Objective Truth without an Objective Reality

I want to talk about a view of objective truth which I've been thinking about for the last few decades. This view equates objectivity of truth with observer-independence.

The truth of a claim is objective precisely to the extent that it is observer-independent.

Suppose I claim that I have an alien (not just visa-less, but extra-terrestrial as well) in my bathroom. Is this statement objectively true? If we send 20 person-off-the-street observers into my bathroom, one at a time, to have a look, and none see any such thing, then the objective truth of my claim will look rather weak. If they all report seeing a short, big-headed grey ectomorph with large eyes, then it must be objectively true. Because the observers all agree, affirmation of the claim appears independent of which observer takes a look.

This view of objective truth makes no assumptions about there existing an external reality which is to be the arbiter of truth or falsehood. You may want to invoke these as part of a causal explanation for observer agreement, but it's not necessary. As Laplace said, "Je n'avais pas besoin de cette hypothèse-là" (I have no need of that hypothesis).

There is a functional reason why we should be interested in objective truths, ie. those observer-independent truths, over others - these are the truths which allow us to share cognitive resources. If a claim is observer-independent among others, then it will probably work for me. If it only works for some other observers, then my expectation of it working for me will be lowered, all other things being equal.

Of course, this account of objective truth can (and probably should) be modelled in a Bayesian way ☺, but I want to keep this posting relatively compact.

A Bayesian Look at Belief and Mind-Changing Argumentation

This posting presents a model of belief and argumentation based on the idea that people are rational Bayesian agents, ie. that they reevaluate beliefs on the basis of Bayes' theorem. It was prompted by a posting on Daniel Midgley's blog Good Reason.

Bayes' Theorem for Beliefs and Arguments

We start with defining some notation and some probability distributions. This will look a little mathematical, but there's really very little maths involved. I'll use B for the belief we're discussing, and Z for the person we're discussing. When we talk about Z's idea of someone else's mental state, we'll refer to that other person as Y. Finally, we'll use A for the argument that Z is considering.

Now here are some basic terms in Bayes' Theorem.

• P(B;Z) - this is the probability Z assigns to B among all possible competing beliefs. For example, maybe there are only 2 possible beliefs: B that the earth is round, and B' that the earth is flat. In that case P(B;Z)+P(B';Z)=1, as any probability that Z assigns to B can't be assigned to B' and vice versa: if Z thinks that there is a 60% chance that the earth is flat, then s/he thinks that there is a 40% chance that it is round.
• P(A|B;Z) - this is the probability that Z would find argument A convincing if s/he believed B to be 100% sure.
• P(A;Z) - this is the probability that Z would find argument A convincing regardless of what they believed.
• P(B|A;Z) - this is the probability that Z would apply to B if they were convinced by argument A.

Now we get to the little bit of maths which is Bayes' Theorem. Don't be frightened.

P(B|A;Z) = P(A|B;Z) P(B;Z) / P(A;Z)

In words, the theorem is saying that the probability you'd assign to a potential belief B if you were convinced by argument A depends on how likely you would be to be convinced by A if you accepted B 100%, multiplied by the probability of believing B, and divided by the probability of being convinced by A no matter what you believed.

The rest of this posting is going to explore some implications of this theorem.

The Bayesian Agent Assumption

In exploring belief and mind-changing in a Bayesian model of this kind, I am making an assumption about the behaviour of the (potential) believer. This assumption is that they adapt their belief to the evidence they see in a rational, Bayesian way. This might sound to be asking to much of people, but there does seem to be some evidence in its favour (eg, see Bayesian Rationality by Oaksford and Chater, 2007).

The Bayesian Theory of Mind

But not only do humans act as rational Bayesian agents, but they also presume others are as well. So Z might have a model of beliefs in another person Y. I'll write Y@Z for Y as imagined by Z. Z will impute Bayes' Theorem to Y.

P(B|A;Y@Z) = P(A|B;Y@Z) P(B;Y@Z) / P(A;Y@Z)

This gives us a theory of mind. We can use it to talk about what Z will be trying to do in changing the belief states of Y.

Certainty and Uncertainty

Now, let's look at the 3 fundamental relationships that Z can have with a belief B.

1. Certainty - P(B;Z) = 1. In this case, Z is certain that B is true in an absolute sense.
2. Uncertainty - 0 < P(B;Z) < 1. In this case Z is neither certain that B is true, or that it is false.
3. Certainty that not - P(B;Z) = 0. In this case, Z is absolutely certain that B is not true.

