# Message to Booleans: It’s an additive world, we just live in it

Boolean models (“it’s either A or (B and C)”) seem to be the natural way that we think, but additive models (“10 points if you have A, 3 points if you have B, 2 points if you have C”) seem to describe reality better—at least, the aspects of reality that I study in my research.

Additive models come naturally to political scientists and economists, including myself. We think of your political attitudes, for example, as a sum of various influences (as for example in this paper with Yair). Similarly for economists’ models of decisions in terms of latent continuous variables. But my impression is that “civilians” think in a much more Boolean way, with different factors being switches that flip you to one state or another.

And, when it comes to statistics, applied people often think Booleanly or lexicographically (“Use rule A, with rule B as a tiebreaker”) and, I think, make mistakes as a result. For example, consider the attitude that seems to be prevalent in econometrics, that you want to use an unbiased estimate and then reduce variance only as a secondary concern. As we’ve discussed elsewhere in this space, such an attitude is incoherent because in practice the only way to get an unbiased estimate is to pool data and thus assume the effect of interest does not vary. Also recall the foolish survey researchers who don’t want to let go of the fiction that they are doing theoretically-justified inference using the principles of probability sampling.

We live in an additive world that our minds try to model Booleanly. Sort of like how Mandelbrot pointed out that mountains and trees are fractals but we like to think of them as triangles, circles, and sticks (as exemplified so clearly in childrens’ drawings).

## 40 thoughts on “Message to Booleans: It’s an additive world, we just live in it”

• In case anyone else wanted to read the broken link, I *painstakingly* tracked it down. ;)

http://cowles.yale.edu/sites/default/files/files/conf/2007/crp_griffith2.pdf

And for those who come after me:
Kalish, M. L., Griffiths, T. L., & Lewandowsky, S. (2007). Iterated learning: Intergenerational knowledge transmission reveals inductive biases. Psychonomic Bulletin & Review, 14(2), 288-294.

1. I absolutely agree that is cool to be additive, :-), thanks for
“We live in an additive world that our minds try to model Booleanly”

2. Is it the same for models in the Physical sciences?

Like, light either travels through an ether or not. The two models don’t work together. Sound is either a transverse wave or a longitudnal wave. Et cetra.

3. I think the confusion arises from the fact that many features of the world are, in fact, boolean (e.g. Mary is pregnant or not), yet our knowledge of the world can only be additive (e.g. I am almost certain Mary is pregnant).

Moreover, many of our actions are contingent on the boolean state of the world (e.g. it only makes sense to seek fertility treatment if Mary is not pregnant). If so, we often find a need to convert an additive scale to a boolean one.

• PS The latter is a common feature to game theoretic solutions that map beliefs to actions.

• PPS To paraphrase Milton Friedman you might argue that, as a statistician, your job is only to update beliefs not take actions. If so you need not leave the additive world.

(Milton argued that his job as an economist was to find ideal solutions, not (politically) feasible ones).

4. I think that more of our social and political world is more booleanly than you account for but that the bits of the world that are booleanly don’t tend to be the bits that social scientists spend as much time on because we like to focus on questions that don’t seem completely obvious.

* In particular, laws are generally pretty boolean at least in their intention:
You can only vote if age>18 & citizenship==”US”
* If you kill someone intentionally, law enforcement is permitted to arrest you
* The party that receives the most votes will win the house race
* If you pay the shop \$5.00 they will give you the \$5 item you asked for
* If your income is below x then you will be eligible for subsidized healthcare

While none of these examples are ironclad, social scientists are likely to take the boolean part as a given and then try to model the deviations from it as the object of interest. e.g. when do ineligible people vote?, what factors increase the probability of a jury making a conviction when there is limited evidence? and what biases are there in recounts of close elections? When do vendors renege on verbal contracts?

I think the final example of healthcare is a particularly good one. If the system was that a bureaucrat had discretion over who was eligible for healthcare on a case by case basis, we would have hundreds of papers modelling the additive decision of that bureaucrat. Instead we have no papers describing the boolean rule that they follow.

This is not to disagree with anything you said above about additive models being better ways of understanding things such as political attitudes, but just that a lot of the world is boolean which is probably why we still think in boolean terms when that isn’t appropriate.

