Paradoxes are interesting phenomena, and they can give us great insight into exactly how we err in interpreting our perceptions about reality. But how exactly do we approach paradoxes to learn about the nature of our perception?

I’m going to approach a selection of famous paradoxes and indicate exactly how the error is indicative of a flaw in our methods of perception, rather than using a paradox to reveal that the universe is somehow unknowable, contradictory, and mysterious. Let’s start with a simple one: the liar’s paradox. Someone who says “I always lie.” Are they lying? If you accept their statement as true, you must therefore accept the statement as false, etc. etc. This is an easy one to disentangle. Is the semantic identity of the word “lie” useful in this context? What do we mean, exactly, when a person “lies”. Do we mean they say something that is literally untrue? Lying is actually a fairly complex set of possibilities, often simplified in this context to mean always saying the opposite of what is true. But even “opposite” is a confusing and entangled term. With a verb, opposite implies a single negation. It could mean active behavior in the opposite, or any number of other possibilities. When applied to a color, somehow it reverses to the opposite side of the human-visible color spectrum. Little children even add incredibly arbitrary constructions, such as “dog” being the opposite of “cat”. Which makes absolutely no sense whatsoever. For the sake of clarity, each context warrants its own predicate, but nobody seems to realize the profound extent of their entanglement. The liar’s paradox points out a clearly evident problem with our contextual interpretation of language, such that “lying” can mean essentially anything. I could even approach the word “always” but I think I’ve made my point. In a practical, logical sense, there is no paradox. The person could be saying that they do not necessarily speak the truth when they say they always lie, not that they always add a single negation to whatever is true. Next!

Nothing is better than eternal bliss, but a slice of bread is better than nothing. Therefore, a slice of bread is better than eternal bliss. This is an obvious one. The word “nothing” is clearly being used to refer to two different semantic entities. In the first case the word nothing is intended to produce a sentence meaning “there exists no thing better than eternal bliss.” In the second case, the logically precise equivalent is “bread is better than the complete absence of substance.” Just because the sequence of letters used is the same is no reason to draw such a conclusion.

The sorieties paradox. You have a heap of sand, and subtract one grain, leaving you with a heap. Presumably this means you could subtract all the grains, one at a time, preserving heap-ness at each step, until you had no sand at all. This is also a fairly rudimentary one, though tougher than the previous one. The resolution is that a heap is an imaginary entity with a vague definition. A “heap” does not actually exist in the same sense that a perfect circle doesn’t exist. In fact, in both cases, by definition they cannot exist because a perfect circle is a two-dimensional object and cannot exist in a three-dimensional world. A heap has a similar problem, but it’s more difficult to articulate. Consider: if you were omnipotent, and could create anything you wanted with a thought, and you wanted to create a heap, just a heap, then how would you do it? You could create numerous other objects and designate that “X is a heap” but you could not create “a heap” in the same way not even an omnipotent being could create a square circle, or a triangle with four sides. However, those exhibit the property of being overdefined, creating contradictions. A heap, however, is underdefined, which means we can automatically fill in the gaps based on context and fool ourselves into thinking such thing as a heap can exist. We can create it in imagination, but it does not therefore exist. I could go further and apply this same idea of imaginary entities to, say, religion, the government, positions of power, etc. etc. ad infinitum. I will restrain myself to move on to another paradox.

