Interpretation

Giving a value (meaning) to mathematical expressions (symbols, formulas, etc.). In mathematics such values are mathematical objects (sets, operations, expressions, etc.). The value itself is called an interpretation of the corresponding expression.

Examples. The value (or interpretation) of the symbol $\cdot$ can be the multiplication operation on real numbers, the addition operation on integers, etc. Suppose that the first of these interpretations is used for $\cdot$. If the symbols $x$ and $y$ denote real numbers (i.e. variables with as possible domain of definition the entire real axis), then the value of the expression $x\cdot y$ is the mapping transforming each pair of real numbers into their product; if the values of $x$, $y$ are, respectively, $6$ and $2.5$, then the value of the expression $x\cdot y$ is the number 15. As the value (interpretation) of a statement of planar Lobachevskii geometry in the Poincaré model, the corresponding statement of planar Euclidean geometry may serve.

The most important interpretations are set-theoretical interpretations of expressions of logical languages. If one discusses the simultaneous interpretations of all expressions of a language, then one has an interpretation of the language. A set-theoretical interpretation of a logical language includes a specification of the values of constants — of the object, function and predicate constants and constants of higher degrees (constants for predicates of predicates, etc.), as well as a specification of the domain of applicability of the variables — of the object, function, etc., variables. In multi-sort interpretations, differing object variables may have differing domains of applicability; the same applies for function variables, etc. The interpretations that are most often used, however, are those for which all object variables, as well as the function variables with identical numbers of arguments, etc., have the same domain of applicability. If the domain of variation of the object variables (sometimes called the domain, or support, of the interpretation) is a set $D_0$, then the domain of variation of $n$-place function variables is a set $D_n$ of $n$-place operations on $D_0$. Often $D_n$ is taken to be the set of all $n$-place operations on $D_0$; in this case the domain of variation of the function variables is often not mentioned. Values of object constants are elements of $D_0$, those of function constants are elements of $D_1,D_2,\ldots$.

In a set-theoretical interpretation of a logical language, the interpretation of a term (i.e. the value of the term in the given interpretation) is the mapping assigning to each choice of values of variables of the language (or, in a slightly different definition, to each choice of values of the variables participating in the term) an element of the domain of the interpretation, by a definite rule. This mapping is usually given by induction on the structure of terms.

In order to obtain an interpretation of the formulas of a language, it is necessary, apart from the components mentioned above, to specify some non-empty set $A$, called the set of logical values. The interpretations of $n$-place predicate constants are mappings from $D_0^n$ into $A$; in particular, zero-place predicate constants are elements of $A$. If there are zero-place, one-place, etc. predicate variables in the language, then their domains of variation are, respectively, the set $A$, some subset of $A^{D_0}$ containing the interpretations of all one-place predicate constants, etc. An interpretation of a formula is defined, analogous to that of a term, as a mapping assigning to each choice of values of object, function and predicate variables of the language an element of $A$. An important kind of set-theoretical interpretations are algebraic interpretations, in which operations on $A$ are taken as values (interpretations) of logical connectives, mappings from the set of subsets of $A$ into $A$ (generalized operations on $A$) as values of quantifiers, and where the interpretation of a formula is defined by induction with respect to the structure. Kripke models are the most important among the other set-theoretical interpretations.

The Boolean-valued algebraic interpretations are characterized by the fact that the set $A$ is a complete Boolean algebra; while the values of connectives and quantifiers are: for the conjunction — intersection; for the existential quantifier — taking the least upper bound, etc. Classical interpretations play an especially important role. They are defined as Boolean-valued interpretations with a two-element Boolean algebra $A$.

The concept of truth of formulas in a given interpretation is defined by distinguishing certain elements in $A$. For example, for classical interpretations it is natural to take the unit of the Boolean algebra as distinguished element (the unit is also called "truth"). A formula is called true in a given interpretation if its interpretation takes only distinguished values. A model (or regular interpretation, or simply an interpretation, cf. Model (in logic)) of a system of formulas of a certain language is an interpretation of this language in which all formulas of the system are true.

The term standard interpretation is used when among all possible values (interpretations) of a certain expression there is a generally accepted one. For example, the standard interpretation of the symbol $=$ in a classical interpretation is that of coincidence of elements, and the standard interpretation of $+$ and $\cdot$ in arithmetic are addition and multiplication of natural numbers. Analogously, one introduces the concepts of the standard interpretations of a language and a standard model. In particular, the classical interpretation of first-order arithmetic with predicate constant $=$ and function constants $+$ and $\cdot$, interpreted as above, is called standard.

Apart from set-theoretical interpretations of logical languages one uses others too. E.g., interpretations in which expressions of one logical language are interpreted as expressions in another logical language (see Immersion operation) are used in proving decidability, undecidability and relative consistency of logical theories. See also Constructive logic.

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Besides Kripke models, Boolean-valued models (cf. Boolean-valued model) are important interpretations. Both can be viewed as special cases of sheaf models, or interpretations in topoi (cf. Topos). See, e.g., [a1].

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