Interpretation (logic)

From Wikipedia, the free encyclopedia

For other uses, see Interpretation (disambiguation).

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

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.^[1]

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. ^[2]

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.^[3]

[edit] See also

[edit] References

^ "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.

v • d • e

Logic

Main articles

Reason · History of logic (Chinese • Greek • Indian • Islamic) · Philosophical logic · Philosophy of logic · Mathematical logic · Metalogic · Logic in computer science

Key concepts
and logics

Reasoning	Deduction · Induction · Abduction
Informal	Proposition · Inference · Argument · Validity · Cogency · Term logic · Critical thinking · Fallacies · Syllogism
Mathematical	Set · Element · Class · Wff · Axiom · Theorem · Proof · Interpretation · Consistency · Soundness · Completeness · Decidability · Satisfiability · Formal system · Set theory · Proof theory · Model theory · Recursion theory · Type theory · Syntax · Semantics · Semantic consequence · Syntactic consequence ·
Zeroth-order	Boolean functions · Monadic predicate calculus · Propositional calculus · Logical connectives · Truth tables
Predicate	First-order · Quantifiers · Second-order
Modal	Deontic · Epistemic · Temporal · Doxastic
Other non-classical	Computability · Fuzzy · Linear · Relevance · Non-monotonic