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| | <pre> | | <pre> |
| − | A "boolean connection" of degree k, also known as a "boolean function"
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| − | on k variables, is a map of the form F : %B%^k -> %B%. In other words,
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| − | a boolean connection of degree k is a proposition about things in the
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| − | universe X = %B%^k.
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| − |
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| − | An "imagination" of degree k on X is a k-tuple of propositions about things
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| − | in the universe X. By way of displaying the various kinds of notation that
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| − | are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given
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| − | as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k. All of
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| − | these features of the typical imagination #f# can be summed up in either one
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| − | of two ways: either in the form of a membership statement, to the effect that
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| − | #f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that
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| − | #f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than
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| − | the former.
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| − | The "play of images" that is determined by #f# and x, more specifically,
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| − | the play of the imagination #f# = <f_1, ..., f_k> that has to with the
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| − | element x in X, is the k-tuple #b# = <b_1, ..., b_k> of values in %B%
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| − | that satisfies the equations b_j = f_j (x), for all j = 1 to k.
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| − |
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| − | A "projection" of %B%^k, typically denoted by "p_j" or "pr_j",
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| − | is one of the maps p_j : %B%^k -> %B%, for j = 1 to k, that is
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| − | defined as follows:
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| − |
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| − | If #b# = <b_1, ..., b_k> in %B%^k,
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| − |
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| − | then p_j (#b#) = p_j (<b_1, ..., b_k>) = b_j in %B%.
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| − |
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| − | The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>.
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| − | A "sentence about things in the universe", for short, a "sentence",
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| − | is a sign that denotes a proposition. In other words, a sentence is
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| − | any sign that denotes an indicator function, any sign whose object is
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| − | a function of the form f : X -> B.
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| − |
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| − | To emphasize the empirical contingency of this definition, one can say
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| − | that a sentence is any sign that is interpreted as naming a proposition,
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| − | any sign that is taken to denote an indicator function, or any sign whose
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| − | object happens to be a function of the form f : X -> B.
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| − |
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| − | ----
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| | I finish out the Subsection on "Propositions & Sentences" with | | I finish out the Subsection on "Propositions & Sentences" with |
| | an account of how I use concepts like "assertion" and "denial". | | an account of how I use concepts like "assertion" and "denial". |
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| − | 1.3.10.3 Propositions & Sentences (cont.)
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| | An "expression" is a type of sign, for instance, a term or a sentence, | | An "expression" is a type of sign, for instance, a term or a sentence, |