Flip It Around

A whirlwind tour through one of the conceptually simplest techniques for better reasoning

Introduction

One of the easiest ways to clarify thinking is to reverse its direction. For example, imagine a friend comes up to you with the claim “the sky is blue”. This seems pretty reasonable, since being blue is a property of the sky. But you decide to check this “logic” by flipping it around: “anything not blue is not the sky”. Wait a minute. During a bad storm last year, the sky turned green! It didn’t cease to be the sky.

Obviously, in a trivial case like this we don’t gain much. It’s easy to see the counterexample from the beginning. This technique is helpful, though, when applied to more complex statements where we’re not exactly sure what all the implications are. In fact, because of the simplicity of the technique, it’s relatively easy to use it for clarification on very complex statements, when more complex techniques tend to be too unwieldy.

Inverse, Converse, Contrapositive, and Obverse

If you hate logic, or already understand what the four terms above mean, feel free to skip to the next section.

We’ll begin by setting off on a brief foray into the mathematical underpinnings of what statement reversal could mean. This means a bit of a dive into predicate logic. Of course, even in basic mathematical terms, there are several candidates for ways to “reverse” a statement. Let’s look at a few: the inverse, converse, and contrapositive.

Name Math English
Statement $P \implies Q$ If P then Q
Inverse $\neg P \implies \neg Q$ If not P then not Q
Converse $Q \implies P$ If Q then P
Contrapositive $\neg Q \implies \neg P$ If not Q then not P

We could build truth tables to check their validity (and I encourage you to do so) but instead we’re just going to use a few counterexamples. In keeping with the long and storied tradition of introductions to logic, we’re going to use examples about rain and stuff getting wet.

The inverse of “if it is raining, the sidewalk is wet” is “if it is not raining, the sidewalk is not wet”. This is false. Perhaps we just poured a bucket of water on the sidewalk.

The converse of “if it is raining, the sidewalk is wet” is “if the sidewalk is wet, it is raining”. Similarly, our mighty water bucket strikes this down.

As for contrapositive…I guess we’ll resort to truth tables after all:

P Q Statement Contrapositive
T T T T
T F F F
F T T T
F F T T

Because the contrapositive has the same truth values as the statement for all possible truth values of P and Q, the statement and its contrapositive are logically equivalent.

There’s more to logic than simple statements though. We also have to consider universal (“for all”) and existential (“there exists”) quantifiers, in both the positive and the negative.

Symbol Name Math English
A universal affirmative $\forall x P(x)$ All x satisfy P(x)
E universal negative $\forall x \neg P(x)$ No x satisfies P(x)
I particular affirmative $\exists x P(x)$ One or more xs satisfy P(x)
O particular negative $\exists x \neg P(x)$ One or more xs don’t satisfy P(x)

We can show that a simple application of contraposition falls apart (in some cases) now! The contrapositive of “no water is dry” is “nothing wet is non-water”. Because of our negative quantifier, a direct contrapositive is now false.

Instead, we want to use the “obverse”. Here, we flip the predicate (P(X)) and also the quality (positive/negative). A table illustrates this:

Statement Obverse
$\forall x P(X)$ $\nexists x \neg P(X)$
$\nexists x P(X)$ $\forall x \neg P(X)$
$\exists x P(X)$ $\exists x \neg(\neg P(X))$
$\exists x \neg P(X)$ $\exists x \neg (\neg (\neg P(X)))$

We can still use the contrapositive under a quantifier but it has to be either A or O. We can also use the converse, but only with E or I quantifiers. Obversion is nice because it works for A, E, I, and O.

These rules are getting a bit complicated for what’s supposed to be a fairly simple technique. And we’re still in the world of pure logic, using math instead of English. How are we supposed to apply all this to statements in the wild?

Correctly Reversing Natural Language

The key to applying reversals to English statements correctly is to actually want to do it. Seriously.

There are just so many ways for humans to do motivated reasoning, that any motivation almost can’t help but affect your reversal. The farther we get from formal statements, the more degrees of freedom we have in any given sentence to massage its reversal a bit and get exactly what we want out of it. The easy way out is to actually not have any motivation other than “I want this reversal to be correct”. Easier said than done, sure, but if your primary motivation is “win the debate” or “hold on to my existing worldview” you’re much more likely to fail. If you can put those other motivations on hold temporarily somehow, maybe you’ve got a shot.

Imagine a discussion between a few individuals on a lunch break from their job inventing new cryptographic schemes. The participants are Aggressive Alice, Banal Bob, and Charitable Carol. Alice, in her malice, uses every underhanded debate tactic imaginable, whereas Bob thinks his job is to accurately represent the facts, more or less. Carol, facing peril, does what her nickname strongly implies and gives the benefit of the doubt to the other side. They all believe the sky is blue, and they all are given the statement “The sky is almost always blue” to reverse.

