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The Pudding Fallacy

Sun 27 Dec 2015 by mskala Tags used: ,

My text for today is from Pink Floyd (Another Brick in the Wall, part II): "If you don't eat your meat, you can't have any pudding!" I'd like to talk about the fallacy embedded in that statement. It's related to many well-known fallacies, but I haven't been able to find an existing name that applies specifically to this fallacy in this form without mixing it up with other things. So I'd like to give it a new name: let this be known as the fallacy of the Pudding.

It has many forms. If you don't eat your meat, you can't have any pudding, but also:

This fallacious argument is invariably stated in a negative form as "(NOT p) IMPLIES (NOT q)." It's never stated in the logically equivalent and simpler form "q IMPLIES p," and there's an important reason why, which I'll get to later. It's also invariably the case that "p" is a proposed course of action and "q" a desired outcome.

Since I am stating a definition here, it is no surprise that things meeting the definition "invariably" do meet the definition. But near misses are rare: we very seldom hear anyone saying anything like "If you can have some pudding, you will have eaten your meat!", which would be the "q IMPLIES p" equivalent form of the statement. I assert that the limited form I am describing is a phenomenologically meaningful category. There is a specific thing that many people do, not created by my arbitrarily slicing off just part of a bigger class of behaviour, and stating the fallacious argument in the form of not-implies-not is part of that thing.

These constraints, which have nothing to do with the underlying logic of the argument, are clues to what's going on here. Although the Pudding Fallacy may resemble a statement in symbolic logic involving propositions that may be exactly true or false and operations like "NOT" and "IMPLIES," it really is not such a statement. Symbolic logic, although I will discuss such an analysis below, is not the intended nor the correct interpretation of the Pudding Fallacy's claim. Saying "If you can have any pudding, you will have eaten your meat!" sounds strange in a way that the original statement does not, even though these two statements' meanings expressed as symbolic logic are exactly equivalent, because their real semantics include information lost when we express them as symbolic formulas with equivalence treated as equality. (I am indebted to Professor Kit Fine of NYU for opening my eyes to the specific issue of logical equivalence not working well for natural language semantics.)

Instead of being formal logic, the Pudding Fallacy is all about informal, unstated, implications, many of which may be only probably true or false, or not exactly true or false, in some complicated way. Modal logic can express some of those distinctions, but the implications are often too vague for even modal logic to really settle the important questions. The informal implications are the other statements we must also believe in order for the openly-stated assertion to be true or even meaningful.

If we accept the Pudding Fallacy as a true statement, then we are necessarily accepting all its built-in assumptions as true, even without consciously being aware that that's what we're doing. And then the person who stated the Pudding Fallacy as true succeeds in convincing us of the unstated assumptions as facts, without needing to justify them by any genuine argument or evidence. That person might not be someone else - accidentally using this weapon against ourselves, to convince ourselves of false implications, is entirely possible, and a frequent and dangerous mistake.

The biggest, albeit not the most direct, hidden implication of the Pudding Fallacy is that you ought to take the proposed course of action. It is telling that this one is left unstated even though it's the main point of the whole exercise. Why don't they say it if it's the point? Because they are, consciously or unconsciously, being deceptive for some reason. Pink's authority figure does not say "Eat your meat!" He just says, well, if you don't. If you don't eat your meat, you can't have any pudding...

Here are some of the hidden assumptions in the Pudding Fallacy. Some of them overlap with each other, some are context-sensitive, and some are logically equivalent to the original statement - but as I mentioned, logical equivalence does not mean they really have the same meaning in natural language, so it can be worth considering the equivalent forms separately. Someone who can convince you of "if you don't eat your meat, you can't have any pudding" without critical thought, will get the benefit of having your agreement to all of the following, too.

