William M. Briggs

Statistician to the Stars!

Tautology Tautology: Day Two Teaching

Tautologies may be like this screwy circle.

Tautologies may be like this screwy circle.

Since probability is the completion—we do not say “branch”—of logic, and logic is the science of the relations between propositions, probability is also the study of the relations between propositions.

A (provable) rule of logic is that you cannot prove a contingent proposition starting from a necessary truth. Since it is a rule of logic, it is a rule of probability, too. Contingent propositions are the ones most familiar to us. “The car will start”, “The price of lemons will increase”, “A proton is made from quarks”, and on and on. Propositions which are themselves not necessarily true, which even if they are true contingently (lemon prices have risen in my neighborhood), don’t have to be true. The world would continue were lemon prices to remain flat, and the universe could have been built with quarkless protons.

In shorthand, if we have some proposition P and evidence E, Pr(P|E) = Pr(P|ET), where T is a necessary truth. T does nothing. It also does nothing when P is contingent and there is no other evidence than T. Thus Pr(P|T) is not a unique number, but because a tacit premise is P’s contingency we know that 0 < Pr(P|T) < 1. Note that the inequalities are not strict.

I think reader JH pointed to whathisname who wrote a book defending Objective Bayesianism (names escape from me faster than female interns from the late Ted Kennedy’s office). Williamson, maybe. Anyway, he and even subjective Bayesians insist on precise probabilities and so would pick 1/2 for Pr(P|T), which is screwy, as an example will prove.

The one that came up in class was the proposition P = “The Patriots win next year’s Super Bowl.” The necessary truth was the tautology T = “The Patriots might win”, which only acknowledges the contingency of P. T is equivalent to T = “The Patriots might win or they might lose.”

It is well at this point to remind that tacit premises about word definitions and grammar always accompany all our written evidence.

Now T is true no matter what, if by “lose” we mean not that they entered the final competition and lost, but that they do not win, such as is the natural state of the Detroit Lions. Any team which does not take the trophy (or whatever it is; I don’t follow football) “loses.”

Students have a difficult time wrapping their heads around (logical, not grammatical) tautologies, not being familiar with them. T is true no matter what, even if the Patriots disassemble their team tomorrow, even if the NFL goes bust or the universe ends.

Again, many like to say Pr(P|T) = 1/2. But we could have also said T2 = “The Lions might win” or T2 = “The Cowboys might win” and it cannot be true that the probability of each of these is 1/2.

Williamson (I think that’s the name; I’ll look it up later) wants probability to be a unique number and says so. He wants that unique number to be, in the face of uncertainty, the maximally equivocal. That’s what he forces the 1/2. But I could have also used the tautology T3 = “Tomorrow it will rain or it won’t” and surely “Pr(P|T) = 1/2″ makes no kind of sense.

Indeed, let Q = “I wore an orange pocket square”, which is true based on observation yesterday. P is the same. Then Pr(P|Q) is undefined or is still the same open interval if we insist the tacit premise of P’s contingency. Q gives no evidence of P, and so there is no possibility of the probability being 1/2.

And there is no problem in asking “What is Pr(P|Q)?” just as there is no problem asking “Given x + y = 7 and w = 13, solve for x.” This is not a typo. The w is not probative or x or y; it adds no information. We cannot come to a unique single number of x, and there is no reason in the world we should insist on one.

Chapter House

That public house, where we usually spend our first night, was mobbed with USA-Ghana watchers, so it was off to Ruloff’s, an indifferent and non-air-conditioned bar in College Town. Greasiest appetizers (a contradiction in terms if ever there was one) you can find. But convenient. They had a small badly tuned television where we were able to celebrate the victory.

I’ll also note for Luis, if he’s reading this—Luis is from Portugal—that suicide is never an option.

Typos

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16 Comments

  1. For your edification about American football: https://en.wikipedia.org/wiki/Vince_Lombardi_Trophy

    The Patriots have shored up their defense, but the receiver corps is still suspect and so reduces their chances of taking the trophy in February. They have a tougher schedule this year, too.

    The Palms was the undergraduate beerhall in College Town when I haunted Ithaca in the late 70s. As a graduate student I rarely went there.

  2. Williamson, maybe. Anyway, he and even subjective Bayesians insist on precise probabilities and so would pick 1/2 for Pr(P|T), which is screwy, as an example will prove.

    Do they insist? Williamson seems to indicate that the Principle of Indifference, which you believe to be axiomatic, does.

    Williamson (I think that’s the name; I’ll look it up later) wants probability to be a unique number and says so/

    No, he doesn’t; see below.

    Also, juts to be clear, the above quotations are not the points I tried to bring up when I referenced Williamson’s book. From Jon Williamson’s book, Ch 2, pp 11-22:

    2.2.3 Logical Probability

    From the point view of Epistemology, the Principle of Indifference tells us almost everything there is to know about logical probabilities. This is a point in favour of Keynes’ theory, in that logical probabilities turn out to be knowable, but a point against inasmuch the logical theory fails to account for other apparent ways of determining probabilities.

    There are two problems here. First, the Principle of Indifference requires a partition of basic alternative over which to equivocate; …

    The second problem arises in the case in which there is an appropriate partition of basic alternatives, but one that is infinite.

    The logical interpretation of probability and the objective Bayesian interpretation are often conflated or thought of as being very closely related (see, e.g., Franklin 2001; Rowbottom 2008). …

    A logical interpretation either takes some probability to have no numerical value (in which case it suffers with respect to the Calculi desideratum) or it takes probability to be fully numerical (in which case the uniqueness of probability required by the logical interpretation becomes implausible). On the other hand, the objective Bayesian interpretation takes probability to be fully numerical but does not insist on uniqueness.

  3. Furthermore, on subjective Bayesian, from Jon Williamson’s book, In Defense of Objective Bayesianism,Ch2:

    2.2.2 Bayesian interpretations

    The Bayesian interpretation of probability should not be confused with Bayesian epistemology (which holds that the strength of our beliefs should satisfy the axioms of probability),…

    Broadly speaking, there are three kinds of Bayesian epistemology and each kind gives rise to a distinct interpretation of probability. Any Bayesian epistemology holds that one’s degrees of belief at a particular time must be probability if they are to be considered rational. We shall call this the Probability norm. Now strictly subjective Bayesian epistemology takes this condition to be sufficient for rationality as well as necessary: if your degrees of belief at a particular time are probabilities then they are rational. (This condition deals with rationality at a particular time; as we see in Chapter 4. Subjective Bayesian also advocate Bayesian conditionalization, a rule for updating degrees of belief which holds that if one learns new evidence e then one’s new degrees of belief should match the old degrees conditional on E…)

  4. Brandon Gates

    17 June 2014 at 1:13 pm

    Briggs,

    Anyway, he and even subjective Bayesians insist on precise probabilities and so would pick 1/2 for Pr(P|T), which is screwy, as an example will prove.

    Thanks for that, you’ve just nuked my probability estimate for the existence of God(s). Are you trying to turn me into an atheist again?

  5. Brandon,

    God would accept your resignation. He accepted mine the minute I was born into my non-theist Buddhist family. Anyway, your comment reminds me of the following, which might help you. :)

    The principle of insufficient reason (PIR, also called the principle of indifference) -http://rationallyspeaking.blogspot.com/2005/10/principle-of-insufficient-reason.html

    The principle of sufficient reason has often been used to argue for the existence of God, basically as underpinning some version of the cosmological argument, and some philosophers have seen it as a generalization of the dictum ex nihilo nihil fit, “nothing comes from nothing.”

    The PIR, on the other hand, would suggest that God is just one among many possibilities, and that the God hypothesis should be granted equal probability against its alternatives. Obviously, the question immediately arises about how many mutually exclusive hypotheses there are here. At minimum, we have “God exists” and “God doesn’t exist,” which would make Leibnitz rather unhappy because the probability of God would drop from 1 (certainty) to 0.5. But of course one could argue that many different kinds of gods have been proposed, and that each one of them should count as a hypothesis. This would make the likelihood of the existence of a particular God (say, the Christian one — of which, of course, there are also several sub-species) pretty low. Indeed, from there it’s a short step to suggest that there is an infinite number of possible gods, and that therefore the probability of existence of any one of them is infinitely small. Leibnitz is getting more and more unhappy! Ironically, then, the Principle of Insufficient Reason becomes a fairly strong anti-theistic (and anti-deistic) argument.

    Of course, things aren’t that simple, because theists can argue that there are in fact other facts known about the universe that make the PIR invalid (e.g., the exquisite apparent “design” of living organisms). Equally reasonably, tough, atheists can argue that the facts actually point even more strongly against the existence of gods (e.g., we have naturalistic explanations of the complexity of living beings, the problem of evil, etc.).

    But these latter arguments go beyond the scope of the PIR, and therefore it is no longer useful in this context. However, the PIR is a simple and yet powerful tool to begin any such discussion, because it at least forces the defenders of one position or the other to state clearly why they think the PIR is not applicable to the issue at hand, i.e. why one thinks that the a priori probability of one hypothesis is significantly higher than the probabilities of all alternatives. And that, of course, is where things become really interesting.

  6. Brandon Gates

    17 June 2014 at 2:40 pm

    JH,

    God would accept your resignation.

    Resignation appeared to be a hyper-link, but there’s no URL?

