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Inverse projections are often referred to as ''extensions'', in spite of the conflict this creates with the ''extensions'' of concepts and terms.

Inverse projections are often referred to as "extensions", in spite of the conflict this creates with the "extensions" of terms, concepts, and sets.

One of the standard turns of phrase that finds use in this setting, not only for translating between ERs and IRs, but for converting both into computational forms, is to associate any set S contained in a space X with two other types of formal objects:  (1) a logical proposition pS known as the characteristic, indicative, or selective proposition of S, and (2) a binary valued function fS: X >B known as the characteristic, indicative, or selective function of S.

One of the standard turns of phrase that finds use in this setting, not only for translating between extensional representations and intensional representations, but for converting both into computational forms, is to associate any set <math>S\!</math> contained in a space <math>X\!</math> with two other types of formal objects:  (1) a logical proposition<math>p_S\!</math> known as the characteristic, indicative, or selective proposition of <math>S,\!</math> and (2) a boolean-valued function <math>f_S : X \to \mathbb{B}\!</math> known as the characteristic, indicative, or selective function of <math>S.\!</math>

Strictly speaking, the logical entity pS is the IR of the tribe, presiding at the highest level of abstraction, while fS and S are its concrete ERs, rendering its concept in functional and geometric materials, respectively.  Whenever it is possible to do so without confusion, I try to use identical or similar names for the corresponding objects and species of each type, and I generally ignore the distinctions that otherwise set them apart.  For instance, in moving toward computational settings, fS makes the best computational proxy for pS, so I commonly refer to the mapping fS: X >B as a "proposition" on X.

Strictly speaking, the logical entity pS is the IR of the tribe, presiding at the highest level of abstraction, while fS and S are its concrete ERs, rendering its concept in functional and geometric materials, respectively.  Whenever it is possible to do so without confusion, I try to use identical or similar names for the corresponding objects and species of each type, and I generally ignore the distinctions that otherwise set them apart.  For instance, in moving toward computational settings, fS makes the best computational proxy for pS, so I commonly refer to the mapping fS: X >B as a "proposition" on X.