this out, perhaps too subtly, in my remarks about the Freihauf example.
The assumption of equal likelihood is a simplifying assumption AND TOTALLY UNWARRANTED except for the fact that you must start somewhere. Without it and lacking any other information that the “object” is 6-sided, all you can say about P(side appearing | everything we know so far) = DON’T KNOW.
As far as it being a “best” bet, keep in mind that “best” varies with requirements and requirements vary with the individual (JH’s point, I think).
I don’t think immortality is possible because it would require an infinite amount of energy to produce that effect.
Speed
TomVonk:
Please give a simple example.
As I said , I did so in the thread of the 6 faced object .
Of course I didn’t mean the Socrates syllogism which is correct .
What I wanted to say is that one has to take EXTREME caution with premises .
Sometimes premises are so loaded with information (it was the case with the 6 faced object) that a contradiction is an almost certainty .
That’s what I meant – thinking to satisfy the premises yet coming to 2 contradictory conclusions with 2 different ways .
DAV
The die problem, the Freihauf problem and in fact all others necessarily carries the assumption that all possible values are equally likely unless there is evidence to the contrary. That, I think, is what most people have in mind when imagining “fair†dice.
Yes !
Even if I would be a bit more general and use invariant under transformations (.eg symetrical) instead of equally likely .
Sometimes when one is not careful enough , like in the 6 faced object case , one actually gives an example where the probabilities are NOT invariant by transformations and the conclusion is no more true .