Ambiguity - AbsoluteAstronomy.com

Ambiguity of words or phrases is the ability to express more than one interpretation. It is distinct from 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...

, which is a statement about the lack of precision contained or available in the information.

Context may play a role in resolving ambiguity. For example the same piece of information may be ambiguous in one context and unambiguous in another.

Linguistic forms

The lexical ambiguity

Polysemy

Polysemy is the capacity for a sign or signs to have multiple meanings , i.e., a large semantic field.Charles Fillmore and Beryl Atkins’ definition stipulates three elements: the various senses of a polysemous word have a central origin, the links between these senses form a network, and ...

of a word or phrase contains in its having more than one meaning in the language to which the word belongs. "Meaning" hereby refers to whatever should be captured by a good dictionary. For instance, the word "bank" has several distinct lexical definitions, including "financial institution

Bank

A bank is a financial institution that serves as a financial intermediary. The term "bank" may refer to one of several related types of entities:...

" and "edge of a river

Bank (geography)

A geographic bank has four definitions and applications:# Limnology: The shoreline of a pond, swamp, estuary, reservoir, or lake. The grade can vary from vertical to a shallow slope....

". Another example is as in apothecary

Apothecary

Apothecary is a historical name for a medical professional who formulates and dispenses materia medica to physicians, surgeons and patients — a role now served by a pharmacist and some caregivers....

. One could say "I bought herbs from the apothecary". This could mean you actually spoke to the apothecary (pharmacist

Pharmacist

Pharmacists are allied health professionals who practice in pharmacy, the field of health sciences focusing on safe and effective medication use...

) or went to the apothecary (pharmacy

Pharmacy

Pharmacy is the health profession that links the health sciences with the chemical sciences and it is charged with ensuring the safe and effective use of pharmaceutical drugs...

).

The context in which an ambiguous word is used often makes it evident which of the meanings is intended. If, for instance, someone says "I deposited $100 in the bank", most people would not think you used a shovel to dig in the mud. However, some linguistic contexts do not provide sufficient information to disambiguate a used word. For example, "Biweekly" can mean "fortnightly" (once every two weeks – 26 times a year), OR "twice a week" (104 times a year). If "biweekly" is used in a conversation about a meeting schedule, it may be difficult to infer which meaning was intended.
Generally, such lexically ambiguous, miscommunication-prone words should be avoided wherever possible, since the user generally has to waste time, effort, and attention span

Attention span

Attention span is the amount of time that a person can concentrate on a task without becoming distracted. Most educators and psychologists agree that the ability to focus one's attention on a task is crucial for the achievement of one's goals....

to define what is meant when they are used.

Lexical ambiguity can be addressed by algorithmic methods that automatically associate the appropriate meaning with a word in context, a task referred to as Word Sense Disambiguation

Word sense disambiguation

In computational linguistics, word-sense disambiguation is an open problem of natural language processing, which governs the process of identifying which sense of a word is used in a sentence, when the word has multiple meanings...

.

The use of multi-defined words requires the author or speaker to clarify their context, and sometimes elaborate on their specific intended meaning (in which case, a less ambiguous term should have been used). The goal of clear concise communication is that the receiver(s) have no misunderstanding about what was meant to be conveyed. An exception to this could include a politician whose "weasel word

Weasel word

A weasel word is an informal term for equivocating words and phrases aimed at creating an impression that something specific and meaningful has been said, when in fact only a vague or ambiguous claim, or even a refutation has been communicated.For example, an advertisement may use a weasel phrase...

s" and obfuscation

Obfuscation

Obfuscation is the hiding of intended meaning in communication, making communication confusing, wilfully ambiguous, and harder to interpret.- Background :Obfuscation may be used for many purposes...

are necessary to gain support from multiple constituents

Electoral district

An electoral district is a distinct territorial subdivision for holding a separate election for one or more seats in a legislative body...

with mutually exclusive

Mutually exclusive

In layman's terms, two events are mutually exclusive if they cannot occur at the same time. An example is tossing a coin once, which can result in either heads or tails, but not both....

conflicting desires from their candidate of choice. Ambiguity is a powerful tool of political science

Political science

Political Science is a social science discipline concerned with the study of the state, government and politics. Aristotle defined it as the study of the state. It deals extensively with the theory and practice of politics, and the analysis of political systems and political behavior...

