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There is usually felt to be a slight but significant distinction between
−the "membership statement" that uses the sign "in" as in Example (1) and
−the "type statement" that uses the sign ":" as in examples (2) and (3).
−The difference that appears to be perceived in categorical statements,
−when those of the form "x in X" and those of the form "x : X" 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 sensed to be
−the more serious matter, 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 may have 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 these Examples, those of the form "x in X => #x# in X'"
−and "x : X => #x# : X'", 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 to abbreviate and
−to summarize what is already stated in the protasis.
−
<pre>
<pre>
−A "boolean connection" of degree k, also known as a "boolean function"
A "boolean connection" of degree k, also known as a "boolean function"
on k variables, is a map of the form F : %B%^k -> %B%. In other words,
on k variables, is a map of the form F : %B%^k -> %B%. In other words,
