The Penrose Triangle of Beliefs

For a long time, I didn’t think it was truly possible to believe contradictory things at the same time. In retrospect, the model I was using of belief-formation was roughly: when people decide that they believe some new claim, it’s because they’ve compared it to their pre-existing beliefs and found no contradictions. Of course, that may be how an ideal-reasoning Artificial Intelligence would build up its set of beliefs*, but we’re not ideal reasoners. It’s not at all difficult to find contradictions in most anyone’s belief set. Just for example,

  1. “The Bible is the word of God.”
  2. “The Bible says you’ll go to hell if you don’t accept Jesus.”
  3. “My atheist friends aren’t going to hell, they’re good people.”

Or – to take an example I’ve witnessed many times:

  1. “The reason it’s not okay to have sex with animals is because they can’t consent to it.”
  2. “Animals can’t consent to being killed and eaten.”
  3. “It’s fine to kill and eat animals.”

Anyway, despite the fact that I had overwhelming evidence demonstrating that, yes, people are quite capable of believing contradictory things, I was having a hard time understanding how they did it. Until I took another look at an old optical illusion called the Penrose Triangle.

The Penrose Triangle

If you look at the whole triangle at once, you can’t see it in enough detail to notice its impossibility. All you can do, at that zoomed-out level, is get a sense of whether it looks, roughly, like a plausible object. And it does.

Alternatively, you can look closely at one part of the picture at a time. Then you can actually check the details of the picture to make sure they make sense, rather than relying on the vague “feels plausible” kind of examination you did at the zoomed-out holistic level. But the catch is that in order to scrutinize the picture in detail, you have to zoom in to one subset of the picture at a time – and each corner of the triangle, on its own, is perfectly consistent.

And I think that the Penrose triangle is an apt visual metaphor for what contradictory beliefs must look like in our heads. We don’t notice the contradictions in our beliefs because we either “examine” our beliefs at the zoomed-out level (e.g., asking ourselves, “Do my beliefs about God make sense?” and concluding “Yes” because no obvious contradictions jump out at us in response to that query)… or we examine our beliefs in detail, but only a couple at a time. And so we never notice that our beliefs form an impossible object.

*(Well, to be more precise, you’d probably want an ideal-reasoning AI to assign degrees of belief, or credence, to claims, rather than a binary “believe”/”disbelieve.” So then the way to avoid contradictions would be to prohibit your AI from assigning credence in a way that violated the laws of probability. So, for example, it would never assign 99% probability to A being true and 99% probability to B being true, but only 5% probability to “A and B being true.”)

8 Responses to The Penrose Triangle of Beliefs

  1. Pingback: Morality 101 « Geoff's Blog

  2. Andrew T says:

    I love that you felt obligated to add the footnote 🙂 Very nice metaphor. Of course, humans too probably don’t believe things 100% – for example a religious person with atheist friends might be a little shaky on the whole “standards for going to hell” thing, allowing for some weakly held opposing beliefs.

  3. Kyle Walsh says:

    If we want to belabor the metaphor, we can take it one step further and discuss how beliefs that appear perfectly congruous are often revealed to be otherwise when approached from a different angle. Consider the Penrose Triangle sculpture in Perth, Australia. When viewed from one perspective, it appears to be a completely possible object. Sure, it’s a Penrose Triangle, but it exists in front of you clear as day. Wander a few meters to your right or left and view it from another angle. You immediately see that your conception of the nature of this object was deceived by your limited perspective. Perhaps it isn’t just how much or how little of the object you are viewing (i.e. zoomed in vs. zoomed out), but also where the viewer resides in relation to the object. This actually fits nicely with Julia’s discussion above.

  4. I think that part of the cause of these discrepancies comes from the fact that people pull their beliefs from multiple locations and typically assent to social norms without analyzing how they relate to already existing beliefs. Take for example your second trio of statements. The first statement really has two components: that it’s not okay to have sex with animals, and that this is because of an issue of consent.

    The first element of that is pretty long-standing. Generally, having sex with animals is considered gross and in my opinion, that feeling is a good instinct, because widespread sexual congress with animals increases the chances of spreading zoonotic infections. However, in today’s set of norms, which are more of an ideological cover for our preconceived notions, consent is the main driver of whether a sexual act is acceptable. Therefore, it’s very easy for people to mix an match to come up with a coherent statement that “[t]he reason it’s not okay to have sex with animals is because they can’t consent to it.” The real reason is because people think it’s gross, but since it can be fit into the consent rubric, and because that sounds more respectable, people do it, with the nonsensical result that other behaviors that are considered perfectly normal, like slaughtering animals specifically raised for their meat, would by the same logic be immoral.

    There are many ways that someone could arrive at the first trio, but one of the most prominent is simply that people have adopted two different ideologies that have contradictions at the margins from childhood. The first is Christian doctrine, which, depending on the denomination, says that nonbelievers will find themselves in hell for their afterlife. The other is American [you could also insert another nationality here] civic religion which is religiously pluralistic. Generally, I suspect a good portion of people simply do not know their denomination’s doctrine on hell and those that do (and have a denomination with a draconian doctrine) and continue to insist that nonbelievers don’t go to hell simply haven’t thought through the implications. Others actually have denominations who have a Biblical interpretation or theological doctrine that does not include throwing unbelievers into the fire and still others are tight-lipped about their denomination’s doctrine around unbelievers (because it would obviously be impolite).

    If I were to overgeneralize concerning these types of contradictions, I’d say that they mainly arise from people’s agreeableness–they take what they are told at face value and don’t extrapolate it out to see if there are errors that such a belief would produce when compared to pre-existing beliefs. Because we live in a pluralistic society as oppose to say North Korea where all beliefs come from a central authority or a small isolated agrarian or hunter gatherer society, where customs do not have to be justified in universal terms, we have multiple competing ideologies that each have appealing elements but often contradict each other, there are plenty of waiting ideological traps for those who mix and match their explanations, which is nearly all of us.

  5. Sam Penrose says:

    If the three propositions which fail to cohere are the corners of the Penrose triangle, what are the sides? I assume they are something like “logic” or “the necessary implications” of the propositions. And here you might stop and think a moment, for does it really illuminate the thought and speech of non-logicians to reveal that their thought and speech, considered as logic, does not hold together? Do you see the tautology? As far as we know, a shared morality composed of a grab-bag of somewhat mutually contradictory tenets is nearly universal across human experience. Only a few very strangely educated folk aspire to an internally consistent morality. Showing lack of internal consistency in others’ beliefs doesn’t, by itself, say much of anything — let alone establish that internal consistency of belief is a desirable quality of a moral sensibility.

    To avoid exemplifying the fallacy I am trying to call out, I’ll ask you to consider the actual human beings who might espouse the first two propositions in your first example: certain devout Christians. The ones I have known are quite aware of the tension between clear maxims and the moral complexity of specific people and their choices. Much of their worship explicitly addresses the challenges of understanding and living by the Word of God — faith that passeth understanding and all that. Now, I personally have no use for their doctrines, but the doctrines’ failings are not measured by their distance from Kant or Russell or whomever. Is it possible that your chosen criticism hides more than it reveals?

    ps no known relation, although the name comes from a very small corner of England.

  6. Max says:

    1. “The reason it’s not okay to have sex with children is because they can’t consent to it.”
    2. “Children can’t consent to being vaccinated or disciplined.”
    3. “It’s fine to vaccinate and discipline children.”

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  8. Scott says:

    I recently read a book called “Why Everyone (Else) Is a Hypocrite” by Robert Kurzban. It talks about the modular nature of the human mind, and how the facility to believe one thing in one part of the brain, while believing something else in a different part (each of which may or may not be “consciously available”), might have some evolutionary value for a social species (e.g., “plausible deniability” in circumstances where social norms demand a personal sacrifice, etc.).

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