.

More problematic are words whose senses express closely related concepts. “Good,” for example, can mean "useful" or "functional" (That’s a good hammer), "exemplary" (She’s a good student), "pleasing" (This is good soup), "moral (a good person versus the lesson to be learned from a story), "righteous", etc. " I have a good daughter" is not clear about which sense is intended. The various ways to apply prefix

Prefix

A prefix is an affix which is placed before the root of a word. Particularly in the study of languages,a prefix is also called a preformative, because it alters the form of the words to which it is affixed.Examples of prefixes:...

es and suffix

Suffix

In linguistics, a suffix is an affix which is placed after the stem of a word. Common examples are case endings, which indicate the grammatical case of nouns or adjectives, and verb endings, which form the conjugation of verbs...

es can also create ambiguity ("unlockable" can mean "capable of being unlocked" or "impossible to lock").

Syntactic ambiguity

Syntactic ambiguity

Syntactic ambiguity is a property of sentences which may be reasonably interpreted in more than one way, or reasonably interpreted to mean more than one thing...

arises when a phrase can be parsed

Parsing

In computer science and linguistics, parsing, or, more formally, syntactic analysis, is the process of analyzing a text, made of a sequence of tokens , to determine its grammatical structure with respect to a given formal grammar...

in more than one way. Such phrases can be assigned different interpretations because different grammatical structures can be assigned to the same string of words. "He ate the cookies on the couch", for example, could mean that he ate those cookies which were on the couch (as opposed to those that were on the table), or it could mean that he was sitting on the couch when he ate the cookies.

Spoken language

Spoken language

Spoken language is a form of human communication in which words derived from a large vocabulary together with a diverse variety of names are uttered through or with the mouth. All words are made up from a limited set of vowels and consonants. The spoken words they make are stringed into...

can contain many more types of ambiguities, where there is more than one way to compose a set of sounds into words, for example "ice cream" and "I scream". Such ambiguity is generally resolved according to the context. A mishearing of such, based on incorrectly resolved ambiguity, is called a mondegreen

Mondegreen

A mondegreen is the mishearing or misinterpretation of a phrase as a result of near homophony, in a way that gives it a new meaning. It most commonly is applied to a line in a poem or a lyric in a song...

.

Semantic ambiguity arises when a word or concept has an inherently diffuse meaning based on widespread or informal usage. This is often the case, for example, with idiomatic expressions whose definitions are rarely or never well-defined, and are presented in the context of a larger argument that invites a conclusion.

For example, "You could do with a new automobile. How about a test drive?" The clause "You could do with" presents a statement with such wide possible interpretation as to be essentially meaningless. Lexical ambiguity is contrasted with semantic ambiguity. The former represents a choice between a finite number of known and meaningful context-dependent interpretations. The latter represents a choice between any number of possible interpretations, none of which may have a standard agreed-upon meaning. This form of ambiguity is closely related to vagueness

Vagueness

.

Linguistic ambiguity can be a problem in law (see Ambiguity (law)

Ambiguity (law)

Ambiguity, in law, is of two kinds, patent and latent.Patent ambiguity is that ambiguity which is apparent on the face of an instrument to any one perusing it, even if he be unacquainted with the circumstances of the parties. In the case of a patent ambiguity parol evidence is admissible to explain...

), because the interpretation of written documents and oral agreements is often of paramount importance.

Intentional application

Philosophers (and other users of logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

) spend a lot of time and effort searching for and removing (or intentionally adding) ambiguity in arguments, because it can lead to incorrect conclusions and can be used to deliberately conceal bad arguments. For example, a politician might say "I oppose taxes which hinder economic growth", an example of a glittering generality

Glittering generality

Glittering generalities are emotionally appealing words so closely associated with highly-valued concepts and beliefs that they carry conviction without supporting information or reason. Such highly-valued concepts attract general approval and acclaim...

