Changes

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 concepts and terms.

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>

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 <math>p_S\!</math> is the intensional representation of the tribe, presiding at the highest level of abstraction, while <math>f_S\!</math> and <math>S\!</math> are its more concrete extensional representations, 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, <math>f_S\!</math> makes the best computational proxy for <math>p_S,\!</math> so I commonly refer to the mapping <math>f_S : X \to \mathbb{B}\!</math> as a proposition on <math>X.\!</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.

Regarded as logical models, the elements of the contension P&Q satisfy the proposition referred to as the "conjunction of extensions" P' and Q'.

Regarded as logical models, the elements of the contension <math>P\!\!\And\!\!Q</math> satisfy the proposition referred to as the ''conjunction of extensions'' <math>P'\!</math> and <math>Q'.\!</math>

Next, the "composition" of P and Q is a dyadic relation R' c XxZ that is notated as R' = P.Q and defined as follows:

Next, the "composition" of P and Q is a dyadic relation R' c XxZ that is notated as R' = P.Q and defined as follows: