Of Liars and Truth-Tellers


(Hidden tribute to the late David Bowie here)

I used to really love logic puzzles when I was a kid. I’ve mentioned the professional logician Raymond Smullyan a few times as a big influence of mine, and I highly recommend his puzzle books. Here’s a complete list of what Amazon carries, and I’ll highlight some, roughly in order of personal preference:

Also noteworthy is his book of philosophical essays, The Tao Is Silent. Smullyan, like fellow mathematician Bertrand Russell, dabbles in philosophy a bit, and this book is a westerner’s perspective on eastern religion. I’m sure it takes a lot of liberties with the subject matter, and I imagine if I reread the whole thing now I’d find I agree with him a lot less than I used to in my teens and twenties. But still, his style is playful and entertaining, and there are a couple of essays in that book which I love to reference: “Is God a Taoist?” and “An Epistemological Nightmare.” The first is one of my favorite speculations I’ve ever read on the nature of the “god” concept.

But I digress. I wanted to talk for a minute about Smullyan’s logic puzzles in order to illustrate a point about religious arguments.

Among many other puzzle frameworks, Smullyan authored numerous little puzzles that take place on an imaginary island where all the residents are known to be either knights, who always tell the truth, or knaves, who always lie. In these stories, some people would come up to you and make various statements, and you would have to deduce what kinds of people you’re really talking to. An example from Wikipedia:

John and Bill are residents of the island of knights and knaves.

John says “We are both knaves.”

In this case, John is a knave and Bill is a knight. John’s statement cannot be true because a knave admitting to being a knave would be the same as a liar telling the lie “I am a liar”, which is known as the liar paradox. Since John is a knave this means he must have been lying about them both being knaves, and so Bill is a knight.

The puzzles get very intricate and fascinating, and I recommend them for anyone who is aspiring to gain a deeper understanding of logic, critical thinking, and the ways that formal mathematical systems can be used to represent the truth of a claim. In fact, I even did a team project for a graduate school computer engineering class, in which we wrote a program to read and parse these puzzles and generate possible solutions.

However, you also have to recognize their limitations. Knight/knave puzzles can be solved only if you accept the premise that everyone you talk to is either a knight or a knave. Characters must be either 100% truthful or 100% liars, no exceptions. Later in the puzzle section, Smullyan would often introduce “normals” who, like any other person, can say true or false things at will. Once those people show up, the rest of the logic mostly goes out the window. A normal can say anything at all, and the truth value can’t be determined by any means. Unlike knights and knaves, a normal can lie and say “I am a knave” and there’s no contradiction. Or a normal can tell the truth and say “I am not a knave,” and that’s also just fine. The only way you can solve any puzzle with normals present is if the author helps you out by providing additional information, such as “this group of people contains at most one normal” or something like that.

The reason I’m mentioning these puzzles is that many bad arguments seem to operate under the assumption that we live in a world of knights or knaves. To give a simple example, a fallacious argument from authority is implicitly based on the assumption that the person is a knight. “A famous person says X, and the famous person is often right, so we can be sure that X is true.” In a world of knights and knaves, that makes perfect sense! Once you can establish that a resident of the island has said at least one true statement, it follows that he is a knight, therefore every sentence he ever utters is true! So X is true, end of discussion.

Religion is full of oversimplifications like that, but the real world is a lot more messy. Since every real person is actually a normal, capable of saying either true or false things at any time, no amount of generalization about their tendency to tell the truth can be enough to guarantee that the last thing that came out of their mouth was still true. Thus, somebody can be a brilliant scientist in their own particular field, and right about most or all things, and yet they can still be a complete crackpot when saying things about a subject they haven’t studied. When an authority figure is talking about the field that they are actually an expert in, we can definitely use their expertise as a guideline to suggest that they are making claims that are very likely true. But that person is still not a knight, and is still capable of saying things that are false — whether through ignorance, malice, or bad communication.

