This article needs attention from an expert in Logic. Please add a reason or a talk parameter to this template to explain the issue with the article. WikiProject Logic may be able to help recruit an expert.

In general, an **interpretation** is the assignment of meanings to linguistic utterances and their representations in written language.

In logic, the written language that is used is a formal language. In a formal language, an **interpretation** is the result of assigning meanings, or semantic values to the language's various formulae and other elements. The study of the interpretations of formal languages is called formal semantics^{[1]}

An interpretation of a first-order formal language designates:

**a)** a non-empty set consisting of the domain of discourse (also called *universe of discourse* or *domain of the interpretation*.) This set forms the range of any variables that occur in any statements in the language; **b)** a unique name for each object in the domain, each of which denotes the particular object to which it refers; **c)** a function (or operation) for each function symbol which assigns a truth-value to the result of any sequence of arguments from the domain; **d)** a property or relation for each predicate variable which is consistent with the sequences of objects in the domain which satisfy the property or hold the relation to each other; and **e)** a truth-value for each sentential letter which represents a statement in the language.^{[2]}

The formulas of first-order logic that are tautologies under any interpretation are called valid formulas. A formula is called satisfiable if it takes at least one true value under some interpretation. A formula whose truth table contains only false under any interpretation is called unsatisfiable.
^{[3]}

The Löwenheim-Skolem theorem establishes that any satisfiable formula of first-order logic is satisfiable in a denumerably infinite domain of interpretation. Hence, domains with a cardinality of aleph-0 are sufficient for interpretation of first-order logic.^{[4]}

A sentence is either **true** or **false** under an **interpretation** which assigns values to the logical variables. We might for example make the following assignments:

**Individual Constants**: These are the members of the domain of discourse (as described above in **a**, and **b**).

- : {a,b,c}

- a: Socrates
- b: Plato
- c: Aristotle

**Logical constants**: The function for each function symbol as described in **c** above .

- : "For all"
- : "There exists"
- : "or"
- : "and"

**Predicates**: These are the relations that apply to the members of the domain of discourse (as described above in **d**).

- Fα: α is sleeping
- Gαβ: α hates β
- Hαβγ: α made β hit γ

**Sentential variables:**: (as described above in **e**)

*p*"It is raining."

Under this interpretation the sentences discussed above would represent the following English statements:

*p*: "It is raining."*F*(*a*): "Socrates is sleeping."*H*(*b*,*a*,*c*): "Plato made Socrates hit Aristotle."*x*(*F*(*x*)): "Everybody is sleeping."*z*(*G*(*a*,*z*)): "Socrates hates somebody."*x**y**z*(*H*(*x*,*y*,*z*)): "Somebody made everybody hit somebody."*x**z*(*F*(*x*)*G*(*a*,*z*)): Everybody is sleeping and Socrates hates somebody.*x**y**z*(*G*(*a*,*z*)*H*(*x*,*y*,*z*)): Socrates hates somebody or somebody made everybody hit somebody.

Typically, a distinction is made between a *standard interpretation* and a *non-standard interpretation*. A standard interpretation of a formal language assigns the set of natural numbers as its *domain of discourse*, zero to "0", addition to "+", etcetera. There are other standard interpretations that are isomorphic to the one just given. The Peano axioms are true on each standard interpretation. There also exist *non-standard interpretations* which do not correlate the numerals one-to-one with domain elements.^{[5]}

It has been suggested that Intended interpretation be merged into this article. (Discuss) Proposed since March 2008.

- Interpretation (model theory)
- First-order logic
- Löwenheim-Skolem theorem
- Model (abstract)
- Model theory
- Satisfiable
- Formal semantics
- Modal logic
- Logical system
- Valuation (mathematics)
- Structure (mathematical logic)
- Structure (mathematics)
- Assignment (mathematical logic)

**^**The Cambridge Dictionary of Philosophy. Cambridge University Press; 1999. ISBN 0-521-63722-8*Formal Semantics***^**"interpretation." The Oxford Dictionary of Philosophy. Oxford University Press, 1994, 1996, 2005. Answers.com 01 Dec. 2007. http://www.answers.com/topic/interpretation**^**Alex Sakharov "Interpretation" From MathWorld--A Wolfram Web Resource.**^**Alex Sakharov "Interpretation" From MathWorld--A Wolfram Web Resource.**^**The Cambridge Dictionary of Philosophy. Cambridge University Press; 1999. ISBN 0-521-63722-8*Formal Semantics*

Major fields |
| ||||
---|---|---|---|---|---|

Foundations | |||||

Lists |
| ||||