. Some will think he opposes taxes in general, because they hinder economic growth. Others may think he opposes only those taxes that he believes will hinder economic growth. In writing, the sentence can be rewritten to reduce possible misinterpretation, either by adding a comma after "taxes" (to convey the first sense) or by changing "which" to "that" (to convey the second sense), or by rewriting it in other ways. The devious politician hopes that each constituent

Electoral district

An electoral district is a distinct territorial subdivision for holding a separate election for one or more seats in a legislative body...

will interpret the statement in the most desirable way, and think the politician supports everyone's opinion. However, the opposite can also be true - An opponent can turn a positive statement into a bad one, if the speaker uses ambiguity (intentionally or not). The logical fallacies of amphiboly and equivocation

Equivocation

Equivocation is classified as both a formal and informal logical fallacy. It is the misleading use of a term with more than one meaning or sense...

rely heavily on the use of ambiguous words and phrases.

In Continental philosophy

Continental philosophy

Continental philosophy, in contemporary usage, refers to a set of traditions of 19th and 20th century philosophy from mainland Europe. This sense of the term originated among English-speaking philosophers in the second half of the 20th century, who used it to refer to a range of thinkers and...

(particularly phenomenology and existentialism

Existentialism

Existentialism is a term applied to a school of 19th- and 20th-century philosophers who, despite profound doctrinal differences, shared the belief that philosophical thinking begins with the human subject—not merely the thinking subject, but the acting, feeling, living human individual...

), there is much greater tolerance of ambiguity, as it is generally seen as an integral part of the human condition. Martin Heidegger

Martin Heidegger

Martin Heidegger was a German philosopher known for his existential and phenomenological explorations of the "question of Being."...

argued that the relation between the subject and object

Subject-object problem

The subject–object problem, a longstanding philosophical issue, is concerned with the analysis of human experience, and of what within experience is "subjective" and what is "objective."...

is ambiguous, as is the relation of mind and body, and part and whole

Mereology

In philosophy and mathematical logic, mereology treats parts and the wholes they form...

. In Heidegger's phenomenology, Dasein

Dasein

Dasein is a German word famously used by Martin Heidegger in his magnum opus Being and Time, which generally translates to being in its ontological and philosophical sense Dasein is a German word famously used by Martin Heidegger in his magnum opus Being and Time, which generally translates to...

is always in a meaningful world, but there is always an underlying background for every instance of signification. Thus, although some things may be certain, they have little to do with Dasein's sense of care and existential anxiety, e.g., in the face of death. In calling his work Being and Nothingness an "essay in phenomenological ontology" Jean-Paul Sartre

Jean-Paul Sartre

Jean-Paul Charles Aymard Sartre was a French existentialist philosopher, playwright, novelist, screenwriter, political activist, biographer, and literary critic. He was one of the leading figures in 20th century French philosophy, particularly Marxism, and was one of the key figures in literary...

follows Heidegger in defining the human essence as ambiguous, or relating fundamentally to such ambiguity. Simone de Beauvoir

Simone de Beauvoir

Simone-Ernestine-Lucie-Marie Bertrand de Beauvoir, often shortened to Simone de Beauvoir , was a French existentialist philosopher, public intellectual, and social theorist. She wrote novels, essays, biographies, an autobiography in several volumes, and monographs on philosophy, politics, and...

tries to base an ethics on Heidegger's and Sartre's writings (The Ethics of Ambiguity

The Ethics of Ambiguity

The Ethics of Ambiguity is Simone de Beauvoir's second major non-fiction work, nearly twice as long as her first, Pyrrhus and Cineas...

), where she highlights the need to grapple with ambiguity: "as long as philosophers and they [men] have thought, most of them have tried to mask it...And the ethics which they have proposed to their disciples has always pursued thre same goal. It has been a matter of eliminating the ambiguity by making oneself pure inwardness or pure externality, by escaping from the sensible world or being engulfed by it, by yielding to eternity or enclosing oneself in the pure moment.". Ethics cannot be based on the authoritative certainty given by mathematics and logic, or prescribed directly from the empirical findings of science. She states: "Since we do not succeed in fleeing it, let us therefore try to look the truth in the face. Let us try to assume our fundamental ambiguity. It is in the knowledge of the genuine conditions of our life that we must draw our strength to live and our reason for acting.". Other continental philosophers suggest that concepts such as life, nature, and sex are ambiguous. Recently, Corey Anton has argued that we cannot be certain what is separate from or unified with something else: language, he asserts, divides what is not in fact separate. Following Ernest Becker

Ernest Becker

Ernest Becker was a cultural anthropologist and interdisciplinary scientific thinker and writer. He is noted for his 1974 Pulitzer Prize-winning book, The Denial of Death.-Early life:...

