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{{Short description|Assignment of meaning to the symbols of a formal language}}
{{Other uses|Interpretation (disambiguation)}}

An '''interpretation''' is an assignment of [[Meaning (philosophy)|meaning]] to the [[symbol (formal)|symbols]] of a [[formal language]]. Many formal languages used in [[mathematics]], [[logic]], and [[theoretical computer science]] are defined in solely [[syntax|syntactic]] terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called [[formal semantics (logic)|formal semantics]].

The most commonly studied formal logics are [[propositional logic]], [[predicate logic]] and their [[modal logic|modal]] analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a [[function (mathematics)|function]] that provides the [[extension (predicate logic)|extension]] of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate ''T'' (for "tall") and assign it the extension {''a''} (for "Abraham Lincoln"). All our interpretation does is assign the extension {a} to the non-logical constant ''T'', and does not make a claim about whether ''T'' is to stand for tall and 'a' for Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like 'and', 'or' and 'not'. Though ''we'' may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.

An interpretation often (but not always) provides a way to determine the [[truth value]]s of [[sentence (mathematical logic)|sentences]] in a language. If a given interpretation assigns the value True to a sentence or [[theory (mathematical logic)|theory]], the interpretation is called a [[model (model theory)|model]] of that sentence or theory.

==Formal languages==
{{main|Formal language}}

A formal language consists of a possibly infinite set of ''sentences'' (variously called ''words'' or ''[[well formed formula|formulas]]'') built from a fixed set of ''letters'' or ''symbols''. The inventory from which these letters are taken is called the ''[[alphabet (computer science)|alphabet]]'' over which the language is defined. To distinguish the strings of symbols that are in a formal language from arbitrary strings of symbols, the former are sometimes called ''[[Well-formed formula|well-formed formulæ]]'' (wff). The essential feature of a formal language is that its syntax can be defined without reference to interpretation. For example, we can determine that (''P'' or ''Q'') is a well-formed formula even without knowing whether it is true or false.

===Example===

A formal language <math>\mathcal{W}</math> can be defined with the
alphabet <math>\alpha = \{ \triangle, \square \}</math>, and with a word being in <math>\mathcal{W}</math> if it begins with <math>\triangle</math> and is composed solely of the symbols <math>\triangle</math> and <math>\square</math>.

A possible interpretation of <math>\mathcal{W}</math> could assign the decimal digit '1' to <math>\triangle</math> and '0' to <math>\square</math>. Then <math>\triangle \square \triangle</math> would denote 101 under this interpretation of <math>\mathcal{W}</math>.

===Logical constants===

In the specific cases of propositional logic and predicate logic, the formal languages considered have alphabets that are divided into two sets: the logical symbols ([[logical constant]]s) and the non-logical symbols. The idea behind this terminology is that ''logical'' symbols have the same meaning regardless of the subject matter being studied, while ''non-logical'' symbols change in meaning depending on the area of investigation.

Logical constants are always given the same meaning by every interpretation of the standard kind, so that only the meanings of the non-logical symbols are changed. Logical constants include quantifier symbols ∀ ("all") and ∃ ("some"), symbols for [[logical connective]]s ∧ ("and"), ∨ ("or"), ¬ ("not"), parentheses and other grouping symbols, and (in many treatments) the equality symbol =.

==General properties of truth-functional interpretations==

Many of the commonly studied interpretations associate each sentence in a formal language with a single truth value, either True or False. These interpretations are called ''truth functional'';{{dubious|reason=The article 'Truth-functional' gives a more restricted definition: the truth-value of a compound sentence should be a function of the truth-value of its sub-sentences.|date=September 2015}} they include the usual interpretations of propositional and first-order logic. The sentences that are made true by a particular assignment are said to be ''[[satisfiable|satisfied]]'' by that assignment.