There are slightly weaker versions of certainty and certainty that not. These are versions I'd call asymptotic certainty and asymptotic certainty that not, with probabilities for P(B;Z) of 1-ε and ε respectively, where ε is sufficiently small that it might as well be 1. (I may expand on this notion in a later blog post.)

Separate Magisteria

If Z is certain of B, P(B;Z) = 1, and so if C is any alternative belief to B, then P(C;Z) = 0. In this situation, P(A|B;Z)=P(A;Z), that is, the likelihood of any argument being judged convincing by Z will not change whether or not Z is asked to take B into account. It's easy to see why: because Z believes B with certainty, asking them to assume that B is true when evaluating A makes no difference; they will assume it anyway.

The separate magisteria arises where agent Z debates with Y, attempting to convert Y to his/her belief B. Argument A is irrelevant for Z, because it cannot affect the probability of B: P(B|A;Z) = P(A|B;Z) P(B;Z) / P(A;Z) = P(B;Z), because P(A|B;Z)/P(A;Z) = 1. It cannot touch their certain worldview. However, if they think they have any chance of converting Y, it must mean that in their view, Y assigns a non-zero probability to B a priori: P(B;Y@Z) > 0. Their goal, then, is to construct arguments A which will likely convince you, but which are particularly likely to be convincing only if belief B is true. In symbols, P(A|B;Y@Z) / P(A;Y@Z) >> 0. For example, argument A might be that ships leaving port disappear hull-first then sail. If the world is round (B), then this is much more likely P(A|B;Y@Z) than if we have no idea P(A;Y@Z).

Why do I say that there are separate magisteria at play here? Because as a round-earth fundamentalist, Z's acceptance of B cannot be influence by argument A, or any other kind of evidence. The evidence is only there to convince the open-minded. So the two magisteria are: the realm of certainties, and the realm of arguables. This accords with Gould's non-overlapping magisteria: religion is the magisterium of non-evidential, a priori certainties, while science is the magisterium of a priori uncertainties that allow experience-based re-evaluation.

"You Haven't Proven B to be False"

One common discourse pattern among the certain proceeds thus:

• interlocutor presents argument A for non-B alternative C,
• arguer says (correctly) but you haven't proven that B is false,
• and this is enough for them to retain their assuredness of B.

Once again, this is a rational attitude, given particular probability assignments in Bayes' theorem. Let Z be the arguer, and Y the interlocutor. To make this work, we need Z to view the prior probability of B as infinitesimally close to 1, without actually being there (Z has an infinitesimally open mind): P(B;Z) = 1-ε, where ε is very, very small but not quite zero. For convenience, we'll abbreviate by a whatever probability P(A|B;Z) would be assigned by Z to the argument A given their belief. Then P(A;Z) = (1-ε)a+εk ≅ a, where k is the probability that Z would assign to A if B wasn't true ... an estimation of no practical consequence, unless a is also infinitesimally small, ie. unless argument A disproves belief B. Unless a is that small, P(A|B;Z)/P(A;Z) ≅ a/a = 1. Consequently, no argument that doesn't conclusively eliminate B will not be sufficient to weaken Z's belief by any noticeable amount.

Hope

I promised some hope for those arguing against beliefs held with certainty. The hope I offer is this: sometimes people lose their faith (in whatever it is they believe in). This may only mean that their belief drops by, say, 1%. Look at the two options:

• Z has never met strong arguments against B, so that (using A! for the combination of all arguments put together that pertain to B) the arguments for and against are evenly weighted P(A!|B;Z)/P(A!;Z) ≅1. In this case, B still comes out the most likely hypothesis after considering the arguments, and self-reinforcement might push that prior probability P(B;Z) back towards 1.
• In the 2nd scenario, all sorts of input has been filling Z's head to the point that P(A!|B;Z) ≅ 0.01, but for the alternative hypothesis C, P(A!|C;Z) ≅ 0.99, in other words, Z has met lots of convincing arguments against his/her views, they just haven't been able to change his/her absolute belief in B. While Z's mind is only infinitesimally open, the arguments are of no consequence, certainty is retained. But with a 10% loss of faith from B to C, (P(B;Z)=0.9, P(C;Z)=0.1) suddenly the probability of the data P(A!;Z) at 0.108 becomes significantly larger than P(A!|B;Z) at 0.01, and so the posterior probability of B, ie P(B|A!;Z), drops to around 0.1, while the posterior of C rises to 0.9. At this point, Z undergoes a conversion experience.