• These are all (well, mostly) examples of boolean thinking imposed on the world, not the other way around right? i’ve often been irritated at policies because of their thresholds and soforth. In particular welfare and subsidy policies, good welfare and subsidy policies would do things like provide a Universal Basic Income (UBI) to everyone, and recover the cost of doing so through a sales tax, or maybe calculate a subsidy from a continuous function of reported income or do something where selling an extra \$20 bouquet of flowers wouldn’t suddenly make you technically totally ineligible for your housing subsidy all at once etc.

Other areas where boolean thinking should be changed relevant to your list: Stop using first-past the post voting and use score voting instead, stop setting thresholds for income tax brackets and specify a simple continuous function, start providing charity not just in terms of giving goods to people but also in terms of access to credit and financial management information (so that people aren’t limited by the “you need \$5 in your pocket now to afford a \$5 item” which is a big part of why it is so expensive to be poor).

so, yes many things in policy/law/society are boolean, but in large part because they’re imposed by boolean thinking, not because they’re inherently boolean.

• OMG!

You are about to kill all the opportunities for regression discontinuity designs!

What are social scientists going to do with their time now? We will live in darkness forever ;-)

• applied mathematical modeler found beaten and bloodied in alley behind federal reserve building. gang suspected of the crime meets at local coffee house, nabbed when droplets of blood on leather arm patches match victim. When asked what the motivation was leader says “No-one messes with the west coast RD mafia”. Gang lieutenant who goes only by “jrc” claims that jacket was one of many random jackets taken off a coat check rack at local museum, and as such the probability is high that any given randomly chosen coffee house patron could on a different day have matching blood stains.

• Well, I realise I’m nitpicking here but those statements are only true conditonally. For example:

‘You can only vote if age>18 & citizenship==”US”’
Only true in the USA. Here it is age => 18 and citizenship == Canadian.

* If you kill someone intentionally, law enforcement is permitted to arrest you.

Millions of solders around the world would be surprised at this. And given the disparity in arms might be a bit resistant to the idea.

* The party that receives the most votes will win the house race.

Presumably true for the US
Not necessarily true in First-Past-The-Post electorial system such as mine here in Canada or in the UK

* If you pay the shop \$5.00 they will give you the \$5 item you asked for.

Assuming you meet other requirements. If you can still find a \$5 package of cigarettes you must be 18 or + where I live

* If your income is below x then you will be eligible for subsidized healthcare.

Assuming you meet other qualifications. Here in Canada depending on what province you live in that is true for refugees but in others it is not (cursed Cdn Govt.)

So it would seem that the world is Boolian, ceteris parabus.

In any case, I hope eveyone has a happy holiday season.

5. My take here is that the university is finite, or at least bounded to us by finite measurements of it (e.g. likelihood being probabilities of what was observed are discrete probabilities).

Representations of the universe (e.g. models) are continuous, between any two representations there is one in between.

So we have no choice but to represent boolean as additive but we can mark out boundaries on our representations and treat them as real.

I think what Andrew is getting at is marking out boundaries on our representations and treating them as real and maybe even fixed.

But then some once told me Darwin once wrote “discontinuities exist only in the minds of men”…

• Keith, my hobby horse is nonstandard analysis, particularly the IST version. In that system, rather than getting a continuous model as a limit of a sequence of discrete models, you get discrete models where the jumps are “limited” and discrete models where the jumps are “infinitesimal”. The infinitesimal models correspond to standard continuous models, but when looked at carefully enough it’s clear that the continuity idea is really just a blurry insufficiently magnified interpretation of the nonstandard infinitesimal-jump model. It gives me a warm and fuzzy feeling to know that this corner of mathematics is out there to be elaborated on by people who build mathematical models of real-world phenomena which are admittedly discrete and have jumps, but generally they are small enough that they might as well be considered infinitesimal for practical purposes (ie. a \$20 purchase relative to the national GDP, a bunch of atoms that get compressed together a little to make a sound wave… etc)

• We want to avoid the superstition that if we are modelling something discrete a discrete model will be better than a continuous one and vice versa. Rather we want to use the models (representations) purposefully to their best effect.

Problems arise, when we project an aspect in the model that is a mis-representation of what we are trying to represent into what we are trying to represent (e.g. that _observations_ from a continuous distribution have 0 probability so anything we have observed had probability 0).

Unfortunately, “an aspect in the model that is a mis-representation” can be very subtle and controversial.

