Interpretation (logic)

In logic, an interpretation is the result of assigning meanings, or semantic values to the various formulae and other elements of formal languages.

An interpretation of a 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.

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.

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.

Examples
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)).
 * $$\mathbb{D}$$: {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."
 * xz(F(x)🇦🇩G(a,z)): Everybody is sleeping and Socrates hates somebody.
 * x'y'z (G(a,z)H(x,y,z)): Either Socrates hates somebody or somebody made everybody hit somebody.