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In this specifically boolean setting, as in the more generally logical context, where ''truth'' under any name is especially valued, it is worth devoting a specialized notation to the ''fiber of truth'' in a proposition, to mark with particular ease and explicitness the set that it indicates.

Fiber of %1% under f_Q  =  (f_Q)^(-1)(%1%)  =  {x in X  :  f_Q (x) = %1%}  =  Q.

In this specifically boolean setting, as in the more generally logical context, where "truth" under any name is especially valued, it is worth devoting a specialized notation to the "fiber of truth" in a proposition, to mark the set that it indicates with a particular ease and explicitness.

For this purpose, I introduce the use of ''fiber bars'' or ''ground signs'', written as a frame of the form <math>[| \, \ldots \, |]</math> around a sentence or the sign of a proposition, and whose application is defined as follows:

For this purpose, I introduce the use of "fiber bars" or "ground signs", written as "[| ... |]" around a sentence or the sign of a proposition, and whose application is defined as follows:

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| <math>\operatorname{If}~ f : X \to \underline\mathbb{B},</math>

| <math>\operatorname{then}~ [| f |] ~=~ f^{-1} (\underline{1}) ~=~ \{ x \in X : f(x) = \underline{1}.</math>

The definition of a fiber, in either the general or the boolean case, is a purely nominal convenience for referring to the antecedent subset, the inverse image under a function, or the pre-image of a functional value.

 

 

The definition of a fiber, in either the general or the boolean case,

is a purely nominal convenience for referring to the antecedent subset,

the inverse image under a function, or the pre-image of a functional value.

The definition of an operator on propositions, signified by framing the signs of propositions with fiber bars or ground signs, remains a purely notational device, and yet the notion of a fiber in a logical context serves to raise an interesting point.  By way of illustration, it is legitimate to rewrite the above definition in the following form:

The definition of an operator on propositions, signified by framing the signs of propositions with fiber bars or ground signs, remains a purely notational device, and yet the notion of a fiber in a logical context serves to raise an interesting point.  By way of illustration, it is legitimate to rewrite the above definition in the following form: