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This article is about a technical term in logic. For other uses see Interpretation(disambiguation)
In logic and mathematics, an interpretation (also logical interpretation, mathematical interpretation, logico-mathematical interpretation, or commonly a model) is a type of formal interpretation which gives meaning to an artificial or formal language by assigning a mathematical denotation to all non-logical constants in that language or in a sentence of that language. For a given formal language L, or a sentence Φ of L, an interpretation assigns a denotation to each non-logical constant occurring in L or Φ. To individual constants it assigns individuals (from some universe of discourse); to predicates of degree 1 it assigns properties (more precisely 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 truth-values.
More precisely, an interpretation of a formal language L or of a sentence Φ of L, consists of a non-empty domain D (i.e. a non-empty 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 truth-values T or F with each n-ary operation or function symbol of L or of Φ an n-ary operation with respect to D (i.e. a function from Dn into D) with each n-ary predicate of L or of Φ an n-ary 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.[1] 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 a formal language include variables, logical-constants, quantifiers and punctuation symbols as well as the non-logical constants. The interpretation of a sentence or language therefore depends on which non-logical constants it contains. Langauges of the sentential (or propositional) calculus are allowed sentential symbols as non-logical constants. Languages of the first order predicate calculus allow in addition individual constants, predicate symbols and operation or function symbols.
In the case of propositional logic, a formal interpretation is a function that maps each propositional variable to one of the truth-values true and false. This is also known as a truth assignment. In the case of first-order logic, a formal interpretation is just a structure (also known as model) of the appropriate signature.
A formula without free variables is called a sentence. In general, the truth-value of a sentence depends on the interpretation. A sentence which is true under every interpretation is called logically valid. A sentence which is false under every interpretation is called unsatisfiable.[2]
Notes
The non-logical constants vary from language to language and sentence to sentence
Any non-empty set may be chosen as the domain of an interpretation
All n-ary relations among the elements of the domain are candidates for assignment to any predicate of degree n
There are a variety of ways of giving or presenting an interpretation
The term interpretation is synonymous with the term structure
The 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 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.
Formal interpretation of a first order formal language
A first-order language L is determined by its non-logical symbols. The set of non-logical 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 =.[3] 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 first-order language, the following information is needed.
- A domain of discourse D, usually required to be non-empty.[4]
- 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, i.e. a function Dn → D, as its interpretation.
- For every n-ary predicate symbol an n-ary relation on D, i.e. a subset of Dn, as its interpretation.
An object carrying this information is known as a structure (of signature σ, or σ-structure, or L-structure), or as a "model".
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.[5]
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.[6] The truth-value of a formula under a given interpretation is intuitively clear; mathematically it is defined recursively by the T-schema, also known as "Tarski's definition of truth".
The Löwenheim-Skolem theorem establishes that any satisfiable formula of first-order 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 first-order logic if one is only interested in a single sentence at a time.[2]
Standard and non-standard models of arithmetic
A distinction is made between standard and non-standard 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 zero, "1" as one, "+" as the addition, etcetera. All models that are isomorphic to the one just given are also called standard; these models all satisfy the Peano axioms. There also exist non-standard models of arithmetic, which contain elements not correlated with any natural number but still satisfy the Peano axioms.[1]
See also
References
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