Changes

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 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.

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An "imagination" of degree k on U is a k�tuple of propositions about things in the universe U.  By way of displaying the kinds of notation that are used to express this idea, the imagination f = <f1, ..., fk> is given as a sequence of indicator functions fj : U �> B, for j = 1 to k.  All of these features of the typical imagination f can be summed up in either one of two ways:  either in the form of a membership statement, to the effect that f C (U �> B)k, or in the form of a type statement, to the effect that f : (U �> B)k, though perhaps the latter form is slightly more precise than the former.

An "imagination" of degree k on U is a k�tuple of propositions about things in the universe U.  By way of displaying the kinds of notation that are used to express this idea, the imagination f = <f1, ..., fk> is given as a sequence of indicator functions fj : U �> B, for j = 1 to k.  All of these features of the typical imagination f can be summed up in either one of two ways:  either in the form of a membership statement, to the effect that f C (U �> B)k, or in the form of a type statement, to the effect that f : (U �> B)k, though perhaps the latter form is slightly more precise than the former.