In [[classical logic]], no sentence can be made both true and false by the same interpretation, although this is not true of glut logics such as LP.<ref>[[Graham Priest|Priest, Graham]], 2008. ''[[An Introduction to Non-Classical Logic|An Introduction to Non-Classical Logic: from If to Is]],'' 2nd ed. Cambridge University Press.</ref> Even in classical logic, however, it is possible that the truth value of the same sentence can be different under different interpretations. A sentence is ''[[Consistency|consistent]]'' if it is true under at least one interpretation; otherwise it is ''inconsistent''. A sentence φ is said to be ''logically valid'' if it is satisfied by every interpretation (if φ is satisfied by every interpretation that satisfies ψ then φ is said to be a ''[[logical consequence]]'' of ψ).

===Logical connectives===

Some of the logical symbols of a language (other than quantifiers) are [[logical connective|truth-functional connectives]] that represent truth functions&nbsp;— functions that take truth values as arguments and return truth values as outputs (in other words, these are operations on truth values of sentences).

The truth-functional connectives enable compound sentences to be built up from simpler sentences. In this way, the truth value of the compound sentence is defined as a certain truth function of the truth values of the simpler sentences. The connectives are usually taken to be [[logical constant]]s, meaning that the meaning of the connectives is always the same, independent of what interpretations are given to the other symbols in a formula.

This is how we define logical connectives in propositional logic:
*¬Φ is True [[iff]] Φ is False.
*(Φ ∧ Ψ) is True iff Φ is True and Ψ is True.
*(Φ ∨ Ψ) is True iff Φ is True or Ψ is True (or both are True).
*(Φ → Ψ) is True iff ¬Φ is True or Ψ is True (or both are True).
*(Φ ↔ Ψ) is True iff (Φ → Ψ) is True and (Ψ → Φ) is True.

So under a given interpretation of all the sentence letters Φ and Ψ (i.e., after assigning a truth-value to each sentence letter), we can determine the truth-values of all formulas that have them as constituents, as a function of the logical connectives. The following table shows how this kind of thing looks. The first two columns show the truth-values of the sentence letters as determined by the four possible interpretations. The other columns show the truth-values of formulas built from these sentence letters, with truth-values determined recursively.

{| class="wikitable" style="text-align:center; margin: 1em auto;"
|+ Logical connectives
!  Interpretation !! Φ !! Ψ !! ¬Φ !! (Φ ∧ Ψ) !! (Φ ∨ Ψ) !! (Φ → Ψ) !! (Φ ↔ Ψ)
|-
|#1|| T || T || F || T || T || T || T
|-
|#2|| T || F || F || F || T || F || F
|-
|#3|| F || T || T || F || T || T || F
|-
|#4|| F || F || T || F || F || T || T
|}

Now it is easier to see what makes a formula logically valid. Take the formula ''F'': (Φ ∨ ¬Φ). If our interpretation function makes Φ True, then ¬Φ is made False by the negation connective. Since the disjunct Φ of ''F'' is True under that interpretation, ''F'' is True. Now the only other possible interpretation of Φ makes it False, and if so, ¬Φ is made True by the negation function. That would make ''F'' True again, since one of ''F''s disjuncts, ¬Φ, would be true under this interpretation. Since these two interpretations for ''F'' are the only possible logical interpretations, and since ''F'' comes out True for both, we say that it is logically valid or tautologous.

==Interpretation of a theory==
{{Main|Theory (mathematical logic)}}

An ''interpretation of a theory'' is the relationship between a theory and some subject matter when there is a [[many-to-one]] correspondence between certain elementary statements of the theory, and certain statements related to the subject matter. If every elementary statement in the theory has a correspondent it is called a ''full interpretation'', otherwise it is called a ''partial interpretation''.<ref>{{cite book |author=Haskell Curry |author-link=Haskell Curry |title=Foundations of Mathematical Logic |url=https://archive.org/details/foundationsofmat0000unse_o5q2 |url-access=registration |publisher=Mcgraw Hill |date=1963 |page=48}}</ref>

==Interpretations for propositional logic==

The formal language for [[propositional logic]] consists of formulas built up from propositional symbols (also called sentential symbols, sentential variables, [[propositional variable]]s) and logical connectives. The only [[non-logical symbol]]s in a formal language for propositional logic are the propositional symbols, which are often denoted by capital letters. To make the formal language precise, a specific set of propositional symbols must be fixed.

