1
A tolerance premise is plausible for each step
If a number of grains is a heap, one grain fewer is still a heap. Each such conditional looks true taken alone, and a chain of them from ten thousand grains to none is valid by modus ponens.
Lesson · Mind, Language and Meaning · Language
Vagueness and the Sorites
One grain is not a heap, and adding one grain never makes a heap, so ten thousand grains are not a heap. Run the chain of tolerance conditionals under classical logic, a sharp cutoff, a borderline range and degrees of truth, say what each treatment denies, and state higher-order vagueness as the problem the cutoff and range inherit.
each step true, the end false: sorites
classical: all true, the end absurd
cutoff: one false, nobody can say which
range: borderline cases, edges also vague
degrees: each 9999/10000, the end 0
Four treatments, four costs
Core ideas
What this lesson establishes
The chain is one you have run before, as a slope in “Slippery Slopes and Small Differences” and as the ship in “The Ship of Theseus”. There the four treatments were four verdicts on an argument. Here the argument is the original heap, and the question is what each treatment says vagueness is, and what each must still say about its own line. The course on paradoxes takes the heap up again.
2
Four treatments, each at a price
Classical logic keeps every conditional and accepts the absurd end. A sharp cutoff makes one conditional false. A borderline range makes some neither true nor false. Degrees make each nearly true and the end wholly false.
3
Higher-order vagueness
Wherever a treatment draws a line, the line is itself hard to place. The cutoff is sharp and unknowable, the edges of the range are vague, and the degrees are exact numbers that no one could have chosen.
The material
The heap, run four ways
The argument, then each treatment on the same chain, then the problem that remains.
Definition · Vagueness
A predicate is vague when it has clear cases, clear non-cases, and cases in between where there seems to be no fact of the matter, and no sharp line between them. “Heap”, “bald” and “tall” are examples.
The argument: ten thousand grains of sand are a heap. For any number k of grains, if k grains are a heap, so are k − 1. So, by ten thousand steps of modus ponens, no grains are a heap. Write F(k) for “k grains are a heap”. The premises are F(10000) and ten thousand conditionals of the form F(k) → F(k − 1), and the conclusion is F(0).
The conclusion is absurd and the form is valid, so one of the premises has to go, or the logic. The conditionals are not individually doubtful. Taking away one grain cannot make the difference between a heap and a non-heap, or if it could, the word would not be the one we use. The lab runs the chain under four treatments, and the same ten thousand steps give four different pictures.
Classical. Every conditional is True, F(10000) is True, and ten thousand applications of modus ponens deliver F(0): the lab prints True for a heap of no grains. The paradox is left standing. This is not a solution but the statement of the problem, and the other three treatments deny a premise.
Cutoff. There is a number at which a heap stops being one, and the conditional at that number is False. The other conditionals are True and the conclusion is False. The cost is that nobody can say where the number is. A defender of this view holds that the boundary exists and that we are ignorant of it for a good reason: our grasp of the word is not fine enough to discriminate adjacent counts, and so any particular count we named would be a guess. The lab accepts any cutoff you type: the first preset puts it at 5000 and reports one false conditional, at k = 5000; the second moves it to 1000, the false conditional moves with it, and the conclusion is False under both.
Range. There is a borderline region, say between 50 and 200 grains, where it is neither true nor false that a count is a heap. The conditionals inside the region are indeterminate: the lab reports 150 of them, and shows the tolerance claim that all of them hold as super-false. That is the supervaluation idea: on every way of drawing a sharp line inside the region, some conditional is false, so the universal claim is false on all of them, though no single conditional is false on all. The cost is that a disjunction can be true while neither disjunct is, and that bivalence fails for the borderline cases.
Degrees. Truth comes in amounts between 0 and 1. F(k) has the value k / 10000, and each conditional F(k) → F(k − 1) has the value 9999/10000, since the antecedent is one step truer than the consequent, which on the Łukasiewicz rule costs 1/10000. Each premise is almost perfectly true. But each application of modus ponens can lose that 1/10000, and after ten thousand the guaranteed value of F(0) is 0. The lab prints 0. The cost is that validity no longer carries near-truth from premises to conclusion: ten thousand almost-true premises, an argument of an unquestioned form, and a wholly false end.
Higher-order vagueness is what remains. The range treatment puts sharp edges at 50 and 200. But is 50 definitely not borderline, and 49 definitely not a heap? If the edges are themselves vague, the treatment needs a borderline region around each, and then around those, and has traded one arbitrary line for several. The cutoff treatment has a sharp line that it concedes cannot be found. Degrees need an exact number for each count, and the choice of k / 10000 is a sharp choice of the same kind. Each treatment says where vagueness is not and then finds that the place where it stops is vague.
So the lab does not show which treatment is right. It shows the verdict each gives and what that verdict denies: a conditional, bivalence, or the transmission of truth by valid argument. The reader chooses by the one they would least mind paying.
Interactive
The sorites, four ways
A chain of tolerance conditionals under four treatments of vagueness
Method
Running the heap under each treatment
Five steps. The last is the one that is not computed.
1
Write the chain
State the clear case, F(10000), the end, F(0), and the tolerance conditional between neighbours. The lab prints the number of steps.
2
Run it classically
Every conditional is True and the end is True. This is the paradox as it stands.
3
Choose a place to cut the chain
A cutoff falsifies one conditional. A range makes a block of them indeterminate. Degrees make every one slightly less than true.
4
Read the conclusion
False for the cutoff, super-false for the range, 0 for degrees. Each denies something, so write down what.
5
Ask where the line is
Ask of the cutoff, the edges of the range and the degrees how anyone could have placed them. That is higher-order vagueness, and it is not a number the lab can print.
What each treatment gives up
Score 0 / 0Choose the answer you can justify.
Ten thousand almost-true conditionals
The chain from 10000 grains to none, under degrees of truth.
steps = 10000 − 0 = 10000
v(F(k)) = k / 10000
v(F(k) → F(k − 1)) = 1 − 1/10000 = 9999/10000
each modus ponens may lose 1/10000
after 10000 steps the guaranteed value of F(0):
1 − 10000 · (1/10000) = 0
Every premise is almost wholly true and the conclusion is wholly false. That is the Łukasiewicz answer to the heap: the argument is valid and not dangerous at each step, and the steps add up.
Common mistakes
1
Taking vagueness to be ignorance of a sharp boundaryThat is one of the four treatments, the cutoff, and it is defensible: the lab marks a False conditional at the count you type, keeps classical logic, and gives the same verdict whichever count that is. What it cannot do is say which count is right, and a reader who takes the boundary to be the obvious default has not yet paid the cost. The other treatments are answers to the thought that no count is the boundary.
2
Concluding that vagueness is a defect that precision removesReplacing “heap” with “at least 1000 grains” settles the cases, and also changes the word. The sorites is about the word as used, and a stipulated cutoff is a decision about what to call a heap, not a discovery about heaps.
3
Thinking degrees make the paradox disappearUnder degrees each premise is almost true and the conclusion is wholly false, from a valid argument. The paradox moves from a false premise to a loss at each step. And the exact number 9999/10000 is itself a sharp choice.
Check your understanding
✓
Finish when you can run the chain four ways and state each price.
Given the heap argument, you should be able to read the lab's verdict under each treatment, say what each denies, and state why the cutoff and the range both face the question where their lines fall.
Note: Vagueness has more treatments than four, and the supervaluation and epistemic views have large literatures that this lesson does not enter. The aim is to see what each treatment does to the chain, and what it asks of you.