My take-away point, if this model can be believed, is as follows. It is worthwhile challenging even those who seem untouchable by reasoned, probabilistic argument. Because one day, they may step away from their certainty, just a little bit, and if your arguments have been strong, even a little step will be enough.

Conclusion

The times of problems I'm dealing with here are the clash between holding beliefs as a priori certainties that are therefore impervious to evidence, and holding them as a priori uncertainties whose fortunes rise and fall as evidence comes to light.

Men vs Women

Once again the Google Ngram Viewer shows us something remarkable about the words we've used in publications. In British English, for much of the 20th century, men are refered to around 4 times as much as women. Then in World War II, this drops to a 3-times difference. From 1970, there has been a steady decline in the use of men, and an increase in uses of women, until the latter outstripped men in the 1980s. To the present day, women tracks consistently at 25% more frequent that men.

Chocolate is a World War Indicator

Graphing the frequency with which chocolate occurred in English in the 20th century, we see peaks approximately aligned with the first and second world wars. What is disturbing is that we have been rising since 1970 towards a new peak.

So if you want to work for peace, stop publishing stories that mention chocolate.

Oops.

Analysing Arguments against Wikileaks (Privacy I)

In this and subsequent postings, I will survey some of the arguments towards the conclusion that Wikileaks, or more precisely, its publication of the Cablegate cables, is a bad thing. For brevity, I'll refer to this publication as WCP from here on. I will deal with these arguments one per blog posting. This is mainly because I expect each discussion to blow out.

I should confess my own position at this point -- I do support Wikileaks and WCP, and consequently will be applying what acid I can to the arguments presented. But I won't be trying to prove that WCP is a good thing, that's a whole other set of arguments.

Argument from Sympathetic Privacy
Argued by: Joanna Bryson (blog), Theodore Dalrymple (City Journal), Dr Wes (Get Better Health), various commenters (Ethika Politika)

Informally, the argument runs as follows. The cables are messages being sent between persons doing their everyday jobs for the US State Department. These cables were not intended by their authors or recipients for general publication, hence their classification. Imagine if all your work emails and conversations were posted on the web for all to see. That would be bad. So WCP is bad.


Not all of the links cited above are to documents that present this argument with the same level of clarity or explicitness, but they all express their concern for privacy in what-if-it-happened-to-you terms.

The argument, like most in informal discussion, is not valid as stated (not a criticism - trying to make arguments valid is time consuming and often tedious). To justify the conclusion that WCP is bad, we need to tighten up its language and express some previously implicit assumptions.

1. The cables are written communications between colleagues which they did not wish published.
2. You are like the authors of of the cables in all relevant ways.
3. You have written communications with your colleagues which you do not wish published.
4. If someone published these communications, that would be bad.
5. If something is bad when it happens to you, then it is bad when it happens to someone like you in all relevant ways.
6. Therefore WCP is bad.

If someone wants to avoid the conclusion that the publications were bad, then they need to deny at least one of these five assumptions. What's more, most of them seem pretty impervious to denial. Premiss (1) is a factual description of the nature of the cables as written communications. The same premiss says that the authors did not intend their publication, which most would accept as an immediate consequence of the classification they were given (SECRET).

Premiss (2) will be left till last as it is the premiss I am going to question.

Premiss (3) is true, I would presume, for all of us. We may not want our communications revealed because (a) they might show we're doing something we shouldn't be, (b) they include necessarily candid evaluations of persons not party to their interests, (c) they relate to trade secrets, (d) they include information which could be used to damage our employer or own business (like passwords), or for many other reasons.

The fourth premiss almost follows from the third, at least if we are sufficiently egotistical. If something happens which we did not wish, then that would be bad. A trifle less subjectively, a leaked email with passwords, for instance, or a hasty remark about a problem client, could lead to substantial business consequences if published widely. So this is also a strong premiss.

The last assumption (before we return to premiss 2) is a version of the well-known Golden Rule (also known as the ethic of reciprocity): but instead of talking about how you treat others, it's more about how you would wish them treated in abstracto. Some may be a little puzzled by the rider in the statement of the rule here like you in all relevant ways. This is to deal with things that should happen to people because of some attribute of their own or their environment which is different to our own. For example, a billionaire should not expect state support while an impoverished person might because we regard the difference between them, their wealth, as relevant to the happening, in this case the provision of state support. Likewise, those of us who have committed no crime would regard our own arrest as bad, while seeing the arrest of a murderer as good. So this limitation to like you in all relevant ways is important.