The standard kind of interpretation in this setting is a function that maps each propositional symbol to one of the [[truth value]]s true and false. This function is known as a ''truth assignment'' or ''valuation'' function. In many presentations, it is literally a truth value that is assigned, but some presentations assign [[truthbearer]]s instead.

For a language with ''n'' distinct propositional variables there are 2<sup>''n''</sup> distinct possible interpretations. For any particular variable ''a'', for example, there are 2<sup>1</sup>=2 possible interpretations: 1) ''a'' is assigned '''T''', or 2) ''a'' is assigned '''F'''. For the pair ''a'', ''b'' there are 2<sup>2</sup>=4 possible interpretations: 1) both are assigned '''T''', 2) both are assigned '''F''', 3) ''a'' is assigned '''T''' and ''b'' is assigned '''F''', or 4) ''a'' is assigned '''F''' and ''b'' is assigned '''T'''.

Given any truth assignment for a set of propositional symbols, there is a unique extension to an interpretation for all the propositional formulas built up from those variables. This extended interpretation is defined inductively, using the truth-table definitions of the logical connectives discussed above.

==First-order logic==

Unlike propositional logic, where every language is the same apart from a choice of a different set of propositional variables, there are many different first-order languages. Each first-order language is defined by a [[signature (mathematical logic)|signature]]. The signature consists of a set of non-logical symbols and an identification of each of these symbols as either a constant symbol, a function symbol, or a [[predicate symbol]]. In the case of function and predicate symbols, a [[natural number]] [[arity]] is also assigned. The alphabet for the formal language consists of logical constants, the equality relation symbol =, all the symbols from the signature, and an additional infinite set of symbols known as variables.

For example, in the language of [[ring (mathematics)|rings]], there are constant symbols 0 and 1, two binary function symbols + and ·, and no binary relation symbols. (Here the equality relation is taken as a logical constant.)

Again, we might define a first-order language '''L''', as consisting of individual symbols a, b, and c; predicate symbols F, G, H, I and J; variables x, y, z; no function letters; no sentential symbols.

===Formal languages for first-order logic===

Given a signature σ, the corresponding formal language is known as the set of σ-formulas. Each σ-formula is built up out of atomic formulas by means of logical connectives; atomic formulas are built from terms using predicate symbols. The formal definition of the set of σ-formulas proceeds in the other direction: first, terms are assembled from the constant and function symbols together with the variables. Then, terms can be combined into an atomic formula using a predicate symbol (relation symbol) from the signature or the special predicate symbol "=" for equality (see the section "[[#Interpreting equality|Interpreting equality"]] below). Finally, the formulas of the language are assembled from atomic formulas using the logical connectives and quantifiers.

===Interpretations of a first-order language===
{{See also|Interpretation function}}

To ascribe meaning to all sentences of a first-order language, the following information is needed.
* A [[domain of discourse]]{{efn|Sometimes called the "universe of discourse"}} ''D'', usually required to be non-empty (see below).
* For every constant symbol, an element of ''D'' as its interpretation.
* For every ''n''-ary function symbol, an ''n''-ary function from ''D'' to ''D'' as its interpretation (that is, a function ''D<sup>n</sup>''&nbsp;→&nbsp;''D'').
* For every ''n''-ary predicate symbol, an ''n''-ary relation on ''D'' as its interpretation (that is, a subset of ''D<sup>n</sup>'').
An object carrying this information is known as a [[structure (mathematical logic)|structure]] ({{not a typo|of}} signature σ), or σ-structure, or ''L''-structure (of language L), or as a "model".

