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In the boolean context, a function ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''' is tantamount to a ''proposition'' about elements of ''X'', and the elements of ''X'' constitute the ''interpretations'' of that proposition.  The fiber ''f''^&ndash;1(1) comprises the set of ''models'' of ''f'', or examples of elements in ''X'' satisfying the proposition ''f''.  The fiber ''f''^&ndash;1(0) collects the complementary set of ''anti-models'', or the exceptions to the proposition ''f'' that exist in ''X''.  Of course, the space of functions (''X''&nbsp;&rarr;&nbsp;'''B''') is isomorphic to the set of all subsets of X, called the ''power set'' of ''X'' and often denoted as <font face="lucida calligraphy">Pow</font>(''X'') or 2^''X''.

In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition.  The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math>

The operation of replacing ''X'' by (''X''&nbsp;&rarr;&nbsp;'''B''') in a type schema corresponds to a certain shift of attitude towards the space ''X'', in which one passes from a focus on the ostensibly individual elements of ''X'' to a concern with the states of information and uncertainty that one possesses about objects and situations in ''X''.  The conceptual obstacles in the path of this transition can be smoothed over by using singular functions (''X''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''') as stepping stones.  First of all, it's an easy step from an element ''x'' of type '''B'''^''n'' to the equivalent information of a singular proposition ''x''&nbsp;:&nbsp;''X''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''', and then only a small jump of generalization remains to reach the type of an arbitrary proposition ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''', perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original ''x''.  I have frequently discovered this to be a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.

The operation of replacing ''X'' by (''X''&nbsp;&rarr;&nbsp;'''B''') in a type schema corresponds to a certain shift of attitude towards the space ''X'', in which one passes from a focus on the ostensibly individual elements of ''X'' to a concern with the states of information and uncertainty that one possesses about objects and situations in ''X''.  The conceptual obstacles in the path of this transition can be smoothed over by using singular functions (''X''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''') as stepping stones.  First of all, it's an easy step from an element ''x'' of type '''B'''^''n'' to the equivalent information of a singular proposition ''x''&nbsp;:&nbsp;''X''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''', and then only a small jump of generalization remains to reach the type of an arbitrary proposition ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''', perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original ''x''.  I have frequently discovered this to be a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.