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The ''fibers'' of truth and falsity under a proposition <math>f : X \to \underline\mathbb{B}</math> are subsets of <math>X\!</math> that are variously described as follows:

The "fibers" of truth and falsity under a proposition f : U �> B are subsets of U that are variously described as follows:

1. The fiber of 1 under f = |f| = f�1(1)

& = & f^{-1} (\underline{1})

& = & \{ x \in X ~:~ f(x) \}.

\text{The fiber of}~ \underline{0} ~\text{under}~ f

& = & \lnot [| f |]

& = & f^{-1} (\underline{0})

& = & \{ x \in X ~:~ \underline{(} f(x) \underline{)} \, \}.

Perhaps this looks like a lot of work for the sake of what seems to be such a trivial form of syntactic transformation, but it is an important step in loosening up the syntactic privileges that are held by the sign of logical equivalence <math>^{\backprime\backprime} \Leftrightarrow \, ^{\prime\prime},</math> as written between logical sentences, and the sign of equality <math>^{\backprime\backprime} = \, ^{\prime\prime},</math> as written between their logical values, or else between propositions and their boolean values.  Doing this removes a longstanding but wholly unnecessary conceptual confound between the idea of an ''assertion'' and notion of an ''equation'', and it allows one to treat logical equality on a par with the other logical operations.

 

 

 

 

 

Perhaps this looks like a lot of work for the sake of what seems to be such a trivial form of syntactic transformation, but it is an important step in loosening up the syntactic privileges that are held by the sign of logical equivalence "<=>", as written between logical sentences, and by the sign of equality "=", as written between their logical values, or else between propositions and their boolean values.  Doing this removes a longstanding but wholly unnecessary conceptual confound between the idea of an "assertion" and notion of an "equation", and it allows one to treat logical equality on a par with the other logical operations.

As a purely informal aid to interpretation, I frequently use the letters "p", "q", and "P", "Q" to denote propositions.  This can serve to tip off the reader that a function is intended as the indicator function of a set, and thus it saves the trouble of declaring the type f : U �> B each time that a function is introduced as a proposition.

As a purely informal aid to interpretation, I frequently use the letters "p", "q", and "P", "Q" to denote propositions.  This can serve to tip off the reader that a function is intended as the indicator function of a set, and thus it saves the trouble of declaring the type f : U �> B each time that a function is introduced as a proposition.