Valuation refers to the assignment of truth values to propositions or formulae in any logical expression.
The term “valuation” normally applies only to propositional logic, and refers to the assignment of truth values to individual propositional variables. For example, if we have the formula (P ∧ Q), then we have four possible valuations. We can express these four valuations by means of a truth table:
|P||Q||P ∧ Q|
Each row of this table corresponds to a single valuation — an interpretation of P (as true or false) and an interpretation of Q (as true or false), and the corresponding result in the truth-functional formula (P ∧ Q). Note how, on the valuation P=True, Q=True, the formula (P ∧ Q) is true. On every other valuation, the formula (P ∧ Q) is false, because P and Q are not both true.
Note that we are also not looking for all possible combinations of truth and falsity among all of the columns — we are only concerned with the possible truth values of the atomic propositional variables P and Q. What happens to (P ∧ Q) is a result of those truth values. It is nonsense to speak of a valuation that makes P false, Q false, but (P ∧ Q) true — such a valuation does not exist.
Without a valuation, (P ∧ Q) has no semantic meaning, it is merely a syntactic expression of an operation on propositional variables. It is only under a valuation that these formulas mean anything.
The term “interpretation” is sometimes used in the place of a valuation, since it adds semantic interpretation to syntactic formulae. The term interpretation is also used for the same concept in first-order logic and higher-order logics, where truth values are applied to predicate symbols by means of sets.