Lexical ambiguity

The lexical ambiguity of a word or phrase applies to it having more than one meaning in the language to which the word belongs.^[4] "Meaning" here refers to whatever should be represented by a good dictionary. For instance, the word "bank" has several distinct lexical definitions, including "financial institution" and "edge of a river". Or consider "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 clearer which of the meanings is intended. If, for instance, someone says "I put $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 make a used word clearer.

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 his or her candidate of choice. Ambiguity is a powerful tool of political science.

More problematic are words whose multiple meanings 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 opened" or "impossible to lock").

Semantic and syntactic ambiguity

Which is wet: the food, or the cat?

Semantic ambiguity occurs when a word, phrase or sentence, taken out of context, has more than one interpretation. In "We saw her duck" (example due to Richard Nordquist), the words "her 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").^[5]

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.^[5] For the notion of, and theoretic results about, syntactic ambiguity in artificial, formal languages (such as computer programming languages), see Ambiguous grammar.

Usually, semantic and syntactic ambiguity go hand in hand. The sentence "We saw her duck" is also syntactically ambiguous. Conversely, a sentence like "He ate the cookies on the couch" is also semantically ambiguous. Rarely, but occasionally, the different parsings of a syntactically ambiguous phrase result in the same meaning. For example, the command "Cook, cook!" can be parsed as "Cook (noun used as vocative), cook (imperative verb form)!", but also as "Cook (imperative verb form), cook (noun used as vocative)!". It is more common that a syntactically unambiguous phrase has a semantic ambiguity; for example, the lexical ambiguity in "Your boss is a funny man" is purely semantic, leading to the response "Funny ha-ha or funny peculiar?"

Spoken language can contain many more types of ambiguities that 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.

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; dealing with complex values, this may cause problems.
• Exponential integral^[6]
• Hermite polynomial^[6]: 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, ${\displaystyle f=f(x)}$. 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 ${\displaystyle f=f(x)}$ 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. In the scientific journal style, one uses roman letters to denote elementary functions, whereas variables are written using italics. For example, in mathematical journals the expression ${\displaystyle sin}$ does not denote the sine function, but the product of the three variables ${\displaystyle s}$, ${\displaystyle i}$, ${\displaystyle n}$, although in the informal notation of a slide presentation it may stand for ${\displaystyle \sin }$.

Commas in multi-component subscripts and superscripts are sometimes omitted; this is also potentially ambiguous notation. For example, in the notation ${\displaystyle T_{mnk}}$, the reader can only infer from the context whether it means a single-index object, taken with the subscript equal to product of variables ${\displaystyle m}$, ${\displaystyle n}$ and ${\displaystyle k}$, or it is an indication to a trivalent tensor.

Examples of potentially confusing ambiguous mathematical expressions

An expression such as ${\displaystyle \sin ^{2}\alpha /2}$ can be understood to mean either ${\displaystyle (\sin(\alpha /2))^{2}}$ or ${\displaystyle (\sin \alpha )^{2}/2}$. Often the author's intention can be understood from the context, in cases where only one of the two makes sense, but an ambiguity like this should be avoided, for example by writing ${\displaystyle \sin ^{2}(\alpha /2)}$ or ${\textstyle {\frac {1}{2}}\sin ^{2}\alpha }$.

The expression ${\displaystyle \sin ^{-1}\alpha }$ means ${\displaystyle \arcsin(\alpha )}$ in several texts, though it might be thought to mean ${\displaystyle (\sin \alpha )^{-1}}$, since ${\displaystyle \sin ^{n}\alpha }$ commonly means ${\displaystyle (\sin \alpha )^{n}}$. Conversely, ${\displaystyle \sin ^{2}\alpha }$ might seem to mean ${\displaystyle \sin(\sin \alpha )}$, as this exponentiation notation usually denotes function iteration: in general, ${\displaystyle f^{2}(x)}$ means ${\displaystyle f(f(x))}$. However, for trigonometric and hyperbolic functions, this notation conventionally means exponentiation of the result of function application.

The expression ${\displaystyle a/2b}$ can be interpreted as meaning ${\displaystyle (a/2)b}$; however, it is more commonly 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 easily 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."^[7]

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 that still can be resolved at the background of statistical noise. See also Accuracy and precision.

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.^[8]