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| | <pre> | | <pre> |
| − | There is usually felt to be a slight but significant distinction between
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| − | the "membership statement" that uses the sign "in" as in Example (1) and
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| − | the "type statement" that uses the sign ":" as in examples (2) and (3).
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| − | The difference that appears to be perceived in categorical statements,
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| − | when those of the form "x in X" and those of the form "x : X" are set
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| − | in side by side comparisons with each other, is that a multitude of
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| − | objects can be said to have the same type without having to posit
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| − | the existence of a set to which they all belong. Without trying
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| − | to decide whether I share this feeling or even fully understand
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| − | the distinction in question, I can only try to maintain a style
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| − | of notation that respects it to some degree. It is conceivable
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| − | that the question of belonging to a set is rightly sensed to be
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| − | the more serious matter, one that has to do with the reality of
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| − | an object and the substance of a predicate, than the question of
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| − | falling under a type, that may have more to do with the way that
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| − | a sign is interpreted and the way that information about an object
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| − | is organized. When it comes to the kinds of hypothetical statements
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| − | that appear in these Examples, those of the form "x in X => #x# in X'"
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| − | and "x : X => #x# : X'", these are usually read as implying some order
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| − | of synthetic construction, one whose contingent consequences involve the
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| − | constitution of a new space to contain the elements being compounded and
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| − | the recognition of a new type to characterize the elements being moulded,
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| − | respectively. In these applications, the statement about types is again
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| − | taken to be less presumptive than the corresponding statement about sets,
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| − | since the apodosis is intended to do nothing more than to abbreviate and
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| − | to summarize what is already stated in the protasis.
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| − |
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| | 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, |