# Ambiguity

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Sir John Tenniel's illustration of the Caterpillar for Lewis Carroll's Alice's Adventures in Wonderland is noted for its ambiguous central figure, whose head can be viewed as being a human male's face with a pointed nose and pointy chin or being the head end of an actual caterpillar, with the first two right "true" legs visible.[1]

Ambiguity is an attribute of any concept, idea, statement or claim whose meaning, intention or interpretation cannot be definitively resolved according to a rule or process consisting of a finite number of steps.

The concept of ambiguity is generally contrasted with vagueness. In ambiguity, specific and distinct interpretations are permitted (although some may not be immediately apparent), whereas with information that is vague, it is difficult to form any interpretation at the desired level of specificity.

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

Structural analysis of an ambiguous Spanish sentence:
'Pepe vio a Pablo enfurecido
Interpretation 1: When Pepe was angry, then he saw Pablo
Interpretation 2: Pepe saw that Pablo was angry.
Here, the syntactic tree in figure represents interpretation 2.

The lexical ambiguity of a word or phrase pertains to its having more than one meaning in the language to which the word belongs. "Meaning" here 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". One could say "I bought herbs from the apothecary". This could mean one actually spoke to the apothecary (pharmacist) or went to the apothecary (pharmacy).

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 buried $100 in the bank", most people would not think someone used a shovel to dig in the mud. However, some linguistic contexts do not provide sufficient information to disambiguate a used word. For example, 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. 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 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 sentence can have two (or more) different meanings because of the structure of the sentence—its syntax. This is often due to a modifying expression, such as a prepositional phrase, the application of which is unclear. "He ate the cookies on the couch", for example, could mean that he ate those cookies that 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. "To get in, you will need an entrance fee of$10 or your voucher and your drivers' license." This could mean that you need EITHER ten dollars OR BOTH your voucher and your license. Or it could mean that you need EITHER ten dollars OR a voucher AND you also need your license. Only rewriting the sentence, or placing appropriate punctuation can resolve a syntactic ambiguity.[2] For the notion of, and theoretic results about, syntactic ambiguity in artificial, formal languages (such as computer programming languages), see Ambiguous grammar.

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 happens when a sentence contains an ambiguous word or phrase—a word or phrase that has more than one meaning. In "We saw her duck" (example due to Richard Nordquist), the word "duck" can refer either

1. to the person's bird (the noun "duck", modified by the possessive pronoun "her"), or
2. to a motion she made (the verb "duck", the subject of which is the objective pronoun "her", object of the verb "saw").[2]

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} 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, 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 which hinder economic growth", an example of a glittering generality. 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 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 rely heavily on the use of ambiguous words and phrases. In Continental philosophy (particularly phenomenology and existentialism), there is much greater tolerance of ambiguity, as it is generally seen as an integral part of the human condition. Martin Heidegger argued that the relation between the subject and object is ambiguous, as is the relation of mind and body, and part and whole.[3] In Heidegger's phenomenology, Dasein 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 follows Heidegger in defining the human essence as ambiguous, or relating fundamentally to such ambiguity. Simone de Beauvoir tries to base an ethics on Heidegger's and Sartre's writings (The Ethics of Ambiguity), 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 the 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.".[4] 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".[5] Other continental philosophers suggest that concepts such as life, nature, and sex are ambiguous.[6] 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.[7] Following Ernest Becker, he argues that the desire to 'authoritatively disambiguate' the world and existence has led to numerous ideologies and historical events such as genocide. 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.[8] In literature and rhetoric, 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. 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). 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. Christianity and Judaism employ the concept of paradox synonymously with 'ambiguity'. Many Christians and Jews endorse Rudolf Otto's description of the sacred as 'mysterium tremendum et fascinans', the awe-inspiring mystery which fascinates humans.{{ safesubst:#invoke:Unsubst||$N=Dubious |date=__DATE__ |\$B= {{#invoke:Category handler|main}}[dubious ] }} The orthodox Catholic writer G. K. Chesterton regularly employed paradox to tease out the meanings in common concepts which he found ambiguous, or to reveal meaning often overlooked or forgotten in common phrases. (The title of one of his most famous books, Orthodoxy, itself employing such a paradox.)

