Ambiguity

Ambiguity is the property of being ambiguous, where a word, term, notation, sign, symbol, phrase, sentence, or any other form used for communication, is called ambiguous if it can be interpreted in more than one way. Ambiguity is different from vagueness, which arises when the boundaries of meaning are indistinct. Ambiguity is context-dependent: the same linguistic item (be it a word, phrase, or sentence) may be ambiguous in one context and unambiguous in another context. For a word, ambiguity typically refers to an unclear choice between different definitions as may be found in a dictionary. A sentence may be ambiguous due to different ways of parsing the same sequence of words.

Linguistic forms
The lexical ambiguity of a word or phrase consists 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” and “edge of a river”. Another example is as in apothecary. You could say "I bought herbs from the apothecary." This could mean you actually spoke to the apothecary (pharmacist) or went to the apothecary (drug store).

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. Many people believe that such lexically ambiguous, miscommunication-prone words should be avoided wherever possible, since the user generally has to waste time, effort, and attention span to define what is meant when they are used.

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 "wiggle words" and obfuscation are necessary to gain support from multiple constituents with mutually exclusive conflicting desires from their candidate of choice. Ambiguity is a powerful tool of political science.

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 prefixes and suffixes can also create ambiguity (“unlockable” can mean “capable of being unlocked” or “impossible to lock”).

Syntactic ambiguity arises when a complex phrase or a sentence can be parsed in more than one way. “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 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.

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.

Linguistic ambiguity can be a problem in law (see Ambiguity (law)), because the interpretation of written documents and oral agreements is often of paramount importance.

Intentional application
Philosophers (and other users of logic) 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 that hinder economic growth.” 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 correct insertion or omission of a comma after “taxes” and the use of "which" can help reduce ambiguity here (for the first meaning, “, which” is properly used in place of “that”), or the sentence can be restructured to completely eliminate possible misinterpretation. The devious politician hopes that each constituent (politics) will interpret the above 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 rely heavily on the use of ambiguous words and phrases.

In literature and rhetoric, on the other hand, ambiguity can be a useful tool. Groucho Marx’s classic joke depends on a grammatical ambiguity for its humor, for example: “Last night I shot an elephant in my pajamas. What he was doing in my pajamas I’ll never know.” Ambiguity can also be used as a comic device through a genuine intention to confuse, as does Magic: The Gathering's Unhinged © Ambiguity, which makes puns with homophones, mispunctuation, and run-ons: “Whenever a player plays a spell that counters a spell that has been played[,] or a player plays a spell that comes into play with counters, that player may counter the next spell played[,] or put an additional counter on a permanent that has already been played, but not countered.” 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).

In narrative, ambiguity can be introduced in several ways: motive, plot, character. F. Scott Fitzgerald uses the latter type of ambiguity with notable effect in his novel The Great Gatsby.

All religions debate the orthodoxy or heterodoxy of ambiguity. Christianity and Judaism employ the concept of paradox 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's description of the sacred as 'mysterium tremendum et fascinans', the awe-inspiring mystery which fascinates humans.

Metonymy involves the use of the name of a subcomponent part as an abbreviation, or jargon, 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 critical semiotics, metonymy encompasses any potentially ambiguous word substitution that is based on contextual contiguity (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. An increasing amount of research is concentrating on how people react and respond to ambiguous situations. Much of this focuses on ambiguity tolerance. 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 David Wilkinson (2006) found strong correlations between an individual leader's reaction to ambiguous situations and the Modes of Leadership they use, the type of creativity (Kirton (2003) and how they relate to others.

Music
In music, pieces or sections which confound expectations and may be or are interpreted simultaneously in different ways are ambiguous, such as some polytonality, polymeter, other ambiguous meters or rhythms, and ambiguous phrasing, or (Stein 2005, p. 79) any aspect of music. The music of Africa is often purposely ambiguous. To quote Sir Donald Francis Tovey (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, which can be interpreted in two ways. Perceptions of such objects remain stable for a time, then may flip, a phenomenon called multistable perception. The opposite of such ambiguous images are impossible objects.

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. Lojban and Loglan are two related languages which have been created with this in mind. 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 and semantic rules are time-consuming and difficult to learn.

