Sunday, September 11, 2005

Abstract and Concrete Probabilities

There seems to be an important distinction between two types of probability. Let's say there's some object O with a property P1, but you don't know what other properties it has. Further suppose that what you're really interested in is whether O has some other property P2. What's the probability that O has P2? We could interpret this as an epistemic question, whereby we abstract away all the unknown details of O and just ask what proportion of P1-objects also have P2. Here we aren't really talking about the specific object O at all. We treat it merely as a token of type "P1-object", without regard for the multiplicity of other (unknown) ways O might be categorized. That's one option. Alternatively, we might focus on the concrete object O, and ask: of this specific object, including its many other properties that we are not yet aware of, how likely is O to have P2? This is a question about the real or metaphysical modal properties of O, rather than being a purely epistemic question. Let's illustrate the distinction with a couple of real-world examples.

The first example came up in the comments to my recent post on ad hominems. Suppose a notoriously unreliable person makes an argument. You are wondering whether the argument (O) is likely to be a sound one (P2), given that it is being made by this notoriously unreliable person (P1). One way to answer this would be to abstract away from the details of the argument itself, and just use your background knowledge of the advocate's unreliability - in particular, that most of his arguments are unsound - to conclude that this argument is therefore likely unsound. This is a classic ad hominem fallacy. Although such abstraction may be rational in a sense, it isn't really fair on the argument in question. After all, you've completely ignored the argument itself, when the dialectical norms of civil discourse recommend that we consider each argument on its own merits.

This latter suggestion is taken seriously by the concrete probabilist. He focuses on the argument (object O) rather than who makes it (property P1). In particular, he recognizes that the validity of the argument is metaphysically independent of the person making it. So he ignores the person, and instead assesses the argument on its own merits, trying to determine whether it is logically valid, and how plausible the premises are. It is from these considerations that he estimates the soundness of the argument.

The second example is provided by stereotyping individuals (e.g. racial profiling). Say you want to know whether Jack (O) committed the crime (P2), when the only other information you have about him is that he is a black male (P1). You can probably tell what comes next. The abstract probabilist ignores all the complexities of Jack the individual, and makes his judgment in light of how many other black males have been known to commit crimes. As with the ad hominems, we can see that this prejudice is rational in a sense, but also incredibly unfair to Jack as an individual. Your judgment of him is based on what you know of others who share the same property as him (i.e. of being a black male), rather than anything concrete and intrinsic to Jack himself. I argue elsewhere that this is morally problematic.

Consider a pair of cross-racial "twins", Alan and Bill, who are identical in all respects except that Alan is white and Bill is black. Clearly this is a difference that makes no difference. As a matter of fact, each is equally likely to make any particular decision, or have any other property or characteristic. These facts are what determine concrete probabilities, and they are unaffected by whether we know about them. But abstract probability is an epistemic notion. If all you knew was that Alan was white and Bill black, then you would judge their abstract probabilities very differently. You might think Alan is likely to be wealthier, better educated, and so forth. But these "probabilities" are metaphysically superficial -- merely reflecting facts about group frequencies and correlations. Concrete probability is much deeper, being grounded in causal and explanatory facts about the concrete individuals themselves.

So, like I said, this strikes me as an important distinction. (If someone with more formal training in statistics can go beyond the intuitive graspings I've offered here, please do leave a comment.) I think the close analogy between ad hominems and sterotyping is also interesting to note.


  1. Judging is always unfair - for example should I be convicted of a crime just because I look like, sound like, and go to all the same places and belong to all the same groups as a criminal?
    But as unfair as that is the alternative is to basicaly give up on catching most criminals.

    Having said that I am not sure exactly where you want to start the debate...

  2. I have, once again, to express profound disagrement with your analysis of ad hominem (AH) arguments. The character and pesonality of an arguer may be intimately connected with the soundness of an argument he/she makes and/or the truthfulness of the statements (facts, if true) used in its support. Consequently the variables P1 and P2 you describe may not be independent. In that circumstance, knowing something about the probability of one variable being true may indeed provide us with information about the probability of the other being true.

    Therefore, not only are we justified in examining the relationship between an argument and its proponent, it is rational to do so in many circumstances. Indeed, not only this, but any inference we draw on this basis is not necessarily fallacious.

    Nothing you've said refutes my comment on your earlier post.

  3. Kofi, that's fine, I was not trying to "refute" your earlier comments, but rather was building on them to explore what I consider an interesting distinction. I granted that it can be rational to assess a statement in the abstract, understood as a mere assertion by that person (and without regard for its individual content). But this is a very superficial approach. The alternative is to assess the individual statement or argument on its own merits -- that is, focusing on the content of the argument rather than its vehicle (i.e. who is making it).