    If someone held a gun to my head and made me choose a religion — in the sense that I must believe in some, any, God(s) or die — it would likely be Taoism, which as I understand is also quite Buddha-friendly.

    Indeed, from there it’s a short step to suggest that there is an infinite number of possible gods, and that therefore the probability of existence of any one of them is infinitely small. Leibnitz is getting more and more unhappy! Ironically, then, the Principle of Insufficient Reason becomes a fairly strong anti-theistic (and anti-deistic) argument.

    I bulk-order irony meters for that very reason.

    But these latter arguments go beyond the scope of the PIR, and therefore it is no longer useful in this context.

    For me at present: At best they (might, maybe, possibly, plausibly) falsify man-made conceptions of God(s). That they are man-made is all but self-evident to me at this point … but with a few BIG buts ….

    i.e. why one thinks that the a priori probability of one hypothesis is significantly higher than the probabilities of all alternatives. And that, of course, is where things become really interesting.

    Understatement alert. And very much helps me better frame my own questions. Thank you.

  7. I think students may have a hard time wrapping their heads around logical tautology because really, when you get into it, it seems redundant until you use it for something controversial, like the cause of the beginning of all motion. ;) The trouble with that, though, is that we really don’t know if there was a beginning, let alone whether, or not (he he), anything caused it or would have to.

    JMJ

  8. Briggs,
    “the universe could have been built with quarkless protons”. Could seems a little strong and so I suggest: as far as I, W.M. Briggs, know it might have been otherwise.

    You have corrupted your followers, who seem compelled to insert religion into every topic now matter how mundane.

    So what is the solution to the quasi spam?

  9. Briggs

    18 June 2014 at 5:07 am

    JH,

    That’s the guy. And that’s his idea: “Equivocate” to a number in the absence of evidence. Sheesh. Reckless, that. Over-certainty guaranteed. Thanks.

  10. Briggs

    18 June 2014 at 6:46 am

    JH,

    Got a good one for Williamson to show that equivocation doesn’t work.

    Premise: E = “There is a bag of 10 marbles with either 2 or 8 white marbles. There cannot be 5 white marbles. One marble will be pulled out.”

    Proposition: P = “The marble pulled is white.”

    The logical probability is {2/10, 8/10}, i.e. the set. Williamson is powerless here. There is no equivocation.

    Of course, my premise was symmetric. It needn’t be. Let there be 33 marbles, with either 3 or 18 or 29 marbles, etc., etc.

  11. Mr. Briggs,

    Williamson does good literature reviews on how probability is assigned via various thoughts, including Principle of Indifference (POI). The flaws of POI (Equivocation), as applied in the absence of calibrating evidence, are summarized in chapter 2.

    I am not sure why you say that Williamson is powerless. Your example is a case when POI is not applicable. It has nothing to do with the flaws of POI. I think Williamson would agree with your assignment of the probability in your example, which can be also done via the classical probability.

    The post by Massimo referenced in my previous comments, (as you know, he is one of my favorite bloggers and philosophers), uses the example of “God’s existence” to show the two problems of POI and to explain it is not applicable when extra relevant information is available.

    My main point of my previous comments is that your descriptions of what Williamson writes/thinks in this post are just wrong.

  12. Am I missing some subtlety of argument here? This seems so straightforward. After you draw the white ball P(w=8) rises to 0.8. In the non-symmetric case P(w=3) falls to 0.06. Starting with a uniform prior when nothing else is known about the bag of marbles is defensible. Starting with a uniform prior when much is known about the Lions and Cowboys is not.

  13. Brandon Gates

    18 June 2014 at 1:24 pm

    Scotian,

    You have corrupted your followers, who seem compelled to insert religion into every topic now matter how mundane.

    Curious critique. I’m probably missing some larger conversation here, but I find myself asking why talking about religion is necessarily a corruption, or indeed why it would have more to do with Briggs than the guy who inserted religion into this thread.

    Mundane, like corrupt, is a matter of opinion as well. I see this topic as abstract, lending itself to numerous applications. “1/2 for Pr(P|T)” speaks to me particularly loudly as an agnostic, for I often ask myself — and am sometimes asked by others — to put a probability on the existence of God(s).

  14. Brandon Gates

    18 June 2014 at 1:29 pm

    JH, thanks for the link. I myself have given many speeches on the subject of the importance of allowing individuals to explore their own (non)relationships with God(s).

  15. Briggs,

    Take the E and P from your comment example with marbles. Add:

    Proposition T: There are two white marbles in the bag.

    Premise F: A coin is tossed and will land with either heads or tails face up. It cannot land any other way.

    Proposition Q: The coin lands heads up.

    What is the difference between the application of the POI to P(Q|F) and to P(T|E) ?

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