• I agree, one needs to evaluate the success of a model regardless of whether it’s got a match or mismatch of discrete-discrete or continuous-discrete. The advantage of using nonstandard analysis is that it helps make plain some of the specific assumptions one makes when creating a continuous model of a discrete system, and also gives you a way forward when trying to improve an insufficient model. I have some examples in continuum mechanics in my dissertation. If you’re interested feel free to look here: http://models.street-artists.org/wp-content/uploads/2014/03/dissertation.pdf

I strongly agree that in general “an aspect in the model that is a mis-representation” can be very subtle and difficult, but there are plenty of times where strict continuity is already OBVIOUSLY a mis-representation, and yet often used to mathematically predict nonsense.

• “My take here is that the university…”
“Representations of the universe…”

For better or worse, it certainly does feel like university = universe sometimes! Or maybe it’s the other way around.

6. So Andrew, can we go farther? Additive is pretty simplistic wouldn’t you think? How about it’s a nonlinear continuous world?

• Daniel:

Sure, we can include nonlinear. But part of what I mean by “linear” is “smooth.” And, I might even add, “monotonic.” Yes of course there are lots of non-monotonic relationships in the world (indeed this was the topic of one of the qualifying exam questions I posted a couple months ago), but I’d argue that non-monotonic expressions can typically be usefully reframed as sums of monotonic nonlinear functions. That’s a story for another day, though.

• What’s the advantage of representing non-monotonics as a sum of monotonics?

In that spirit couldn’t most, “practical” non-linear functions also be modeled as piece-wise linear functions?

• Rahul:

This is really the subject of a separate post, but the quick answer is that I think this sort of additive re-expression can make sense for conceptual reasons, In examples I’ve seen in social science where f(x) goes up and then down, I find it helpful to think of two mechanisms: an increasing function that dominates at the low end of the scale, and a decreasing function that dominates at the high end. It’s the mixture modeling idea. This is, for example, how I think about political attitudes as a function of education.

• Yes, that makes sense.

In many situations though, an aggregate, non-monotonic, non-linear model is very convenient to handle. Especially, when your model is one of a larger set of diverse models that go into modeling some other larger phenomenon.

• Sums and/or products perhaps. Lots of non-monotonic things are products of stuff that grow and decay at different rates. Like the gamma distribution pdf, grows polynomially initially, and then decays exponentially at large x.

I think the biggest message is really that there’s a smooth continuous range for many/most measurements. Even things that are discrete, like counts, when measured as a fraction of a large total look continuous and smooth.

7. Professor Gelman, I am an admirer of your work and think you are one of the clearest writers on research methods in the social sciences. That said, I am a bit confused by your post as it seems to imply that sets of events or discrete events do not in fact exist or that the perception of such is illusory. But, surely we do engage in behaviors that have both conceptually and reality based changes that are appropriately considered in binary terms. For example, a researcher can either use latent variable modeling or not. Now we can certainly think in terms of the variations of latent variable modeling that exist and think in terms of a continuum of complexity of the models, a continuum of quality of the collected data based on a variety of criteria — but, does this negate the binary nature of whether or not latent variable modeling was used? Or to put it in a healthcare setting — do the multiple variables that are truly continuous in nature negate a very important criterion of survive versus not?

• James:

I agree that we make discrete decisions, and I also agree with previous commenter Jon that various aspects of the natural world such as court decisions, economic transactions, elections, and death itself represent discrete aspects of reality that we must model. My point (perhaps obscured by my exaggerated statement of it) is that many many aspects of life are essentially continuous but people want to model them discretely, and many many tradeoffs are essentially continuous but people want to consider them discretely.

For an important example, consider the nature of statistical evidence. Many people seem to want to make a sharp decision, to say that if a comparison is statistically significant and published in a legitimate journal, that we should treat as true some claim about the world. Instead I’d like to say that this may be weak evidence (as in my “power = .06” graph). Someone might reply that, no, we need to make decisions, we need a sharp rule. But, when it comes to scientific inference, I don’t see it. If we have to decide whether or not to build some bridge, then, sure, it’s a sharp decision and we must treat it as such. But all the time I see people imposing discreteness where it is not needed and not appropriate. I suspect this in part due to human nature, to a way in which our brains like to formulate things. That’s why I say we’re Boolean brains in a linear world.