These four assumptions seem unassailable. So any contesting of this argument can only be had in the second premiss: that the authors of the cables are like us in all relevant ways. In the context of WCP, relevance refers to the reasons for publishing confidential written communications. In these terms, we can flesh out the assumption a little more: US State Department employees are just like me with regards to any reason for publishing confidential written communications.


So in order to avoid the conclusion that WCP is bad, while remaining consistent, I have to maintain that US State Department employees are not just like me with regards to any reason for publishing confidential written communications. So how might I think we differ in ways relevant to publication?

Let me suggest two:

1. They are projecting the might of a superpower through influence and negotiations that affect the lives of thousands and sometimes millions, while I am not.
2. While acting as representatives of very many people, they routinely keep their acts secret from those people, while I do not.

For me, these differences are sufficient to break down the ethic of reciprocity between me and US State Department representatives. There is a lineto be drawn between the casual and ruthless invasions of individual privacy (the stuff of traditional tabloid media) and the pursuit of information about the influences great powers are exerting over our lives.

But there's another question to be asked here. Is Wikileaks itself (or at least Julian Assange) immune from the argument from sympathetic privacy? Or if it was explained to them would they respond: Damn! I wish I'd thought of that. I better stop what I'm doing and pack up shop. The question boils down to whether they differentiate themselves categorically from the people they leak about.

When asked (during this Ted Talk about 10min in) whether there have been any leaks from Wikileaks yet, Assange responds: We don't have dissidents. That is at least one difference between them and persons party to the US State Department secrets, and it is a difference crucially relevant to the process of leaking.

So this argument doesn't get me to the conclusion that WCP is bad, because I don't see my role as similar to that of the people whose communications have been leaked.

Optimism Bias - A Better than Average Rant

I have been exercised to write because of a misapprehension, or at least sloppy terminology, seen in a recent psychology talk at a university, and on Australia's national science TV show Catalyst (this story). The phenomenon being mentioned in both cases is something known as the optimism bias, among other names. You can read about it on this wikipedia page.

The basic idea is that for some quality X, an excessive number (a majority or a very big majority, called a supermajority) of people assume they have above the norm values of X. Now to be properly a fallacy, in the sense of being a logical impossibility rather than a simple factual error, we really need to find that a majority of people assume they have above-median levels of X. More than a majority claiming above-median levels of X mean that someone is wrong.

The usual presentation of the fallacy, and this is the form in the Catalyst story, is that for some quality X a majority of people believe they have above-average levels of X. This statement doesn't work as a logical fallacy. The wikipedia page (in the section on skewed distributions) points out that most people have an above-average number of legs (since we have a large majority of 2-legged people, and people with fewer than 2 legs). Similarly, most people: have above-average numbers of kidneys; have committed fewer-than-average crimes; have practiced less-than-average cannibalism; have names more-than-averagely non-identical with Donald Duck (at least one non-cartoon Scot holds this name); and most people in Australia hold more than the average number of drivers licenses.

Let's look at the drivers license in detail. Many people in the Australia do not hold a drivers license (eg minors), many people hold one drivers license, very few hold two or more. The average number of drivers licenses per person in this country is between 0 and 1. Since more people have a drivers license than not, the majority of Australians hold more-than-average numbers of drivers licenses.

Now, it is entirely possible that people are more optimistic about their abilities as drivers or share traders than they should be. On a reasonable scale of abilities, maybe too many believe they are above average. But this is not a logical fallacy as presented, merely an error of fact, where the fact in error is the skew in the distribution.

Extend Trading Hours ... into the Morning!

I had another mad idea on Friday. First, a bit of background. Many W.A. people are morning people, perhaps more than elsewhere, and social values are a little bit more oriented towards doing stuff in the morning than late at night.

Anyway, there's been a lot of discussion about extended trading hours recently. Once again, for those unacquainted with our dear state in the west, W.A. does not have extended trading hours: supermarkets are obliged to shut on Sundays and weekdays after 6pm.

My big idea is that here in W.A., we should extend trading hours into the morning ... as opposed to into the evening. Want fresh bread for your breakfast? Just nip to the supermarket at 6am before everyone else is up (ok, 5am). This way, the state can remain morning-oriented, family-focussed, and all the rest, while enjoying trading hours that would no doubt soon be the envy of the rest of the country if not the free world.