Sorites paradox
Encyclopedia
The sorites paradox is a paradox
that arises from vague
predicate
s. The paradox of the heap is an example of this paradox which arises when one considers a heap of sand
, from which grains are individually removed. Is it still a heap when only one grain remains? If not, when did it change from a heap to a non-heap?
. The paradox goes as follows: consider a heap
of sand from which grain
s are individually removed. One might construct the argument, using premise
s, as follows:
Repeated applications
of Premise 2 (each time starting with one fewer grains), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand (and consequently, if one grain of sand is still a heap, then removing that one grain of sand to leave no grains at all still leaves a heap of sand; indeed a negative number of grains must also form a heap).
On the face of it, there are some ways to avoid this conclusion. One may object to the first premise by denying 1,000,000 grains of sand makes a heap. But 1,000,000 is just an arbitrarily large number, and the argument will go through with any such number. So the response must deny outright that there are such things as heaps. Peter Unger
defends this solution. Alternatively, one may object to the second premise by stating that it is not true for all heaps of sand that removing one grain from it still makes a heap. Or one may accept the conclusion by insisting that a heap of sand can be composed of just one grain, and solely deny the further conclusions regarding zero-grain or negative-grain-number heaps.
argued that all of natural language, even logical connectives, is vague; moreover, representations of propositions are vague. However, most views do not go that far, but it is an open question.
However, such solutions are unsatisfactory as there seems little significance to the difference between 9,999 grains and 10,000 grains. The boundary, wherever it may be set, remains as arbitrary and so its precision is misleading. It is objectionable on both philosophical and linguistic grounds: the former on account of its arbitrariness, and the latter on the ground that it is simply not how we use natural language. A more acceptable solution is to call any collection of multiple grains (two or more) a heap.
and Roy Sorensen hold an approach that there are fixed boundaries but that they are necessarily unknowable.
s and vagueness
. Consider the sentence 'Pegasus
likes licorice' in which the name 'Pegasus' fails to refer. What should its truth value be? There is nothing in the myth that would justify any assignment of values to it. On the other hand, consider 'Pegasus likes licorice or Pegasus doesn't like licorice' which is an instance of the valid schema
(i.e. '
or not-
'). Shouldn't it be true regardless of whether or not its disjuncts have a truth value? According to supervaluationism, it should be.
Precisely, let
be a classical valuation
defined on every atomic sentence
of the language
, and let 
be the number of distinct atomic sentences in 
. Then there are at most 
classical valuations defined on every sentence 
. A supervaluation 
is a function from sentences to truth values such that, 
is super-true (i.e. 
) if and only if 
for every 
; likewise for super-false. Otherwise, 
is undefined—i.e. exactly when there are two valuations 
and 
such that 
and 
.
For example, let
be the formal translation of 'Pegasus likes licorice'. Then there are exactly two classical valuations 
and 
on 
, viz. 
and 
. So 
is neither super-true nor super-false.
. From this point of view, the problem is with the principle of bivalence
: the sand is either a heap or is not a heap, without any shades of gray. Instead of two logical states, heap and not-heap, a three value system can be used, for example heap, unsure and not-heap. However, three valued systems do not truly resolve the paradox as there is still a dividing line between heap and unsure and also between unsure and not-heap. The third truth-value can be understood either as a truth gap or as a truth glut.
Alternatively, fuzzy logic
offers a continuous spectrum of logical states represented in the unit interval
of real numbers [0,1]—it is a many-valued logic with infinitely-many truth-values, and thus the sand moves smoothly from "definitely heap" to "definitely not heap", with shades in the intermediate region. Fuzzy hedges are used to divide the continuum into regions corresponding to classes like definitely heap, mostly heap, partly heap, slightly heap, and not heap.
