Be sure to check that very, very fine print.
At first glance, this week's thought experiment might seem silly and easy to dismiss. But seeing as how it undermines the entire field of logic and currently has no accepted resolution, I'd say it's definitely worth spending some time to consider in more detail. For those of us taking an evolutionary perspective on the real world, we see the consequences of this theoretical problem showing up in the species problem, whereby despite all the confident day-to-day talk of various species in the world, it's actually very, very difficult to define them with full accuracy. Charles Darwin noted this in On the Origin of Species when he wrote,
I was much struck how entirely vague and arbitrary is the distinction between species and varieties.
To philosophers in search of perfect rigour, such an ultimate consequence is indeed vexing. But let's start with a simpler introduction and then see where that takes us.
--------------------------------------------------
A Political Broadcast by the Chancellor of the Exchequer, Lord Sorites
These are taxing times for our country. The last government left us with run-down public finances and the need to raise extra revenue. But you, the people, do not want to have to foot the bill. So how can we raise the money we need without making you feel pain?
The answer is simple. Focus groups, opinion polls, and economists have shown that charging an extra 0.01 percent tax has a negligible effect on personal income. No one who is comfortably off is made to struggle, no one rich is made poor, no one already struggling is made to struggle more, by paying 0.01 percent extra on their tax bill.
So today we are raising income tax by 0.01 percent. And logically, since this small amount makes as little difference to the person who earns 0.01 percent less than you as it does to you, we can repeat the step tomorrow, when you are in the position of that insignificantly poorer person. And so the next day, and the next, for the next 300 days.
Each time we raise taxes, we do so in a way that makes no difference to your quality of life. And so your quality of life will not be affected. Yet, miraculously, the net effect of these measures will be a large increase in government revenue, which we intend to use to cut the national debt and still have enough change left to buy everyone in the country a drink. We hope you will use it to toast to our ingenuity.
Source: The ancient Sorites paradox, attributed to Eubulides of Miletus, 4th century BCE.
Baggini, J., The Pig That Wants to Be Eaten, 2005, p. 280.
---------------------------------------------------
This example of the Sorites Paradox relies on playing with various lengths of time, so I think it's a bit easier to deal with. To the citizens of this government, they're simply facing a 3% tax raise during the year, which is an easily measurable unit of time to discuss the implications of the policy. And as new reader Tina said in a comment on my post on Monday, " taxes are due all at once." So even if you were to slice the units of time for the accrual of this tax all the way down to every one of the 25,920,000 seconds that exist in 300 days, you would still arrive at a 3% tax raise eventually. And this reminds me of my Response to Zeno's Paradox where I pointed out how time waits for no man and it will not stand still no matter how much we think about it. There may be uncertainty about when precisely the tax raise becomes noticeable, but in the end it certainly does.
There is a deeper problem here, however, which is that philosophers would very much like to speak about everything very precisely. And in the original Sorites paradox, this included not just precise moments in time, but any physical definitions as well. If, for example, you remove grains of sand from a pile one at a time, when is it no longer "a heap?" Answers to this question are either arbitrary or unimportant, but that leads to a profound problem, as Baggini noted in his comments on this thought experiment:
Many argue that the way out requires us to accept the vagueness of many concepts, such as rich and poor, tall or short, heap or pile. The problem with that solution is that, if we allow too much vagueness into language and logic, reason itself becomes vague. The alternative — that tiny changes really can make the difference between being rich and being poor — preserves the rigour of logic and language, but seemingly at the cost of realism.
So we can either be logical or we can be realistic — that's quite a difficult decision! Is it really necessary to choose? Over at the Stanford Encyclopedia of Philosophy (SEP), there's an extensive entry on the Sorites paradox, which (among other things) outlines the various responses that have been attempted to answer that question. Under section 3, it goes through the following possible solutions in great detail:
3.1) Ideal Language Approaches (i.e. the vagueness of natural language can be eliminated using more precision)
3.2) The Epistemic Response (i.e. sharp semantic boundaries are essentially unknowable)
3.3) Supervaluationism (i.e. a proposition can have a truth value even when its components do not)
3.4) Many-Valued Logic (i.e. some number of non-classical truth-values can be used to model vagueness)
3.5) Contextualism (i.e. context-dependent interpretations just mask the relevant boundaries)
3.6) Embracing the Paradox (i.e. radical incoherence — no amount of grains of sand makes a heap)
All of these, however, are found in turn to be seriously lacking. The only conclusion that the SEP arrives at is:
The sorites paradox presents a serious logical challenge.
