Teaching Journal: Day 2

As might have been obvious from yesterday, the truth, falsity, or the somewhere-in-betweenness of any conclusion-hypothesis-proposition can only be assessed with reference to a list of assumptions, premises, data. That is, you cannot know the status of any proposition without some list of premises. Different premises lead to different statuses.

In particular, this means that you cannot ask, “What is the probability of X?”, where X is some proposition. For example, you cannot ask, X = “What is the probability that I roll a six on a die?” This probability does not exist. Similarly, you cannot ask, Y = “What is the probability that Socrates is mortal?” This probability also does not exist. There are no unconditional probabilities, no unconditional arguments of any kind.

If I assume that E = “All men are mortal and Socrates is a man” then I can claim that “It is certain, given E, that Y”, or that “If I assume it is true, regardless whether or not it is or that I can know it is, that all men are mortal and Socrates is a man, then the probability that Socrates is mortal is 1.” Or I can write:

    Pr( Y | E ) = 1.

But I cannot write:

    Pr( Y ) = something,

for that is forever unknown. There just is no such thing as an unconditional probability, just as there are no such things as unconditional logical arguments, just as there are no such things as unconditional mathematical theorems, and so on. If you find yourself disagreeing, have a go at creating the probability of some hypothesis that does not make reference to any assumptions/premises/data.

(It’s only true that in most textbooks probability is written as if it were unconditional. While this makes life for the author and for typesetters, it ends up producing confusion about the nature of probability.)

It’s important to understand that E is only that which we assume is true. It matters not one whit whether E—with respect to some other set of premises—really is true, or false, or somewhere in between. Logic concerns itself only with the connections between premises and conclusions. The premises and conclusions are something exterior, something given to us.

Another hoary example. Let Ed = “A six-sided object, just one side of which is labeled 6, will be tossed and only one side can show.” Then if X = “A 6 shows”,

    Pr( X | Ed ) = 1/6.

We have deduced—just as we do with all probabilities—the probability that X will be true. Notice that this says nothing about real dice in any real situation. This is just a logical argument, no different in nature from the premise “All Martians wear hats and George is a Martian” which lets us deduce that “George wears a hat.” This conclusion with respect to this evidence is true, it’s probability is 1; and this is so even though we know, with respect to observational evidence, that there are no Martians.

Now if we write X = “A Buick shows”, we can write

    0 < Pr( X | Ed ) < 1

We are stuck because our evidence says nothing about a Buick. There may be a Buick on one of the other five sides, there may not. The evidence is mostly mute on this subject. Except if we suppose there is an implicit call to the contingent nature of this object being tossed. If we assume that, then we can at least say the probability is not 0 and not 1, but it may be anywhere in between. But we can also make the argument that Ed should be interpreted more strictly. If it is the case, then the best we can do is this:

    Pr( X | Ed ) = unknown.

Probability cannot be relative frequency. For example, given “Half of all Martians wear hats and George is a Martian” which lets us deduce that the probability “George wears a hat” is 0.5. But there is no relative frequency of this “experiment.” This one counter-example is enough to show that the relative frequency interpretation of probability is false (it doesn’t show it has things backwards; for that, read the book, paying attention to the references).

Probability cannot be subjective in the following sense. If we accept that “All men are mortal and Socrates is a man” then the probability that “Socrates is mortal” is 1. Even if we don’t want it to be. And above the probability that “George wears a hat” cannot be anything but 0.5. Probability only appears to be subjective in some instances because we often are bad at listing the premises we hold when assessing probabilities. However, if we agree on the exact list of premises (and on the rules of logic) then we must agree on the probabilities deduced.

Probability cannot always be quantified. If we accept that “Some men are mortal and Socrates is a man” then the probability that “Socrates is mortal” is something between 0 and 1, but the exact number cannot be deduced.

Homework

Why doesn’t adding “the six-sided object is weighted” or “the six-sided object is fair” to Ed change the probability that Pr( X | Ed ) = 1/6?

List the exact premises you hold in your Pr (“Barack Obama will be re-elected” | your premises).

Change the evidence “All men are mortal and Socrates is a man” so that the probability of “Socrates is mortal” is bounded between two fixed numbers.

In a tearing hurry today. Did not have time to check for typos!