—that is, knowledge of what the collection of sand started as. Equivalent amounts of sand may be called heaps or not based on how they got there. If a large heap (indisputably described as a heap) is slowly diminished, it preserves its "heap status" to a point, even as the actual amount of sand is reduced to a smaller number of grains. For example, let's say 500 grains is a pile and 1,000 grains is a heap. There will be an overlap for these states. So if you are reducing it from a heap to a pile, it is a heap going down until, say, 750. At that point you would stop calling it a heap and start calling it a pile. But if you replace one grain, it would not instantly turn back into a heap. When going up it would remain a pile until, say, 900 grains. The numbers picked are arbitrary; the point is that the same amount can be either a heap or a pile depending on what it was before the change. A common use of hysteresis would be the thermostat for air conditioning: the AC is set at 77 °F and it then cools down to just below 77 °F, but does not turn on again instantly at 77.001 °F—it waits until almost 78 °F degrees, to prevent instant change of state over and over again.
that any collection is considered a heap is the expected value
of the distribution of the group's views.
A group may decide that:
Between the two extremes, individual members of the group may disagree with each other over whether any particular collection can be labelled a "heap". The collection can then not be definitively claimed to be a "heap" or "not a heap". This can be considered an appeal to descriptive linguistics
rather than prescriptive linguistics, as it resolves the issue of definition based on how the population uses natural language. Indeed, if a precise prescriptive definition of "heap" is available then the group consensus will always be unanimous and the paradox does not arise.
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
that arises from vague
Vagueness
The term vagueness denotes a property of concepts . A concept is vague:* if the concept's extension is unclear;* if there are objects which one cannot say with certainty whether belong to a group of objects which are identified with this concept or which exhibit characteristics that have this...
predicate
Predicate (grammar)
There are two competing notions of the predicate in theories of grammar. Traditional grammar tends to view a predicate as one of two main parts of a sentence, the other being the subject, which the predicate modifies. The other understanding of predicates is inspired from work in predicate calculus...
s. The paradox of the heap is an example of this paradox which arises when one considers a heap of sand
Sand
Sand is a naturally occurring granular material composed of finely divided rock and mineral particles.The composition of sand is highly variable, depending on the local rock sources and conditions, but the most common constituent of sand in inland continental settings and non-tropical coastal...
, from which grains are individually removed. Is it still a heap when only one grain remains? If not, when did it change from a heap to a non-heap?
Paradox of the heap
The word "sorites" derives from the Greek word for heap. The paradox is so named because of its original characterization, attributed to Eubulides of MiletusEubulides of Miletus
Eubulides of Miletus was a philosopher of the Megarian school, and a pupil of Euclid of Megara. He is famous for his paradoxes.-Life:Eubulides was a pupil of Euclid of Megara, the founder of the Megarian school. He was a contemporary of Aristotle, against whom he wrote with great bitterness...
. The paradox goes as follows: consider a heap
Heap
Heap may refer to:Computer science* Heap , a data structure commonly used to implement a priority queue* Heap , an area of memory used for dynamic memory allocationMathematics...
of sand from which grain
GRAIN
GRAIN is a small international non-profit organisation that works to support small farmers and social movements in their struggles for community-controlled and biodiversity-based food systems. Our support takes the form of independent research and analysis, networking at local, regional and...
s are individually removed. One might construct the argument, using premise
Premise
Premise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...
s, as follows:
- 1,000,000 grains of sand is a heap of sand (Premise 1)
- A heap of sand minus one grain is still a heap. (Premise 2)
Repeated applications
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
of Premise 2 (each time starting with one fewer grains), eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand (and consequently, if one grain of sand is still a heap, then removing that one grain of sand to leave no grains at all still leaves a heap of sand; indeed a negative number of grains must also form a heap).
On the face of it, there are some ways to avoid this conclusion. One may object to the first premise by denying 1,000,000 grains of sand makes a heap. But 1,000,000 is just an arbitrarily large number, and the argument will go through with any such number. So the response must deny outright that there are such things as heaps. Peter Unger
Peter Unger
Peter K. Unger is a contemporary American philosopher and professor at New York University. His main interests lie in the fields of metaphysics, epistemology, ethics, and the philosophy of mind. He attended Swarthmore College at the same time as David Lewis, earning a B.A. in philosophy in 1962,...
defends this solution. Alternatively, one may object to the second premise by stating that it is not true for all heaps of sand that removing one grain from it still makes a heap. Or one may accept the conclusion by insisting that a heap of sand can be composed of just one grain, and solely deny the further conclusions regarding zero-grain or negative-grain-number heaps.