Indeed. In a related SEP entry on The Philosophical Challenge Posed By Vagueness, we learn that "most philosophers doubt whether precise analytical tools fit vague arguments. H. G. Wells was amongst the first to suggest that we must moderate the application of logic." In his philosophical text of 1908, First and Last Things, H. G. Wells said this:
Every species is vague, every term goes cloudy at its edges, and so in my way of thinking, relentless logic is only another name for stupidity—for a sort of intellectual pigheadedness. If you push a philosophical or metaphysical enquiry through a series of valid syllogisms—never committing any generally recognized fallacy—you nevertheless leave behind you at each step a certain rubbing and marginal loss of objective truth and you get deflections that are difficult to trace at each phase in the process.
Is that the answer then? That we must only apply logic and rigour at a macro level, never pushing it "too far"? If so, doesn't that answer also fall prey to the Sorites paradox? Won't this ultimately leave us with nothing that we can precisely judge as TRUE or FALSE? In my opinion, yes.
As I discussed in my most important blog post on epistemology, we’ve relied on Plato’s definition of knowledge for millennia as being: 1) justified, 2) true, 3) belief. But that was built on an ancient’s view of the universe as an unchanging and eternal thing. Once we discovered evolution in 1859 and the Big Bang was confirmed by background radiation in the 1960’s, our cosmological revolutions should have led to epistemological revolutions as well, but so far they have not. Gettier challenged Plato’s JTB theory of knowledge, but was unable to replace it. I say that in this changing universe, however, there is no such thing that is eternally TRUE. Therefore, knowledge can only ever be: 1) justified, 2) beliefs, that 3) are surviving.
This is all part of an inherent problem that I now see lies behind many philosophical difficulties. It comes from trying to impose binary TRUE/FALSE logic on an analog world, which is really the result of seeing the universe as something static and unchanging rather than as the dynamic and fluid thing that it is. This may be analogous (though unrelated) to how physics is currently split between quantum mechanics and the theory of relativity. One works at the smallest scale; the other works at the macro level. I currently think the applicability of philosophy may also be split like this between static and dynamic views of the world. Call it the Static-Dynamic Problem. In the choice between being logical and being realistic, I think one can only apply logic to a static picture where TRUE or FALSE definitions can remain valid. Once you move to the dynamic realm, however, it seems that classical logic breaks down when it is pressed to its limits. I'd like to develop this theory in much greater detail someday, perhaps 0.01% at a time, but that's where I currently think I'm headed with a solution to the Sorites paradox.
What do you think? Does that sound like a promising direction? Until then, can you live with the vagueness?
I was much struck how entirely vague and arbitrary is the distinction between species and varieties.
To philosophers in search of perfect rigour, such an ultimate consequence is indeed vexing. But let's start with a simpler introduction and then see where that takes us.
--------------------------------------------------
A Political Broadcast by the Chancellor of the Exchequer, Lord Sorites
These are taxing times for our country. The last government left us with run-down public finances and the need to raise extra revenue. But you, the people, do not want to have to foot the bill. So how can we raise the money we need without making you feel pain?
The answer is simple. Focus groups, opinion polls, and economists have shown that charging an extra 0.01 percent tax has a negligible effect on personal income. No one who is comfortably off is made to struggle, no one rich is made poor, no one already struggling is made to struggle more, by paying 0.01 percent extra on their tax bill.
So today we are raising income tax by 0.01 percent. And logically, since this small amount makes as little difference to the person who earns 0.01 percent less than you as it does to you, we can repeat the step tomorrow, when you are in the position of that insignificantly poorer person. And so the next day, and the next, for the next 300 days.
Each time we raise taxes, we do so in a way that makes no difference to your quality of life. And so your quality of life will not be affected. Yet, miraculously, the net effect of these measures will be a large increase in government revenue, which we intend to use to cut the national debt and still have enough change left to buy everyone in the country a drink. We hope you will use it to toast to our ingenuity.
Source: The ancient Sorites paradox, attributed to Eubulides of Miletus, 4th century BCE.
Baggini, J., The Pig That Wants to Be Eaten, 2005, p. 280.
---------------------------------------------------
This example of the Sorites Paradox relies on playing with various lengths of time, so I think it's a bit easier to deal with. To the citizens of this government, they're simply facing a 3% tax raise during the year, which is an easily measurable unit of time to discuss the implications of the policy. And as new reader Tina said in a comment on my post on Monday, " taxes are due all at once." So even if you were to slice the units of time for the accrual of this tax all the way down to every one of the 25,920,000 seconds that exist in 300 days, you would still arrive at a 3% tax raise eventually. And this reminds me of my Response to Zeno's Paradox where I pointed out how time waits for no man and it will not stand still no matter how much we think about it. There may be uncertainty about when precisely the tax raise becomes noticeable, but in the end it certainly does.