Other examples of faulty knight/knave assumptions are all over the place in religious logic, once you start looking for it. Here are a few examples I encounter regularly, off the top of my head.

  • CS Lewis’ “Liar, Lunatic, or Lord” trilemma. Jesus Christ claimed to be God; therefore either he is God (being a knight), or he is lying or crazy (being a knave who, having said something untrue, must say or believe untrue things at all times). The problem is, even if we assume that the reports of Jesus making such a claim are accurate (which we can’t really be sure of), he could have been partially or temporarily deluded, mistaken, or lying; and yet he could still have made many other true statements.
  • The Bible mentions some cities that don’t exist today; but archaeologists discovered the remains of those cities. Therefore the Bible is accurate that those cities existed. Therefore (can you see the logical transition coming?) the Bible is accurate, so we can trust anything else we read in it.
  • Ray Comfort’s favorite apologetic, “Are you a good person?” is quite obviously based on proving that you are a knave. “Have you ever lied? Yes? What does that make you? A liar.” The sleight of hand is that a liar is not a knave, but a normal. Lying some of the time doesn’t make you a knave, any more than telling the truth some of the time makes you a knight.
  • Stating that the pope, or anyone else, is infallible — making him a knight.

Here’s an example that’s a little bit trickier: the ontological argument for God basically starts by conditionally assuming that God exists. Then it broadly defines the properties of this hypothetical God, and leverages the implications of these properties to prove that God does, in fact, exist. In a world of knights and knaves, proofs like that aren’t always just wheel spinning. In fact, Smullyan’s books include a number of puzzles in which the very fact that somebody said something about a proposition, makes that proposition true. For instance, we can prove that when a knight or knave says “If I am a knight, then God exists,” it logically follows that the speaker is a knight, and that God does exist. [Note: See the comments section for this proof.]

If the speaker is definitely a knight or a knave. Which, again, is not a condition we ever have to accept in reality.

In reality, not only do people frequently say things that are unpredictably true or false, they can even say things that are neither true nor false. For instance, when I say the well known paradoxical statement, “This sentence is false,” is it true or false? It’s only an impossible statement under a strictly binary logical system: if the sentence is false then it’s true, and if it’s true then it’s false. But the real answer is, in fact, that the statement is neither true nor false. It just occupies a space where those words don’t apply or have meaning.

In the world of apologetics, many arguments are con games, in the most literal sense of the word: Someone is trying to give you “confidence” in the reliability of a source, and then relying on what the source says to prove things true. By proving that the authority has said at least one true thing, they hope you’ll accept their claim that all the rest of it is true. “Hi, I’m Russell. The sky is blue. Also, God exists.” Two of those sentences are true. That doesn’t tell us anything about the third sentence. In a world of normal people, it is frequently a waste of time to try to find things out by evaluating whether the speaker tells the truth or not.

Comments

  1. favog says

    One more thing that you touched on obliquely, but I didn’t see explicitly stated: the “knight” can only tell you what he believes to be the truth, and the “knave” can only tell you what he believes to be a lie. But either of them can in fact be wrong, which is another reason that, just like with “normals”, it’s best to have other independent information as well.

  2. Russell Glasser says

    Knights in these puzzles are usually assumed to be 100% reliable. But Smullyan also has a separate set of puzzles, based in Transylvania, where everyone is either human (truth teller) or vampire (liar); and furthermore, they are also sane (believe only true things) or insane (believe only false things).

    An insane vampire will believe a true statement is false, but will accurately tell you that it is true, because he is lying. Those get a little complicated.

  3. says

    “For instance, we can prove that when a knight or knave says “If I am a knight, then God exists,” it logically follows that the speaker is a knight, and that God does exist.”
    I don’t see how, and I’d be so grateful if you’d explain.

  4. Russell Glasser says

    I was hoping someone would ask. 🙂 The explanation seemed too long to put in the post itself, but the proof is still interesting.