Achilles and his tortoise is an even tougher one, but still resolvable. This paradox is about the division of space, where if Achilles shoots arrows at a tortoise as it walks away, halving the distance his shots hit from the tortoise with every shot, will he ever hit the tortoise? Barring the practical considerations that Achilles would have to be a damn good shot, and also that there comes a time when it’s “close enough” and you’ve hit the tortoise anyway, this paradox raises some issues. In the mathematical sense, no, Achilles will never hit the tortoise. This is, as you may have noticed, just like taking any number c and dividing it by 2^n, where n is the number of shots Achilles is firing– this equation is asymptotic to the line x=0, and therefore Achilles should never hit the tortoise. However, the practical and mathematical interpretations both fall prey to the same issue. Where exactly is the tortoise? Are we measuring the position of the tortoise from its center, or from the point on the tortoise closest to Achilles? Clearly, if we’re measuring from the center of the tortoise, he’s going to hit it because presumably we’re not dealing with an infinitely small tortoise. While this seems like a trivial point, I imagine if you’ve heard of this paradox before, you were told that Achilles could never hit the tortoise and you believed it because there was sufficient leeway in your interpretation of the word “position” that you could rearrange the position to the point closest to Achilles. This post-interpretive semantic shift is what I’m aiming at, where dissonance in a conclusion can be resolved by rearranging the premises more easily than questioning the conclusion.

Now we can really get started. Let’s look at Newcomb’s Paradox. I’m now realizing that I may have to do multiple posts on paradoxes– this one grows long already, and there are many paradoxes to get to. Anyway, Newcomb’s Paradox is a game theory problem. You are presented with two boxes, one of which contains $1,000 and the other is a mystery which you are told contains either nothing, or $1 million. The host tells you that you may take either the opaque box, or both boxes. However, if the host predicted you would take both boxes, the mystery box is empty, and if they predicted you’d take just the mystery box then it has $1 million inside. Should you take both, or just the mystery box? Now, simple greed tells us to just take both– either the black box has the money, or it doesn’t. The difficulty is in the fact that because this is the obvious choice, the mystery box probably has nothing in it. Consider that logic loop for a moment– it’s stable. However, if you allow for the fact that the host figured you’d take the mystery box, thinking that because you choose the mystery box it retroactively alters prior conditions, you end up with $1 million. This situation has to do with the application of intent. The host knows you are trying to maximize your winnings, and can figure you either for believing in the retroactive action, or for skepticism. If the host guesses you’ll go for retroactivity, he’ll put the money in the mystery box. However, the fact that he has done so disqualifies his retroactive prediction because choosing both will net the mystery box as well. In short, this puzzle seems more confusing than it is. Actually, the case reduces to a single, stable state where you always choose both boxes. However, if you were to repeat the test, then the ideal strategy is to always select the mystery box only, leading the host to predict you will only take the mystery box. So we arrive at the Prisoner’s Dilemma (slightly modified), easy as pie. Language used purely for the sake of obfuscation.

To wrap up this post, I’ll bring in a case of where fake logic, rigorously applied, can be used to obfuscate. The St. Petersburg Paradox is extremely interesting, especially in that it is actually logically justifiable. Consider the statement “all crows are black”. What evidence would you need to confirm or contradict this statement? Clearly, finding a white crow would nullify that statement, but no matter how many black crows you find you’ll never be able to prove this statement authoritatively– you’re using inductive logic. However, consider the logically equivalent statement “all non-black things are non-crows.” According to this phrasing, finding an object that is not black and that is also not a crow is evidence for the fact that all crows are black. So, in rigorous logical thought, finding a purple cow provides evidence for the fact that all crows are black. Logic is a fantastic tool, but just like any tool it can be easily misused, or broken. In this case, the distinction is similar to saying that “John is not telling the truth” which indicates that John is not saying a statement which is true, but not necessarily that they are saying something that is false. This is a difficult problem of perception to enunciate clearly, but consider the statement “I don’t like Jean”. I have not indicated that I dislike Jean, only that I don’t like her. I may not know Jean, or I may be neutral. The state of not-liking X is not equivalent to the state of disliking X, although dislike satisfies the condition of not-liking. If I disliked Jean, I could truthfully say I didn’t like her, but I would be equally truthful if I hadn’t met her.

Just some interesting paradoxes revealing interesting features of the mind that tend to get entangled. I may do another post like this some other time. This is a pretty good sampling of basic and ubiquitous errors of thinking. This could get you to the point where you can start identifying more on your own. Mind your mind, please.

January 3, 2008 at 11:40 am

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