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Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems (view source)
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The "fiber" of a codomain element y in Y under a function f : X -> Y is the subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X. In other language that is often used, the fiber of y under f is called the "antecedent set", the "inverse image", the "level set", or the "pre-image" of y under f. All of these equivalent concepts are defined as follows:
−Fiber of y under f = f^(-1)(y) = {x in X : f(x) = y}.
−In the special case where f is the indicator function f_Q of the set Q c X, the fiber of 1 under fQ is just the set Q back again:
−Fiber of 1 under fQ = fQ-1(1) = {x in X : fQ(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 "[| ... |]" around a sentence or the sign of a proposition, and whose application is defined as follows:
−If f : X -> %B%,
−then [| f |] = f^(-1)(%1%) = {x in X : f(x) = %1%}.
−----
−Some may recognize here fledgling efforts
−to reinforce flights of Fregean semantics
−with impish pitches of Peircean semiotics.
−Some may deem it Icarean, all too Icarean.
−1.3.10.3 Propositions & Sentences (cont.)
−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:
−If f : X -> %B%,
−then [| f |] = f^(-1)(%1%) = {x in X : f(x)}.
−The set-builder frame "{x in X : ... }" requires a grammatical sentence or
−a sentential clause to fill in the blank, as with the sentence "f(x) = %1%"
−that serves to fill the frame in the initial definition of a logical fiber.
−And what is a sentence but the expression of a proposition, in other words,
−the name of an indicator function? As it happens, the sign "f(x)" and the
−sentence "f(x) = %1%" represent the very same value to this context, for
−all x in X, that is, they will appear equal in their truth or falsity
−to any reasonable interpreter of signs or sentences in this context,
−and so either one of them can be tendered for the other, in effect,
−exchanged for the other, within this context, frame, and reception.
−
<pre>
<pre>
−The sign "f(x)" manifestly names the value f(x).
The sign "f(x)" manifestly names the value f(x).
This is a value that can be seen in many lights.
This is a value that can be seen in many lights.