Ambiguity

"Ambiguous" redirects here. For the film, see Ambiguous (film).

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 a type of uncertainty of meaning in which several interpretations are plausible. It is thus an attribute of any idea or statement whose intended meaning cannot be definitively resolved according to a rule or process with a finite number of steps. (The ambi- part of the name reflects an idea of "two" as in two meanings.)

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 your license AND you need EITHER ten dollars OR a voucher. 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 which are called phonological 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

to the person's bird (the noun "duck", modified by the possessive pronoun "her"), or
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. 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

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 for this, 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 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,^[15]
Hermite polynomial,^[15]^:775

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 $f=f(y+1)$ , there is no way to distinguish whether it means $f=f(x)$ 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++ 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, $a/bc$ 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 $\sin[\alpha]$ .

A 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 trivalent 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. 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, $X=Y$ 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.^[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:

~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.

References

↑ "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.
1 2 Critical Thinking, 10th ed., Ch 3, Moore, Brooke N. and Parker, Richard. McGraw-Hill, 2012
↑ Heidegger, Martin. "The Origin of the Work of Art". Poetry, Language, Thought. Trans. Albert Hofstadter. NY: Harper Collins, 1971, pg. 18.
↑ de Beauvoir, Simone. The Ethics of Ambiguity. Trans. Bernard Frechtman. New York: Citadel Press, 1976 [1948], pg. 8.
↑ de Beauvoir, Ethics, pg. 9.
↑ Foucault, Michel. The History of Sexuality, An Introduction (Vol. 1). Trans Robert Hurley. New York: Vintage Books, 1978.
↑ Anton, Corey. Sources of Significance: Worldly Rejuvenation and Neo-Stoic Heroism. West Lafayette: Purdue University Press, 2010, pg. 35-63.
↑ Corey, Anton. "Authoritative Disambiguation". Professoranton, 2009.
↑ "CSI: Sim8". Chass.utoronto.ca. 1942-11-14. Retrieved 2013-01-23.
↑ Veale, Tony (2003): "Metaphor and Metonymy: The Cognitive Trump-Cards of Linguistic Humor"
↑ in Motivational Styles in Everyday life: A guide to reversal Theory. M.J. Apter (ed) (2001) APA Books
↑ Wilkinson, D.J. (2006) The Ambiguity Advantage: What great leaders are great at. New York Palgrave Macmillan.
↑ Kirton, M.J. (2003)Adaption-Innovation: In the Context of Diversity and Change. Routledge.
↑ Prefixes for binary multiples
1 2 Abramovits, M.; Stegun, I. Handbook on mathematical functions. p. 228.
↑ Wittgenstein, Ludwig (1999). Tractatus Logico-Philosophicus. Dover Publications Inc. p. 39. ISBN 0-486-40445-5.
↑ Russell/Whitehead, Principia Mathematica
↑ Goldstein, Laurence (1996). "Reflexivity, Contradiction, Paradox and M. C. Escher". Leonardo (The MIT Press) 29 (4): 299–308. doi:10.2307/1576313. JSTOR 1576313
↑ Haug, H.; Koch, S. Quantum Theory of the Optical and Electronic Properties of Semiconductors.

External links

Look up ambiguity in Wiktionary, the free dictionary.

Ambiguity entry in the Stanford Encyclopedia of Philosophy
Ambiguity at the Indiana Philosophy Ontology Project
Ambiguity at PhilPapers
Collection of Ambiguous or Inconsistent/Incomplete Statements
Leaving out ambiguities when writing

Fallacies of relevance

Ad nauseam (Argument from repetition)
Argumentum ad ignorantiam (Argument from ignorance)
Argumentum ad lapidem / Proof by assertion
Argumentum e silentio (Argument from silence)
Ignoratio elenchi (Irrelevant conclusion)
Invincible ignorance
Moralistic / Naturalistic
Non sequitur
Rationalization
Red herring (Appeal to worse problems
Two wrongs make a right)
Special pleading
Straw man
Cliché
I'm entitled to my opinion

Appeals to emotion

Genetic fallacies

Ad hominem	Abusive Association (Reductio ad Hitlerum) Circumstantial (Appeal to motive Bulverism) Poisoning the well Tu quoque

Authority (Accomplishment Ipse dixit Poverty / Wealth) Nature Tradition / Novelty (Chronological snobbery)

