Dictionary Definition
interpretation
Noun
2 the act of interpreting something as expressed
in an artistic performance; "her rendition of Milton's verse was
extraordinarily moving" [syn: rendition, rendering]
3 an explanation that results from interpreting
something; "the report included his interpretation of the forensic
evidence"
4 an explanation of something that is not
immediately obvious; "the edict was subject to many
interpretations"; "he annoyed us with his interpreting of
parables"; "often imitations are extended to provide a more
accurate rendition of the child's intended meaning" [syn: interpreting, rendition, rendering]
User Contributed Dictionary
Noun
 The act of interpreting; explanation of what is obscure; translation; version; construction; as, the interpretation of a foreign language, of a dream, or of an enigma.
 The sense given by an interpreter; exposition or explanation given; meaning; as, commentators give various interpretations of the same passage of Scripture.
 The power or explaining.
 An artist's way of expressing his thought or embodying his conception of nature.
 The act or process of applying general principles or formulae to the explanation of the results obtained in special cases.
Translations
act of interpreting
 Portuguese: interpretação
sense given by an interpreter
 Portuguese: interpretação
power or explaining
artist's way of expressing his thought
 Portuguese: interpretação
process of applying general principles to the
explanation of the results
 Portuguese: interpretação
 ttbc French: interprétation
 ttbc Spanish: apreciación
External links
Extensive Definition
In logic
an interpretation gives meaning to an artificial or formal
language or to a
sentence of such a language by assigning a denotation
(extension of a set) to each Nonlogical
symbol (sometimes called nonlogical constant) in that language
or in that sentence. For a given formal language L, or a sentence Φ
an interpretation assigns a denotation to each nonlogical constant
occurring in L or Φ. It specifies a set as the domain or universe
of discourse; to individual constants it assigns elements from the
domain; to predicates of degree 1 it assigns properties
(more precisely extensions of sets); to predicates of degree 2 it
assigns binary relations of individuals; to predicates of degree 3
it assigns ternary relations of individuals, and so on; and to
sentential letters it assigns truthvalues.
More precisely, an interpretation of a formal
language L or of a sentence Φ of such a language, consists of a
nonempty domain D (i.e. a nonempty set) as the universe of
discourse together with an assignment that associates with each
individual constant of L or of Φ an element of D with each
sentential symbol of L or of Φ one of the truthvalues T or F with
each nary operation or function symbol of L or of Φ an nary
operation with respect to D (i.e. a function from Dn into D) with
each nary predicate of L or of Φ an nary relation among elements
of D and (optionally) with some binary predicate I of L, the
identity
relation among elements of D
In this way an interpretation provides meaning or semantic values to the terms
or formulae of the language.
The study of the interpretations of formal languages is called
formal
semantics.. In mathematical
logic an interpretation is a mathematical object that contains
the necessary information for an interpretation in the former
sense.
The symbols used in formal languages include
variables, logicalconstants, quantifiers and punctuation symbols
as well as the nonlogical constants. (For an explanation of these
terms see Firstorder
logic.) The interpretation of a sentence or language therefore
depends on which nonlogical constants it contains. Languages of
the sentential
(or propositional) calculus are allowed sentential symbols as
nonlogical constants. Languages of the first
order predicate calculus allow in addition individual
constants, predicate symbols and operation or function
symbols.
Nomenclature
The term interpretation is synonymous with the term structureThe term model applied to a language is
synonymous with the term interpretation applied to a formal
language
If a sentence is true under an interpretation then
that interpretation is called a model of that sentence
A formula without free variables is called a
sentence.
A sentence which is true under every
interpretation is called logically valid.
A sentence which is false under every
interpretation is called unsatisfiable.
A signature
lists and describes the nonlogical symbols of a formal
language.
In universal algebra and in model theory, a
structure is a type of formal interpretation which consists of
an underlying set along with a collection of finitary functions and
relations which are defined on it.
In mathematical
logic an
assignment can be regarded as an auxiliary notion, an important
step in a specific way for defining the concept of truth formally
(e.g. for firstorder theories). It enables us to give meanings to
terms (truth to sentences) of a language which deals with (free)
variables.
A formal
language is a set of words, i.e. finite strings of letters, or
symbols. The inventory from which these letters are taken is called
the alphabet over which the language is defined. A formal language
is often defined by means of a formal grammar. Formal languages are
a purely syntactical notion, i.e. a priori there is no meaning
associated with them. To distinguish the words of a language from
arbitrary words over its alphabet, they are sometimes called
wellformed words or (in logic) wellformed formulas.
Mathematical
logic is a subfield of logic and mathematics. It consists both
of the mathematical study of logic and the application of this
study to other areas of mathematics. Mathematical logic has close
connections to computer science and philosophical logic, as well.
Unifying themes in mathematical logic include the expressive power
of formal logics and the deductive power of formal proof
systems.
