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For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus.  This rule of interpretation has exceptions, though.  There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation.  It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus.  Just to provide a hint of what's at stake:  In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about.  Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information.  Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation ''f''^–1 ⊆ '''B''' × '''B'''^''n'', or what is the same thing, ''f''^–1 : '''B''' → ''Pow''('''B'''^''n''), and the ''fibers'' or inverse images ''f''^–1(0) and ''f''^–1(1), associated with each boolean function ''f'' : '''B'''^''n'' → '''B''' that we use.  In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets ''f''^–1(''b''), for ''b'' ∈ '''B''', is part and parcel of understanding the denotative uses of each propositional function ''f''.

For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus.  This rule of interpretation has exceptions, though.  There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation.  It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus.  Just to provide a hint of what's at stake:  In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about.  Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information.  Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use.  In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math>

===Special Classes of Propositions===

===Special Classes of Propositions===