A WOMAN once approached me with a curious problem concerning her husband. Like most people who choose to get married, she had promised to love her spouse to the exclusion of all others. But there was a problem: according to her, the man she married simply wasn’t the same person any more. He had the same name and career, the same memories and skills. But over many years, an accumulation of small changes had, she felt, made her husband a completely different person.
This woman had approached me not because I’m an expert in matters of the heart, but because I had just given a talk about paradoxes. These puzzles have entertained and perplexed us for millennia. They force us to grapple with some of the deepest matters of logic and meaning. What does it mean for something to be “the same”, for instance?
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I couldn’t offer the woman any simple answers. I reminded her that she had probably changed quite a bit since her youth too. And I pointed out that sometimes our intuitions about concepts like identity can be unhelpful.
In fact, the point goes well beyond relationships. Chewing over paradoxes can show us places where our intuitions need tweaking, and this applies everywhere from the foundations of mathematics to social media and our efforts to live more sustainable lives. Paradoxes have helped thinkers resculpt our understanding of key concepts and attain fresh scientific insights time and again. Now, a new way of thinking through paradoxes is emerging, one that holds promise because it puts our mushy human intuition front and centre.
One reasonable way to define a paradox is as “a set of mutually inconsistent claims, each of which appears to be true”. One of the oldest and most famous of these puzzles is Zeno’s dichotomy paradox, developed by Zeno of Elea, a thinker who lived in Greece in the 5th century BC. Imagine a person walking from point A to point B. To reach point B, they first have to walk half the distance, and this takes some finite amount of time. When they get halfway, they still have to walk halfway between where they are now and point B, and this also takes a finite amount of time, albeit a little less. Zeno carried on this argument to apparently show that, no matter how far you have travelled towards point B, you will always still have to walk halfway from your current position to point B and this takes at least some amount of time. The conclusion is that all journeys should take an infinite amount of time – yet clearly that isn’t true.
When we encounter a paradox, our instinct is that something has gone wrong. Contradictions like this shouldn’t exist and we want to solve or explain away the inconsistencies. In the case of Zeno’s paradox, mathematicians have since discovered that it is possible to divide up a distance into an infinite number of portions – but also to add up that infinite series and get a result that is finite. For most of us, that seems counter-intuitive, but it is a truth showing that reality doesn’t always conform to our preconceptions.
In my work, I have identified a number of general strategies that logicians use to tackle paradoxes. For instance, the aforementioned solution to Zeno’s paradox is an example of what I call the “odd guy out” strategy, where we spot that one of the premises of the paradox is dodgy. Sometimes, paradoxes are extremely difficult to explain away, and in these cases logicians can employ what I call the “detour”, where you admit that the paradox can’t be solved on its own terms, but propose that some deep assumption about reality needs revising.
This kind of thinking has been helpful on many occasions, not least through the famous Schrödinger’s cat paradox, which has helped us interrogate the true meaning of quantum theory for decades. But using logic alone to analyse paradoxes ignores an important element of what makes them so engrossing in the first place. A lot of the language used to discuss paradoxes uses terms like “seem”, “apparently” or “appears to show”, which subtly demonstrates that human understanding is at the core of how paradoxes work. Paradoxes don’t exist in a vacuum, they are puzzles that take shape in our minds.
This is what prompted me to begin developing a new way of analysing paradoxes a few years ago. My method puts us, the readers of the paradox, at the heart of things. It hinges on an old idea called subjective probability, often used in maths but not applied to paradoxes before. Subjective probability is the degree to which a rational person will believe something. Take a statement like “two plus two equals four”. This would have a subjective probability of 1, in other words it is certain. I would put “The bus will be late” at be 0.75; more likely than not. “The next coin toss will be a heads” would get a value of 0.5. And “two plus two equals seven” would be 0. Subjective probabilities can vary from person to person, but not usually by much. In this story, the values I give are my choices, but they are likely to be similar to what you would assign.