Model theory
studies the models of various formal theories. Here a theory is a
set of sentences in a particular formal language (signature),
while a model is a structure whose interpretation of the symbols of
the signature cause the sentences of the theory to be true. Model
theory is closely related to universal algebra and algebraic
geometry, although the methods of model theory focus more on
logical considerations than those fields.
Notes
The nonlogical constants vary from language to
language and sentence to sentence
Any nonempty set may be chosen as the domain of
an interpretation
All nary relations among the elements of the
domain are candidates for assignment to any predicate of degree
n
A sentence of a formal language is either true
under an interpretation in that language or it is false under that
interpretation in that language
A sentence of a formal language is neither true
nor false except under an interpretation
An interpretation does not associate a predicate
with a property but with its denotation, the elements which have
that property; in other words interpretations are extensional not intensional.
In the case of propositional
logic, a formal interpretation is a function that maps each
propositional variable to one of the truthvalues
true and false. This is also known as a truth assignment.
In the case of firstorder
logic, a formal interpretation is just a
structure (also known as model) of the appropriate signature.
Truthvalue of a sentence depends on the
interpretation.
Nonempty domain requirement
It is stated above that an interpretation must specify a nonempty domain as the universe of disourse. An important reason for this is so that equivalences like:
 (\phi \lor \exists x \psi) \leftrightarrow \exists x (\phi \lor \psi),

 \forall y ( y = y) \lor \exists x ( x = x)

 \exists x ( \forall y ( y = y) \lor x = x)
Empty relations, however, don't cause this
problem since there is no similar notion of passing a relation
symbol across a logical connective, enlarging its scope in the
process.
Methods of presenting an interpretation
There are a variety of ways of giving or presenting an interpretation; the method to be used is not part of the definition of a language.Formal interpretation of a first order formal language
A firstorder
language L is determined by its nonlogical
symbols. The set of nonlogical symbols, together with
information identifying each symbol as a constant symbol or as a
function symbol or predicate symbol of a certain "arity", is also
known as its signature
σ. Terms are assembled from the constant and function symbols
together with the variables. Terms can be combined into an atomic
formula using a predicate symbol (relation symbol) from the
signature or the special predicate symbol =. Finally, the formulas
of the language are assembled from atomic formulas using the
logical connectives and quantifiers.
To ascribe meaning to all sentences of a
firstorder language, the following information is needed.
 A domain of discourse D, usually required to be nonempty.
 For every constant symbol an element of D as its interpretation.
 For every nary function symbol an nary function from D to D, i.e. a function Dn → D, as its interpretation.
 For every nary predicate symbol an nary relation on D, i.e. a subset of Dn, as its interpretation.
Some authors also admit propositional
variables in firstorder logic, which must then also be
interpreted. A propositional variable can stand on its own as an
atomic formula. The interpretation of a propositional variable is
one of the two truthvalues true and false.
The domain of discourse forms the range
of any variables that
occur in any statements in the language. As for structures, the
cardinality of an
interpretation is defined as the cardinality of the domain. The
truthvalue of a formula under a given interpretation is
intuitively clear; mathematically it is defined recursively by the
Tschema,
also known as "Tarski's definition of truth".
The
LöwenheimSkolem theorem establishes that any satisfiable
formula of firstorder logic is satisfiable in a denumerably
infinite domain of interpretation. Hence, countable domains
(i.e. domains whose cardinality is countable) are sufficient for
interpretation of firstorder logic if one is only interested in a
single sentence at a time.
Standard and nonstandard models of arithmetic
A distinction is made between standard and nonstandard models of Peano arithmetic, which is intended to describe the addition and multiplication operations on the natural numbers. The canonical standard model is obtained by taking the set of natural numbers as the domain of discourse, and interpreting "0" as 0, "1" as 1, "+" as the addition, and "x" as the multiplication. All models that are isomorphic to the one just given are also called standard; these models all satisfy the Peano axioms. There also exist nonstandard models of the Peano axioms, which contain elements not correlated with any natural number.See also
 Interpretation (model theory)
 Herbrand interpretation
 Firstorder logic
 LöwenheimSkolem theorem
 Model (abstract)
 Model theory
 Satisfiable
 Formal semantics
 Modal logic
 Logical system
 Valuation (mathematics)
 Structure (mathematical logic)
 Structure (mathematics)
 Assignment (mathematical logic)
 Formal language#Formal interpretations
 Formal interpretation
 Stanford Enc. Phil: Classical Logic, 4. Semantics
References
Synonyms, Antonyms and Related Words
accomplishment, analysis, answer, ascertainment, clarification, clearing
up, construal,
construction,
cracking, decipherment, decoding, definition, denouement, determination, diagnosis, disentanglement,
elucidation,
end, end result, examination, exegesis, explanation, explication, expose, exposition, finding, findingout, illustration, inference, issue, outcome, paraphrasing, reading, reason, rendering, rendition, resolution, resolving, result, riddling, simplification, solution, solving, sorting out, translation, understanding, unraveling, unriddling, unscrambling, unspinning, untangling, untwisting, unweaving, upshot, version, working,
workingout