    First of all, you have to understand that in formal logic, any statement of the form “If P then Q” is logically equivalent to “Either Q is true, or P is false, or both.” This is something first time readers often struggle with, because it doesn’t quite sound like the way that people use “if-then” sentences in casual conversation.

    I hope this example will help clarify. I say: “If it’s raining, then I’m carrying an umbrella.” There are three possible ways this statement could be true:
    1. It’s raining and I’m carrying an umbrella. (Obvious.)
    2. It’s not raining and I’m not carrying an umbrella. (Also probably obvious.)
    3. It’s not raining and I AM carrying an umbrella.

    That last one seems counter-intuitive to most people because they assume the two things are mutually dependent on each other. But they’re not. Maybe I always carry an umbrella, every day. When it rains, I’m still carrying an umbrella, so my statement was still technically true (though maybe misleading).

    The only way my statement could be false is if it rains but I’m not carrying an umbrella. Then I would be lying (or confused). So my statement:
    “If it’s raining, then I’m carrying an umbrella”
    is logically identical to this statement:
    “It is not the case that it is raining and I’m NOT carrying an umbrella.”

    Anyway. An island dweller who says
    “If I’m knight then God exists”
    is also saying the equivalent of:
    “Either I am a knave, or God exists.”

    Could he be possibly be a knave? Well, if he is a knave then the statement must be true. Because being a knave, the statement “I am a knave OR God exists” is automatically true whether God’s existence is true or not. But knaves never say true things, so this is a contradiction. So he can’t be a knave, he must be a knight.

    Now we know that he is a knight who just said “Either I am a knave, or God exists.” He is not a knave, so the first half is untrue. If God doesn’t exist, then both “I am a knave” and “God exists” are false, so the whole statement (that one of them is true) is incorrect. So the statement would be false.

    But knights can’t make false statements, so we have another contradiction. The only way to solve this problem is by concluding that the speaker is a knight, and God exists. QED.

    I know that’s a little hard to follow. I still recommend reading the books I linked if you’re interested in learning more about how to approach the logic. I started reading them when I was ten, so you should have no problem with them.

  5. says

    Thank you, that was helpful, and don’t worry, your explanation is very understandable. I read it wrong, stupid me.
    By the way: I can’t comment with my preferred mail address because I get the message “Possible imposter, please try to log in via WordPress” (because that address is linked to a WordPress-account), and when I try that, I get an empty window with “error” on top. My WordPress login works perfectly well, so I guess the problem must be somewhere else.
    Do you have any idea what could be done about that?

  6. L.Long says

    I find most of these types of puzzles have little to do with reality, truth, or lies. They simply are some form of puzzle. If you ask someone a question it will lie/truth as necessary to accomplish its ends it care little for logic of puzzles. You can be an expert at solving them and useless as an aid for dealing with reality, such as determining truth or lies. Besides EVERYBODY LIES and they sometimes tell the truth, if it is perceived to be in their advantage to do so.

  7. Russell Glasser says

    @L.Long, As a software engineer, I completely disagree. Not everything in the world can be reduced down to a set of simple true and false statements, but you can model stuff that very much impacts the real world with a very high degree of accuracy. We build abstract models of the real world all the time using information that is basically a set of true or false conditions, or numerical values, or descriptions stored as bits in a database. And that stuff isn’t just imaginary; it reaches out and affects things that happen in physical reality, like when Amazon successfully sends you the book you wanted without you leaving your house, or how stocks get traded. Business logic is written with a tremendous amount of information about reality and truth. If that didn’t work, we wouldn’t have modern communication infrastructure as we know it.

  8. says

    Muriel

    Up top of the page there’s a bar of links just beneath the site header reading “Advertise”, “Privacy Policy”, “Tech Issues”, FTB Shop”, and “Recent Posts”.

    Click on “Tech Issues”.