Eating your meat is the only way to have pudding

The Pudding Fallacy is powerful because its real meanings are hidden below the surface, but the surface meaning is usually false to begin with. Flipping it around to the form "If you can have any pudding, you will have eaten your meat!", which has a logically equivalent surface meaning, may make the case clearer. It is physically possible for a person to have some pudding without having eaten meat first. The strongest statement the authority figure can truthfully make is "If you don't eat your meat, then I intend to forbid you from having any pudding!", and confusing the exercise of subjective temporal authority with operation of objective eternal fact is a classic tactic of abusive authorities everywhere. Equating authority (prescriptive rules) with physics (descriptive rules) is not logically valid even if it is an effective way for them to retain practical power.

Note that in the Pink Floyd song they hammered it in place with a rhetorical question: "If you don't eat your meat, you can't have any pudding/How can you have any pudding if you don't eat your meat?" In fact, we can imagine many answers to how such a thing could occur, but it's strong tactics to deny any possibility of there existing an answer to the rhetorical question, and sweepingly dismiss any consideration of the fallacious preceding statement maybe not quite being literally and universally true. These two lines about abuse of authority contribute to the artistic point: with their false implications dressed up as logic, they're just bricks in the wall of the adult Pink's nihilism.

In symbolic logic, the surface meaning of the Pudding Fallacy could be part of a valid argument - understanding that "valid" here has a specific technical meaning and does not mean the conclusion is true. The logic runs like this, where "p" represents "You eat your meat" and "q" represents "You can have some pudding":

  1. (NOT p) IMPLIES (NOT q) ["If you don't eat your meat, you can't have any pudding."]
  2. THEREFORE q IMPLIES p ["If you can have pudding, then you will have eaten your meat."]
  3. q [(you want it to be the case that) "You can have pudding."]
  4. THEREFORE p ["Eat the damn meat."]

Step 2 follows perfectly correctly from step 1 by the laws of symbolic logic. Step 4 is a correct application of modus ponens to steps 2 and 3. This is a valid argument. The parenthetical comment on step 3 - that this is something we want to be true, not something that is true in a grander sense - is a clue to the fact that symbolic logic does not address everything that's important here. We've also glossed over the time sequence of events hinted at by the future perfect tense will have eaten. But we could at least imagine using some kind of modal logic (incorporating things we want to be true as well as things that really are true, and distinguishing these kinds of truth, and maybe even with a temporal element to capture the idea of which thing has to happen first) to handle those distinctions in a more satisfying way. We can build a tight, valid argument out of it.

The more important problem is that the premise in step 1, which is the surface meaning of the Pudding Fallacy assertion, is not true. You actually can, at least possibly, have some pudding without eating your meat. Eating the meat is not the only possible way to have pudding; that was at most a true statement of the authority figure's wishes, not an objective truth. Then the argument, although valid, cannot be expected to give a true conclusion. The strongest conclusion validly argued from true premises might be something like "You are forbidden to have pudding," which is quite different from really not having it.

Global climate change could conceivably be mitigated by massive geo-engineering without anybody changing their diet, or at some time in the future without any action today, or by everyone except a small minority becoming vegetarian, and you might be in that minority. Even if we might argue details and probabilities and fairness, the sweeping black-and-white statement about vegetarianism and climate change in Pudding Fallacy form, is false on its face. Maybe atheists do get to go to Heaven after all. Some people make money on software that is not open source. The cute boy or girl could ask you out. It's hard to literally win the lottery without a ticket, but you could be given one without having bought it yourself, and anyway there are many ways to obtain large amounts of money randomly, without literally winning it in a lottery. In general terms, it's nearly always at least remotely possible to have that without necessarily doing this. Very few paths are the only ones to their destinations.