  4. Apologies, Richard, that I misunderstood your intent.

    However, I still don't think I agree with your approach. I don't think one can undertake a rational assessment of the content of a statement independently of an assessment of the person uttering it in many (perhaps even most) circumstances. Moreover, doing so strikes me as irrational, because one is ignoring information pertinent to any complete assessement of the statement content. The medium is not necessarily the message, but one cannot assess the worth (eg, truth-value) of the message without simultaneously seeking to assess the worth (eg, reliability) of the medium.

    As it happens, this is the crux of the criticism made in the 1960s of formal logic by what became the informal logic community, which led to the modern revival of the study of informal reasoning (so-called), argumentation and rhetoric since then.

  5. I think it's confusing to talk about "abstract and concrete probabilities", because probabilities are just like truth values and attach to propositions. I think it's better to talk about abstract and concrete propositions, though I'm not sure it's a useful distinction, because as Kofi says, what matters is whether the pieces of evidence are dependent or independent of one another, not whether they're abstract or concrete.

  6. Another way of looking at it is to consider the de dicto / de re distinction with regard to "Bob's argument". The superficial de dicto reading corresponds to what I called 'abstract' above. That is, you conceptualize an argument as belonging to Bob and so assess it in this light. But on the de re reading, you look beyond the identifying description ("Bob's argument") to the thing itself -- a thing which is entirely independent of Bob. This same argument could just as easily be made by Sue, or God, or be coincidentally observed in a random swirl of leaves blown around in the wind.

    So, while we want to judge the probability that "Bob's argument is sound" is true, the answer will be dependent on whether we conceive of "Bob's argument" in an abstract/de dicto or concrete/de re fashion.

  7. I don't see these as different sorts of probability: rather as probabilities relative to different evidence.

    One can attach a probability to the proposition "Argument A is sound" based on who makes the argument. We do it all the time, and we do it precisely because we think some people are more likely (through training, experience or whatever) to make sound arguments or reach sound conclusions. If my doctor tells me I am well, and my mother tells me I am sick, I attach a greater probability to my doctor being right than my mother. I do so because (1) I believe on reasonable grounds that my doctor is trained in such a way as to make her diagnoses relatively reliable. And (2) I am not in a position to evaluate the argument or conclusions for myself.

    Of course, if I am in a position to get and evaluate evidence other than the mere fact that "X says it is so", then I can evaluate a probability relative to that evidence. Often I will regard that probability as more reliable than one based merely on authority. (Not always, though: Suppose I am myself an inexperienced novice, and the person who has expressed the view is the doyen of the relevant profession.)

    As I see it, the distinction is not between "abstract" and "concrete" probabilities, but between probabilities based on not-very-much evidence, and probabilities based on a lot of evidence. The "argument" example is superficially attractive because we assume that it is possible to evaluate the soundness of the argument independently of the person making it. So it often is, and where it is we will normally do so--because even experts make mistakes. But if we can't do so (for instance, because we do not have the expertise to evaluate the argument ourselves, or the time) there's nothing irrational with attaching a high positive probability to an "expert' being right and a low positive probability to a "non-expert" being right.

    Assuming that it makes any sense to speak of probability at all (which would not be the case with some arguments: consider mathematical proofs) the relevant distinction is simply one of the evidence to which we attend in making our probability judgments. If H is the proposition "the argument A is sound", and E1 is the proposition "X made the argument" while E2 is some statement about the argument itself, then P(H|E1) may be different from P(H|E2). But these are not different types of probability: just probabilities relative to different evidence. In each case they are epistemic in the sense that they reflect the degree to which we are inclined to believe the proposition under analysis given the evidence we have analysed.

    The same applies to the racial profiling example: it's largely a matter of how good or bad the evidence is. Even if we knew, say, that a given crime is committed only by one group of people in the population and never by another (e.g., that rape is committed by men and not by women), the probability that a given man committed a given crime remains vanishingly small: there are so many men out there! It's not merely "unfair to X as an individual" to single him out on this basis: it is simply wrong as a matter of probability to attach more than the most minimal probability to X's guilt in these circumstances. No metaphysical magic is required.

    That said, theorists of law and probability have worried about "merely statistical evidence", along the lines you discuss. The classic instance is the gatecrasher example. Suppose 1000 spectators attend an event, but only 400 pay for tickets. There are 600 gatecrashers. Suppose the organisers sue X alleging he is a gatecrasher. The only evidence they adduce is that he was at the event: from which (they ask the court to infer) there is a .6 probability that he did not pay, making it "more likely than not" that he was a gatecrasher. Who wins?

    Most lawyers feel the organisers lose, if this is the only evidence. But why? Is it because the evidence "does not point specifically at him": some certainly think so. Or is it because the evidence, even viewed in its own terms, is so inadequate and incomplete. In real life one would ask: What is his story? Does he say he paid? If so, is he an honest sort of person? Can he prove he paid with a credit card receipt or something like that? If not, why not? In other words, in real life there always would be a lot of extra evidence and we would never "just stop" at the pure statistical probability.