Mathematical notation
Mathematical notation, widely used in physics and other sciences, avoids many ambiguities compared to expression in natural language. However, for various reasons, several lexical, 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, 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 to Mathematica, 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, 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, $$~f=f(x)~$$. Then, if one sees $$~g=f(y+1)~$$, there is no way to distinguish, does it mean $$~f=f(x)~$$ multiplied by $$~(y+1)~$$, or function $$~f~$$ evaluated at argument equal to $$~(y+1)~$$. 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++, MATLAB, Fortran) require the character * as symbol of multiplication. The language Mathematica 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 languages, 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, $$~a/bc~$$ is interpreted as $$~a/(bc)~$$; 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 style, the expression $$~ s i n \alpha~$$ means product of variables $$~s~$$, $$~i~$$, $$~n~$$ and $$~\alpha~$$, although in a slideshow, it may mean $$~\sin[\alpha]~$$.

Comma in subscripts and superscripts sometimes is omitted; it is also ambiguous notation. If it is written $$~T_{mnk}~$$, the reader should guess from the context, does it mean a single-index object, evaluated while the subscript is equal to product of variables $$~m~$$, $$~n~$$ and $$~k~$$, or it is indication to a three-valent tensor. The writing of $$~T_{mnk}~$$ instead of $$~T_{m,n,k}~$$ 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 $$F_{x}$$.

Examples of potentially confusing ambiguous mathematical expressions
$$\sin^2\alpha/2\,$$, which could be understood to mean either $$(\sin(\alpha/2))^2\,$$ or $$(\sin(\alpha))^2/2\,$$. In addition, $$\sin^2(x)$$ may mean $$\sin(\sin(x))$$, as $$\exp^2(x)$$ means $$\exp(\exp(x))$$ (see tetration).

$$~\sin^{-1} \alpha$$, which by convention means $$~\arcsin(\alpha) ~$$, though it might be thought to mean $$(\sin(\alpha))^{-1}\,$$ since $$~\sin^{n} \alpha$$ means $$(\sin(\alpha))^{n}\,$$.

$$a/2b\,$$, which arguably should mean $$(a/2)b\,$$ but would commonly be understood to mean $$a/(2b)\,$$

Notations in quantum optics and quantum mechanics
It is common to define the coherent states in quantum optics with $$~|\alpha\rangle~ $$ and states with fixed number of photons with $$~|n\rangle~$$. Then, there is an "unwritten rule": the state is coherent if there are more Greek characters than Latin characters in the argument, and $$~n~$$photon state if the Latin characters dominate. The ambiguity becomes even worse, if $$~|x\rangle~$$ is used for the states with certain value of the coordinate, and $$~|p\rangle~$$ means the state with certain value of the momentum, which may be used in books on quantum mechanics. Such ambiguities easy lead to confusions, especially if some normalized adimensional, dimensionless variables are used. Expression $$ |1\rangle $$ 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, 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.

A highly confusing term is gain. 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, luminous intensity, radiant intensity, or radiance, depending on the background of the person using the term.

Also, confusions may be related with the use of atomic percent as measure of concentration of a dopant, or resolution 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 and its talk.

The Berry paradox 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 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, $$X=Y$$ leaves open what the value of X is – while its opposite is a self-contradiction, also called inconsistency, paradoxicalness, or oxymoron, in an overdetermined system – such as $$X=2, X=3$$, which has no solution – see also underdetermination.

Logical ambiguity and self-contradiction is analogous to visual ambiguity and impossible objects, such as the Necker cube and impossible cube, or many of the drawings of M. C. Escher.

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:
 * $$~f(\omega)=\int f(t) \exp(i\omega t) {\rm d}t $$.

Rigorously speaking, such an expression requires that $$~ f=0 ~$$; even if function $$~ f ~$$ is a self-Fourier function, the expression should be written as $$~f(\omega)=\frac{1}{\sqrt{2\pi}}\int f(t) \exp(i\omega t) {\rm d}t $$; however, '''it is assumed that the shape of the function  (and even its norm $$\int |f(x)|^2 {\rm d}x $$) 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 $$~t~$$ or $$~\tau~$$, than characters $$~\omega~$$, and the second function is assumed in the opposite case. Expressions like $$~f(\omega t)~$$ or $$~f(y)~$$ contain symbols $$~t~$$ and $$~\omega~$$ in equal amounts; they are ambiguous and should be avoided in serious deduction.