There is a deeper problem here, however, which is that philosophers would very much like to speak about everything very precisely. And in the original Sorites paradox, this included not just precise moments in time, but any physical definitions as well. If, for example, you remove grains of sand from a pile one at a time, when is it no longer "a heap?" Answers to this question are either arbitrary or unimportant, but that leads to a profound problem, as Baggini noted in his comments on this thought experiment:
Many argue that the way out requires us to accept the vagueness of many concepts, such as rich and poor, tall or short, heap or pile. The problem with that solution is that, if we allow too much vagueness into language and logic, reason itself becomes vague. The alternative — that tiny changes really can make the difference between being rich and being poor — preserves the rigour of logic and language, but seemingly at the cost of realism.
So we can either be logical or we can be realistic — that's quite a difficult decision! Is it really necessary to choose? Over at the Stanford Encyclopedia of Philosophy (SEP), there's an extensive entry on the Sorites paradox, which (among other things) outlines the various responses that have been attempted to answer that question. Under section 3, it goes through the following possible solutions in great detail:
3.1) Ideal Language Approaches (i.e. the vagueness of natural language can be eliminated using more precision)
3.2) The Epistemic Response (i.e. sharp semantic boundaries are essentially unknowable)
3.3) Supervaluationism (i.e. a proposition can have a truth value even when its components do not)
3.4) Many-Valued Logic (i.e. some number of non-classical truth-values can be used to model vagueness)
3.5) Contextualism (i.e. context-dependent interpretations just mask the relevant boundaries)
3.6) Embracing the Paradox (i.e. radical incoherence — no amount of grains of sand makes a heap)
All of these, however, are found in turn to be seriously lacking. The only conclusion that the SEP arrives at is:
The sorites paradox presents a serious logical challenge.
Indeed. In a related SEP entry on The Philosophical Challenge Posed By Vagueness, we learn that "most philosophers doubt whether precise analytical tools fit vague arguments. H. G. Wells was amongst the first to suggest that we must moderate the application of logic." In his philosophical text of 1908, First and Last Things, H. G. Wells said this:
Every species is vague, every term goes cloudy at its edges, and so in my way of thinking, relentless logic is only another name for stupidity—for a sort of intellectual pigheadedness. If you push a philosophical or metaphysical enquiry through a series of valid syllogisms—never committing any generally recognized fallacy—you nevertheless leave behind you at each step a certain rubbing and marginal loss of objective truth and you get deflections that are difficult to trace at each phase in the process.
Is that the answer then? That we must only apply logic and rigour at a macro level, never pushing it "too far"? If so, doesn't that answer also fall prey to the Sorites paradox? Won't this ultimately leave us with nothing that we can precisely judge as TRUE or FALSE? In my opinion, yes.
As I discussed in my most important blog post on epistemology, we’ve relied on Plato’s definition of knowledge for millennia as being: 1) justified, 2) true, 3) belief. But that was built on an ancient’s view of the universe as an unchanging and eternal thing. Once we discovered evolution in 1859 and the Big Bang was confirmed by background radiation in the 1960’s, our cosmological revolutions should have led to epistemological revolutions as well, but so far they have not. Gettier challenged Plato’s JTB theory of knowledge, but was unable to replace it. I say that in this changing universe, however, there is no such thing that is eternally TRUE. Therefore, knowledge can only ever be: 1) justified, 2) beliefs, that 3) are surviving.
This is all part of an inherent problem that I now see lies behind many philosophical difficulties. It comes from trying to impose binary TRUE/FALSE logic on an analog world, which is really the result of seeing the universe as something static and unchanging rather than as the dynamic and fluid thing that it is. This may be analogous (though unrelated) to how physics is currently split between quantum mechanics and the theory of relativity. One works at the smallest scale; the other works at the macro level. I currently think the applicability of philosophy may also be split like this between static and dynamic views of the world. Call it the Static-Dynamic Problem. In the choice between being logical and being realistic, I think one can only apply logic to a static picture where TRUE or FALSE definitions can remain valid. Once you move to the dynamic realm, however, it seems that classical logic breaks down when it is pressed to its limits. I'd like to develop this theory in much greater detail someday, perhaps 0.01% at a time, but that's where I currently think I'm headed with a solution to the Sorites paradox.
What do you think? Does that sound like a promising direction? Until then, can you live with the vagueness?
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