    A statistician would add: your example only works even in theory if you assume that X has been selected *at random* from the spectators. If he has not been randomly selected, the probabilities may be very different. For instance, the organisers are likely to sue someone who has money to meet a verdict, a fixed address, is an adult, etc. That skews the selection so that one cannot even as a matter of statistics make the example work.

    Whichever way you look at it, though, it seems to me to be more about evidence than about different kinds of probability.

  8. This sounds to me like the distinction between subjective and objective probabilities, or epistemic and alethic, or whatever. Alan Hájek has quite a good discussion of the varieties of each in his article on interpretations of probability in the Stanford Encyclopedia of Philosophy.

    I actually incline towards the opinion that subjective probabilities make much more sense than objective ones. Subjective probabilities are the ones you criticize, but it seems to me that equally good criticism can be made in the instances you cite by pointing out that one has an obligation to gather a large amount of data before doing certain actions. After all, it's never clear what the right reference class is for a particular argument or a particular person.

  9. Paul, that's okay, I was mainly wanting to point to a distinction between two ways of making probability judgments. Evidence is certainly a part of that, so my general point stands if we find that the fundamental difference lies in the different types of evidence that are considered in either case. We can have "abstract" evidence, about group characteristics and the chances that result when we consider the object as a randomly chosen member of the group. Alternatively, we can have "concrete" evidence, which attaches to the individual object itself.

    Still, I suspect that the distinction does run deeper than this. As suggested in my main post, it seems that concrete evidence speaks not only to epistemic probabilities (i.e. what is likely to be actually true, given what we know), but also to modal properties of the individual, i.e. how they are likely to act across a range of possible worlds.

    Update: just saw Kenny's comment - yes, objective vs. subjective probability is the idea I'm getting at here. I wasn't wanting to suggest that subjective probabilities are necessarily bad; just that they leave something to be desired. I think it isn't just a matter of weak evidence either. Even if we imagine that the vast majority of black people were criminals, it would still be morally problematic (despite being quite rational) to judge an individual on that basis, or so I argue in the posts linked above. I think it really is a problem that the evidence "does not point specifically at him" (as Paul put it), no matter how compelling the evidence is in its own terms.

  10. So I think we have two separate issues here: First, there's the matter of abstract vs. concrete evidence, i.e. whether the evidence points specifically to the individual. I think this is probably what's important for various normative purposes, e.g. the ethics of stereotyping, and the dialectical norms which frown upon ad hominems, etc.

    But then there's also the issue of subjective vs. objective probabilities. This mirrors the distinction above, but is perhaps important in different ways. For example, it might be possible to have concrete evidence that only affected subjective (and not objective) probabilities, and I think this would be morally unproblematic.

  11. I think you are right there are two separate questions, but I'm not sure they are so closely related.

    The subjective/objective distinction is one thing. If I grasp your argument it is along these lines: sometimes when we say something is "probable" we mean simply to express our degree of belief in it. Sometimes, on the other hand, we mean to capture and quantify something we regard as inherent in the thing we are talking about, something along the lines of Popper's propensity. (The notion that, say, a fair coin has a "propensity" to land heads 1/2 the time.)

    That may be right, though it's controversial. And if that's how you think about probabilities I can see the objection to talk of the "probability that X is right": in propensity terms it tends to suggest that the rightness of the argument in some way depends upon or is caused by it being X who makes it, whereas the rightness of the argument does not depend on who makes it.

    But if one thinks epistemically (in terms of partial or uncertain beliefs) I may reasonably be inclined to accept an argument just because X makes it, and in that (different) it makes sense to say "If X says so, it's probably true." That comment doesn't mean to suggest that it is true because X says it; only that because X says it I have reasons to accept it as true which I might otherwise lack.

    The evidence problem is different and relates (I think) only to epistemic or subjective probabilities. There you want to distinguish between "abstract" and "concrete" evidence. I'm simply not sure this is a stable distinction; or, at least, I doubt there is a clear line.

    Consider the following example. Evidence from a burglary points to the following facts: the burglar was probably tall; the burglar was probably left-handed; the burglar was probably an insider; the burglar probably had previous experience of burglary. Each of those items of evidence is "abstract" in the sense that it points at a group of persons defined abstractly. And put together the evidence still remains abstract (it points to the abstract group: "tall, left-handed insiders with previous convictions for burglary"). But, depending on other facts it may point very specifically at one person.

    Is evidence like that "concrete" or "abstract"? Can the judgment be made of any particular item of evidence, or only of the mass as a whole? Does the evidence somehow "become" concrete when it turns out that there is only one person left in the relevant group? If so, how does that contingent fact affect the nature of the evidence as evidence? These seem to me to be quite difficult questions.


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