If you do eat your meat, you can have some pudding

Even as a child, Pink probably knew better than to assume that if he did eat his meat he necessarily could have pudding. It was at best a possibility, that could be extinguished by his not eating the meat but also perhaps by random forces outside his control. We can easily imagine this conversation:

The authority figure's promise was only implied, and only informally (not logically) implied. It is literally true that he didn't promise pudding under any circumstances; only no pudding if Pink didn't eat his meat. The Pudding Fallacy only makes a statement about what happens if you don't take the proposed action. It is carefully silent about what happens if you do, leaving that to the listener's imagination. It only makes a statement about not achieving the desired result; it is carefully silent about any circumstances under which the desired result might ever be achieved, leaving that to the listener's imagination. The Pudding Fallacy makes a promise, but lays the groundwork to deny having done so.

The child Pink's logic, if he were not already too jaded at that point in the album to depend on it, would run something like this, where "p" represents "You eat your meat" and "q" represents "You can have some pudding":

  1. (NOT p) IMPLIES (NOT q) ["If you don't eat your meat, you can't have any pudding."]
  2. *THEREFORE p IMPLIES q ["If you eat your meat, you can have some pudding."]
  3. p ["I ate my meat."]
  4. THEREFORE q ["I can have some pudding."]

As an exercise in symbolic logic, the problem is clear: step 1 does not prove step 2, because that would be an incorrect application of the law of the contrapositive (which descends from de Morgan's Laws and a little bit of algebra, if you like that way better). After that mistake, step 4 is a perfectly correct modus ponens argument. The logically correct contrapositive statement from step 1 would be "q IMPLIES p," as I said before: if you can have some pudding, then you will have eaten your meat, which is not particularly inspirational.

But the authority figure's hands are not clean here; the child is not entirely wrong. The promise "p IMPLIES q" really was an intended part of the meaning conveyed by the original statement about meat and pudding. You could think of it as directly included, or an indirect consequence of another implied promise, such as "It is possible, and under your control, for a circumstance to occur under which you would be allowed to have pudding." The fact that this promise was made only by informal implication and can be plausibly denied because it is not a logical implication of the surface meaning of the sentence, only makes the whole business even more dishonest.

The climate may change so that it kills us all, regardless of what you eat. Maybe nobody goes to Heaven. Maybe your software isn't worth any money even with the source code. Maybe the cute girl or boy will say "no," and punish you for asking. Maybe your ticket won't be a winner... and anybody who raises any of those points can reasonably expect them to be met with the Pudding Fallacy stated again, louder than before. It's hard for me to even read those few sentences without mentally hearing the responses: Maybe nobody goes to Heaven but you can't if you don't believe. Maybe your ticket won't win but you can't win without a ticket. Maybe there is no pudding, but, but if you don't eat your meat, you can't have any pudding!

An article I recently read from an anti-GMO activist made the point that in his experience, every time he tried to talk to his opponents about the possible risks of deploying genetically modified organisms, they would switch to talking about the benefits. Any talk of or thought about risks was just automatically set aside. I'm not familiar enough with that fight to verify the claim, but something similar certainly happens when someone deploys the Pudding Fallacy. Ask about meat, and the answer will be about pudding.

When you are confronted with the Pudding Fallacy, I dare you to raise a serious question about circumstances under which you do take the proposed action, or do achieve the desired result. You'll get a response (typically just a loud restatement of the original Pudding Fallacy assertion) about the opposite case, the one where you don't. You are supposed to conclude that you should do the thing, based on statements carefully limited to the case where you don't do the thing. If someone has a plate of meat to sell, and especially if they have already sold it to themselves, then you can wind them right up into an hysterical freak-out just by quietly and repeatedly pointing them back at whatever case they were using the Pudding Fallacy to avoid talking about. I narrowly avoided losing a long-time friend over that recently, with a Pudding Fallacy discussion of dating on Facebook that spiralled out of control.

That's why the Pudding Fallacy is stated in negative terms. It is not a "q IMPLIES p" statement but a "(NOT p) IMPLIES (NOT q)" statement, despite the logical equivalence to the simpler form, because its rhetorical purpose is to shift the focus onto the safe "NOT p" case, when the relevant and important case is the other one. The Pudding Fallacy is best invoked when the proposed course of action is very much more expensive and risky than the speaker wants to admit, as a way of avoiding consideration of its real risks and costs. Especially if, as I mentioned, they have already sold the meat to themselves, then making explicit any question about the expensive and risky case and whether it's truly worth it, becomes emotionally unacceptable to the speaker.