“Paradoxes don’t exist in a vacuum, they are puzzles that take shape in our minds”
When we encounter a paradox, we can break it down into a set of claims and a conclusion and then evaluate the subjective probability of each. To see how this works, let’s try it with a deceptively simple puzzle called the liar paradox, which comes in the form of a single sentence: “This sentence is false.” We will call this sentence “L”. If L is true, then L, which claims that L is false, must be false. And if L is false, then it is false that what L says is false, so it would then be true. And since L is a declarative sentence, and should therefore be either true or false, it seems to follow that L is both true and false. When we look at the paradox as a set of claims and give their subjective probabilities in parentheses we get the following:
1. If L is true, then it is false (1.0)
2. If L is false, then it is true (1.0)
3. L is either true or false (0.9)
Conclusion: Therefore, L is both true and false (0)
The subjective probability of the parts of this paradox are high, yet the conclusion seems like nonsense. The first claim, that if the sentence is true, then it is false, follows from the meaning of the sentence. So does the second claim. As for the third, it is an oft-cited feature of declarative sentences that they are true or false, and L takes the form of a declarative sentence. Yet the conclusion is an outright contradiction.
One advantage of my method is that it helps us see how strong a paradox is. Assuming we are using valid logical reasoning, then the greater the disparity between the subjective probabilities of the claims and the final conclusion, the stronger the paradox. The liar paradox is a strong paradox.
Another benefit of this approach is that it suggests a way to tackle paradoxes – by finding a way to lower the subjective probability of some of the claims. In the liar paradox, we can see that premise 3 has the lowest subjective probability. Premises 1 and 2 are true as a direct result from the meaning of L. Premise 3, though, assumes that every statement is either true or false. This seems intuitively right, but there is that word – intuition. Perhaps we are on to something here.
One legitimate response to this paradox is to say it highlights a problem with our intuition about truth and falsehood. The philosopher Alfred Tarski pointed out that a peculiar feature of languages like English is that they don’t separate out the expressions used to speak about everyday facts, such as “the book is on the table”, from the expressions that refer to the language itself, like “the sentence about the book is true”. Tarski didn’t think this would be a practical problem for most of us. But he did realise that it would cause trouble in certain circumstances, such as when a linguist is analysing the structure of a language. He proposed that in those situations we should employ a separate artificial meta-language, typically based in mathematics, to help us talk about the language being analysed.
This might seem abstract, but it tells us two interesting things. First, sometimes it is helpful to separate out logical arguments to avoid self-reference problems. Second, it hints that we may need to question our intuition about the fundamental nature of truth (see “Both true and false”,).
Paradoxes aren’t solely relevant to semantics and logic, nor are they always ancient puzzles. As I have explored paradoxes, I have seen how they occur in everyday life and draw attention to recurring issues with how we think about the world.
One puzzle that keeps rearing its head is known as the Jevons paradox. It was discovered in the 1850s by the economist and logician William Stanley Jevons in the context of coal. It seems to show that by increasing the fuel efficiency of machines, you end up using more fuel overall. Here is how it looks, point by point, together with subjective probabilities:
1. Increasing the fuel efficiency of a piece of technology will allow for less consumption of fuel for the same amount of work (0.9)
2. A piece of technology that will allow for less consumption of fuel for the same amount of work will be used more often (0.8)
3. Increasing the use of a piece of technology will increase the amount of overall fuel consumption (0.9)
Conclusion: Therefore the increasing fuel efficiency of a piece of technology will increase the amount of overall fuel consumption (0.3)
Even on a first read, this paradox feels surprising, but less deep than the liar paradox. It turns out there is an way to increase the subjective probability of the conclusion. Think of what has happened as internet speeds have gone up over the past two decades. We might have assumed that faster speeds would mean that people take less time to complete tasks and so overall time spent online would decrease. However, the increased speed actually opened up new opportunities and people ended up spending more time online. This paradox teaches us that changes to part of a system don’t necessarily get multiplied at a constant rate as that improvement rolls out across the whole system.
