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→‎1.3.10.3. Propositions and Sentences: merge variants & delete duplicates
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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.
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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. 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:
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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:
      
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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.
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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:
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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:
      
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The "fiber" of a codomain element y ? Y under a function f : X �> Y is the subset of the domain X that is mapped onto y, in short, f�1(y) ? X.  In other language that is often used, the fiber of y under f is called the "antecedent set", "inverse image", "level set", or "pre�image" of y under f.  All of these equivalent concepts are defined as follows:
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Fiber of y under f  =  f�1(y)  =  {x ? X : f(x) = y}.
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In the special case where f is the indicator function fX of the set X, the fiber of 1 under fX is just the set X back again:
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Fiber of 1 under fX  =  fX�1(1)  =  {u ? U : fX(u) = 1}  =  X.
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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 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:
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If f : U �> B,
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then |f| = f�1(1) = {u ? U : f(u) = 1}.
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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:
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If f : U �> B,
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then |f| = f�1(1) = {u ? U : f(u)}.
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The set�builder frame "{u ? U : ... }" requires a sentence to fill in the blank, as with the sentence "f(u) = 1" that finishes it up in the initial definition of a logical fiber.  And what is a sentence but the expression of a proposition, that is, the name of an indicator function?  As it happens, the sign "f(u)" and the sentence "f(u) = 1" represent the very same value to this context, for all u ? U, that is, they are 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, or exchanged for the other, within this frame.
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The sign "f(u)" manifestly names the value f(u).  This is a value that can be seen in many lights.  It is, at turns:  (1) the value that the proposition f has at the point u, in other words, that it bears at the point where it is evaluated, and that it takes on with respect to the argument or the object that the whole proposition is taken to be about, (2) the value that the proposition f not only takes up the point u, but that it carries, conveys, transfers, or transports into the setting "{u ? U : ... }" or into any other context of discourse where f is meant to be evaluated, (3) the value that the sign "f(u)" has in the context where it is placed, that it stands for in the context where it stands, and that it continues to stand for in this context just so long as the same proposition f and the same object u are borne in mind, and last but not least, (4) the value that the sign "f(u)" represents to its full interpretive context as being its own logical interpretant, namely, the value that it signifies as its canonical connotation to any interpreter of the sign that is cognizant of the context in which it appears.
 
The sign "f(u)" manifestly names the value f(u).  This is a value that can be seen in many lights.  It is, at turns:  (1) the value that the proposition f has at the point u, in other words, that it bears at the point where it is evaluated, and that it takes on with respect to the argument or the object that the whole proposition is taken to be about, (2) the value that the proposition f not only takes up the point u, but that it carries, conveys, transfers, or transports into the setting "{u ? U : ... }" or into any other context of discourse where f is meant to be evaluated, (3) the value that the sign "f(u)" has in the context where it is placed, that it stands for in the context where it stands, and that it continues to stand for in this context just so long as the same proposition f and the same object u are borne in mind, and last but not least, (4) the value that the sign "f(u)" represents to its full interpretive context as being its own logical interpretant, namely, the value that it signifies as its canonical connotation to any interpreter of the sign that is cognizant of the context in which it appears.
  
Jon Awbrey
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