This article first appeared in Issue 13 of our free digital magazine CURIOUS.

If I asked whether the Earth was spherical, most readers would say yes. Only the pedantic would point out it is an oblate spheroid, but technically speaking those who agreed with the proposition would be wrong. Clearly, however, they’re nowhere near as wrong as anyone claiming the Earth is flat. There are degrees of wrongness, which are often very important; arguably this implies there are also degrees of truth. Yet we often rely on classical logic that assumes only two values: true and false, which can lead to paradoxes when either option leads to contradiction.

Logic, by one definition is "A method of reasoning that involves a series of statements, each of which must be true if the statement before it is true.” Put like that, logic seems unimpeachable. Any failures can only be because a preliminary statement is wrong, or one of the subsequent statements doesn’t follow as universally as is claimed.

When we talk about “flawed logic” we usually mean one of these things: someone starts with an incorrect assumption, and builds a pile of conclusions on these shaky foundations. It’s the equivalent of “garbage in, garbage out”, where a computer’s errors are not the fault of the system, but of the data it was fed. Alternatively, the original premise is right, but at some point one of a number of potential implications was treated as the only one possible.

The solution to errors of each is obvious, although not always easy to implement. In the first case, we just need to check our premises better. In the second we need to examine each step to see if it's as watertight as stated.

Some philosophical traditions, particularly those from Asia, are more comfortable with the idea that truth can be a spectrum.

Yet, thousands of years ago, Greek philosophers were posing a more fundamental challenge to logic in the form of paradoxes. Paradoxes are no doubt much older still, but it was thinkers like Zeno of Elea who gave us both the name and entrenched the idea of self-contradictory conclusions from apparently sound premises as a challenge to the nature of reality.

The original paradoxes were (as far as we can tell) created to challenge our sense of reality, not to question logic. Many paradoxes, both original and more recent, have eventually turned out to have quite logical solutions that simply weren’t obvious at the time. Others, however, present a deeper challenge.

According to one definition, a paradox is “An argument that apparently derives self-contradictory conclusions by valid deduction from acceptable premises.”

Breaking the binary

Other paradoxes, however, present a more fundamental challenge, sufficient to raise questions about logic at its core. These, known as antinomies, collide not merely with our observations, but with their own internal logic.

Some, perhaps all antinomies, arise from a binary view of the world. For example, one of the most famous, the Liar’s Paradox, includes examples such as “This statement is false”.       

If we believe that things can only be either true or false, like the ones and zeros on a computer, then the statement generates a sort of recursive loop that has fascinated people for centuries.

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However, some philosophical traditions, particularly those from Asia, are more comfortable with the idea that truth can be a spectrum than Europe’s, which built heavily on classical Greece.

Centuries before Aristotle was formalizing the rules of logic the West has largely relied on ever since, Jains were developing the doctrine of anekāntavāda, in which all statements contain both truth and falsehood.

The famous question of whether a glass is half full or half empty hints at another example. Both of these things are true, but at what point can we drop the “half” and say, “My glass is empty”? 

Arguably, the rise of quantum computers, in which superpositions of both one and zero, in shifting ratios, replace binary bits, has vindicated these alternative views.

Underlying many logical paradoxes is the belief that something is either one thing or another. This breaks down almost entirely at the subatomic level, with photons being both particles and waves, but it doesn’t always stack up that well in more familiar contexts, as the initial example of the shape of the Earth may illustrate. 

The famous question of whether a glass is half full or half empty hints at another example. Both of these things are true, but at what point can we drop the “half” and say, “My glass is empty”? Does it need to be bone dry, or do a few drops still count as “empty”? 

Fuzzy logic was invented to try to capture this complexity, allowing the truth value of a proposition to be anywhere between one and zero. 

Arguably, actual human decision-making has more in common with fuzzy logic than the Boolean logic of ones and zeroes.

However, while fuzzy logic has been used in artificial intelligence and medical image analysis, among other uses, it has not come close to displacing the classical version in most education systems or applications.

Perhaps that’s because ingrained cultural traditions can be very hard to shift. The furious reaction to the idea that gender can be anything other than binary might be seen as an example of this. On the other hand, it might be argued that traditional logic still has plenty going for it.

Solved paradoxes – victories for logic?

Paradoxes come in many forms, and not all of these pose a problem for logic, indeed many of them have vindicated it. It’s more common for paradoxes’ contradictions to be with observed reality than with themselves, and these have often been resolved by combining logic with deeper investigation.

For example, some paradoxes have turned out to be useful ways to expose that reality is more complex than our naïve assumptions. Others involve a sort of sleight of hand, such as dividing by zero to “prove” that 1=2.

One of the original paradoxes presented by Zeno (as paraphrased by Aristotle and others), imagined a deeply unfair race between Achilles and a tortoise. Achilles sportingly gives his reptilian opponent a head start. We know that, provided the race is long enough, this will be insufficient, the faster runner will eventually overtake the slower one. However, Zeno argued, by the time Achilles reaches the point the tortoise started from the tortoise will have gained some ground, and by the time Achilles crosses that secondary distance, the tortoise will have moved a still smaller distance. Achilles constantly gains on the tortoise, but according to this view, can never overtake it.

The paradox of why the night sky is dark was used to prove the universe is not infinite, although we have since learned it is expanding.

Aristotle provided an answer but concerns it was inadequate helped drive the study of infinite series, which has proven exceptionally valuable for science.

Not everyone is satisfied with such mathematical solutions, with some seeing it as a still unsolved metaphysical problem. Nevertheless, classical logic comes out of a challenge like this rather well – far from being discredited, it has led us to something true and useful.

Similarly, the paradox of why the night sky is dark was used to prove the universe is not infinite, although we have since learned it is expanding.      

There are other paradoxes we don’t yet have complete solutions to, but scientists are well on the chase. For example, the Faint Young Sun Paradox notes that stars with masses like the Sun’s emit less heat and light in their first billion years than they do at the Sun’s current age.

Consequently, the initial form of the paradox went that the Earth should have been much colder during the Hadean eon, preventing the presence of liquid water. Yet we know the Earth’s oceans date back 4.4 billion years. More abundant carbon dioxide, and therefore a stronger greenhouse effect, has largely resolved this problem for Earth, but it continues to trouble Mars researchers. Explaining what gasses could have made up the Martian atmosphere during its Noachian era, and where they went is an ongoing problem, but it’s one we can reasonably expect to solve without throwing out the book of logic.

Maybe we need a word (meta-paradox perhaps) for the fact that paradoxes can both expose the flaws in classical logic, and demonstrate its strength.

CURIOUS magazine is a digital magazine from IFLScience featuring interviews, experts, deep dives, fun facts, news, book excerpts, and much more. Issue 16 is out now.