• “Continuous” vs “discrete” makes more sense to me than “additive” vs “Boolean.” I tend to think continuously, but rarely additively. Both additive and discrete seem to me to be “models,” approximations to the (usually) continuous reality. (e.g., in grading, I might use an additive model for what is really a continuous amount of variability, just so I don’t go crazy)

But thinking about it, I sometimes see a continuous model as a useful model for an additive or discrete situation, but where there are lots of summands/options — far from binary. I guess that’s the way I think of meiosis (the crossover stage) — there aren’t really infinite possibilities, but the number is so large that members of the same species are individuals, that can vary in a large enough number of ways that “infinite’ does seem like a good model.

• You may also enjoy some of the IST/nonstandard analysis topics I discuss in my dissertation http://models.street-artists.org/wp-content/uploads/2014/03/dissertation.pdf

My impression is that mathematicians have historically shunned nonstandard analysis because … well it’s nonstandard and it’s kind of seen as a hack to get around having to take limits, which mathematicians don’t think of as all that valuable a “hack”… but in the context of building models, the language of nonstandard analysis maps much better onto the process of approximating real world things, the limit seems more like a “hack” to ensure mathematical rigor. Many of these real world situations are technically discrete (such as assorting genes) but make up such a large spectrum of possibilities that they “might as well be” treated as infinite, and the mathematical language for that is very naturally IST in my humble opinion.

• I think a lot of social science is about understanding ways in which humans do or do not make the world (what Andrew is calling) Boolean and why it happens or doesn’t happen in different times/places/cultures and also how ambiguity is managed and what it all means for daily lives. Gender and race would be the two most obvious examples, but clean and unclean would be another, not to mention social class. In other words the social science question is why and when do people do that.

Is it years of education that matter or whether you are a high school or college graduate? Yes. Yes because social life is complicated. Is income (continuous) good enough or do we have to know if you own the means of production (dichotomous)? They are really very different things.

I have to agree that I find calling this Boolean versus additive not very helpful. Sometimes things are A and not A and that’s interesting, but sometimes they are A and B and that’s interesting too. And when they are one or the other is really interesting. I’m not quite sure if what Andrew is saying is about true and false, discrete versus continuous or dichotomous versus continuous or some combination.

And then there is the issue of what we think about as people analyzing data and I think Andrew is trying to make a kind of social psychological argument about how disciplinary culture relates to this.

• Elin:

I agree that there are Boolean aspects of the social and natural world that are worth studying. Almost everybody is distinctly male or female, most politicians in the U.S. are distinctly Democratic or Republican, and various psychiatric conditions seem essentially discrete (you have it or you don’t) even if there is some continuity around the edges.

So let me revise my post to say, not that we live in a purely continuous world, but that there are many aspects of the world that are continuous but which people want to model as Boolean.

8. Are lay-people instinctively / biologically wired to think more in a descrete / Boolean manner? I wonder. Our life seems designed to favor those approaches.

e.g. Tax codes come in brackets, Speeding fines are graduated on a discrete scale. Interest rates in banks (at least back in India) came in fat tables & schedules with pages & pages of combinations where a single equation may have sufficed. Utility bills come in slabs of tariffs. Even discount coupons use step jump thresholds.

Maybe the cognition / vocabulary needed for continuous / additive thinking isn’t accessible to most people? Perhaps that makes the Boolean abstraction a useful one.

• “Are lay-people instinctively / biologically wired to think more in a descrete / Boolean manner? I wonder. Our life seems designed to favor those approaches.”

Yes and so are non-lay (??) people if you trust most of the Tversky and Kahneman work plus others’ work. If I am reading Kahneman’ Thinking Fast and Slow more or less correctly I think you could say that in many cases we do little real thinking and so probably default to a Boolean viewpoint a lot–yhis is not what he says but I think it is a reasonably interpretaion of many daily activities.

In cases like Income Cut-Offs etc it may just be a failure to have lawmakers (and many of their bureaucrats) able to count above 10 when wearing close-toed shoes. They may just not realise it is possible to do something with a more sophisticated approach.

9. It is interesting to note that in a very direct sense Boolean algebra is fundamentally additive; any boolean function can be represented by the OR (the logical counterpart to addition) of variables and their complements through the “Disjunctive Normal Form”.

Additionally, I am amazed how a linear algebraic (additive) structure creeps into almost everything in physics (and beyond): Quantum Mechanics, Classical Mechanics, etc. The ubiquity of vector spaces is mind blowing.