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Your Nibling Knows the Logic of Possibilities: Visualizing the Modal Fallacy

Posted in freedom, introductory, logic, math, and philosophy

Description: We can solve problems that arise when thinking about what is and isn’t possible. To do so, we need to learn how to visualize the logic of possibilities.

Estimated reading time: 8:34 (238 wpm)
Estimated reading time range: 5:58 (134 wpm) to 15:13 (342 wpm)

cw: allusion to suicidal ideation near the beginning, discussion of notorious documented history and institutional racism

Coulda

You’re watching a rather problematic but classic film from 1946. In it, a man finds his spirits asunder on an escalator down the Great Depression. Like an infamous horse, he resigns to feeling the weak breeze whisper nothing as the water screams sublime, and in just a moment from now, his body crashing against wave and stone.

But a divine beast shows him a world in which he never came to be. The lives he saved now in oblivion, the evils he prevented now unfettered, and the four people he made and raised now no people at all.

“HEY WAIT A MINUTE!” the man, George, protests. “That’s a contradiction!”

“What’s the contradiction?” Clarence, the divine beast snorts back.

“Mary Hatch! Mary Hatch not having any piglets! That’s logically impossible! You’re quite simply showing me something contradictory, a way it logically could never have been!”

Clarence found this perplexing. Intuitively, it’s just wrong! So they prompted an explanation out of George.

George explained. “Well, Mary Hatch and I had at least four piglets, right?”

“Right.”

“And if Mary Hatch and I had at least four piglets, then Mary Hatch and I logically must have had at least two piglets, right?”

“Right.”

“Therefore, Mary Hatch and I logically must have had at least two piglets! That’s just modus ponens!”1

“Oh my,” Clarence said, putting a chin on their snout. “Well, I certainly can’t argue against a strict modus ponens here. So I suppose you’re right, even without you, you and Mary Hatch would have had at least two piglets. Apologies, carry on!” and they faded into the wind just as they came, leaving George once more gazing upon violent waves.

Logicoulda

That conversation should strike you as bizarre and even implausible. So bizarre that your nibling—a child of your siblings or cousins—would find it suspect. Niblings can be any age, can have any background, but we are born with the tools to understand the logic of possibility. But arguments like this are made all the time. For example, in free will, causation, and in everyday life. But no logical contradiction arises if Mary Hatch and George Bailey never had any piglets at all. We need to articulate and generalize what the mistake was here. Then we can avoid George’s mistake in those other topics where they’re more common.

The task becomes much easier once we simply rephrase the sentences in this argument. First, let’s take the argument as given, shortening “Mary Hatch and George Bailey” to just “MHGB.”

The Given

  1. MHGB had at least four piglets.
  2. If MHGB had at least four piglets, then MHGB logically must have had at least two children.
  3. MHGB logically must have had at least two piglets.

Now, let’s try rephrasing it.

The Rephrase

  1. In the way the situation actually played out: MHGB had at least four piglets.
  2. In every way this situation logically could have played out: if MHGB had at least four piglets, then MHGB had at least two piglets.
  3. In every way the situation logically could have played out: MHGB had at least two piglets.

Wording it in this way, it is hopefully easier to visualize it in the way I’ve drawn below.

We’re starting to think of the argument in terms of the possible situations.2 That gets us closer to why the argument is wrong-headed. Here, we can see that the argument doesn’t work. We’ve made a visual in which the first two premises are true. In the actual situation, there are at least four piglets. All the situations with at least four piglets have at least two pigs.

And yet, it is logically unnecessary for there to be at least two piglets. The final two circles represent an entire class of possible situations.3 So we can say there are two classes of possible situations where “MHGM have at least two piglets” is false.

But now let’s move to an argument about free will that people without a background in logic actually make. This way, we can understand how this mistake arises.

Free Will

Here, ‘free will‘ will be used in perhaps the most common way. It refers to that property which allows opportunity costs and benefits to exist. That property which makes sense of how we deliberate between our choices. This is often also called ‘the ability to do otherwise.’

The laws of physics are the regularities of a situation from beginning to end. It’s the patterns of everything. I’ll call the actual laws of physics L0 for short.

History is all the events prior to some time. I’ll call the history up to the infamous point that Woodrow Wilson ardently defended the Ku Klux Klan H0 for short.

