Intuitively the scope of an operator is the part of an expression over which it holds its effect. The scope of ‘+’ in ‘(3+5)×7’ would be the sum in the brackets, whereas the scope of ‘×’ is the whole expression. In a formal system the scope of an operator is the smallest well-formed formula in which it occurs. A scope ambiguity arises when there is insufficient indication of the scope of an operator, meaning that an expression can be evaluated in two quite different ways; for example (3+5×7) might refer to 56 or to 38.
In a logical calculus, formation rules prevent scope ambiguities, which are common in natural language. In ‘The Master of Balliol College used to be a priest’ the ambiguity can be represented as one of the respective scope of the description and the tense operator: it used to be so that the Master of Balliol College was a priest (e.g. in the 14th century) versus: take the Master of Balliol College, it used to be so that he was a priest. See also de re/de dicto.