The information specified in the interpretation provides enough information to give a truth value to any atomic formula, after each of its [[free variable]]s, if any, has been replaced by an element of the domain. The truth value of an arbitrary sentence is then defined inductively using the [[T-schema]], which is a definition of first-order semantics developed by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, {{nowrap|φ ∧ ψ}} is satisfied if and only if both φ and ψ are satisfied.

This leaves the issue of how to interpret formulas of the form {{nowrap|∀ ''x'' φ(''x'')}} and {{nowrap|∃ ''x'' φ(''x'')}}. The domain of discourse forms the [[quantifier (logic)#Range of quantification|range]] for these quantifiers. The idea is that the sentence {{nowrap|∀ ''x'' φ(''x'')}} is true under an interpretation exactly when every substitution instance of φ(''x''), where ''x'' is replaced by some element of the domain, is satisfied. The formula {{nowrap|∃ ''x'' φ(''x'')}} is satisfied if there is at least one element ''d'' of the domain such that φ(''d'') is satisfied.

Strictly speaking, a substitution instance such as the formula φ(''d'') mentioned above is not a formula in the original formal language of φ, because ''d'' is an element of the domain. There are two ways of handling this technical issue. The first is to pass to a larger language in which each element of the domain is named by a constant symbol. The second is to add to the interpretation a function that assigns each variable to an element of the domain. Then the T-schema can quantify over variations of the original interpretation in which this variable assignment function is changed, instead of quantifying over substitution instances.

Some authors also admit [[propositional variable]]s in first-order 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 truth values ''true'' and ''false.''<ref>{{Citation
 | last = Mates
 | first = Benson
|authorlink = Benson Mates
 | title = Elementary Logic, Second Edition
 | publisher = [[Oxford University Press]]
 | year = 1972
 | location = New York
 | pages = [https://archive.org/details/elementarylogic00mate/page/56 56]
 | isbn = 0-19-501491-X
 | url-access = registration
 | url = https://archive.org/details/elementarylogic00mate/page/56
 }}</ref>

Because the first-order interpretations described here are defined in [[set theory]], they do not associate each predicate symbol with a property{{efn|The extension of a property (also called an attribute) is a set of individuals, so a property is a unary relation. E.g. The properties "yellow" and "prime" are unary relations.}} (or relation), but rather with the extension of that property (or relation). In other words, these first-order interpretations are [[Extensional definition|extensional]]{{efn|See also [[Extension (predicate logic)]]}} not [[Intensional definition|intensional]].

===Example of a first-order interpretation===

An example of interpretation <math>\mathcal{I}</math> of the language '''L''' described above is as follows.
* Domain: A chess set
* Individual constants: a: The white King, b: The black Queen, c: The white King's pawn
* F(x): x is a piece
* G(x): x is a pawn
* H(x): x is black
* I(x): x is white
* J(x, y): x can capture y

In the interpretation <math>\mathcal{I}</math> of L:
* the following are true sentences: F(a), G(c), H(b), I(a), J(b, c),
* the following are false sentences: J(a, c), G(a).

===Non-empty domain requirement=== <!--Please preserve this section heading; it is being used in another article -->

