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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, is interpreted as ; in this case,... |
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 |
does not denote the sine function, but the |
product of the three variables |
, |
, |
, although in the informal notation of a slide presentation it may stand for . |
Commas in multi-component subscripts and superscripts are sometimes omitted; this is also potentially ambiguous notation. |
For example, in the notation , the reader can only infer from the context whether it means a single-index object, taken with the subscript equal to product of variables , and , or it is an indication to a trivalent tensor. |
Examples of potentially confusing ambiguous mathematical expressions |
An expression such as can be understood to mean either or . 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 or . |
The expression means in several texts, though it might be thought to mean , since commonly means . Conversely, might seem to mean , as this exponentiation notation usually denotes function iteration: in general, means . However, for trigonometric and hyperbolic functions, this notation conventionally means exponen... |
The expression can be interpreted as meaning ; however, it is more commonly understood to mean . |
Notations in quantum optics and quantum mechanics |
It is common to define the coherent states in quantum optics with and states with fixed number of photons with . Then, there is an "unwritten rule": the state is coherent if there are more Greek characters than Latin characters in the argument, and photon state if the Latin characters dominate. The ambiguity becomes e... |
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 li... |
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. |
Mathematical interpretation of ambiguity |
In mathematics and logic, ambiguity can be considered to be an instance of the logical concept of underdetermination—for example, leaves open what the value of X is—while its opposite is a self-contradiction, also called inconsistency, paradoxicalness, or oxymoron, or in mathematics an inconsistent system—such as , wh... |
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. |
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 ... |
Biology |
In structural biology, ambiguity has been recognized as a problem for studying protein conformations. The analysis of a protein three-dimensional structure consists in dividing the macromolecule into subunits called domains. The difficulty of this task arises from the fact that different definitions of what a domain is... |
Christianity and Judaism |
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. The orthodox Catholic writer G. K. Chesterton regularly employed parado... |
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 ambigu... |
Visual art |
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? |
Social psychology and the bystander effect |
In social psychology, ambiguity is a factor used in determining peoples' responses to various situations. High levels of ambiguity in an emergency (e.g. an unconscious man lying on a park bench) make witnesses less likely to offer any sort of assistance, due to the fear that they may have misinterpreted the situation a... |
Computer science |
In computer science, the SI prefixes kilo-, mega- and giga- were historically used in certain contexts to mean either the first three powers of 1024 (1024, 10242 and 10243) contrary to the metric system in which these units unambiguously mean one thousand, one million, and one billion. This usage is particularly preval... |
Subsequently, the Ki, Mi, and Gi prefixes were introduced so that binary prefixes could be written explicitly, also rendering k, M, and G unambiguous in texts conforming to the new standard—this led to a new ambiguity in engineering documents lacking outward trace of the binary prefixes (necessarily indicating the new ... |
See also |
References |
External links |
Collection of Ambiguous or Inconsistent/Incomplete Statements |
Leaving out ambiguities when writing |
Semantics |
Mathematical notation |
Concepts in epistemology |
Barriers to critical thinking |
Formal semantics (natural language) Abel is a Biblical figure in the Book of Genesis within Abrahamic religions. He was the younger brother of Cain, and the younger son of Adam and Eve, the first couple in Biblical history. He was a shepherd who offered his firstborn flock up to God as an offering. God accepted his off... |
According to Genesis, this was the first murder in the history of mankind. |
Genesis narrative |
Interpretations |
Jewish and Christian interpretations |
According to the narrative in Genesis, Abel ( Hébel, in pausa Hā́ḇel; Hábel; , Hābēl) is Eve's second son. His name in Hebrew is composed of the same three consonants as a root meaning "breath". Julius Wellhausen has proposed that the name is independent of the root. Eberhard Schrader had previously put forward the A... |
In Christianity, comparisons are sometimes made between the death of Abel and that of Jesus, the former thus seen as being the first martyr. In Jesus speaks of Abel as "righteous", and the Epistle to the Hebrews states that "The blood of sprinkling ... [speaks] better things than that of Abel" (). The blood of Jesus i... |
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