Hopefully none of this is objectionable—it’s unwieldy and unreadable to say either of these things over and over in the middle of a sentence.

  1. In this situation, L0 and H0 are the case.
  2. If L0 and H0 are the case, it is logically necessary that Woodrow Wilson ardently defends the Ku Klux Klan.
  3. It is logically necessary that Woodrow Wilson ardently defends the Ku Klux Klan.
  4. If an action is logically necessary, the agent has no choice but to do that action.
  5. Woodrow Wilson has no choice but to ardently defend the Ku Klux Klan.
  6. If an agent has no choice but to do an action, then there is no opportunity cost to that action.
  7. Woodrow Wilson has no opportunity to do better than what he does.

This is an argument for incompatibilism, the position that free will and determinism are incompatible. The opposite is compatibilism, the position most people start from.

1 and 2 are true. It would be a contradiction to think that in the Universe with our past, our past didn’t occur, for instance.

4 is true. You have no ability to do something such that, if you’d done it, there would be a logical contradiction.

6 is true. Having multiple opportunities available is necessary for the existence of opportunity benefits and costs.

Economists will need a stronger argument before giving up opportunity costs and benefits. Not just economists either. Any reasonable person would think Woodrow Wilson had the opportunity to do better. If that’s not apparent, feel free to think of another very bad decision. From the Dulles brothers helping Nazi Germany as much as they could while forming the CIA. Mark Zuckerberg inciting multiple ethnic cleansings. History is full of them.

What’s more is that no expert (e.g. logicians) thinks this argument works either. This is regardless of their position on free will. But sometimes it does persuade people that free will and determinism are incompatible. What’s going on? Are biased logicians clinging to their initial compatibilism because they’re afraid of the truth? Well we know that can’t be right. Incompatibilists also all think this argument is invalid.

To understand, let’s lay out three different arguments for comparison.

Argument one

  1. In this situation, L0 and H0 are the case.
  2. If L0 and H0 are the case, it is logically necessary that Woodrow Wilson ardently defends the Ku Klux Klan.
  3. It is logically necessary that Woodrow Wilson ardently defends the Ku Klux Klan.

Argument two

  1. It is logically necessary that L0 and H0 are the case.
  2. If L0 and H0 are the case, it is logically necessary that Woodrow Wilson ardently defends the Ku Klux Klan.
  3. It is logically necessary that Woodrow Wilson ardently defends the Ku Klux Klan.

Argument three

  1. In this situation, L0 and H0 are the case.
  2. It is logically necessary that if L0 and H0 are the case, Woodrow Wilson ardently defends the Ku Klux Klan.
  3. Woodrow Wilson ardently defends the Ku Klux Klan.

Argument two and argument three work. The premises necessitate the conclusion, even in argument two where the premises are all false.

But more importantly, notice here how easy it is to confuse these arguments for one another. This is in part due to quirks of language. The sentence “If L0 and H0 are the case, it is logically necessary that Woodrow Wilson ardently defends the Ku Klux Klan” sounds as though it is saying that if L0 and H0 are actually the case, then Woodrow Wilson’s ardent defense is a necessary event. But, upon examination, the actual structure of the assertion, beneath the surface structure, is closer to “This is logically necessary: If L0 and H0 are the case, Woodrow Wilson ardently defends the Ku Klux Klan.”4

We can rephrase each of these arguments to clear them up even further.

Argument one (reprise)

  1. In the way the situation actually played out: L0 and H0 are the case.
  2. In every way the situation logically could have played out: if L0 and H0 are the case, Woodrow Wilson ardently defends the Ku Klux Klan.
  3. In every way the situation logically could have played out: Woodrow Wilson ardently defends the Ku Klux Klan.

The first two premises are true (provided physical determinism), but the conclusion is false. We can see in the final circle that despite the first two premises being true—L0 and H0 in the actual situation and Wilson being racist in every situation where L0 and H0 is—the final circle has a consistent situation where Wilson isn’t racist.

Argument two (reprise)

  1. In every way the situation could have played out: L0 and H0 are the case.
  2. In every way the situation logically could have played out: if L0 and H0 are the case, Woodrow Wilson ardently defends the Ku Klux Klan.
  3. In every way the situation logically could have played out: Woodrow Wilson ardently defends the Ku Klux Klan.
Visualization of what goes on given our laws and past.
[ID: The first two circles of the previous image with the four labeled circles.]