As stated above, a first-order interpretation is usually required to specify a nonempty set as the domain of discourse. The reason for this requirement is to guarantee that equivalences such as
<math display="block">(\phi \lor \exists x \psi) \leftrightarrow \exists x (\phi \lor \psi),</math>
where ''x'' is not a free variable of φ, are logically valid. This equivalence holds in every interpretation with a nonempty domain, but does not always hold when empty domains are permitted. For example, the equivalence
<math display="block">[\forall y (y = y) \lor \exists x ( x = x)] \equiv \exists x [ \forall y ( y = y) \lor x = x]</math>
fails in any structure with an empty domain. Thus the proof theory of first-order logic becomes more complicated when empty structures are permitted. However, the gain in allowing them is negligible, as both the intended interpretations and the interesting interpretations of the theories people study have non-empty domains.<ref>{{Citation | last1=Hailperin | first1=Theodore | title=Quantification theory and empty individual-domains |mr=0057820 | year=1953 | journal=[[The Journal of Symbolic Logic]] | volume=18 | pages=197–200 | doi=10.2307/2267402 | issue=3 | publisher=[[Association for Symbolic Logic]] | jstor=2267402| s2cid=40988137 }}</ref><ref>{{Citation | last1=Quine | first1=W. V. |author1link = Willard Quine| title=Quantification and the empty domain |mr=0064715 | year=1954 | journal=The Journal of Symbolic Logic | volume=19 | pages=177–179 | doi=10.2307/2268615 | issue=3 | publisher=Association for Symbolic Logic | jstor=2268615| s2cid=27053902 }}</ref>

Empty relations do not cause any problem for first-order interpretations, because there is no similar notion of passing a relation symbol across a logical connective, enlarging its scope in the process. Thus it is acceptable for relation symbols to be interpreted as being identically false. However, the interpretation of a function symbol must always assign a well-defined and total function to the symbol.

===Interpreting equality===

The equality relation is often treated specially in first order logic and other predicate logics. There are two general approaches.

The first approach is to treat equality as no different than any other binary relation. In this case, if an equality symbol is included in the signature, it is usually necessary to add various axioms about equality to axiom systems (for example, the substitution axiom saying that if ''a'' = ''b'' and ''R''(''a'') holds then ''R''(''b'') holds as well). This approach to equality is most useful when studying signatures that do not include the equality relation, such as the signature for [[set theory]] or the signature for [[second-order arithmetic]] in which there is only an equality relation for numbers, but not an equality relation for set of numbers.

The second approach is to treat the equality relation symbol as a logical constant that must be interpreted by the real equality relation in any interpretation. An interpretation that interprets equality this way is known as a ''normal model'', so this second approach is the same as only studying interpretations that happen to be normal models. The advantage of this approach is that the axioms related to equality are automatically satisfied by every normal model, and so they do not need to be explicitly included in first-order theories when equality is treated this way. This second approach is sometimes called ''first order logic with equality'', but many authors adopt it for the general study of first-order logic without comment.

There are a few other reasons to restrict study of first-order logic to normal models. First, it is known that any first-order interpretation in which equality is interpreted by an [[equivalence relation]] and satisfies the substitution axioms for equality can be cut down to an [[elementary substructure|elementarily equivalent]] interpretation on a subset of the original domain. Thus there is little additional generality in studying non-normal models. Second, if non-normal models are considered, then every consistent theory has an infinite model; this affects the statements of results such as the [[Löwenheim–Skolem theorem]], which are usually stated under the assumption that only normal models are considered.

===Many-sorted first-order logic===

A generalization of first order logic considers languages with more than one ''sort'' of variables. The idea is different sorts of variables represent different types of objects. Every sort of variable can be quantified; thus an interpretation for a many-sorted language has a separate domain for each of the sorts of variables to range over (there is an infinite collection of variables of each of the different sorts). Function and relation symbols, in addition to having arities, are specified so that each of their arguments must come from a certain sort.

One example of many-sorted logic is for planar [[Euclidean geometry]]{{clarification needed|date=June 2022|reason=This should probably refer to a particular axiomatization that the author has in mind. Tarski's axiomatization uses only a single sort, namely points.}}. There are two sorts; points and lines. There is an equality relation symbol for points, an equality relation symbol for lines, and a binary incidence relation ''E'' which takes one point variable and one line variable. The intended interpretation of this language has the point variables range over all points on the [[Euclidean plane]], the line variable range over all lines on the plane, and the incidence relation ''E''(''p'',''l'') holds if and only if point ''p'' is on line ''l''.