Etymological fallacy

Appeals to consequences

Argumentum ad baculum (Appeal to force)
Wishful thinking

Other appeals

List of fallacies
Other types of fallacy
Philosophy portal

Formal fallacies

Masked man fallacy

In propositional logic	Affirming a disjunct Affirming the consequent Denying the antecedent Argument from fallacy False dilemma

In quantificational logic	Existential fallacy Illicit conversion Proof by example Quantifier shift

Syllogistic fallacy	Affirmative conclusion from a negative premise Exclusive premises Existential Necessity Four-term fallacy Illicit major Illicit minor Negative conclusion from affirmative premises Undistributed middle

Other types of formal fallacy List of fallacies

Informal fallacies

Correlative-based fallacies	False dilemma (Perfect solution fallacy) Denying the correlative Suppressed correlative

Fallacies of illicit transference	Composition Division

Deductive fallacies	Accident Converse accident

Inductive fallacies	Sampling bias (Cherry picking McNamara fallacy) Base rate fallacy / Conjunction fallacy False analogy Hasty generalization / Slothful induction Misleading vividness Overwhelming exception

Questionable cause	Animistic (Furtive) Correlation proves causation (Cum hoc ergo propter hoc) Gambler's (inverse) Post hoc Regression Single cause Slippery slope Texas sharpshooter Third-cause Wrong direction

Vagueness / ambiguity	Accent Amphibology Continuum fallacy / Sorites paradox False precision Slippery slope

Equivocation	Equivocation False attribution Quoting out of context Loki's Wager No true Scotsman Reification

Question-begging fallacies	Circular reasoning / Begging the question Loaded language (Leading question) Compound question / Loaded question No true Scotsman

List of fallacies Other types of fallacy Philosophy portal

Philosophical logic

Critical thinking & informal logic	Analysis Ambiguity Argument Belief Bias Credibility Evidence Explanation Explanatory power Fact Fallacy Inquiry Opinion Parsimony (Occam's razor) Premise Propaganda Prudence Reasoning Relevance Rhetoric Rigor Vagueness

Theories of deduction	Constructivism Dialetheism Fictionalism Finitism Formalism Intuitionism Logical atomism Logicism Nominalism Platonic realism Pragmatism Realism

Philosophy of language

Philosophers	Plato (Cratylus) Gorgias Confucius Xunzi Aristotle Stoics Pyrrhonists Scholasticism Ibn Rushd Ibn Khaldun Thomas Hobbes Gottfried Wilhelm Leibniz Johann Herder Ludwig Noire Wilhelm von Humboldt Fritz Mauthner Paul Ricœur Ferdinand de Saussure Gottlob Frege Franz Boas Paul Tillich Edward Sapir Leonard Bloomfield Zhuangzi Henri Bergson Lev Vygotsky Ludwig Wittgenstein Philosophical Investigations Tractatus Logico-Philosophicus Bertrand Russell Rudolf Carnap Jacques Derrida Of Grammatology Limited Inc Benjamin Lee Whorf Gustav Bergmann J. L. Austin Noam Chomsky Hans-Georg Gadamer Saul Kripke A. J. Ayer G. E. M. Anscombe Jaakko Hintikka Michael Dummett Donald Davidson Paul Grice Gilbert Ryle P. F. Strawson Willard Van Orman Quine Hilary Putnam David Lewis John Searle Joxe Azurmendi Scott Soames Stephen Yablo John Hawthorne Stephen Neale

Theories	Causal theory of reference Contrast theory of meaning Contrastivism Conventionalism Cratylism Deconstruction Descriptivist theory of names Direct reference theory Dramatism Expressivism Linguistic determinism Logical atomism Logical positivism Mediated reference theory Nominalism Non-cognitivism Phallogocentrism Quietism Relevance theory Semantic externalism Semantic holism Structuralism Supposition theory Symbiosism Theological noncognitivism Theory of descriptions Verification theory

Concepts	Ambiguity Linguistic relativity Meaning Language Truth-bearer Proposition Use–mention distinction Concept Categories Set Class Intension Logical form Metalanguage Mental representation Principle of compositionality Property Sign Sense and reference Speech act Symbol Entity Sentence Statement more...

Related articles	Analytic philosophy Philosophy of information Philosophical logic Linguistics Pragmatics Rhetoric Semantics Formal semantics Semiotics

Category Task Force Discussion

This article is issued from Wikipedia - version of the Wednesday, January 20, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.