It’s February 2015 and Kanye’s back on the radar after a long hiatus. Gone is the abrasive, industrial sound of Yeezus. Instead he’s making pretty little ditties with the help of Paul McCartney: first Only One and then FourFive Seconds with Rihanna.

Album-hype is building. No hints on the name yet but we’ve been told it’s “80% done” and that it’s gonna be “cookout music.” Naturally, we’re expecting more of the same from his BRITs performance. It’ll push the boundaries – sure – but it’ll be gentle, wholesome even.

These people were listening to a weepy Sam Smith ballad before this! They were watching Ed Sheeran hum and slap his guitar! These poor people are sitting cabaret-style! And in comes Kanye with that distorted bass and those flamethrowers. Even Taylor Swift is left wide-eyed and slack-jawed.


Far from being a modern aberration, getting money has been a concern since time immemorial. Even philosophers are not immune. Here’s David Hume in An Enquiry Concerning the Principles of Morals in 1751:

“Perhaps to your second question, why he desires health, he may also reply, that it is necessary for the exercise of his calling. If you ask, why he is anxious on that head, he will answer, because he desires to get money. If you demand why? It is the instrument of pleasure, says he. And beyond this it is an absurdity to ask for a reason.”
– David Hume, An Enquiry Concerning the Principles of Morals

He desires to get money.
The question is, at what point can you truly say that you’re getting money? It seems pretty clear that Adrian – making $7.75-an-hour at Jimmy Johns – is not getting money. And it seems undeniable that Carlos Tevez – making £615,000-a-week at Shanghai Shenua – is getting money. But at what point does one make the switch from not-getting-money to getting-money?

This is one instance of a problem that affects almost every word in almost every language: the problem of vagueness. So many of the words we use – rich, tall, great, bald, old, green, small – are vague. Their meaning is not precise. There’s no clear boundary between cases where the word applies and cases where it doesn’t.

You’ve probably recognised this phenomenon before. It's not just a problem for philosophers but a difficulty that arises in day-to-day life. Say, for instance, there’s a lull in conversation when you’re with your friends at the pub and someone says, ‘Hey, how much would you have to be paid to drink a pint of ketchup?’ You’ve got a weak stomach and a vivid imagination so you say, ‘£1000.’ Then, like clockwork, your friend asks, ‘Oh, so you wouldn’t do it for £999?’

What do you say? If you’re being honest, you say, ‘Yes, I would,’ but that only pushes the problem a step further. This is already boring but your friend is in it for the long haul. ‘Would you do it for £998?’

The thing is, there seems to be no good place to draw the line. After all, £998 is a decent amount of money, and taking away £1 doesn’t suddenly make it non-decent. It’s just £1! So before you know it you’re answering 'Yeah I guess' and 'Sure, why not?' to question after question. Then all of a sudden it’s closing time and you’ve got ketchup dripping from your chin. Your friends stopped laughing 200 millilitres ago. You go home with a pat on the back and £0.

You’ve just fallen victim to a version of the Sorites Paradox, otherwise known as the Paradox of the Heap. The Greeks who came up with it were super concerned with how many grains of sand you needed to make a heap. It seems like 1000 definitely makes a heap, and it seems like taking one grain away doesn’t stop it being a heap, but if you agree with both of those statements then at some point you have to say that one grain of sand is a heap and even (whisper it) zero grains of sand is a heap.

When you put it in these terms the problem doesn’t seem very important, but the problem of vagueness also crops up in plenty of other places. For instance, when does a foetus become a person? How visually-impaired do you have to be to be declared legally blind? For how long can you leave someone alone in a room before it becomes torture? How old should you have to be before you’re allowed to drink? In all of these instances, a line needs to be drawn. The problem is, where do we draw it?

Kanye’s All Day line is one way of responding to the problem of vagueness. ‘You ain’t getting money unless you got eight figures’ is a definite boundary. Ask Kanye if your $9,999,999-salary counts as getting money and he can quite happily tell you no.

The problem with this response is that it departs from the way we ordinarily use words. Sure Kanye can use the eight-figure-definition if he wants to, but most people are happy to describe a seven-figure-salary as getting money. The problem of vagueness applies to words as we use them so our solution has to apply to words as we use them.

With this condition in place, there are a few possible responses to the problem. The first is to say that there is a nice clean boundary between getting money and not getting money, but we can never know exactly where it is. Call this the ‘unknowable boundary’ (UB) response. It’s got a few advantages this one. The first is that it respects the law of the excluded middle: the law that says every proposition must be either true or false. For any dollar-value, it’s either true or false that the recipient is getting money. The second advantage is that it corresponds with our intuitive feelings. Kanye aside, we don’t feel like we know exactly where the boundary is. The UB response says this is no surprise: we just can’t know exactly where it is.

But the UB response must answer a big question: what determines exactly where the boundary lies? Getting money may well have been a concern since time immemorial, but it’s nevertheless a human concept. Someone made it up and began using it in certain cases. It has no life of its own – no intrinsic nature – outside of the cases in which it is used. So how can this concept have a boundary that no one can know?

This objection draws on a principle in the philosophy of language: that meaning is determined by use. That is, you can determine the meaning of a word or phrase by observing the contexts in which it is used. The UB response violates this principle because, according to them, you could hear all kinds of people use the phrase ‘getting money’ and still not know where exactly the boundary of getting money lies. This seems crazy to me so I’m not a fan of UB.

But it only gets wilder from here on out. Next up is Peter Unger’s theory.

I like to imagine Unger stumbling across the Sorites paradox in his reading and passing over it without much thought. ‘This is a trifling problem,’ he tells himself, ‘It shan’t concern me.’ Later that night he’s tossing and turning in bed, tormented by visions of sand piling up into heaps and hairs plucked one by one from men’s heads. ‘Heap?’, a voice from the dream asks, ‘Heap?’. Then a second, deeper voice, ‘Bald? Bald?’.

Three nights go by before Unger decides something has to be done. He sits down at his desk. ‘C’mon Peter,’ he tells himself, ‘let’s put this thing to bed. A nice, measured response from a nice, measured man.’ He puts pen to paper and immediately it’s as if some unknown force has taken control of his hand. Slowly but surely, the title etches itself out. Unger looks down at the page in horror.

I want to take a moment to consider just how nutty this article is. Peter Unger, an adult human in possession of all his critical faculties, has put pen to paper and written 27 pages arguing that THERE ARE NO ORDINARY THINGS: no tables, no chairs, no footballs, no houses, no hotdogs, no stones, no rocks, no twigs, no tumbleweeds, no fingernails. He says it right there on the first page! ‘I believe that none of these things exist.’ He believes that! The Sorites paradox – the silly heap puzzle – made him believe that.

Okay, now that’s out of the way: the argument. Unger asks us to consider four sentences:

(1)   There exists at least one table.
(2)   If something is a table, then it consists of a finite number of atoms.
(3)   If something is a table, then removing one atom will not prevent it from being a table.
(4)   No table can be made up of 0 atoms.

Each of the four sentences, at first look, seems true. None of them can easily be denied. But the four sentences together are inconsistent. They can’t all be true at the same time. So we have to deny one of them, and Unger says we should deny (1): There exists at least one table.

Why (1)? Because denying (2) would mean claiming there are infinite tables, and ‘THERE ARE INFINITE TABLES’ is even more nutty as an article title than ‘THERE ARE NO ORDINARY THINGS’. You could try deny (3), but then you’d have to specify the precise number of atoms that marks the difference between a table and a not-table. This is where you might try the Kanye-manoeuvre – say a number loudly and confidently and hope nobody asks too many questions – but, as we’ve already seen, that doesn’t work.  And denying (4) would mean claiming you can have non-material tables. Obvious no-go.

Looking at it this way, Unger’s conclusion doesn’t seem that unreasonable after all. Imagine it as a choice between four paper-titles:


Which one would you be least ashamed to hand in to your professor?

Nevertheless, Unger’s theory has a big weakness. Like the Kanye-manoeuvre, it fails the ordinary language test. Sure, you can say that a collection of atoms in the shape of a table is not a table but everyone else is going to look at you funny. That’s not how people speak! People talk about tables all the time! The Sorites paradox is a puzzle about the way we use words, so solving the paradox by using words in a completely different way is always going to feel like cheating. Unger ‘solved’ the paradox in the same way that my little brother ‘solved’ his Rubix cube when he took all the coloured stickers off and rearranged them.

One final response to the Sorites paradox is to get rid of the principle of bivalence. This is the logical principle which says that every sentence has to be either true or false. If we deny it, we can say things like ‘It’s mostly true that that’s a heap’ and ‘It’s kinda true that Kanye is getting money.’ When we say these things, we are using a multi-valued logic. Multi-valued logics come in all kinds of forms. For instance, one kind could have these as options: definitely false, mostly false, slightly false, neither true nor false, slightly true, mostly true, definitely true. So instead of umming and ahhing about whether 5’9’’ is tall, we can say it’s slightly tall and move on with our lives.

But hold up! Multi-valued logics do not so much solve the problem as multiply it. Before we only had one worry – when does short become tall? Now we have six! – (1) when does definitely short become mostly short? (2) when does mostly short become slightly short? (3) when does slightly short become neither short nor tall? (4) when does neither short nor tall become slightly tall? (5) when does slightly tall become mostly tall? (6) and when does mostly tall become definitely tall? Vagueness is back, baby! And it’s vaguer than ever!

Now it would be easy to back down at this point. We could slink back, tail between our legs, and beg the principle of bivalence to take us back. The world is crueller and lonelier than we thought, and is one point of vagueness really so bad? We could make it work!

But if we’re too proud to go back, we could always double down. Fuzzy logic lets us do just that. It’s not just a multi-valued logic, it’s a no-holds-barred, hell-for-leather kind of multi-valued logic. It says that there are not just two truth-values – true and false – but an infinite number. These are represented by all the numbers between 0 and 1, where 0 is totally false and 1 is totally true.

This idea finally rids us of vagueness. There are no longer any lines to be discovered, just one continuous spectrum from true to false, from heap to not-heap, from getting money to not-getting-money.

Unfortunately, though, fuzzy logic suffers from other problems. Chief among them is this one.

Imagine a pair of twins. Call them Fred and Ted. Imagine that Fred and Ted are both going bald in exactly the same way. Every time a hair vacates Fred’s head, a corresponding hair vacates Ted’s head too. Eventually there will come a point, according to fuzzy logic, where both Fred and Ted are 0.5 bald. That is, there will come a point where it is half-true and half-false that Fred is bald and half-true and half-false that Ted is bald. But that would mean – bear with me here – that it is half-true that Fred is bald and Ted isn’t. But this seems crazy! Remember that we specified that Fred and Ted always have exactly the same number of hairs. So how can it be even slightly true that one of them is bald and one of them isn’t? They’re exactly the same!

I don’t think this objection is as damaging as it’s often made out to be. Although it sounds kinda weird to say it’s half-true that Fred is bald and Ted isn’t, I think it makes sense! After all, it sounds weird because the two halves of the conjunction don’t correspond. Hence, half-true. But, of course, many people will disagree.

It’s hard to say whether the problem of vagueness is a big one in philosophy – it might just be slightly big or mostly big – but it’s probably not going away any time soon.

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