==Higher-order predicate logics==

A formal language for [[higher-order logic|higher-order predicate logic]] looks much the same as a formal language for first-order logic. The difference is that there are now many different types of variables. Some variables correspond to elements of the domain, as in first-order logic. Other variables correspond to objects of higher type: subsets of the domain, functions from the domain, functions that take a subset of the domain and return a function from the domain to subsets of the domain, etc. All of these types of variables can be quantified.

There are two kinds of interpretations commonly employed for higher-order logic. ''Full semantics'' require that, once the domain of discourse is satisfied, the higher-order variables range over all possible elements of the correct type (all subsets of the domain, all functions from the domain to itself, etc.). Thus the specification of a full interpretation is the same as the specification of a first-order interpretation. ''Henkin semantics'', which are essentially multi-sorted first-order semantics, require the interpretation to specify a separate domain for each type of higher-order variable to range over. Thus an interpretation in Henkin semantics includes a domain ''D'', a collection of subsets of ''D'', a collection of functions from ''D'' to ''D'', etc. The relationship between these two semantics is an important topic in higher order logic.

==Non-classical interpretations==

The interpretations of propositional logic and predicate logic described above are not the only possible interpretations. In particular, there are other types of interpretations that are used in the study of [[non-classical logic]] (such as [[intuitionistic logic]]), and in the study of modal logic.

Interpretations used to study non-classical logic include [[topological model]]s, [[Boolean-valued model]]s, and [[Kripke model]]s. [[Modal logic]] is also studied using Kripke models.

==Intended interpretations==

Many formal languages are associated with a particular interpretation that is used to motivate them. For example, the first-order signature for set theory includes only one binary relation, ∈, which is intended to represent set membership, and the domain of discourse in a first-order theory of the natural numbers is intended to be the set of natural numbers.

The intended interpretation is called the ''standard model'' (a term introduced by [[Abraham Robinson]] in 1960).<ref>{{cite book|editor=Anthonie Meijers|title=Philosophy of technology and engineering sciences|year=2009|publisher=Elsevier|isbn=978-0-444-51667-1|series=Handbook of the Philosophy of Science|volume=9|author=Roland Müller|chapter=The Notion of a Model}}</ref> In the context of [[Peano arithmetic]], it consists of the natural numbers with their ordinary arithmetical operations. All models that are [[isomorphic]] to the one just given are also called standard; these models all satisfy the [[Peano axioms]]. There are also [[Peano axioms#Nonstandard models|non-standard models of the (first-order version of the) Peano axioms]], which contain elements not correlated with any natural number.

While the intended interpretation can have no explicit indication in the strictly formal [[deductive system|syntactical rules]], it naturally affects the choice of the [[formal grammar|formation]] and [[transformation rules]] of the syntactical system. For example, [[primitive notion|primitive signs]] must permit expression of the concepts to be modeled; [[sentential formula]]s are chosen so that their counterparts in the intended interpretation are [[Meaning (linguistics)|meaningful]] [[declarative sentence]]s; [[axiom|primitive sentences]] need to come out as [[Truth|true]] [[sentence (mathematical logic)|sentences]] in the interpretation; [[rules of inference]] must be such that, if the sentence <math>\mathcal{I}_j</math> is directly [[formal proof|derivable]] from a sentence <math>\mathcal{I}_i</math>, then <math>\mathcal{I}_i \to \mathcal{I}_j</math> turns out to be a true sentence, with {{imp}} meaning [[material conditional|implication]], as usual. These requirements ensure that all [[formal proof|provable]] sentences also come out to be true.<ref>{{cite book|author=Rudolf Carnap|author-link=Rudolf Carnap|title=Introduction to Symbolic Logic and its Applications |url=https://archive.org/details/introductiontosy00carn |url-access=registration |publisher=Dover publications| location=New York |date=1958 |isbn=9780486604534}}</ref>

