An interpretation is an assignment of meaning to the
symbols of a
formal language. Many formal languages used in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
logic, and
theoretical computer science are defined in solely
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.
The most commonly studied formal logics are
propositional logic,
predicate logic and their
modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a
function that provides the
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ext ...
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 (for "Abraham Lincoln"). Note that all our interpretation does is assign the extension 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 values of
sentences in a language. If a given interpretation assigns the value True to a sentence or
theory, the interpretation is called a
model of that sentence or theory.
Formal languages
A formal language consists of a possibly infinite set of ''sentences'' (variously called ''words'' or ''
formulas'') built from a fixed set of ''letters'' or ''symbols''. The inventory from which these letters are taken is called the ''
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 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
can be defined with the
alphabet
, and with a word being in
if it begins with
and is composed solely of the symbols
and
.
A possible interpretation of
could assign the decimal digit '1' to
and '0' to
. Then
would denote 101 under this interpretation of
.
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 constants) 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 connectives ∧ ("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''; 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 ''
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. 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 ''
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
truth-functional connectives that represent truth functions — 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 constants, 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.
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
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''.
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 variables) and logical connectives. The only
non-logical symbols 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 values 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
truthbearers instead.
For a language with ''n'' distinct propositional variables there are 2
''n'' distinct possible interpretations. For any particular variable ''a'', for example, there are 2
1=2 possible interpretations: 1) ''a'' is assigned T, or 2) ''a'' is assigned F. For the pair ''a'', ''b'' there are 2
2=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
A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
. The signature consists of a set of non-logical symbols and an identification of each of these symbols as 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
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" below). Finally, the formulas of the language are assembled from atomic formulas using the logical connectives and quantifiers.
Interpretations of a first-order language
To ascribe meaning to all sentences of a first-order language, the following information is needed.
* A
domain 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
n'' → ''D'').
* For every ''n''-ary predicate symbol, an ''n''-ary relation on ''D'' as its interpretation (that is, a subset of ''D
n'').
An object carrying this information is known as a
structure ( 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 variables, 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, is satisfied if and only if both φ and ψ are satisfied.
This leaves the issue of how to interpret formulas of the form and . The domain of discourse forms the
range for these quantifiers. The idea is that the sentence 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 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 variables 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.''
Because the first-order interpretations described here are defined in set theory, they do not associate each predicate symbol with a property (or relation), but rather with the extension of that property (or relation). In other words, these first-order interpretations are
extensional not
intensional.
Example of a first-order interpretation
An example of interpretation
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
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
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
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