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, 20:00, 15 January 2009→Propositions and Sentences: delete reconciled variants
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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,
−a boolean connection of degree k is a proposition about things in the
−universe X = %B%^k.
−An "imagination" of degree k on X is a k-tuple of propositions about things
−in the universe X. By way of displaying the various kinds of notation that
−are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given
−as a sequence of indicator functions f_j : X -> %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# is in (X -> %B%)^k, or in the form of a type statement, to the effect that
−#f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than
−the former.
−The "play of images" that is determined by #f# and x, more specifically,
−the play of the imagination #f# = <f_1, ..., f_k> that has to with the
−element x in X, is the k-tuple #b# = <b_1, ..., b_k> of values in %B%
−that satisfies the equations b_j = f_j (x), for all j = 1 to k.
−A "projection" of %B%^k, typically denoted by "p_j" or "pr_j",
−is one of the maps p_j : %B%^k -> %B%, for j = 1 to k, that is
−defined as follows:
−If #b# = <b_1, ..., b_k> in %B%^k,
−then p_j (#b#) = p_j (<b_1, ..., b_k>) = b_j in %B%.
−The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>.
−A "sentence about things in the universe", for short, a "sentence",
−is a sign that denotes a proposition. In other words, a sentence is
−any sign that denotes an indicator function, any sign whose object is
−a function of the form f : X -> B.
−To emphasize the empirical contingency of this definition, one can say
−that a sentence is any sign that is interpreted as naming a proposition,
−any sign that is taken to denote an indicator function, or any sign whose
−object happens to be a function of the form f : X -> B.
−----
−1.3.10.3 Propositions & Sentences (cont.)
<pre>
<pre>
−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".
−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,