Most formal systems have many more models than they were intended to have (the existence of [[non-standard model]]s is an example). When we speak about 'models' in [[empirical science]]s, we mean, if we want [[reality]] to be a model of our science, to speak about an ''intended model''. A model in the empirical sciences is an ''intended factually-true descriptive interpretation'' (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same [[domain of discourse]] as the intended one, but other [[value assignment|assignments]] for [[non-logical constant]]s.<ref>{{cite book|editor=[[Hans Freudenthal]] |title=The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (Colloquium proceedings)|publisher=Springer |isbn= 978-94-010-3669-6 |date=Jan 1960}}</ref>{{page needed|reason=Every chapter is written by a different author.|date=September 2015}}

===Example===

Given a simple formal system (we shall call this one <math>\mathcal{FS'}</math>) whose alphabet α consists only of three symbols <math>\{ \blacksquare, \bigstar, \blacklozenge \}</math> and whose formation rule for formulas is:
: 'Any string of symbols of <math>\mathcal{FS'}</math> which is at least 6 symbols long, and which is not infinitely long, is a formula of <math>\mathcal{FS'}</math>. Nothing else is a formula of <math>\mathcal{FS'}</math>.'

The single [[axiom schema]] of <math>\mathcal{FS'}</math> is:

: " <math>\blacksquare \ \bigstar \ast \blacklozenge \ \blacksquare \ast</math> " (where " <math>\ast</math> " is a [[metasyntactic variable]] standing for a finite string of " <math>\blacksquare</math> "s )

A formal proof can be constructed as follows:
# <math>\blacksquare \ \bigstar \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare</math>
# <math>\blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare</math>
# <math>\blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare</math>

In this example the theorem produced " <math>\blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare</math> " can be interpreted as meaning "One plus three equals four." A different interpretation would be to read it backwards as "Four minus three equals one."<ref>{{Hunter 1996}}</ref>{{Failed verification|date=December 2024}}

==Other concepts of interpretation==

There are other uses of the term "interpretation" that are commonly used, which do not refer to the assignment of meanings to formal languages.

In [[model theory]], a structure ''A'' is said to interpret a structure ''B'' if there is a definable subset ''D'' of ''A'', and definable relations and functions on ''D'', such that ''B'' is isomorphic to the structure with domain ''D'' and these functions and relations. In some settings, it is not the domain ''D'' that is used, but rather ''D'' modulo an equivalence relation definable in ''A''. For additional information, see [[Interpretation (model theory)]].

A theory ''T'' is said to interpret another theory ''S'' if there is a finite [[extension by definition]]s ''T''′ of ''T'' such that ''S'' is contained in ''T''′.

==See also==

*[[Conceptual model]]
*[[Free variables and bound variables|Free variables]] and [[Name binding]]
*[[Formal semantics (natural language)]]
*[[Herbrand interpretation]]
*[[Interpretation (model theory)]]
*[[Logical system]]
*[[Löwenheim–Skolem theorem]]
*[[Modal logic]]
*[[Model theory]]
*[[Satisfiable]]
*[[Truth]]

==Notes==
{{notelist}}

==References==

{{reflist}}

==External links==

* [http://plato.stanford.edu/entries/logic-classical/#4 Stanford Enc. Phil: Classical Logic, 4. Semantics]
* {{mathworld|id=formallanguage|title=Formal Language|author=Sakharov, Alex}}
* {{mathworld|id=Connective|title=Connective}}
* {{mathworld|id=interpretation|title=Interpretation|author=Sakharov, Alex}}
* {{mathworld|id=PropositionalCalculus|title=Propositional Calculus|author=Sakharov, Alex}}
* {{mathworld|id=First-OrderLogic|title=First Order Logic|author=Sakharov, Alex}}

{{Mathematical logic}}
{{Metalogic}}

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[[Category:Interpretation (philosophy)]]
[[Category:Model theory]]
[[Category:Philosophy of language]]
[[Category:Semantics]]