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MyWikiBiz, Author Your Legacy — Friday November 22, 2024
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If <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> is a boolean function on <math>k\!</math> variables, then it is possible to define a mapping <math>F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}),</math> in effect, an operation that takes <math>k\!</math> propositions into a single proposition, where <math>F^\$</math> satisfies the following conditions:
 
If <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> is a boolean function on <math>k\!</math> variables, then it is possible to define a mapping <math>F^\$ : (X \to \underline\mathbb{B})^k \to (X \to \underline\mathbb{B}),</math> in effect, an operation that takes <math>k\!</math> propositions into a single proposition, where <math>F^\$</math> satisfies the following conditions:
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<pre>
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{| align="center" cellpadding="2" width="90%"
F$(f1, ..., fk) : U -> B
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|
:
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<math>\begin{array}{lcl}
F$(f1, ..., fk)(u) = F(f(u))
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F^\$ (f_1, \ldots, f_k)
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& : &
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X \to \underline\mathbb{B}
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\\
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\\
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F^\$ (f_1, \ldots, f_k) (x)
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& = &
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F(\underline{f} (x))
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\\
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& = &
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F((f_1, \ldots, f_k) (x))
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\\
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& = &
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F(f_1 (x), \ldots, f_k (x)).
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\\
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\end{array}</math>
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|}
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= F(<f1, ..., fk>(u))
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Thus, <math>F^\$</math> is what a propositional connective denotes, a particular way of connecting the propositions that are denoted by a number of sentences into a proposition that is denoted by a single sentence.
 
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= F(f1(u), ..., fk(u)).
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Thus, F$ is what a propositional connective denotes, a particular way of connecting the propositions that are denoted by a number of sentences into a proposition that is denoted by a single sentence.
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<pre>
 
Now "fX" is sign that denotes the proposition fX, and it certainly seems like a sufficient sign for it.  Why is there is a need to recognize any other signs of it?
 
Now "fX" is sign that denotes the proposition fX, and it certainly seems like a sufficient sign for it.  Why is there is a need to recognize any other signs of it?
 
If one takes a sentence as a type of sign that denotes a proposition and a proposition as a type of function whose values serve to indicate a set, then one needs a way to grasp the overall relation between the sentence and the set as taking place within a "higher order" (HO) sign relation.
 
If one takes a sentence as a type of sign that denotes a proposition and a proposition as a type of function whose values serve to indicate a set, then one needs a way to grasp the overall relation between the sentence and the set as taking place within a "higher order" (HO) sign relation.
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