Ambiguity

Ambiguity is the property of words, terms, notations and concepts (within a particular context) as being undefined, undefinable, or without an obvious definition and thus having an unclear meaning.

A word, phrase, sentence, or other communication is called “ambiguous” if it can be interpreted in more than one way. Ambiguity is distinct from vagueness, which arises when the boundaries of meaning are indistinct. Ambiguity is in contrast with definition, and typically refers to an unclear choice between standard definitions, as given by a dictionary, or else understood as common knowledge.

Linguistic forms

Lexical ambiguity arises when context is insufficient to determine the sense of a single word that has more than one meaning. For example, the word “bank” has several meanings, including “financial institution” and “edge of a river,” but if someone says “I deposited $100 in the bank,” the intended meaning is clear. 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” (He is a good person), and probably other similar things. “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 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 also contain such 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 based on 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.

Physics and mathematics

The mathematical notations, widely used in physics and other sciences, are supposed to avoid any ambiguity. However, the application of mathematics require all possible simplifications. This may lead to the lexical, syntactic and semantic ambiguities mentioned above.

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.

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.

Creators of algorithmic languages try to avoid ambiguities. Many algorithmic languages (C++, MATLAB, Fortran, Maple) 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 $~sin\alpha ~$ 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 fee), or aims to increase number of publications without considering readers. The same may apply to any other use of ambiguous notations.

Examples of potentially confusing ambiguous expressions

$\sin ^{2}\alpha /2\,$, which could be understood to mean either $(\sin(\alpha /2))^{2}\,$ or $(\sin(\alpha ))^{2}/2\,$.

$~\sin ^{-1}\alpha$, 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 $a/(2b)\,$

Citations

Some scientific journals required, that all the references are marked, as if they would be exponential functions, for example: ..number of partial lasers does not exceed 10⁹"(can you guess that it is reference number 9, not 1000000000 lasers?). Recently, OSA journals improved the style to avoid such ambiguity; since 2007, February 14, the cites appear in squared parentheses .

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.

Ambiguity of 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.

Examples of ambiguous terms

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.

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).

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.

Many terms are ambiguous. Each use of an ambiguous term should be preceded by the definition, suitable for a specific case.

Psychology and Management

An increasing amount of research is concentrating on how people react and respond to ambiguous and uncertain 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 organised and less casual lifestyle, greater care and precision in descriptions, a lower sensitivity to emotional and unpleasant words, a less acute sense of humour, 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 behaviour.

In the field of leadership Wilkinson (2006) found strong correlations between an individual leaders reaction to ambiguous situations and the Leadership modes they use, the type of creativity (Kirton (2003) and how they relate to others.

Applications

Philosophers (and other users of logic) spend a lot of time and effort searching for and removing 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.” Some will think he opposes taxes in general because they hinder economic growth; others will think he opposes only those taxes that he believes will hinder economic growth (although in writing, the correct insertion or omission of a comma after “taxes” removes ambiguity here - in addition, for the latter meaning, “that” is properly used in place of “which”). The politician hopes that each will interpret the statement in the way he wants, and both will think the politician is on his side. The logical fallacies of amphiboly and equivocation also rely 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, such as 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 'contradicition'. 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.

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 natural languages, although historically, such attempts at language improvement have been criticized.

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.”

There was also a popular rock band in the early 2000's called "Ambiguous." They were based out of Nashville, TN and had a large following until their mysterious break-up in 2005.

There is also a rising indie rock band out of Western Massachusetts called The Ambiguities <http://ambig.org/>. This eclecticism has been years in the making, starting with their previous incarnation as SHArQ. Lead singer, guitarist, and primary songwriter Daniel Hales formed SHArQ in 1998 in his Northampton basement apartment. The band released their first album Future Artifact in 2001. Soon thereafter, (then lead guitarist, now bassist) Leo Hwang-Carlos joined, and keyboard player and vocalist Carrie Ferguson made guest appearances on SHArQ’s 2002 release, Cactus In A Fishbowl Blues.

As the sound and lineup continued to evolve, the name did too. In 2004, the first incarnation of The Ambiguities released eau de ambiguity, of which Daniel Oppenheimer wrote “a defiantly anxious, eclectic and insistent work. Punk riffs, lovely melodies, distortion, hip-hop beats bizarre “psalmbytes,” high flown speculations and earthier songs like “Griddle Good n’ Greasy...not a record made for casual listening. It’s too demanding for that, but justifiably so.”

With the addition of drummer Steven Freiman and harmony vocalist Hilary Weiner, the current Ambiguous ensemble came together in 2004. Based in a Greenfield, MA basement, The Ambiguities continue to “rock and unrock...audiences throughout the Pioneer Valley”(Bill Waltz, Conduit magazine)

Abbreviations

Abbreviations form, perhaps, the richest field of ambiguiguity, see List_of_classical_abbreviations, which is still far from complete. For example, AU may mean Atomic Unit, Astronomical unit, as well as Arbitrary Unit, American University, and a lot of other things. Simple transmutation of the same letters gives University of Arizona (which is 200 km away from the Arizona State University), United Airlines, Unidad Administrativa (Spanish) and so on.

Sometimes, an abbreviation, which looks pretty innocent in one language, allows sexual or dirty interpretation in other language; especially if an abbreviation constructed of several words is used as URL. The interpretation of ambiguous abbreviation should be extremely careful. Better to say, that interpretation of ANY abbreviation should be careful: one never knows, how many meanings may have an apparently obvious abbreviation.

Examples:

• http://www.opticsexpress.org; how an automatic filter can guess, that it does not refer to some editorial publishing materials about some kind of optic sex?
• http://xxx.lanl.gov; (one system manager condemned one "investigador" for intents to access this URL, and the director also was pretty sure that this site refers to an adult material, he neither clicked this link, nor discussed it with the researcher.)

References

See also

• Abbreviation
• Amphibology
• Double entendre
• Imprecise language
• Fallacy

• Formal fallacy
• Informal fallacy

• Semantics
• Ambiguity tolerance
• Essentially contested concept
• Self reference
• Disambiguation

External links

• Collection of Ambiguous or Inconsistent/Incomplete Statements