Contents 1 Linguistic forms 2 Music 3 Visual art 4 Constructed language 5 Computer science 6 Mathematical notation 6.1 Names of functions 6.2 Expressions 6.3 Examples of potentially confusing ambiguous mathematical expressions 6.4 Notations in quantum optics and quantum mechanics 6.5 Ambiguous terms in physics and mathematics 7 Mathematical interpretation of ambiguity 8 See also 9 References 10 External links Linguistic forms[edit] 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". 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 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.
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.
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] 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.
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"
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (September 2016) (Learn how and when to remove this template message) 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[edit] This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (September 2016) (Learn how and when to remove this template message) Some languages have been created with the intention of avoiding
ambiguity, especially lexical ambiguity.
Sinc function
Elliptic integral of the third kind; translating elliptic integral
form
Expressions[edit] 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 ) displaystyle f=f(x) . Then, if one sees f = f ( y + 1 ) displaystyle f=f(y+1) , there is no way to distinguish whether it means f = f ( x ) displaystyle f=f(x) multiplied by ( y + 1 ) displaystyle (y+1) , or function f displaystyle f evaluated at argument equal to ( y + 1 ) 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 (
a / b c displaystyle a/bc is interpreted as a / ( b c ) 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 s i n α displaystyle sinalpha means product of variables s displaystyle s , i displaystyle i , n displaystyle n and α displaystyle alpha , although in a slideshow, it may mean sin [ α ] displaystyle sin[alpha ] . A comma in subscripts and superscripts sometimes is omitted; it is also ambiguous notation. If it is written T m n k 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 m displaystyle m , n displaystyle n and k displaystyle k , or it is indication to a trivalent tensor. The writing of T m n k displaystyle T_ mnk instead of T m , n , k 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 F x displaystyle F_ x . Examples of potentially confusing ambiguous mathematical expressions[edit] sin 2 α / 2 displaystyle sin ^ 2 alpha /2 , which could be understood to mean either ( sin ( α / 2 ) ) 2 displaystyle (sin(alpha /2))^ 2 or ( sin ( α ) ) 2 / 2 displaystyle (sin(alpha ))^ 2 /2 . In addition, sin 2 ( x ) displaystyle sin ^ 2 (x) may mean sin ( sin ( x ) ) displaystyle sin(sin(x)) , as exp 2 ( x ) displaystyle exp ^ 2 (x) means exp ( exp ( x ) ) displaystyle exp(exp(x)) (see tetration). sin − 1 α displaystyle sin ^ -1 alpha , which by convention means arcsin ( α ) displaystyle arcsin(alpha ) , though it might be thought to mean ( sin ( α ) ) − 1 displaystyle (sin(alpha ))^ -1 , since sin n α displaystyle sin ^ n alpha means ( sin ( α ) ) n displaystyle (sin(alpha ))^ n . a / 2 b displaystyle a/2b , which arguably should mean ( a / 2 ) b displaystyle (a/2)b but would commonly be understood to mean a / ( 2 b ) displaystyle a/(2b) . Notations in quantum optics and quantum mechanics[edit] It is common to define the coherent states in quantum optics with
α ⟩ displaystyle ~alpha rangle ~ and states with fixed number of photons with
n ⟩ displaystyle ~nrangle ~ . Then, there is an "unwritten rule": the state is coherent if there are more Greek characters than Latin characters in the argument, and n displaystyle ~n~ photon state if the Latin characters dominate. The ambiguity becomes even worse, if
x ⟩ displaystyle ~xrangle ~ is used for the states with certain value of the coordinate, and
p ⟩ displaystyle ~prangle ~ 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
1 ⟩ displaystyle 1rangle 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[edit]
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."[4]
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
The
In mathematics and logic, ambiguity can be considered to be an instance of the logical concept of underdetermination—for example, X = Y displaystyle X=Y 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 X = 2 , X = 3 displaystyle X=2,X=3 , which has no solution.
Logical ambiguity and self-contradiction is analogous to visual
ambiguity and impossible objects, such as the
Abbreviation
Essentially contested concept Fallacy Formal fallacy Golden hammer Informal fallacy Self reference Semantics Uncertainty Volatility, uncertainty, complexity and ambiguity Word-sense disambiguation References[edit] ^ "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
External links[edit] Look up ambiguity in Wiktionary, the free dictionary. Zalta, Edward N. (ed.). "Ambiguity". Stanford Encyclopedia of
Philosophy.
v t e Fallacies of relevance
Appeal to worse problems Two wrongs make a right
Appeals to emotion Fear
Flattery
Novelty
Pity
Ridicule
Think of the children
In-group favoritism
Genetic fallacies Ad hominem Abusive Association reductio ad Hitlerum Godwin's law reductio ad Stalinum Circumstantial Appeal to motive Bulverism Poisoning the well Tone Tu quoque Authority Accomplishment Ipse dixit Poverty / Wealth Etymology Nature Tradition / Novelty Chronological snobbery Appeals to consequences Argumentum ad baculum Appeal to force Wishful thinking Other appeals
List of fallacies Other types of fallacy Philosophy portal v t e Formal fallacies Masked man fallacy Non sequitur In propositional logic Affirming a disjunct
Affirming the consequent
Denying the antecedent
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 v t e Informal fallacies Equivocation Equivocation False equivalence False attribution Quoting out of context Loki's Wager No true Scotsman Reification Question-begging fallacies
Correlative-based fallacies
Fallacies of illicit transference Composition Division Deductive fallacies Accident Converse accident Inductive fallacies Anecdotal evidence
Accent
Amphibology
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 List of fallacies Other types of fallacy Philosophy portal v t e Philosophical 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 v t e Philosophy of language Philosophers
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 Roger Gibson 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 Paul Watzlawick Theories Causal theory of reference
Contrast theory of meaning
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Quietism
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 Tas |