  9. Crip Dyke, Right Reverend Feminist FuckToy of Death & Her Handmaiden says

    @Russel:

    CS Lewis’ “Liar, Lunatic, or Lord”

    He always forgets to put in that two crucial “L” categories. The 4th category? Lyricist.

    For instance, we learn from Our Lady of Rock & Roll, Saint Joan Jett, the definitely non-virginal GodMother of Punk that she’s a cherry bomb.

    Was she lying about being a small explosive, typically painted red? That seems unlikely when she’s given the titles “GodMother” and “Saint Joan Jett” (BTW: in neither case was I the one to bestow those titles).

    Was she actually committable mentally ill? That’s unlikely as small explosives neither play musical instruments nor sing, and she wasn’t so deluded as to preclude her doing either of those things.

    Was she, in fact, a small explosive, typically painted red? That’s unlikely as, using the above example, small explosives neither play musical instruments nor sing, and the extent of her being-a-small-explosive problem was not so severe as to preclude her doing either of those things.

    Thus she must actually be a magical small explosive, typically painted red, that used its divine power to play musical instruments and sing with a voice normally beyond the capacities of small explosives, typically painted red, right? Right????

    Or is there the possibility that she was using metaphor to say something that was neither lying, nor delusional, nor the kind of farcically simple fact that would lead one to conclude that one shouldn’t get matches anywhere near Joan Jett.

    What’s a metaphor, after all? More than just keeping your cows in, that’s for sure. As a writer of near-allegorical fiction CS Lewis should have know that.

    Of course, this is all conceding (for argument’s sake) that Jesus was anything at all beyond the 5th “L”: legend.
    ========================
    As an aside, after all the times I’ve seen Lewis say that trite phrase, you’d think he would have learned better by now.

  10. says

    the ways that formal mathematical systems can be used to represent the truth of a claim.

    I also recommend Bayesian Reasoning, here’s a video of Richard Carrier teaching about it at Skepticon:

  11. says

    @ Crip Dyke #12
    As you climb out of that metaphorical frying pan, be careful not to fall into the Poe Pit right next to it. I hear it has a pendulum made from Occam’s razor.

  12. Monocle Smile says

    @Russell
    These puzzles are pretty great for software guys, I imagine. I typically solve them by using return values of 0 and 1 instead of thinking Boolean.

  13. Daniel Schealler says

    @Monocle Smile

    Be careful with that!

    A binary value of 1 or 0 carries no inherent meaning, and the meaning must be prescribed by the application processing those values. This distinguishes single-digit binary numbers from boolean values such as ‘true’ or ‘false’.

    This is a particularly key concept to grasp if you ever get stuck into multivalued or plurivalent logic. 🙂

  14. yiab says

    I’m curious, Russell, if you’ve encountered the “hardest logic puzzle ever” (probably not actually, but it was published under that name). See the Wikipedia article of that name for a description if you’re not familiar. Basically it amounts to the liar/truth-teller puzzle but with 3 people (the third decides randomly whether to tell the truth), but with the added twist that you don’t understand their language though they understand yours.

  15. Russell Glasser says

    @Yiab: Wikipedia credits the “hardest logic puzzle ever” to Raymond Smullyan’s What is the Name of this Book, which is on the list of books I recommended. It comes near the end.

  16. says

    May I suggest you make a pre show call in and out of the studio before broadcasting. Such a shame the technical problems you are having. I love your show I wish I lived close to you to help you out with the technology. Keep it up guys you will get there.

  17. Kao says

    It took me a while to understand what I felt was wrong with Comfort’s “Are you a good person?” shtick. I’ve come to see it has to do with a singular action defining a person in totality. Much like “Are you a writer?” and having it be true because I wrote a short essay in high school. People do lots of things to varying frequency and magnitude. Comfort’s world must be really easy to work with when things are simple black or white divisions.

  18. EnlightenmentLiberal says

    > An Epistemological Nightmare
    I read it. I think mistakes were made. The worst part about it is that I think that most of it makes sense, in a way that is obtuse. I need some brain bleach.