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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>

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(<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.

 

 

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.

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.