There is no promise that you can ever achieve the desired result at all, except there really is a promise. It's implied, but implications are real. The implied promise is that you can have the desired result by taking the proposed course of action. But since the promise is only informally implied and not logically implied by the surface meaning of the assertion, this false promise is shielded from rational examination.

Ignoring the possibility that the pudding might be denied regardless of Pink's actions also has the useful consequence of making it sound like he has a choice about what will happen - which is particularly useful when, as in most of my examples, the real outcome is primarily or entirely determined by the choices of others over whom the listener does not have power.

Pink's access to pudding depends on the authority figure's whim. Saving the climate depends on everybody else agreeing to do so, including many persons who would be forced to make greater sacrifices than you would. Eternal salvation is a matter between you and God (depending on questions of fact that are at issue and unprovable), but that's at least another agent if not another human being, and you certainly don't hold the final power there. Whether people are willing to pay you for software, or go out with you, is decided by them, not by you. Really the lottery is the only plain honest deal on that list, and it's still determined primarily by chance - you can decide whether to buy a ticket, but you cannot meaningfully decide whether to win.

Shifting responsibility onto someone for events not under their control is another of these classic abusive-authority tactics. Promising that someone has control over a situation when they do not is a method of dishonestly securing their cooperation, and of blaming the victims instead of anyone else (especially, instead of yourself or your own ingroup) for any bad outcomes that do occur. And those things are done in many applications of the Pudding Fallacy.

Misvalued costs

Eating your meat has a cost.

I like eating meat in general, but some people do not, and even I do not always want to eat all the meat I can, at every possible opportunity. Maybe at the moment I'm more interested in pudding; or maybe this is bad meat. It might taste bad, or it might make me ill. In addition to being a nonzero cost, the cost of eating the meat may be uncertain, and some imaginable outcomes of eating it are very bad indeed. No amount or quality of pudding is worth it if the meat kills me first, and people do regularly die from eating bad meat.

The Pudding Fallacy dismisses all consideration of the cost and risk of the proposed action: we are asked to assume that it costs literally nothing, or so little that pretending it costs literally nothing is a reasonable way to analyse the situation. And we are asked to assume that the cost is known with literal certainty, or so close that pretending we know it with literal certainty is reasonable.

Often, as in the lottery, there is a game going on with vastly different scales of numbers. Paying the price of a lottery ticket is a certainty if we're going to buy one in the normal way, but the price is a small amount of money, so we multiply certainty times small, get small, and are asked to operate on the assumption that the cost of buying a ticket is practically zero. On the other side, the chance of winning is small, but the value of the jackpot appears large, so we multiply a small chance by a large amount of money, get an amount of money that is (necessarily, because the game runs at a profit for its operators) less than the price of a ticket... and now we are asked to assume that that small amount is more than negligible in a way worth using as the basis for a decision, never mind that we already were asked to agree that the greater small amount corresponding to the ticket price, was negligible.

There are more complicated ways to do the calculation, having to do with the subtleties of how to put a fair price on uncertainty, but it's a Hell of a stretch to come up with a way for buying the ticket to be the mathematically right move, with realistic numbers on the prices and probabilities, as long as potentially winning the jackpot is the only thing you hope to get out of it. Playing the lottery only makes sense if you very much enjoy the excitement of gambling for its own sake (so there is relevant value other than the money value of winning), or if you want to support a cause that's funded by the lottery (so that you get positive value from losing), and even then, most people who have such interests could serve their interests better by playing other gambling games or by making donations in other ways. The math is relatively easy to do with lotteries, and yet they remain popular. How unsurprising that in my other examples, which are harder to evaluate objectively and accurately, people will try to misdirect us from considering the costs of proposed actions as important parts of the analysis.

