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 Prefix es and Suffix es 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 Notation s, widely used in Physics and other Science s, 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.

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 Language 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, $~a/bc~$ 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 $~\sin$ {Link without Title} ~.

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 Journal s 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
H. Haug, S. Koch. Quantum Theory of the Optical and Electronic Properties of Semiconductors, http://www.allbookstores.com/book/9812387560

give the same name to the function and to its Fourier Transform :
: $~f(\omega)=\int f(t) \exp(i\omega t) {
m d}t$ .
Rigorously speaking, such an expression requires that $~ f=0 ~$ ;
even if function $~ f ~$ is a Self-Fourier Function , the expression should be written as
$~f(\omega)=rac{1}{\sqrt{2\pi}}\int f(t) \exp(i\omega t) {
m d}t$ ; however, it is assumed that
the shape of the function (and even its norm

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