MyWikiBiz, Author Your Legacy — Friday November 22, 2024
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, 15:08, 18 January 2009
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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: |
| | | |
− | <pre> | + | {| align="center" cellpadding="2" width="90%" |
− | F$(f1, ..., fk) : U -> B | + | | |
− | :
| + | <math>\begin{array}{lcl} |
− | F$(f1, ..., fk)(u) = F(f(u)) | + | F^\$ (f_1, \ldots, f_k) |
| + | & : & |
| + | X \to \underline\mathbb{B} |
| + | \\ |
| + | \\ |
| + | F^\$ (f_1, \ldots, f_k) (x) |
| + | & = & |
| + | F(\underline{f} (x)) |
| + | \\ |
| + | & = & |
| + | F((f_1, \ldots, f_k) (x)) |
| + | \\ |
| + | & = & |
| + | F(f_1 (x), \ldots, f_k (x)). |
| + | \\ |
| + | \end{array}</math> |
| + | |} |
| | | |
− | = F(<f1, ..., fk>(u))
| + | 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. |
− | | |
− | = F(f1(u), ..., fk(u)).
| |
− | | |
− | 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.
| |
| | | |
| + | <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. |