The child Pink may really hate eating meat, either in general or this particular meat for some reason, and if the authority figure cares or pretends to care what Pink wants, he may easily be unaware of how much the child wants to avoid eating the meat. Not eating meat has a cost in nutrition and enjoyment usually underestimated by environmentalists. Believing in God has important costs and risks, especially if you are asked to choose just one specific God and risk offending the others. Open-sourcing your software limits your ability to extract trade-secret and other value from it, as well as letting you in for ongoing support obligations, and the risk of evil people using it for evil purposes. Extroverts always underestimate how much it really costs an introvert to approach another person for any reason (let's politely say "underestimate," but they don't estimate it to be any more than zero), and lots of people have political reasons to pretend that the innocent are never, ever, at any nonzero risk of being unfairly punished in the dating arena. I've already talked about lotteries. It is a systematic tendency that someone who says "if you don't do the thing, you can't have the result" will hang an inaccurate price tag on doing the thing, and inaccurately assess its built-in risks.

Misvalued benefits

Having pudding might not really be so good after all.

The plain pudding instantly cast its shadow over the deepening gloom of our young minds.

"I wonder how plain she'll make it?" Dicky said.

"As plain as plain, you may depend," said Oswald. "A here-am-I-where-are-you pudding - that's her sort."

- E. Nesbit, The Conscience-Pudding

I do not just want to "have" pudding, but to eat and enjoy it - my ability to do which may be impaired if I fill up on meat first. The value of having some pudding may thus depend on what I did to get it; and that change in value of the benefit is distinct from the already-mentioned cost and risk of eating the meat in the first place. It also might well turn out to be one of those punitive traditional British puddings. Uncertainty in the value of the benefit is often ignored in the Pudding Fallacy. The Pudding Fallacy asks us to misvalue the purported benefit by operating on the assumption it is a best case as well as a certainty; even if we are optimists, that demand is a problem because the Fallacy previously asked us to make contrary assumptions elsewhere.

Climate change has many complicated and unpredictable effects, some of which may be good for some people, maybe even you and me, such that preventing it seems a desirable goal primarily from general arguments about unknown risks and principles of fairness. Those arguments may not be strong. Some descriptions of the Christian Heaven (those put forward by Scholastic philosophers like Thomas Aquinas, in particular) sound not much different from nor better than methamphetamine. Being paid for software is a lousy dog's way to make a living. I do not actually want a date with the cute girl, but to have sexual intercourse with her, and the claim I can't have that without undergoing a date is another Pudding Fallacy. People who buy lottery tickets do not want to win for its own sake. They want the things they imagine they can buy with the money, and they usually imagine it as a lump sum, either unaware of the fact or unaware of its consequences, that lottery jackpots are paid out as annuities.

Infinities and zeroes

Something like the example about believing in God is famous under the name "Pascal's Wager," but when Blaise Pascal proposed it in the 17th Century, he added a wrinkle that is important in that particular case but that I don't think is essential or important in all cases of the Pudding Fallacy. That's why I distinguish the Pudding Fallacy as a larger category instead of just identical to Pascal's Wager applied to non-religious contexts.

The interesting wrinkle in Pascal's Wager has to do with arguing finite quantities against infinite quantities. Although some religious persons claim that belief in God has zero cost and even positive benefit in itself, and some others would argue whether that's true or not, Pascal said that at the very least, we all ought to be able to agree that the cost of belief in God is finite. There is some limit to how much you can possibly lose by doing that. He also claimed that the benefit of having eternal pudding in Heaven is infinite. Then if on top of those two questionable statements we are willing to also believe that the probability of going to Heaven in case of belief is not zero (existence of God in the form described has to be both possible, and a matter meaningfully subject to probability quantified by a constant real number), and the probability of belief incurring an infinite cost is zero, we can say "Infinite times not zero equals infinite, which is greater than finite, and therefore the benefit outweighs the cost." This isn't a classic Pudding Fallacy because it is on the positive side (promising a good outcome in the case of doing the thing, instead of a lack of good outcome if you don't), but a similar argument can be made on the other side, considering a supposed infinite punishment in Hell for lack of belief.