Alice wants to do a fake reversal, so complicates things a bit. It’s in her interest to, after logical reductions, basically end up with the original statement again, but for the new statement to seem reversed. She says “The sky is almost never not blue”.

Bob does the straightforward thing and inserts a single “not” in the right place to negate the whole statement but not negate the qualifying subject: “The sky is not almost always blue”.

Carol, bless her heart, goes too far in giving away ground. Instead of negating the whole pertinent statement, she negates only a single property within it. She ends up with “The sky is almost always not blue”.

Each of us has, at different times, been Alice, Bob, and Carol. We’re often Alice when we really care about winning a debate, whether by technicality or not, and maybe don’t have such a strong incentive to represent the truth accurately. It’s easiest to be Bob in simple, non-emotionally-charged situations e.g. technical discussions about the facts of a matter with lots of evidence and clearly understood theory behind it. And Carol comes out when we take the principle of charity too far in a debate, and instead of responding to a charitable representation of what others have said, begin responding to something they didn’t even say.

Although we can sit here and universally pan Alice’s actions as dishonest, this is where a lot of the incentives are in real life, and incentives are a hell of a drug. Sometimes it really is valuable, for some external instrumental purpose, to win a debate rather than to represent the facts accurately.

Being Bob is fine, albeit boring, but his domain is limited. He probably won’t seem to win as much as Alice, and he probably won’t be seen as charitably as Carol. He tends to get his facts right though, which in the question of “how do we perform reversals correctly” is kind of the key point.

Being Carol is valuable to some extent in human debates around fuzzy topics (so, almost every topic). People make mistakes, definitions are fuzzy, and so on. Carol’s domain has to end though, where facts override it. You cannot reasonably be charitable to an opponent about something strictly factually correct. (You can allow them to make a counterfactual assumption—what if the primes weren’t infinite?—but you can’t let the assumption pollute the global environment.)

Devilish Daniel and Excellent Edna walk up and sit down at the table, asking what’s going on. After learning about the blue-sky argument, they decide to try it out themselves on a few statements. Daniel is going to use his devilish ways to come up with bad reversals (whether more like Alice or Carol, doesn’t matter), while Edna with her typical excellence is going to reverse the arguments as well as she knows how, being as balanced as Fair Fred and as graceful as Graceful Grace (who are gracefully sitting back at the office, unaware of this commotion).

A few onlookers come up and stand over the table: Habitual Harold and Intellectual Isabelle. They’re looking for general patterns they can easily apply to their respective lifestyles.

Statement Excellent Devilish Devilish Technique
The sky is almost always blue The sky is not almost always blue Never is this true / The sky be not blue Bad poetry, removing a qualifier
The primes are infinite The primes are finite The non-primes are infinite Reversing the subject instead of the object
On that coaster, up is down I did not feel disoriented on that coaster On that coaster, up is up. Tautologically! Disrespecting a figure of speech
People who like jazz like music People who like jazz don’t like all music People who like jazz don’t like any music Assuming the wrong implicit qualifier

Harold and Isabelle scan the list, and begin coming up with rules of thumb that might work. “Only perform one negation per statement”. “Negate as much as you can while not affecting the domain of discourse”. “Don’t be too literal or too figurative, but try to decipher the author’s intent”. (The remainder of Harold and Isabelle’s discoveries are left as an exercise to the reader).

Finally, Jovial Justin, Kidding Kristen, and Laughing Larry show up and everyone spends the remainder of the lunch break laughing until they have to go back to the office and yell at people still using MD5 for passwords.

Opposite Day

If you’ve ever been a child, you may have heard of “Opposite Day”, a day in which everything suddenly becomes its opposite. A common game is to try to decipher whether today is opposite day by asking a friend some logical puzzle.

You can’t just ask “is today opposite day?” because whether it is or isn’t, the correct answer is “no”. You’ve got to catch your friend in some other way (or more commonly, get them to contradict themselves). For example, if they reply yes to both “is today opposite day?” and “is today not opposite day?”, regardless of what day it is you know that your friend is untrustworthy (or at the least, a little confused about logic).

This can all be sidestepped quite easily by asking your friend a question of simple truth. If you ask “does 2 + 2 = 4?” they might answer “no” on opposite day and “yes” on other days. You can now deduce what day it is.

But wait, there’s more! If it really is opposite day, your statement must have meant “does two plus two NOT equal four?”. Or did it perhaps mean “does a number other than two plus a number other than two not equal a number other than four?”?

The problem here is not knowing when to stop negating when taking the complement of a statement. If we arbitrarily stop after an even number of iterations, we get back the original statement (the nots cancel out). If we stop after an odd number, we get back the opposite of the original statement.