, he argues that the desire to 'authoritatively disambiguate' the world and existence has led to numerous ideologies and historical events such as genocide

Genocide

Genocide is defined as "the deliberate and systematic destruction, in whole or in part, of an ethnic, racial, religious, or national group", though what constitutes enough of a "part" to qualify as genocide has been subject to much debate by legal scholars...

. On this basis, he argues that ethics must focus on 'dialectically integrating opposites' and balancing tension, rather than seeking a priori validation or certainty. Like the existentialists and phenomenologists, he sees the ambiguity of life as the basis of creativity.

In literature

Literature

Literature is the art of written works, and is not bound to published sources...

and rhetoric

Rhetoric

Rhetoric is the art of discourse, an art that aims to improve the facility of speakers or writers who attempt to inform, persuade, or motivate particular audiences in specific situations. As a subject of formal study and a productive civic practice, rhetoric has played a central role in the Western...

, ambiguity can be a useful tool. Groucho Marx

Groucho Marx

Julius Henry "Groucho" Marx was an American comedian and film star famed as a master of wit. His rapid-fire delivery of innuendo-laden patter earned him many admirers. He made 13 feature films with his siblings the Marx Brothers, of whom he was the third-born...

’s classic joke depends on a grammatical ambiguity for its humor, for example: "Last night I shot an elephant in my pajamas. How he got in my pajamas, I’ll never know". Songs and poetry often rely on ambiguous words for artistic effect, as in the song title "Don’t It Make My Brown Eyes Blue" (where "blue" can refer to the color, or to sadness

Sadness

Sadness is emotional pain associated with, or characterized by feelings of disadvantage, loss, despair, helplessness, sorrow, and rage. When sad, people often become outspoken, less energetic, and emotional...

).

In narrative, ambiguity can be introduced in several ways: motive, plot, character. F. Scott Fitzgerald

F. Scott Fitzgerald

Francis Scott Key Fitzgerald was an American author of novels and short stories, whose works are the paradigm writings of the Jazz Age, a term he coined himself. He is widely regarded as one of the greatest American writers of the 20th century. Fitzgerald is considered a member of the "Lost...

uses the latter type of ambiguity with notable effect in his novel The Great Gatsby
The Great Gatsby
The Great Gatsby is a novel by the American author F. Scott Fitzgerald. First published in1925, it is set on Long Island's North Shore and in New York City from spring to autumn of 1922....

.

All religions debate the orthodoxy

Orthodoxy

The word orthodox, from Greek orthos + doxa , is generally used to mean the adherence to accepted norms, more specifically to creeds, especially in religion...

or heterodoxy

Heterodoxy

Heterodoxy is generally defined as "any opinions or doctrines at variance with an official or orthodox position". As an adjective, heterodox is commonly used to describe a subject as "characterized by departure from accepted beliefs or standards"...

of ambiguity. Christianity

Christianity

Christianity is a monotheistic religion based on the life and teachings of Jesus as presented in canonical gospels and other New Testament writings...

and Judaism

Judaism

Judaism ) is the "religion, philosophy, and way of life" of the Jewish people...

employ the concept of paradox

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...

synonymously with 'ambiguity'. Ambiguity within Christianity (and other religions) is resisted by the conservatives and fundamentalists, who regard the concept as equating with 'contradiction'. Non-fundamentalist Christians and Jews endorse Rudolf Otto

Rudolf Otto

Rudolf Otto was an eminent German Lutheran theologian and scholar of comparative religion.-Life:Born in Peine near Hanover, Otto attended the Gymnasium Andreanum in Hildesheim and studied at the universities of Erlangen and Göttingen, where he wrote his dissertation on Martin Luther's...

's description of the sacred as 'mysterium tremendum et fascinans', the awe-inspiring mystery which fascinates humans.