13 Comments

  1. I have never understood why Ed = “A six-sided object, just one side of which is labeled 6, will be tossed and only one side can show” is sufficient to give a probability of Pr( X | Ed ) = 1/6, where X = “A 6 shows”.

    I can think of countless six-sided objects where this is simply not the case – I have to add “regular” to the properties of the six sided object to believe that this probability holds – so I presume I would fail the Homework task.

    I feel we are in the realm of the spheres, without gravity and goodness knows what else and where “tossed” means something different from everyday usage to understand Prof Briggs’ point.

    I also don’t really get his requirement to list the exact presmises for Obama’s reelection. How can my premises warrant a numerical precision better than 0<p<1?

    Sure I can say, given that a coin toss is as reliable as any other method I'll say its 50:50.

    If that gets me a mark for effort, well and good.

    But I feel I am missing the point. That given is one iffy given, and surely that is where the debate lies? How do you get reasonable givens, not the consequences of what those givens give.

  2. I noticed the same error as Chinahand in the definition of the six sided die. The sides must be of equal area. A pyramid with its top cut off would be an example with unequal probabilities.

  3. Chinahand,

    Surely one of your premises in the Obama election is that he hasn’t died before the election.** There must be others that amount to “all bets are off”.

    I think you’ve got a point on the 6-sided object. A square washer is 6-sided and I wouldn’t assign an equal probability to all sides. Brigg’s might have been thinking “cube” for his object.

    ** Let’s hope not. There must be a reason why the VP was the #2 party choice. Could be he’s a #2 kinda guy.

  4. CH: “I have never understood why Ed = “A six-sided object, just one side of which is labeled 6, will be tossed and only one side can show” is sufficient to give a probability of Pr( X | Ed ) = 1/6, where X = “A 6 shows”.”

    I see two possibilities:

    1) All other evidence not mentioned is assumed to have no bearing on the case – which is why it is not mentioned – so all faces are equally likely since there is no evidence to say they’re not. This could easily be made explicit.

    2) The evidence supplied is to be treated as incomplete and must be supplemented with statements about the area of the faces, the distribution of matter within the die, an accurate description of the force vector that results in the toss, a description of the force and direction of any impinging draughts, and so on until the cosmos has been described in detail with respect to this dice throw.

    I assume most people work with 1).

  5. What is the probability of getting a 6 on a six-sided die with each side showing a different number from 1 to 6? When there is no other information, some people may answer “I don’t know.” Others may resort to a supposedly reasonable answer of 1/6, i.e., using the uniform distribution by applying the principle of indifference.

    Wouldn’t it be simpler (at least for me, anyway) to just say that a probability satisfying certain axioms is assigned based on given evidence/premises (a measure of conditional uncertainty) and that objective means people tend to come to an agreement, one way or another, over time and with more acceptable evidence?

  6. “True” is a verb. Sorta. X cannot simply be true; it must be true to something. Hence, a husband is true to his spouse; a student may be (as the Beach Boys taught) be true to his school. Fiction ought to be true to life; a scientific theory, true to the facts. A theorem in Euclid is true to the postulates and axioms of Euclid.
    If I understand correctly, P(X)=p must be “true to” E, the oft-times unspoken or even unexamined assumptions; hence P(X|E)=p is a more precise way of expression. But E must be “true-worthy” (trustworthy), that is, it must merit the faith placed in it.
    “Truth” relates to “faith” in that “to be faithful” means “to hold true” to something. One has faith that the dice are not loaded. A proposition is then true to the extent that it is faithful, not necessarily to the extent that it is factual. (Factual comes from the verbal participle factum est, “that which has the property of having been accomplished.” Cognate with “feat”. See German Tatsache, lit. deed-stuff. So: the proposition about Martian George is true but not factual. So is the fable of Beauty and the Beast.
    ‘Nuff musing.

  7. Ed = A six-sided object non-uniform density with irregular faces, just one side of which is labeled 6, will be tossed and only one side can show.

    We don’t know the probability of each of the faces showing
    There is enough information to say that it is unlikely to be the same for all 6 faces.
    And we know that the sum of those probabilities must =1.
    We also don’t know if the 6 is on the larges face or the smallest face, on the light end or the heavy end.

