referentially transparent An occurrence of a singular term t in a sentence ‘. . . . . .’ is referentially transparent (or purely referential) if and only if the truth-value of ‘. . . . . .’ depends on whether the referent of t satisfies the open sentence ‘. . . . . .’; the satisfaction of ‘. . . . . .’ by the referent of t would guarantee the truth of ‘. . . . . .’, and failure of this individual to satisfy ‘. . . . . .’ would guarantee that ‘. . . . . .’ was not true. ‘Boston is a city’ is true if and only if the referent of ‘Boston’ satisfies the open sentence ‘x is a city’, so the occurrence of ‘Boston’ is referentially transparent. But in ‘The expression ‘Boston’ has six letters’, the length of the word within the quotes, not the features of the city Boston, determines the truth-value of the sentence, so the occurrence is not referentially transparent. According to a Fregean theory of meaning, the reference of any complex expression (that is a meaningful unit) is a function of the referents of its parts. Within this context, an occurrence of a referential term t in a meaningful expression ‘. . . . . .’ is referentially transparent (or purely referential) if and only if t contributes its referent to the reference of ‘. . . . . .’. The expression ‘the area around Boston’ refers to the particular area it does because of the referent of ‘Boston’ (and the reference or extension of the function expressed by ‘the area around x’). An occurrence of a referential term t in a meaningful expression ‘. . . . . .’ is referentially opaque if and only if it is not referentially transparent. Thus, if t has a referentially opaque occurrence in a sentence ‘. . . . . .’, then the truth-value of ‘. . . . . .’ depends on something other than whether the referent of t satisfies ‘. . . . . .’.
Although these definitions apply to occurrences of referential terms, the terms ‘referentially opaque’ and ‘referentially transparent’ are used primarily to classify linguistic contexts for terms as referentially opaque contexts. If t occurs purely referentially in S but not in C(S), then C ( ) is a referentially opaque context. But we must qualify this: C( ) is a referentially opaque context for that occurrence of t in S. It would not follow (without further argument) that C( ) is a referentially opaque context for other occurrences of terms in sentences that could be placed into C( ).
Contexts of quotation, propositional attitude, and modality have been widely noted for their potential to produce referential opacity. Consider: (1) John believes that the number of planets is less than eight. (2) John believes that nine is less than eight. If (1) is true but (2) is not, then either ‘the number of planets’ or ‘nine’ has an occurrence that is not purely referential, because the sentences would differ in truth-value even though the expressions are co-referential. But within the sentences: (3) The number of planets is less than eight. (4) Nine is less than eight. the expressions appear to have purely referential occurrence. In (3) and (4), the truth-value of the sentence as a whole depends on whether the referent of ‘The number of planets’ and ‘Nine’ satisfies ‘x is less than eight’. Because the occurrences in (3) and (4) are purely referential but those in (1) and (2) are not, the context ‘John believes that ( )’ is a referentially opaque context for the relevant occurrence of at least one of the two singular terms. Some argue that the occurrence of ‘nine’ in (2) is purely referential because the truth-value of the sentence as a whole depends on whether the referent, nine, satisfies the open sentence ‘John believes that x is less than eight’. Saying so requires that we make sense of the concept of satisfaction for such sentences (belief sentences and others) and that we show that the concept of satisfaction applies in this way in the case at hand (sentence (2)). There is controversy about whether these things can be done. In (1), on the other hand, the truth-value is not determined by whether nine (the referent of ‘the number of planets’) satisfies the open sentence, so that occurrence is not purely referential. Modal contexts raise similar questions. (5) Necessarily, nine is odd. (6) Necessarily, the number of planets is odd. If (5) is true but (6) is not, then at least one of the expressions does not have a purely referential occurrence, even though both appear to be purely referential in the non-modal sentence that appears in the context ‘Necessarily, ———’. Thus the context is referentially opaque for the occurrence of at least one of these terms. On an alternative approach, genuinely singular terms always occur referentially, and ‘the number of planets’ is not a genuinely singular term. Russell’s theory of definite descriptions, e.g., provides an alternative semantic analysis for sentences involving definite descriptions. This would enable us to say that even simple sentences like (3) and (4) differ considerably in syntactic and semantic structure, so that the similarity that suggests the problem, the seemingly similar occurrences of co-referential terms, is merely apparent. See also DE DICTO, QUANTIFYING IN, SUB- STITUTIVITY SALVA VERITAT. T.M.