Negating an even number of times is fairly natural: it leads us right back to where we started. No logical issues arise from this, we’re just doing a bit of unnecessary work. Negating an odd number of times, though, leads to nasty results like the “does a number other than…” statement above. If we can randomly negate any statement once (1 is odd) then we can negate it twice by negating once then negating once again. This breaks logic. If we have a statement “the primes are infinite” and negate once we get “the primes are not infinite”. Even if this statement is taken as “true” (it is opposite day after all), the statement “the primes are not not infinite” is also allowed. Instant contradictions emerge. This is opposite day, not non-logical day.

The inevitable conclusion is that we can treat opposite day exactly the same as any other day, and get the same results. Whether everything is being opposite-ed under the hood or not, there’s no way to know, but we can use the same logic we use on regular days, and it works just as well.

Putting Yourself in the Other Side’s Shoes

Empathy is a form of reversal. You’re swapping your current position for what you imagine the other person is going through. If done correctly, this can lead to a lot of ethical insight, but of course if done incorrectly, can lead to not much at all.

A key word above is “imagine”. When we imagine what someone is going through, or what their experience feels like to them, it’s quite difficult to get all the details just right. Maybe they don’t feel like sharing their burdens with others, so you don’t know the true extent of their pain. On the other hand, maybe they’re a complaining crybaby, and are really not all that bad off. (Note: for maximally increased empathy, try not to call people “complaining crybabies”.)

Like so much else, this is a question of degree, with essentially infinite gradations. The way to find the right one is to use all the tools at your disposal to narrow in around the correct point, without forgetting your fundamental ignorance (so don’t throw your uncertainty out the window). Our imaginations can be overactive, so you’ve got to be careful here, but if you’re really just going for the emotional tug of “what that might feel like at its worst”, imagine away.

One way to perform reversal via empathy is to think of each of your situations as being handed down by God in some kind of cosmic lottery. Given the same background, the same education, the same parents and the same friends, the same environment etc. etc., would you act more like the other person and less like your current self? What kinds of things would differ about you, if only your history looked more like their history?

It may also be helpful to think of yourself as swapped with your debate partner in any sort of debate. What evidence would be convincing that you should switch to this person’s viewpoint of things? What evidence do you think they’ve seen and have been convinced by? If there is no such evidence, perhaps your stance is too hardline. Think of how it must feel to argue against someone so closed-minded that there’s no evidence that would convince them whatsoever. You might as well just go home.

If this explanation of reversal seems more wishy-washy than the others, that’s because it is. I’m not great at writing about this kind of stuff, and there’s definitely no hard technical backing I can put behind it. It’s really more of an argument for niceness. Do unto others as you would have them do unto you, or better yet, as the idealized version of you would have done. I think this sort of reversal is most effective when you have almost no idea how someone could be thinking the things they currently think, and you need to dig into the basis of their personality to really find out.

Bostrom’s Reversal

The Reversal Test is a way to use a “flip it around”-style argument to prevent ourselves from sticking too strongly to the status quo. Here’s the statement of it from that paper:

Reversal Test: When a proposal to change a certain parameter is thought to have bad overall consequences, consider a change to the same parameter in the opposite direction. If this is also thought to have bad overall consequences, then the onus is on those who reach these conclusions to explain why our position cannot be improved through changes to this parameter. If they are unable to do so, then we have reason to suspect that they suffer from status quo bias.

In other words, “if it breaks when you try to fix it, why ain’t it broke?”

Of course, some status quos are good and exist for a reason. Although not normally a consideration since things tend to change so slowly anyways, it also wouldn’t really be tenable for everybody to go around messing with every possible parameter all the time. This would break too many of our assumptions about the world. This argument for certain status quos is explicitly covered as a case of the reversal test though. If there is a good argument for why the position cannot be bettered, then maybe it truly is better to stay at the status quo.

This test is interesting because it takes on a slightly different flavor than many of the other reversals we’ve covered. Often when reversing an argument, you want to see the negation of the original argument. Good becomes bad and bad becomes good. In the reversal test, however, we actually want to see negative consequences on both sides of a potential parameter shift before proceeding. (If we see positive consequences in one of the directions, we can clearly exit the algorithm early and begin implementing those consequences.)

Rather, if this test shows negative results in both directions, then it is most effective. It’s an argument against the middle, just sitting at a point because we happen to see negativity on either side. Strangely, we want to jump into the more-negative waters surrounding us, because there’s so little reason that the waters we’re currently in are positive. We’ve just gotten attached to our little spot in the ocean because that’s where we were before, dammit!