    But with the information we have the 6 could be on any of the faces, and we do not have a better model. 1/6 is still our best estimate of the likelihood of a 6 based on the information we have.

    Obama-Romney, I can say that I think that Obama has a 55% chance of winning the election in November based on my information set. But, what does that mean? Does it mean that I would be willing to accept an even money wager? Not-necessarily. However, us fincancial types will take the reverse and flip it around. If a bookmaker is sets the betting line such that he will pay 82 cents for every dollar wagered. The fincancier would say that the bookmaker (or the market) is implying a 55% chance of victory for O.

  8. The best I can come up with, is that P(one side labeled 6…) = 1/6 is that, Briggs is avoiding putting an assumption about this experiment into evidence and causing wailing and gnashing of teeth. Correct me if I’m wrong…

    There exists an infinite number of real-number measured six-sided objects (SSOs). Choose 1 six sided object from the universe of all six-sided objects. The probability of the side marked six being chosen with exact probabilty 1/6 = 0, because the probabality that X = x in any continuous distribution = 0. Also, there are equally many SSOs (infinite!) where the Pr(X|E) 1/6. Therefore, using CLT, the “exptected” P(X|E) converges to 1/6.

    Am I close?

  9. It deleted my less thans and greater thans. Last line should say that there are equal number of SSOs with Pr less than 1/6 and Pr greater than 1/6.

  10. Swade016 – your explanation looks a lot like frequentism to me (though what do I know!). My understanding is Prof Briggs really, really dislikes frequentism.

    This conversation rapidly gets caught up in empiricism vrs platonism.

    Is it logical that a SSO will land with one particular side at a probability of 1/6? Or is this the long term average of an empirical experiment?

    For me IT DEPENDS – which I suppose is sort of the point Prof Briggs seems to be making!

    If you live in a universe with regular gravity, where regular objects keep their regular shape and where the creator doesn’t have a real thing about the number 6 and hence doesn’t intervene to stop someone tossing (whatever that actually means – can a macro-scale process ever really be totally random!) a 6 on a particular SSO. I think it is reasonable to assume a uniform regular SSO can be modelled with a probability of 1/6 for any particular side.

    These assumptions aren’t based on logic, but on empirical observation and if you lived in E(a universe with a hexaphobic intervening deity) you would be happy to say it was reasonable to assume the p(6|E) was zero.

    I don’t think there is any logic in that, which does make me a little unsure about Prof Brigg’s insistance that this process is entirely logically based. Axioms aren’t logical, they are just presumed … but I think that is the point Prof Briggs is making – logic, empiracism, whatever – you are making presumptions and they bound your model; be concious of that! Your presumptions may not be empirically valid, even if your model is rigorously internally consistent.

  11. Doug M,

    The valid answer for a die or an irregular die is “I can’t know for sure, since there is insufficient information.” The extra word “irregular” doesn’t add any relevant information.

    However, if you have to assign probabilities, e.g., you have to assign a prior in Bayesian analysis, then the principle of indifference is a heuristic rule to follow. It’s not a probability axiom. The principle is also called principle of insufficient reason for obvious reasons. It doesn’t determine the correct prior, but so-called an objective one.

    —-
    The problems with the principle of indifference are well-known. Our dear friend Google is always here for us when there are problems.

  12. I agree with Chinahand. I’ve posed this question more than once on this site and have yet to receive an unambiguous answer:

    Let E = “I have an object whose six sides are labelled 1 to 6 AND the probability of rolling a 6 is NOT 1/6”.

    Let X = “I roll a 6”.

    Then, according to my understanding of our host’s argument, P(X | E) = 1/6 even though this directly contradicts E!

    Note that I am accepting here the notion of probability-as-a-state-of-knowledge rather than probability-as-a-state-of-reality.

  13. Ralph Becket,

    From any false or self-contradictory premise you may derive anything. Your E is not unlike those that run, “A six-sided object, etc. which is ‘fair’.” If C = “A 6 shows” then Pr(C|E) = 1/6, but it is a tautology. “Fair” just means “The probability of C given the probability of C is 1/6 is 1/6” which is true but circular.

    For you Pr (X | E) does not equal 1/6. You have constructed a self-contradictory argument, also not unlike the “liar.” For you, Pr(X|E) = undefined.

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