Metonymy

Metonymy

Metonymy is a figure of speech used in rhetoric in which a thing or concept is not called by its own name, but by the name of something intimately associated with that thing or concept...

involves the use of the name of a subcomponent part as an abbreviation, or jargon

Jargon

Jargon is terminology which is especially defined in relationship to a specific activity, profession, group, or event. The philosophe Condillac observed in 1782 that "Every science requires a special language because every science has its own ideas." As a rationalist member of the Enlightenment he...

, for the name of the whole object (for example "wheels" to refer to a car, or "flowers" to refer to beautiful offspring, an entire plant, or a collection of blooming plants). In modern vocabulary

Vocabulary

A person's vocabulary is the set of words within a language that are familiar to that person. A vocabulary usually develops with age, and serves as a useful and fundamental tool for communication and acquiring knowledge...

critical semiotics

Semiotics

Semiotics, also called semiotic studies or semiology, is the study of signs and sign processes , indication, designation, likeness, analogy, metaphor, symbolism, signification, and communication...

, metonymy encompasses any potentially ambiguous word substitution that is based on contextual contiguity

Contiguity

A contiguity is a continuous mass, or a series of things in contact or proximity. In a different meaning, contiguity is the state of being contiguous...

(located close together), or a function or process that an object performs, such as "sweet ride" to refer to a nice car. Metonym miscommunication is considered a primary mechanism of linguistic humour.

Psychology and management

In sociology and social psychology, the term "ambiguity" is used to indicate situations that involve uncertainty

Uncertainty

Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...

. An increasing amount of research is concentrating on how people react and respond to ambiguous situations. Much of this focuses on ambiguity tolerance

Ambiguity tolerance

Ambiguity tolerance is the ability to perceive ambiguity in information and behavior in a neutral and open way.Ambiguity tolerance is an important issue in personality development and education...

. A number of correlations have been found between an individual’s reaction and tolerance to ambiguity and a range of factors.

Apter and Desselles (2001) for example, found a strong correlation with such attributes and factors like a greater preference for safe as opposed to risk-based sports, a preference for endurance-type activities as opposed to explosive activities, a more organized and less casual lifestyle, greater care and precision in descriptions, a lower sensitivity to emotional and unpleasant words, a less acute sense of humor, engaging a smaller variety of sexual practices than their more risk-comfortable colleagues, a lower likelihood of the use of drugs, pornography and drink, a greater likelihood of displaying obsessional behavior.

In the field of leadership

Leadership

Leadership has been described as the “process of social influence in which one person can enlist the aid and support of others in the accomplishment of a common task". Other in-depth definitions of leadership have also emerged.-Theories:...

David Wilkinson

David Wilkinson (ambiguity expert)

David John Wilkinson is considered to be one of the foremost experts and authors on how people deal with Ambiguity and Emotional Resilience / Psychological resilience...

(2006) found strong correlations between an individual leader's reaction to ambiguous situations and the Modes of Leadership

Modes of Leadership

David Wilkinson described four modes of leadership in his 2006 book, The Ambiguity Advantage.-Mode vs style:In situational leadership theory, styles of leadership refer to behaviors that a leader should engage with in different situations...

they use, the type of creativity

Creativity

Creativity refers to the phenomenon whereby a person creates something new that has some kind of value. What counts as "new" may be in reference to the individual creator, or to the society or domain within which the novelty occurs...

, Kirton (2003) and how they relate to others.

Music

In music

Music

Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, pieces or sections which confound expectations and may be or are interpreted simultaneously in different ways are ambiguous, such as some polytonality

Polytonality

The musical use of more than one key simultaneously is polytonality . Bitonality is the use of only two different keys at the same time...

, polymeter, other ambiguous meters

Metre

The metre , symbol m, is the base unit of length in the International System of Units . Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole , its definition has been periodically refined to reflect growing knowledge of metrology...

or rhythm

Rhythm

Rhythm may be generally defined as a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions." This general meaning of regular recurrence or pattern in time may be applied to a wide variety of cyclical natural phenomena having a periodicity or...

s, and ambiguous phrasing

Phrase (music)

In music and music theory, phrase and phrasing are concepts and practices related to grouping consecutive melodic notes, both in their composition and performance...