Here, we go through what would follow provided that L0 and H0 are the case in every single situation. And here, naturally, the argument works out nicely. Of course, since nobody would defend the first premise, this means this argument isn’t very useful!

Argument three (reprise)

  1. In the way the situation actually played out: L0 and H0 are the case.
  2. In every way the situation logically could have played out: if L0 and H0 are the case, Woodrow Wilson ardently defends the Ku Klux Klan.
  3. In the way the situation actually played out: Woodrow Wilson ardently defends the Ku Klux Klan.
Visualization of what goes on given our laws and past.
[ID: Same image as the previous one.]

While this looks the same, what we’re doing now is we’re only looking at the situations where L0 and H0 are the case. This is because the second premise is only about those situations. In the end, we still end up with a possibility space wherein every situation has the same laws and history. And because the laws are physically deterministic, the events are the same. Here, the first two premises are true, but now the final premise has no bearing on free will!

Notice that acknowledging that this argument is flawed does not mean affirming compatibilism! I’ve only shown how you can intuitively explain to your nibling how this argument for doesn’t work. This particular popular argument for incompatibilism, not incompatibilism in general! In other words, this is bipartisan. Wherever you stand on the compatibilist consensus, all logicians and experts agree that this argument won’t work!

Hope for our niblings

The visuals for each argument make it clear that they are different arguments. This is very important. I encounter someone who confuses argument one for modus ponens or for argument three daily. And not just in free will! And we can do more than distinguish them. Now we can now see why argument one doesn’t work.

So, it can be tempting when encountering this mistake to think that most brains just aren’t wired for this stuff. But notice that we aren’t totally bad. That first argument between George and Clarence seems to have some kind of mistake. It’s easy for most niblings of most ages and backgrounds to see that it doesn’t work. It’s just not obvious how to generalize that to avoid that mistake in other topics.

But these visuals offer a way to generalize thinking about the logic of possibility to avoid these mistakes. So you may catch your nibling may make an argument like this. Save yourself a few frustrating hours by simply showing George’s argument. Your nibling can understand it is faulty, and then you can generalize from there! Start thinking in terms of ways a situation could be, and this all becomes much clearer!

The rest of the series

I made this series (and this blog) with a set of goals.

  • To give people the logical tools they need to deal with very common problems.
  • To resist pessimism about people’s ability to deal with those problems.
  • To generalize from what most of us already know to help deal with those common problems.
  • To enter into the Summer of Math Exposition 1 (SoME1) competition.

Here is a link to the rest of the series.

Endnotes

  1. A more technical note: I’m aware that modus ponens is technically invalid, in particular with indicative conditionals! This is just a silly little line. 🙂
  2. Or:
    • possible worlds,
    • possible cases,
    • possible states,
    • whatever you’d like to call them.
  3. You can think of all the situations with minute differences where it’s still the case that they only have one piglet. A situation with one atom moved North, a situation where electrons generally behave like Mary’s electrons except near the end of the situation, and so on. There’s an infinite number of possible situations, so each circle is a class of logically possible situations.
  4. Here’s one handy trick for detecting when the surface and deep structure of a sentence don’t match. Notice that the sentence seems true, but the sentence on the surface is false. So, the sentence must actually mean something other than how it looks.

    “If L0 and H0 are the case, it is logically necessary that Woodrow Wilson ardently defends the Ku Klux Klan” is, on the surface, false. But it kinda seems like it should be true. From there, it just takes a bit of linguistic savvy to build up what the actual sentence is. I know, for instance, that possibility-related words like ‘can,’ ‘must,’ ‘should,’ etc. tend to be placed in deceptive ways so that’s a good candidate.

    There are better ways to do this, but this is a quick and handy trick for dealing with sentences like this.

Further resources

  • As of this writing, “Can We Define ‘Must’? The Semantics of Modality” by The Ling Space has just under seven thousand views, which is so wild to me! I haven’t found very many other videos introducing possible worlds semantics that were very good. The other videos I’ve found got…weird and misleading. This one is a real gem!

Acknowledgements

Thank you to Amadanny Borvito for his helpful notes!

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