න්‍යාය හා පරස්පර

 න්‍යාය හා පරස්පර

 

2023 ජනවාරි 14 නිව් සයන්ටිස්ට් සඟරාව වෙන් වී ඇත්තේ දැනුමේ සීමා (Limits of Knowledge) සඳහා. ඒ සම්බන්ධ ලිපි පහක් පළ වී තිබෙනවා. අප ඊයේ එක් ලිපියක් පළ කළා. අද න්‍යාය (Logic) ගැන වූ ලිපියක්.

මෙයත් මෙරට දර්ශනය හදාරන්නන්ට වැදගත් වනවාට සැකයක් නැහැ. මා දන්නවා කැලණිය විශ්වවිද්‍යාලයේ දර්ශන අධ්‍යනංශයේ න්‍යාය හා විරුද්ධාභාස (Paradoxes) ගැන උනන්දුවක් දක්වන අය ඉන්න බව. අවශ්‍ය අයට ආචාර්ය ධරණීතගේ ආචාර්ය උපාධි නිබන්ධනය පරිශීලනය කරන්න පුළුවන්. සංජීවනී රුපසිංහ මහත්මිය විරුද්ධාභාස ගැන අධ්‍යයනය කරනවා. 

 

Logic underpins knowledge – but what if logic itself is flawed?

We use logic to build facts into systems of thought, but paradoxes force us to question what we think we know. And it could be worse, because logic may not be sufficient to comprehend reality

 

MIND 10 January 2023

By Abigail Beall

 

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’t 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 “heap”). It is often used as evidence that classical logic might be insufficient to describe the world around us.

 

That is troubling because, though we don’t 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:

How can we understand quantum reality if it is impossible to measure?

How AI is shifting the limits of knowledge imposed by complexity

Why maths, our best tool to describe the universe, may be fallible

Why some aspects of physical reality must be experienced to be known

 

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 “Will we ever solve the liar paradox?”, below).

The back of a bald head, painted with a rainbow symbol

‘Baldness’ turns out to be a slippery concept

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 “not bald” 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’s Watson.

 

Supervaluationism

 

Another approach, called supervaluationism, provides a way to discuss vague concepts by categorising some statements as “true” and others as “supertrue”. Imagine a minor character in a story, for example. If we aren’t told how many siblings they have, we can say that it isn’t supertrue that they have three siblings. It is true, however, as long as there is no information in the story to tell us otherwise. “The bold claim is that truth – ordinary truth – really is supertruth, and falsity is superfalsity,” says philosopher Brian Weatherson 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 David Wolpert 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 some higher mode of logic that could be used to understand the universe, but that human minds wouldn’t be able to grasp.

Read more: A new kind of logic: How to upgrade the way we think

Just think of that humble linguistic device, the question. Wolpert says there are creatures – things like a single-celled paramecium – that couldn’t 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? “We are the paramecia,” says Wolpert. “What is beyond us?”

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? “I can’t conceive of it,” says Wolpert. “But that’s the whole point.”

 

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WILL WE EVER SOLVE THE LIAR PARADOX?

 

What are we to make of the sentence “This sentence is false”? 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 Margaret Cuonzo, a philosopher at Long Island University in New York. “One is to say that our logic needs to be revised to handle more complex phenomena.” 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 “how claiming something can be true and false doesn’t lead to a system in which you can prove anything”, 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.