The paper also introduces a “Double Reversal Test”. Imagine that we have some status quo, and some natural force pushes a parameter away from that set point. Would we have reason to enact force to move the parameter back to where it was originally? To reverse the reversal?

It seems to me this Double Reversal is more powerful for shaking people out of their status-quo-preserving sensibilities. It’s not appealing to take action when that action has negative consequences (even if inaction is somewhat more negative). The Double Reversal is a clever way to swap inaction and action, so we can separate our preference for the way things are from our preference for not changing the way things are.

With both Reversal and Double Reversal on hand, it’s somewhat easier to discover ideas that need a good shaking. There are many opportunities to do so. If you first apply a reversal, and you see positive consequences in the opposite direction of the originally intended change, then by all means pursue that direction instead. If you see negative consequences, then maybe the fault lies in the general region we’re in, and not in small movements away from the center of that region. Apply double reversal then to find out whether we’re just sticking to inaction because it’s safe and easy (and often convincing) or whether we have good reason for the point we’re at at all.

Flipping Probabilities

The equation $P(X) + P(\neg X) = 1$ is a gift that keeps on giving. Okay, maybe it only gives a few presents, but they’re powerful ones indeed. It gives us the Law of the Excluded Middle (that we can either have a proposition X be true, or be false, but not some other middle-ground third option). It also tells us that if we have a probability $P(X)$ of some event happening, the chance of its not happening is simply $1 - P(X)$. This is fertile ground for another reversal!

What’s this reversal’s special power? It just seems like an extremely straightforward bit of math.

This probability reversal shows its stuff when you apply it to your beliefs. Let $P(X)$ be how likely you think something is to be true. For example, let’s say we’re 99% sure the sky is blue, because whenever we think of “the sky” we think of a rich blue sky with clouds lazily drifting across its face, on a warm summer’s day, in the middle of the afternoon….

Stop right there! You just asserted a 1% chance that the sky is not blue. First of all, ever heard of nighttime? Or sunset? Or cloudy days? Let’s assume that what you meant is “during the daytime and ignoring cloud cover, the sky is blue”. Why only 1% chance?

You remember the sky being green due to a really bad storm a few times, and so said to yourself “1% is about right”. How often do such events really happen though? If the sky changes color at most once per year, your probability should be 0.27%, not 1%. Thus making your certainty that the sky is blue 99.7%—a bit higher than you naively said before examining the reversed probability.

There are reasons to beware high levels of certainty. What if Descartes’ daemon is deceiving us into believing this? What if Maxwell’s daemon had a day off from its job in the negentropy factory and is joining in on the fun? What if this is all a simulation and “sky” doesn’t exist? What if you have a strange condition that causes you to misremember the color of the sky?

What if some completely banal error in reasoning (in this case, we’ve already seen a misremembering of the evidence) is affecting the results? We can likely adjust away from 100% certainty a bit, at any rate. However, the point stands. By reversing the probability and examining the probability of the opposite belief, we were able to achieve a bit more clarity.

There’s another sense in which we can swap around probabilities: change their form. For example, if we have odds, we can convert to a percentage, or to a fraction, etc. Sometimes this kinds of form change can make things more clear.

For example, a one-in-a-million event has the same likelihood as three 1% events all happening. If it’s hard to deal with very small (or very large) chances, maybe breaking them down into manageable pieces this way could help.

A Geometric Diversion

Once you’re in the habit of reversing arguments, it can be hard to stop. You’ll keep asking “What if things were the other way?” Many times, if you spend too much effort on questioning your most basic assumptions, you’ll just be spinning your wheels. Sometimes, though, an idea for a different way things could be comes along and provides a huge amount of progress. See: relativity, heliocentrism, the ShamWow.

One such idea that got upended in this way was…all of geometry? Euclid’s fifth postulate, the “parallel postulate”, means that if you have two lines in a plane intersecting a third line with less than two $90^{\circ}$ angles on one side, the lines must intersect somewhere on that side of the third line.

It turns out that if you reverse this postulate such that your geometry doesn’t force lines to intersect before infinity, you can make some interesting things happen. These non-Euclidean geometries allow for all kinds of fancy stuff. In hyperbolic geometries, the angles of a triangle sum to less than $180^{\circ}$, whereas in elliptic geometries, those angles sum to more.

In fact, in hyperbolic geometry you can have a triangle whose angles sum to $0^{\circ}$, and what’s more, it’s called an “ideal triangle”! These ideas would not have come about if geometers blindly accepted Euclid’s ideas forevermore and didn’t bother turning them on their head.

This is all just to say that reversal is a powerful tool. Use it wisely, and only when appropriate. As much fun as reversing anything and everything can be, make sure you’re keeping after your truly important goals. But most of all, when you encounter an argument, never be afraid to flip it around.