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Because this notation is only used in settings where the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> are distinguished by their types, it does not really matter whether one writes <math>{}^{\backprime\backprime} F ~\$~ \underline{f} {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \underline{f} ~\$~ F {}^{\prime\prime}</math> for the initial composition.

Because this notation is only used in settings where the imagination <math>\underline{f} : (X \to \underline\mathbb{B})^k</math> and the connection <math>F : \underline\mathbb{B}^k \to \underline\mathbb{B}</math> are distinguished by their types, it does not really matter whether one writes <math>^{\backprime\backprime} F ~\$~ \underline{f} \, ^{\prime\prime}</math> or <math>^{\backprime\backprime} \underline{f} ~\$~ F \, ^{\prime\prime}</math> for the initial composition.

Just as a sentence is a sign that denotes a proposition, which thereby serves to indicate a set, a propositional connective is a provision of syntax whose mediate effect is to denote an operation on propositions, which thereby manages to indicate the result of an operation on sets.  In order to see how these compound forms of indication can be defined, it is useful to go through the steps that are needed to construct them.  In general terms, the ingredients of the construction are as follows:

Just as a sentence is a sign that denotes a proposition, which thereby serves to indicate a set, a propositional connective is a provision of syntax whose mediate effect is to denote an operation on propositions, which thereby manages to indicate the result of an operation on sets.  In order to see how these compound forms of indication can be defined, it is useful to go through the steps that are needed to construct them.  In general terms, the ingredients of the construction are as follows:

1.  An imagination of degree k on U, in other words, a k-tuple of propositions fj : U -> B, for j = 1 to k, or an object of the form f = <f1, ..., fk> : (U -> B)k.

# An imagination of degree k on U, in other words, a k-tuple of propositions fj : U -> B, for j = 1 to k, or an object of the form f = <f1, ..., fk> : (U -> B)k.

 

# A connection of degree k, in other words, a proposition about things in Bk, or a boolean function of the form F : Bk -> B.

2.  A connection of degree k, in other words, a proposition about things in Bk, or a boolean function of the form F : Bk -> B.

From these materials, it is required to construct a proposition P : U -> B such that P(u) = F(f1(u), ..., fk(u)), for all u C U.  The desired construction is determined as follows:

From these materials, it is required to construct a proposition P : U -> B such that P(u) = F(f1(u), ..., fk(u)), for all u C U.  The desired construction is determined as follows: