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There is usually felt to be a slight but significant distinction between the membership statement of the form <math>^{\backprime\backprime} x \in X \, ^{\prime\prime}</math> as in example (1), and the type statement of the form <math>^{\backprime\backprime} x : X \, ^{\prime\prime}</math> as in examples (2) and (3).  The difference that is perceived in categorical statements, those of the form <math>^{\backprime\backprime} x \in X \, ^{\prime\prime}</math> or <math>^{\backprime\backprime} x : X \, ^{\prime\prime},</math> is that a multitude of objects can be said to have the same type without necessarily positing the existence of a set to which they all belong.  Without trying to decide whether I share this feeling or fully understand this distinction, I can only try to maintain a style of notation that respects it to some degree.  It is conceivable that the question of belonging to a set is rightly sensed to be the more serious issue, one that has to do with the reality of an object and the substance of a predicate, than the question of falling under a type, that has more to do with the way that a sign is interpreted and the way that information about an object is organized.  When it comes to the kinds of hypothetical statements that appear in the present instance, those of the form <math>^{\backprime\backprime} x \in X ~\Leftrightarrow~ \underline{x} \in \underline{X} \, ^{\prime\prime}</math> or <math>^{\backprime\backprime} x : X ~\Leftrightarrow~ \underline{x} : \underline{X} \, ^{\prime\prime},</math> these are usually read as implying some kind of synthesis, whose contingent consequences are the construction of a new space to contain the elements as compounded and the recognition of a new type to characterize the elements as listed, respectively.  In this application, the statement about types is again taken to be weaker than the corresponding statement about sets, since the apodosis is only meant to abbreviate and to summarize what is already stated in the protasis.

There is usually felt to be a slight but significant distinction between a ''membership statement'' of the form <math>^{\backprime\backprime} x \in X \, ^{\prime\prime}</math> and a ''type indication'' of the form <math>^{\backprime\backprime} x : X \, ^{\prime\prime},</math> for instance, as they are used in the examples above.  The difference that appears to be perceived in categorical statements, when those of the form <math>^{\backprime\backprime} x \in X \, ^{\prime\prime}</math> and those of the form <math>^{\backprime\backprime} x : X \, ^{\prime\prime}</math> are set in side by side comparisons with each other, is that a multitude of objects can be said to have the same type without having to posit the existence of a set to which they all belong.  Without trying to decide whether I share this feeling or even fully understand the distinction in question, I can only try to maintain a style of notation that respects it to some degree.  It is conceivable that the question of belonging to a set is rightly regarded as the more serious matter, one that concerns the reality of an object and the substance of a predicate, than the question of falling under a type, that may depend only on the way that a sign is interpreted and the way that information about an object is organized.  When it comes to the kinds of hypothetical statements that appear in the present instance, those of the forms <math>^{\backprime\backprime} x \in X ~\Leftrightarrow~ \underline{x} \in \underline{X} \, ^{\prime\prime}</math> and <math>^{\backprime\backprime} x : X ~\Leftrightarrow~ \underline{x} : \underline{X} \, ^{\prime\prime},</math> these are usually read as implying some order of synthetic construction, one whose contingent consequences involve the constitution of a new space to contain the elements being compounded and the recognition of a new type to characterize the elements being moulded, respectively.  In these applications, the statement about types is again taken to be less presumptive than the corresponding statement about sets, since the apodosis is intended to do nothing more than abbreviate and summarize what is already stated in the protasis.

A ''boolean connection'' of degree <math>k,\!</math> also known as a ''boolean function'' on <math>k\!</math> variables, is a map of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math>  In other words, a boolean connection of degree <math>k\!</math> is a proposition about things in the universe <math>X = \underline\mathbb{B}^k.</math>

A ''boolean connection'' of degree <math>k,\!</math> also known as a ''boolean function'' on <math>k\!</math> variables, is a map of the form <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}.</math>  In other words, a boolean connection of degree <math>k\!</math> is a proposition about things in the universe <math>X = \underline\mathbb{B}^k.</math>