Variations
This paradox can be reconstructed for a variety of predicates, for example, with "tall", "rich", "old", "blue," "bald," and so on. Bertrand RussellBertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
argued that all of natural language, even logical connectives, is vague; moreover, representations of propositions are vague. However, most views do not go that far, but it is an open question.
Setting a fixed boundary
A common first response to the paradox is to call any set of grains that has more than a certain number of grains in it a heap. If one were to set the "fixed boundary" at, say, 10,000 grains then one would claim that for fewer than 10,000, it's not a heap; for 10,000 or more, then it is a heap.However, such solutions are unsatisfactory as there seems little significance to the difference between 9,999 grains and 10,000 grains. The boundary, wherever it may be set, remains as arbitrary and so its precision is misleading. It is objectionable on both philosophical and linguistic grounds: the former on account of its arbitrariness, and the latter on the ground that it is simply not how we use natural language. A more acceptable solution is to call any collection of multiple grains (two or more) a heap.
Unknowable boundaries (or Epistemicism)
Timothy WilliamsonTimothy Williamson
Timothy Williamson is a British philosopher whose main research interests are in philosophical logic, philosophy of language, epistemology and metaphysics....
and Roy Sorensen hold an approach that there are fixed boundaries but that they are necessarily unknowable.
Supervaluationism
Supervaluationism is a semantics for dealing with irreferential singular termSingular term
There is no really adequate definition of singular term. Here are some definitions proposed by different writers:# A term that tells us which individual is being talked about....
s and vagueness
Vagueness
The term vagueness denotes a property of concepts . A concept is vague:* if the concept's extension is unclear;* if there are objects which one cannot say with certainty whether belong to a group of objects which are identified with this concept or which exhibit characteristics that have this...
. Consider the sentence 'Pegasus
Pegasus
Pegasus is one of the best known fantastical as well as mythological creatures in Greek mythology. He is a winged divine horse, usually white in color. He was sired by Poseidon, in his role as horse-god, and foaled by the Gorgon Medusa. He was the brother of Chrysaor, born at a single birthing...
likes licorice' in which the name 'Pegasus' fails to refer. What should its truth value be? There is nothing in the myth that would justify any assignment of values to it. On the other hand, consider 'Pegasus likes licorice or Pegasus doesn't like licorice' which is an instance of the valid schema
(i.e. ' or not-'). Shouldn't it be true regardless of whether or not its disjuncts have a truth value? According to supervaluationism, it should be.Precisely, let
be a classical valuationValuation (logic)
In logic and model theory, a valuation can be:*In propositional logic, an assignment of truth values to propositional variables, with a corresponding assignment of truth values to all propositional formulas with those variables....
defined on every atomic sentence
Atomic sentence
In logic, an atomic sentence is a type of declarative sentence which is either true or false and which cannot be broken down into other simpler sentences...
of the language
, and let be the number of distinct atomic sentences in . Then there are at most classical valuations defined on every sentence . A supervaluation is a function from sentences to truth values such that, is super-true (i.e. ) if and only if for every ; likewise for super-false. Otherwise, is undefined—i.e. exactly when there are two valuations and such that and .For example, let
be the formal translation of 'Pegasus likes licorice'. Then there are exactly two classical valuations and on , viz. and . So is neither super-true nor super-false.Truth gaps, gluts, and many-valued logics
Another approach is to use a multi-valued logicMulti-valued logic
In logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...
. From this point of view, the problem is with the principle of bivalence
Principle of bivalence
In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...
: the sand is either a heap or is not a heap, without any shades of gray. Instead of two logical states, heap and not-heap, a three value system can be used, for example heap, unsure and not-heap. However, three valued systems do not truly resolve the paradox as there is still a dividing line between heap and unsure and also between unsure and not-heap. The third truth-value can be understood either as a truth gap or as a truth glut.
Alternatively, fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
offers a continuous spectrum of logical states represented in the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
of real numbers [0,1]—it is a many-valued logic with infinitely-many truth-values, and thus the sand moves smoothly from "definitely heap" to "definitely not heap", with shades in the intermediate region. Fuzzy hedges are used to divide the continuum into regions corresponding to classes like definitely heap, mostly heap, partly heap, slightly heap, and not heap.
Hysteresis
Another approach is to use hysteresisHysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...