, or (Stein 2005, p. 79) any aspect of music

Aspect of music

An aspect of music is any characteristic, dimension, or element taken as a part or component of music.-European music:The traditional musicological or European-influenced aspects of music often listed are those elements given primacy in European-influenced classical music, so 7 basic elements of...

. The music of Africa

Music of Africa

Africa is a vast continent and its regions and nations have distinct musical traditions. The music of North Africa for the most part has a different history from sub-Saharan African music traditions....

is often purposely ambiguous. To quote Sir Donald Francis Tovey

Donald Francis Tovey

Sir Donald Francis Tovey was a British musical analyst, musicologist, writer on music, composer, conductor and pianist...

(1935, p. 195), “Theorists are apt to vex themselves with vain efforts to remove uncertainty just where it has a high aesthetic value.”

Visual art

In visual art, certain images are visually ambiguous, such as the Necker cube

Necker cube

The Necker Cube is an optical illusion first published as a rhomboid in 1832 by Swiss crystallographer Louis Albert Necker.-Ambiguity:The Necker Cube is an ambiguous line drawing....

, which can be interpreted in two ways. Perceptions of such objects remain stable for a time, then may flip, a phenomenon called multistable perception

Multistable perception

Multistable perceptual phenomena are a form of perceptual phenomena in which there are unpredictable sequences of spontaneous subjective changes...

.
The opposite of such ambiguous images are impossible object

Impossible object

An impossible object is a type of optical illusion consisting of a two-dimensional figure which is instantly and subconsciously interpreted by the visual system as representing a projection of a three-dimensional object although it is not actually possible for such an object to exist An impossible...

s.

Pictures or photographs may also be ambiguous at the semantic level: the visual image is unambiguous, but the meaning and narrative may be ambiguous: is a certain facial expression one of excitement or fear, for instance?

Constructed language

Some languages have been created with the intention of avoiding ambiguity, especially lexical ambiguity

Polysemy

. Lojban

Lojban

See also discussed by Arthur Protin, Bob LeChevalier, Carl Burke, Doug Landauer, Guy Steele, Jack Waugh, Jeff Prothero, Jim Carter, and Robert Chassell, as well as , the concepts which "average English speakers won't recognize" because most of them "have no exact English counterpart".Like most...

and Loglan

Loglan

Loglan is a constructed language originally designed for linguistic research, particularly for investigating the Sapir–Whorf Hypothesis. The language was developed beginning in 1955 by Dr James Cooke Brown with the goal of making a language so different from natural languages that people learning...

are two related languages which have been created with this in mind, focusing chiefly on syntactic ambiguity as well. The languages can be both spoken and written. These languages are intended to provide a greater technical precision over big natural languages, although historically, such attempts at language improvement have been criticized. Languages composed from many diverse sources contain much ambiguity and inconsistency. The many exceptions to syntax

Syntax

In linguistics, syntax is the study of the principles and rules for constructing phrases and sentences in natural languages....

and semantic rules are time-consuming and difficult to learn.

Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...

, widely used in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

and other science

Science

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

s, avoids many ambiguities compared to expression in natural language. However, for various reasons, several lexical

Lexical (semiotics)

In the lexicon of a language, lexical words or nouns refer to things. These words fall into three main classes:*proper nouns refer exclusively to the place, object or person named, i.e...

, syntactic and semantic ambiguities remain.

Names of functions

The ambiguity in the style of writing a function should not be confused with a multivalued function

Multivalued function

In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...

, which can (and should) be defined in a deterministic and unambiguous way. Several special functions still do not have established notations. Usually, the conversion to another notation requires to scale the argument and/or the resulting value; sometimes, the same name of the function is used, causing confusions. Examples of such underestablished functions:

Sinc function
Elliptic integral of the Third Kind; translating elliptic integral form MAPLE
Maple
Acer is a genus of trees or shrubs commonly known as maple.Maples are variously classified in a family of their own, the Aceraceae, or together with the Hippocastanaceae included in the family Sapindaceae. Modern classifications, including the Angiosperm Phylogeny Group system, favour inclusion in...

to Mathematica
Mathematica
Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...

