I AM not bald. At least, not as I write this. Yet if a malevolent philosopher were to pluck the hairs out of my head, one by one, I would end up bald. But how many would have to be removed before I went from having a lustrous head of hair to being bald? It is tricky, if not impossible, to say. And if we can鈥檛 identify the transition to baldness, am I actually bald at all?
This is a version of a thought experiment favoured by philosophers, first described with reference to grains of sand in a heap, called the sorites paradox (from the Greek word for 鈥渉eap鈥�). It is often used as evidence that classical logic might be insufficient to describe the world around us.
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That is troubling because, though we don鈥檛 pay it much attention, logic runs through human knowledge as if it were a stick of rock. We assume that we can build up a sequence of facts into systems of thought. But if logic itself is lacking, where does that leave us?
This article is part of a special series on the limits of knowledge, in which we explore:
Paradoxes start with a premise that seems true, apply reasoning that also seems valid, but end up at a false or contradictory conclusion. As a result, many paradoxes force us to question what we think we know. They come in different varieties, some more difficult to explain away than others. One of the most confounding takes the form of a simple sentence (see 鈥淲ill we ever solve the liar paradox?鈥�, below).
One solution to the sorites paradox is to admit that terms are sometimes too vague to be useful outside of everyday conversation. But some philosophers argue that logic itself needs a refresh. One approach is to say that there are different degrees of truth. Take the case of my hair removal. Halfway through the process of plucking, I am still not bald, but I am less 鈥渘ot bald鈥� than I was at the start. Fuzzy logic, a kind of computing using degrees of truth rather than 1s and 0s, was introduced by computer scientist Lotfi Zadeh in 1965. It is still used in some artificial intelligence systems today, like IBM鈥檚 Watson.
Supervaluationism
Another approach, called supervaluationism, provides a way to discuss vague concepts by categorising some statements as 鈥渢rue鈥� and others as 鈥渟upertrue鈥�. Imagine a minor character in a story, for example. If we aren鈥檛 told how many siblings they have, we can say that it isn鈥檛 supertrue that they have three siblings. It is true, however, as long as there is no information in the story to tell us otherwise. 鈥淭he bold claim is that truth 鈥� ordinary truth 鈥� really is supertruth, and falsity is superfalsity,鈥� says philosopher at the University of Michigan.
But there is a deeper question here: can we be sure that logic, even a reformed kind, is enough to understand the universe in all its fullness?
It is a question that at the Santa Fe Institute in New Mexico has been thinking about for decades. In a recent monograph, he spelled out his argument that it is more likely than not that there is that could be used to understand the universe, but that human minds wouldn鈥檛 be able to grasp.
Just think of that humble linguistic device, the question. Wolpert says there are creatures 鈥� things like a single-celled paramecium 鈥� that couldn鈥檛 conceive of the idea of a question. In fact, according to our standards of intelligence, every other species on Earth is limited in some regard in the way it understands the world around it. Why should we be any different? 鈥淲e are the paramecia,鈥� says Wolpert. 鈥淲hat is beyond us?鈥�
Wolpert thinks there are ways we could potentially get at higher systems of thought that go beyond logic as we know it. Perhaps it will be a super Turing machine that can transcend the normal rules of computing or an intelligent form of extraterrestrial life that shares its wisdom with us. Perhaps it will be something different altogether. And what will this new plane of understanding be like? 鈥淚 can鈥檛 conceive of it,鈥� says Wolpert. 鈥淏ut that鈥檚 the whole point.鈥�
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WILL WE EVER SOLVE THE LIAR PARADOX?
What are we to make of the sentence 鈥淭his sentence is false鈥�? If we say the sentence is true, then it is false. If we say it is false, then it is true. Whichever we choose, there is a violation of a common-sense rule: declarative sentences ought to be either true or false, not both. This kind of paradox rarely crops up in everyday life, of course, and some have suggested that we can avoid problems by having better linguistic rules that exclude paradoxes.
But there are other ways to respond to this puzzle, says , a philosopher at Long Island University in New York. 鈥淥ne is to say that our logic needs to be revised to handle more complex phenomena.鈥� This is the route a few philosophers have taken. For instance, an approach called dialetheism says that some things can be true and false at the same time. But this gets complicated. Proponents of dialetheism have to figure out 鈥渉ow claiming something can be true and false doesn鈥檛 lead to a system in which you can prove anything鈥�, says Cuonzo. If things can be true and false, facts start to lose meaning. This is one paradox that remains deeply challenging to explain away.