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,[9] 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.[10]

### 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)[11] 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)[12] 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)[13] 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

The Necker cube, an ambiguous image

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

## Computer science

In computer science, the SI prefixes kilo-, mega- and giga- are used ambiguously to mean either the first three powers of 1000 (1000, 10002 and 10003) or the first three powers of 1024 (1024, 10242 and 10243), respectively.[14]

## 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:

### 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, ${\displaystyle f=f(x)}$. Then, if one sees ${\displaystyle f=f(y+1)}$, there is no way to distinguish whether it means ${\displaystyle f=f(x)}$ multiplied by ${\displaystyle (y+1)}$, or function ${\displaystyle f}$ evaluated at argument equal to ${\displaystyle (y+1)}$. 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++ and Fortran) require the character * as symbol of multiplication. The Wolfram language used in 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, ${\displaystyle a/bc}$ is interpreted as ${\displaystyle a/(bc)}$; 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 ${\displaystyle sin\alpha }$ means product of variables ${\displaystyle s}$, ${\displaystyle i}$, ${\displaystyle n}$ and ${\displaystyle \alpha }$, although in a slideshow, it may mean ${\displaystyle \sin[\alpha ]}$.

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

### Examples of potentially confusing ambiguous mathematical expressions

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

### Notations in quantum optics and quantum mechanics

It is common to define the coherent states in quantum optics with ${\displaystyle ~|\alpha \rangle ~}$ and states with fixed number of photons with ${\displaystyle ~|n\rangle ~}$. Then, there is an "unwritten rule": the state is coherent if there are more Greek characters than Latin characters in the argument, and ${\displaystyle ~n~}$photon state if the Latin characters dominate. The ambiguity becomes even worse, if ${\displaystyle ~|x\rangle ~}$ is used for the states with certain value of the coordinate, and ${\displaystyle ~|p\rangle ~}$ 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 ${\displaystyle |1\rangle }$ 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. Just like Ludwig Wittgenstein states in Tractatus Logico-Philosophicus: "... Only in the context of a proposition has a name meaning." [16]

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.[17]

## Mathematical interpretation of ambiguity

The Necker cube and impossible cube, an underdetermined and overdetermined object, respectively.

In mathematics and logic, ambiguity can be considered to be an underdetermined system (of equations or logic) – for example, ${\displaystyle X=Y}$ 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 ${\displaystyle X=2,X=3}$, 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.[18]

## Pedagogic use of ambiguous expressions

Ambiguity can be used as a pedagogical trick, to force students to reproduce the deduction by themselves. Some textbooks[19] give the same name to the function and to its Fourier transform:

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

Rigorously speaking, such an expression requires that ${\displaystyle ~f=0~}$; even if function ${\displaystyle ~f~}$ is a self-Fourier function, the expression should be written as ${\displaystyle ~f(\omega )={\frac {1}{\sqrt {2\pi }}}\int f(t)\exp(i\omega t){\rm {d}}t}$; however, it is assumed that the shape of the function (and even its norm ${\displaystyle \int |f(x)|^{2}{\rm {d}}x}$) 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 ${\displaystyle ~t~}$ or ${\displaystyle ~\tau ~}$, than characters ${\displaystyle ~\omega ~}$, and the second function is assumed in the opposite case. Expressions like ${\displaystyle ~f(\omega t)~}$ or ${\displaystyle ~f(y)~}$ contain symbols ${\displaystyle ~t~}$ and ${\displaystyle ~\omega ~}$ in equal amounts; they are ambiguous and should be avoided in serious deduction.

## References

1. "And do you see its long nose and chin? At least, they look exactly like a nose and chin, that is don't they? But they really are two of its legs. You know a Caterpillar has got quantities of legs: you can see more of them, further down." Carroll, Lewis. The Nursery "Alice". Dover Publications (1966), p 27.
2. Critical Thinking, 10th ed., Ch 3, Moore, Brooke N. and Parker, Richard. McGraw-Hill, 2012
3. Heidegger, Martin. "The Origin of the Work of Art". Poetry, Language, Thought. Trans. Albert Hofstadter. NY: Harper Collins, 1971, pg. 18.
4. de Beauvoir, Simone. The Ethics of Ambiguity. Trans. Bernard Frechtman. New York: Citadel Press, 1976 [1948], pg. 8.
5. de Beauvoir, Ethics, pg. 9.
6. Foucault, Michel. The History of Sexuality, An Introduction (Vol. 1). Trans Robert Hurley. New York: Vintage Books, 1978.
7. Anton, Corey. Sources of Significance: Worldly Rejuvenation and Neo-Stoic Heroism. West Lafayette: Purdue University Press, 2010, pg. 35-63.
8. Template:Cite web
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10. Veale, Tony (2003): "Metaphor and Metonymy: The Cognitive Trump-Cards of Linguistic Humor"[1]
11. in Motivational Styles in Everyday life: A guide to reversal Theory. M.J. Apter (ed) (2001) APA Books
12. Wilkinson, D.J. (2006) The Ambiguity Advantage: What great leaders are great at. New York Palgrave Macmillan.
13. Kirton, M.J. (2003)Adaption-Innovation: In the Context of Diversity and Change. Routledge.
14. Prefixes for binary multiples
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