Continuum fallacy
Encyclopedia
The continuum fallacy (also called the fallacy of the beard, line drawing fallacy, bald man fallacy, fallacy of the heap, and the sorites fallacy) is an informal logical fallacy closely related to the sorites paradox
, or paradox of the heap. The fallacy causes one to erroneously reject a vague
claim
simply because it is not as precise as one would like it to be. Vagueness alone does not necessarily imply invalidity.
The fallacy appears to demonstrate that two states or conditions cannot be considered distinct
(or do not exist
at all) because between them there exists a continuum
of states. According to the fallacy, differences in quality
cannot result from differences in quantity
.
There are clearly reasonable and clearly unreasonable cases in which objects either belong or do not belong to a particular group of objects based on their properties. We are able to take them case by case and designate them as such even in the case of properties which may be vaguely defined. The existence of hard or controversial cases does not preclude our ability to designate members of particular kinds of groups.
, and while Newtonian physics models the world as continuous, in modern quantum physics, notions of continuous length break down at the Planck length, and thus what appear to be continua may, at base, simply be very many discrete states.
For the purpose of the continuum fallacy, one assumes that there is in fact a continuum, though this is generally a minor distinction: in general, any argument against the sorites paradox
can also be used against the continuum fallacy. One argument against the fallacy is based on the simple counterexample
: there do exist bald people and people who aren't bald. Another argument is that for each degree of change in states, the degree of the condition changes slightly, and these "slightly"s build up to shift the state from one category to another. For example, perhaps the addition of a grain of rice causes the total group of rice to be "slightly more" of a heap, and enough "slightly"s will certify the group's heap status – see fuzzy logic
.
:
Sorites paradox
The sorites paradox is a paradox that arises from vague predicates. 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...
, or paradox of the heap. The fallacy causes one to erroneously reject a 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...
claim
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
simply because it is not as precise as one would like it to be. Vagueness alone does not necessarily imply invalidity.
The fallacy appears to demonstrate that two states or conditions cannot be considered distinct
Difference
Difference may refer to:* Difference , a 2005 power metal album* Difference , a concept in computer science* Difference , any systematic way of distinguishing similar coats of arms belonging to members of the same family* Difference , a statement about the relative size or order of two objects**...
(or do not exist
Exist
Exist may refer to:* eXist, an open source database management system built on XML* Existence* Energetic X-ray Survey Telescope, a proposed hard X-ray imaging all-sky deep survey mission...
at all) because between them there exists a continuum
Continuum (theory)
Continuum theories or models explain variation as involving a gradual quantitative transition without abrupt changes or discontinuities. It can be contrasted with 'categorical' models which propose qualitatively different states.-In physics:...
of states. According to the fallacy, differences in quality
Quality (philosophy)
A quality is an attribute or a property. Attributes are ascribable, by a subject, whereas properties are possessible. In contemporary philosophy, the idea of qualities and especially how to distinguish certain kinds of qualities from one another remains controversial.-Background:Aristotle analyzed...
cannot result from differences in quantity
Quantity
Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more" or "less" or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation...
.
There are clearly reasonable and clearly unreasonable cases in which objects either belong or do not belong to a particular group of objects based on their properties. We are able to take them case by case and designate them as such even in the case of properties which may be vaguely defined. The existence of hard or controversial cases does not preclude our ability to designate members of particular kinds of groups.
Relation with sorites paradox
Narrowly speaking, the sorites paradox refers to situations where there are many discrete states (classically between 1 and 1,000,000 grains of sand, hence 1,000,000 possible states), while the continuum fallacy refers to situations where there is (or appears to be) a continuum of states, such as temperature – is a room hot or cold? Whether any continua exist in the physical world is the classic question of atomismAtomism
Atomism is a natural philosophy that developed in several ancient traditions. The atomists theorized that the natural world consists of two fundamental parts: indivisible atoms and empty void.According to Aristotle, atoms are indestructible and immutable and there are an infinite variety of shapes...
, and while Newtonian physics models the world as continuous, in modern quantum physics, notions of continuous length break down at the Planck length, and thus what appear to be continua may, at base, simply be very many discrete states.
For the purpose of the continuum fallacy, one assumes that there is in fact a continuum, though this is generally a minor distinction: in general, any argument against the sorites paradox
Sorites paradox
The sorites paradox is a paradox that arises from vague predicates. 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...
can also be used against the continuum fallacy. One argument against the fallacy is based on the simple counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
: there do exist bald people and people who aren't bald. Another argument is that for each degree of change in states, the degree of the condition changes slightly, and these "slightly"s build up to shift the state from one category to another. For example, perhaps the addition of a grain of rice causes the total group of rice to be "slightly more" of a heap, and enough "slightly"s will certify the group's heap status – see 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...
.
Fred can never be called bald
Fred can never be called bald. Fred isn't bald now, however if he loses one hair, that won't make him go from not bald to bald either. If he loses one more hair after that, then this one loss, also does not make him go from not bald to bald. Therefore, no matter how much hair he loses, he can never be called bald.The heap
The fallacy can be described in the form of a conversationConversation
Conversation is a form of interactive, spontaneous communication between two or more people who are following rules of etiquette.Conversation analysis is a branch of sociology which studies the structure and organization of human interaction, with a more specific focus on conversational...
:
- Q: Does one grain of wheat form a heap?
- A: No.
- Q: If we add one, do two grains of wheat form a heap?
- A: No.
- Q: If we add one, do three grains of wheat form a heap?
- A: No.
- ...
- Q: If we add one, do one hundred grains of wheat form a heap?
- A: No.
- Q: Therefore, no matter how many grains of wheat we add, we will never have a heap. Therefore, heaps don't exist!
Others
Other uses of this fallacy seem to prove that:- No man has a beardBeardA beard is the collection of hair that grows on the chin, cheeks and neck of human beings. Usually, only pubescent or adult males are able to grow beards. However, women with hirsutism may develop a beard...
, no matter how long it is (or every post-pubescentPubertyPuberty is the process of physical changes by which a child's body matures into an adult body capable of reproduction, as initiated by hormonal signals from the brain to the gonads; the ovaries in a girl, the testes in a boy...
male has a beard, no matter how cleanly shaveShaveShave may refer to:*to shave refers to the act of shaving*"Shave," a song by Enon from their 2003 album Hocus Pocus*Shave , a periodical magazine...
n) because a beard can have varying lengths. - A room is never either "hot" or "cold", because of the continuum of temperatureTemperatureTemperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
s.
See also
- 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:...
- 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...
- Closed conceptClosed conceptA closed concept is a concept where all the necessary and sufficient conditions required to include something within the concept can be listed. For example, the concept of a triangle is closed because a three-sided polygon, and only a three-sided polygon, is a triangle...