The Jevons paradox is crucially important today as we drive to reduce our carbon emissions. As a technology becomes more fuel efficient, it is tempting to use it more often. But this can result in what is called a rebound effect. All of us – especially those in power – must understand this as we pursue net-zero emissions targets. Simply increasing fuel efficiency is at best an insufficient measure. We also need zero-carbon technologies and interventions such as taxes that prod human behaviour in the right direction.
An even more prescient paradox today is the paradox of tolerance, expressed most famously by the philosopher Karl Popper in 1945. It begins with the seemingly unproblematic idea that being tolerant means tolerating all views, but trouble quickly follows:
1. To be completely tolerant, a society must allow the expression of all views (0.9)
2. Intolerant views are views (1.0)
3. To be completely tolerant, a society must allow the expression of intolerant views (0.9)
4. The expression of intolerant views creates intolerance, either in thought or action (0.9)
Conclusions: Therefore, to be completely tolerant, a society must allow the creation of intolerance (0)
If we apply subjective probability, we see this is a deep paradox. There is no straightforward way of lowering the probability of any of the claims, which suggests we might need to modify our intuitions about tolerance at a higher level. One way we can do this draws from Tarski’s analysis of the liar paradox. We might say that tolerance operates at two levels: individual views and a meta-level notion of tolerance that floats above those views and says that both tolerant and intolerant views are tolerated. However, what wouldn’t be tolerated under the meta-notion is any intolerant view that seeks to restrict the ground rule of tolerance itself. A claim like “The xs are bad” may be an intolerant view of any x, but such a view can exist in this tolerant system without paradox. However, “The xs should not have the right to express their views” cannot, because it violates the meta-level rules of tolerant discourse.
“The Jevons paradox is important as we drive to reduce carbon emissions”
We see the fallout of this paradox all around us in the way that social media companies grapple with intolerant views – racism, sexism, xenophobia and more – on their platforms. Many social media sites, citing the need for the free expression of ideas, have been hesitant to ban those who espouse conspiracy theories, misinformation and intolerance. A fuller understanding of the paradox of tolerance helps us see why this is reasonable but also where the boundaries ought to lie. When the views promoted restrict the abilities of others to express their views, then this is a violation of the ground rule of tolerance.
Identifying and rethinking paradoxes can help us all understand the world around us a little better. Engaging with them may cause us to question our intuitions and sometimes that can feel frightening. But next time you encounter one it is worth stopping to ponder for a while. When you come away you might not be quite the same person – whatever that means.
Both true and false
It feels like common sense to say that all statements must be true or false. Aristotle called it “the most certain of principles”. But is it? Some philosophers hold the radical view that statements can be both true and false, which is called dialetheism. This view is gaining traction among logicians because of the way it can help with paradoxes.
What are we to make of this strange idea? One big problem for dialetheism is that it allows direct contradictions to exist and this leads to a well-known difficulty in logic called the problem of explosion. If it is fine to say it is raining and not raining, then our entire basis for belief and action blows up.
One reason this is so tricky is that you can use a contradiction to prove anything you like. To see how this works, let’s take the sentence “Alice is pregnant” and call it A. According to dialetheism, we can say that both A and Not A are true. Now we can construct a statement of logic in which two options are present, A and B, where A is the sentence about Alice and B can be absolutely anything, such as “bread is expensive”. We can say that either A or B must be true, because we have already assumed that A is true as part of our starting assumptions. If A or B is true and so is Not A (which, again, was part of our starting point) then, according to a rule of inference called disjunctive syllogism, we can conclude that B is true.
All this seems like an indictment of dialetheism. Surely, we shouldn’t be able to use a contradiction to prove some unrelated fact chosen more or less at random? But on the other hand, strong mind-benders like the liar paradox (see main story) force us to give ideas like this a chance. Plus, dialetheists have developed clever ways of denying that disjunctive syllogism is always valid.
Toying with the foundations of logic in this way is a worthwhile pursuit. Much of science is based on logical reasoning and there is no guarantee that the classical rules are the perfect or only tools we need. The human brain works in a more malleable fashion most of the time, with grey areas and contradictions. Perhaps our logic would be better if it did likewise.