—that is, knowledge of what the collection of sand started as. Equivalent amounts of sand may be called heaps or not based on how they got there. If a large heap (indisputably described as a heap) is slowly diminished, it preserves its "heap status" to a point, even as the actual amount of sand is reduced to a smaller number of grains. For example, let's say 500 grains is a pile and 1,000 grains is a heap. There will be an overlap for these states. So if you are reducing it from a heap to a pile, it is a heap going down until, say, 750. At that point you would stop calling it a heap and start calling it a pile. But if you replace one grain, it would not instantly turn back into a heap. When going up it would remain a pile until, say, 900 grains. The numbers picked are arbitrary; the point is that the same amount can be either a heap or a pile depending on what it was before the change. A common use of hysteresis would be the thermostat for air conditioning: the AC is set at 77 °F and it then cools down to just below 77 °F, but does not turn on again instantly at 77.001 °F—it waits until almost 78 °F degrees, to prevent instant change of state over and over again.
Group consensus
One can establish the meaning of the word "heap" by appealing to group consensus. This approach claims that a collection of grains is as much a "heap" as the proportion of people in a group who believe it to be so. In other words, the probabilityProbability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that any collection is considered a heap is the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the distribution of the group's views.
A group may decide that:
- One grain of sand on its own is not a heap.
- A large collection of grains of sand is a heap.
Between the two extremes, individual members of the group may disagree with each other over whether any particular collection can be labelled a "heap". The collection can then not be definitively claimed to be a "heap" or "not a heap". This can be considered an appeal to descriptive linguistics
Descriptive linguistics
In the study of language, description, or descriptive linguistics, is the work of objectively analyzing and describing how language is spoken by a group of people in a speech community...
rather than prescriptive linguistics, as it resolves the issue of definition based on how the population uses natural language. Indeed, if a precise prescriptive definition of "heap" is available then the group consensus will always be unanimous and the paradox does not arise.
See also
- Coastline paradoxCoastline paradoxThe coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. It was first observed by Lewis Fry Richardson....
- Continuum fallacyContinuum fallacyThe continuum fallacy is an informal logical fallacy closely related to the sorites paradox, or paradox of the heap...
- False dilemmaFalse dilemmaA false dilemma is a type of logical fallacy that involves a situation in which only two alternatives are considered, when in fact there are additional options...
- Fuzzy logicFuzzy logicFuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
- I know it when I see itI know it when I see itThe phrase "I know it when I see it" is a colloquial expression within the United States by which a speaker attempts to categorize an observable fact or event, although the category is subjective or lacks clearly defined parameters. The phrase was famously used by United States Supreme Court...
- Imprecise languageImprecise languageOften, infomal, spoken language, "everyday language" is less precise than any more formals or academic languages.Language might be said to be imprecise because it exhibits one or more of the following features:...
- Multi-valued logicMulti-valued logicIn logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...
- Nirvana fallacyNirvana fallacyThe nirvana fallacy is the logical error of comparing actual things with unrealistic, idealized alternatives. It can also refer to the tendency to assume that there is a perfect solution to a particular problem...
- Philosophical InvestigationsPhilosophical InvestigationsPhilosophical Investigations is, along with the Tractatus Logico-Philosophicus, one of the most influential works by the 20th-century philosopher Ludwig Wittgenstein...
- Principle of bivalencePrinciple of bivalenceIn logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...
- Ship of TheseusShip of TheseusThe Ship of Theseus, also known as Theseus' paradox, or various variants, notably grandfather's axe and Trigger's Broom is a paradox that raises the question of whether an object which has had all its component parts replaced remains fundamentally the same object.The paradox is most notably...
- Slippery slopeSlippery slopeIn debate or rhetoric, a slippery slope is a classic form of argument, arguably an informal fallacy...
- Three men make a tigerThree men make a tiger"Three men make a tiger" is a Chinese proverb or chengyu . Three men make a tiger refers to an individual's tendency to accept absurd information as long as it is repeated by enough people...
- VaguenessVaguenessThe term vagueness denotes a property of concepts . A concept is vague:* if the concept's extension is unclear;* if there are objects which one cannot say with certainty whether belong to a group of objects which are identified with this concept or which exhibit characteristics that have this...