, one should replace the second argument to its square, see Talk:Elliptic integral#List of notations; dealing with complex values, this may cause problems.
Exponential integral
Exponential integral
In mathematics, the exponential integral is a special function defined on the complex plane given the symbol Ei.-Definitions:For real, nonzero values of x, the exponential integral Ei can be defined as...

, page 228 http://www.math.sfu.ca/~cbm/aands/page_228.htm
Hermite polynomial, page 775 http://www.math.sfu.ca/~cbm/aands/page_775.htm

Expressions

Ambiguous expressions often appear in physical and mathematical texts.
It is common practice to omit multiplication signs in mathematical expressions. Also, it is common, to give the same name to a variable and a function, for example,

. Then, if one sees

, there is no way to distinguish, does it mean

multiplied by

, or function

evaluated at argument equal to

. In each case of use of such notations, the reader is supposed to be able to perform the deduction and reveal the true meaning.

Creators of algorithmic languages try to avoid ambiguities. Many algorithmic languages (C++

C++

C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

, Fortran

Fortran

Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...

) require the character * as symbol of multiplication. The language Mathematica

Mathematica

Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...

allows the user to omit the multiplication symbol, but requires square brackets to indicate the argument of a function; square brackets are not allowed for grouping of expressions. Fortran, in addition, does not allow use of the same name (identifier) for different objects, for example, function and variable; in particular, the expression f=f(x) is qualified as an error.

The order of operations may depend on the context. In most programming language

Programming language

A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....

s, the operations of division and multiplication have equal priority and are executed from left to right. Until the last century, many editorials assumed that multiplication is performed first, for example,

is interpreted as

; in this case, the insertion of parentheses is required when translating the formulas to an algorithmic language. In addition, it is common to write an argument of a function without parenthesis, which also may lead to ambiguity.
Sometimes, one uses italics letters to denote elementary functions.
In the scientific journal

Scientific journal

In academic publishing, a scientific journal is a periodical publication intended to further the progress of science, usually by reporting new research. There are thousands of scientific journals in publication, and many more have been published at various points in the past...

style, the expression

means
product of variables

and

, although in a slideshow, it may mean

.

Comma in subscripts and superscripts sometimes is omitted; it is also ambiguous notation.
If it is written

, the reader should guess from the context, does it mean a single-index object, evaluated while the subscript is equal to product of variables

and

, or it is indication to a three-valent tensor.
The writing of

instead of

may mean that the writer either is stretched in space (for example, to reduce the publication fees, or aims to increase number of publications without considering readers. The same may apply to any other use of ambiguous notations.

Subscripts are also used to denote the argument to a function, as in

Examples of potentially confusing ambiguous mathematical expressions

, which could be understood to mean either

. In addition,

may mean

, as

means

(see tetration

Tetration

In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

, which by convention means

, though it might be thought to mean

since

means

, which arguably should mean

but would commonly be understood to mean

Notations in quantum optics and quantum mechanics

It is common to define the coherent states in quantum optics

Quantum optics

Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter.- History of quantum optics :...

with

and states with fixed number of photons with

. Then, there is an "unwritten rule": the state is coherent if there are more Greek characters than Latin characters in the argument, and

photon state if the Latin characters dominate. The ambiguity becomes even worse, if

is used for the states with certain value of the coordinate, and

means the state with certain value of the momentum, which may be used in books on quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. Such ambiguities easy lead to confusions, especially if some normalized adimensional, dimensionless variables are used. Expression

may mean a state with single photon, or the coherent state with mean amplitude equal to 1, or state with momentum equal to unity, and so on. The reader is supposed to guess from the context.

Ambiguous terms in physics and mathematics

Some physical quantities do not yet have established notations; their value (and sometimes even dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

, as in the case of the Einstein coefficients) depends on the system of notations. Many terms are ambiguous. Each use of an ambiguous term should be preceded by the definition, suitable for a specific case. Just like Ludwig Wittgenstein

Ludwig Wittgenstein

Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...

states in Tractatus Logico-Philosophicus

Tractatus Logico-Philosophicus

The Tractatus Logico-Philosophicus is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein in his lifetime. It was an ambitious project: to identify the relationship between language and reality and to define the limits of science...

: "... Only in the context of a proposition has a name meaning."