There are of course many reasons not to eat this particular dish of meat, but the one I'd like to highlight is that doing math on abstract non-numerical quantities like infinite and not zero is subtle, prone to error, unlikely to convince anyone, and hard to reconcile with any factual statements about the real world. In Pudding Fallacy applications, we are usually asked to consider quantities that really are finite, but that are claimed to be so large we're supposed to agree they "might as well" be infinite. The rewards are claimed to be in that category. Then we are supposed to believe (usually only as an implication, because the proponents of the Fallacy will not willingly discuss this case at any level of seriousness) that the chance of a good outcome if we take the proposed action is not zero, weighed against the exactly zero probability of success if we don't take the proposed action, and that's supposed to be an overwhelmingly strong argument in favour of the proposed action because infinite times not zero equals infinite.

Really, the argument depends more on the supposed zero probability of success (the original assertion: zero probability of having any pudding if you don't eat your meat), because we can factor out the value of the reward on both sides. That is, if we believe factoring out infinite quantities is a legitimate thing to do, which it is not in general, but that's not the worst problem in this calculation. Zero chance of success without taking the proposed action may often be as implausible as infinite claimed benefit, and then we are down to arguing which small nonzero probability is smaller than the other, under the handicap of refusing to discuss one of them directly.

I'm not really interested in cases like Pascal's original Wager where quantities are claimed to really be literally infinite, except to warn again that math on infinite quantities is tricky even in the abstract, and absurdly hard to reconcile with the real world. What interests me more are the cases where quantities definitely not zero or infinite are asserted to nonetheless be equivalent to zeroes and infinities, often (as in the lottery analysis) with inconsistent assumptions about how large or small the numbers should be in order to count. That happens a lot in applications of the Pudding Fallacy, such as the ones I described in earlier sections. It happens enough to be a characteristic even if not universal part of the complex of incorrect assertions that make up the Pudding Fallacy. When people under- or over-estimate quantities, they often carry it to the extreme of making the estimates zero and infinite, at which point it becomes a way to simplify away the important parts and draw false conclusions.

Final thoughts

I've described the basic form of the Pudding Fallacy, given some examples, and examined in detail the many ways in which it is incorrect and dishonest. I did this mostly for my own amusement and future reference, so that I can have something online to point at next time I see a Pudding Fallacy in action somewhere. But I'm sure that after reading these thoughts, you'll be able to recognize the Pudding Fallacy when it occurs too, and think of other examples.

Remember, if you don't agree with me, you can't comment intelligently.

4 comments

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What a brilliant ending!
Abdul M - 2016-01-13 07:12
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"Any every fool knows a dog needs a home, a shelter from pigs on the wing."
A) Does every fool know this?
b) Is a shelter from a non-existent entity actually required?

But,sreiusly Matthew, do you have a similar paper on bisuit conditionals I could read?
Philip - 2020-10-25 18:07
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There's a decent article in Language Log, findable through Google.
Matt - 2020-10-26 09:11
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> "If you can have any pudding, you will have eaten your meat!" sounds strange in a way that the original statement does not

That's awkward on its own merits... Any love for "If you'll have dessert, you'll eat your veggies first"? (Hmm, that still sounds archaic...)

> There is no promise that you can ever achieve the desired result at all, except there really is a promise. It's implied, but implications are real. The implied promise is that you can have the desired result by taking the proposed course of action.

I'm reminded of the story where a tyrant King summons his Jester in retaliation for a prank based on a logic puzzle, and presents the Jester a logic puzzle of his own, concluding with "if you choose the wrong box, I'll cut off your head" -- leaving unsaid "if you choose the right box, I'll cut off your head anyway"...
Liam I. - 2024-07-17 11:42


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