A highly confusing term is gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...

. For example, the sentence "the gain of a system should be doubled", without context, means close to nothing.

It may mean that the ratio of the output voltage of an electric circuit to the input voltage should be doubled.

It may mean that the ratio of the output power of an electric or optical circuit to the input power should be doubled.

It may mean that the gain of the laser medium should be doubled, for example, doubling the population of the upper laser level in a quasi-two level system (assuming negligible absorption of the ground-state).

The term intensity is ambiguous when applied to light. The term can refer to any of irradiance

Irradiance

Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...

, luminous intensity

Luminous intensity

In photometry, luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle, based on the luminosity function, a standardized model of the sensitivity of the human eye...

, radiant intensity

Radiant intensity

In radiometry, radiant intensity is a measure of the intensity of electromagnetic radiation. It is defined as power per unit solid angle. The SI unit of radiant intensity is watts per steradian . Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity...

, or radiance

Radiance

Radiance and spectral radiance are radiometric measures that describe the amount of radiation such as light or radiant heat that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from...

, depending on the background of the person using the term.

Also, confusions may be related with the use of atomic percent

Atomic percent

The atomic ratio is a measure of the ratio of atoms of one kind to another kind . A closely related concept is the atomic percent , which gives the percentage of one kind of atom relative to the total number of atoms...

as measure of concentration of a dopant

Dopant

A dopant, also called a doping agent, is a trace impurity element that is inserted into a substance in order to alter the electrical properties or the optical properties of the substance. In the case of crystalline substances, the atoms of the dopant very commonly take the place of elements that...

, or resolution

Optical resolution

Optical resolution describes the ability of an imaging system to resolve detail in the object that is being imaged.An imaging system may have many individual components including a lens and recording and display components...

of an imaging system, as measure of the size of the smallest detail which still can be resolved at the background of statistical noise. See also Accuracy and precision

Accuracy and precision

In the fields of science, engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value. The precision of a measurement system, also called reproducibility or repeatability, is the degree to which...

and its talk.

The Berry paradox

Berry paradox

The Berry paradox is a self-referential paradox arising from the expression "the smallest possible integer not definable by a given number of words". Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G...

arises as a result of systematic ambiguity in the meaning of terms such as "definable" or "nameable". Terms of this kind give rise to vicious circle

Virtuous circle and vicious circle

A virtuous circle and a vicious circle are economic terms. They refer to a complex of events that reinforces itself through a feedback loop. A virtuous circle has favorable results, while a vicious circle has detrimental results...

fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal.

Mathematical interpretation of ambiguity

In mathematics and logic, ambiguity can be considered to be an underdetermined system (of equations or logic) – for example,

leaves open what the value of X is – while its opposite is a self-contradiction, also called inconsistency, paradoxicalness, or oxymoron

Oxymoron

An oxymoron is a figure of speech that combines contradictory terms...

, in an overdetermined system

Overdetermined system

In mathematics, a system of linear equations is considered overdetermined if there are more equations than unknowns. The terminology can be described in terms of the concept of counting constraints. Each unknown can be seen as an available degree of freedom...

– such as

, which has no solution – see also underdetermination

Underdetermination

In scientific theory, underdetermination refers to situations where the evidence available is insufficient to identify which belief we should hold about that evidence...

.

Logical ambiguity and self-contradiction is analogous to visual ambiguity and impossible object

Impossible object

s, such as the Necker cube and impossible cube, or many of the drawings of M. C. Escher

M. C. Escher

Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...

Pedagogic use of ambiguous expressions

Ambiguity can be used as a pedagogical trick, to force students to reproduce the deduction by themselves. Some textbooks
give the same name to the function and to its Fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

.
Rigorously speaking, such an expression requires that

;
even if function

is a self-Fourier function, the expression should be written as

; however, it is assumed that
the shape of the function (and even its norm

) depend on the character used to denote its argument.
If the Greek letter is used, it is assumed to be a Fourier transform of another function,
The first function is assumed, if the expression in the argument contains more characters

, than characters

, and the second function is assumed in the opposite case. Expressions like

contain symbols

and

in equal amounts; they are ambiguous and should be avoided in serious deduction.

External links

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