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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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In this approach to propositional logic, with a view toward computational realization, one begins with a space <math>X,\!</math> called a ''universe of discourse'', whose points can be reasonably well described by means of a finite set of logical features. Since the points of the space <math>X\!</math> are effectively known only in terms of their computable features, one can assume that there is a finite set of computable coordinate projections <math>x_i : X \to \mathbb{B},\!</math> for <math>i = 1 ~\text{to}~ n,\!</math> for some <math>n,\!</math> that can serve to describe the points of <math>X.\!</math> This means that there is a computable coordinate representation for <math>X,\!</math> in other words, a computable map <math>T : X \to \mathbb{B}^n\!</math> that describes the points of <math>X\!</math> insofar as they are known. Thus, each proposition <math>F : X \to \mathbb{B}\!</math> can be factored through the coordinate representation <math>T : X \to \mathbb{B}^n\!</math> to yield a related proposition <math>f : \mathbb{B}^n \to \mathbb{B},\!</math> one that speaks directly about coordinate <math>n\!</math>-tuples but indirectly about points of <math>X.\!</math> Composing maps on the right, the mapping <math>f\!</math> is defined by the equation <math>F = T \circ f.\!</math> For all practical purposes served by the representation <math>T,\!</math> the proposition <math>f\!</math> can be taken as a proxy for the proposition <math>F,\!</math> saying things about the points of <math>X\!</math> by means of <math>X\!</math>'s encoding to <math>\mathbb{B}^n.\!</math>
In this approach to propositional logic, with a view toward computational realization, one begins with a space <math>X,\!</math> called a ''universe of discourse'', whose points can be reasonably well described by means of a finite set of logical features. Since the points of the space <math>X\!</math> are effectively known only in terms of their computable features, one can assume that there is a finite set of computable coordinate projections <math>x_i : X \to \mathbb{B},\!</math> for <math>i = 1 ~\text{to}~ n,\!</math> for some <math>n,\!</math> that can serve to describe the points of <math>X.\!</math> This means that there is a computable coordinate representation for <math>X,\!</math> in other words, a computable map <math>T : X \to \mathbb{B}^n\!</math> that describes the points of <math>X\!</math> insofar as they are known. Thus, each proposition <math>F : X \to \mathbb{B}\!</math> can be factored through the coordinate representation <math>T : X \to \mathbb{B}^n\!</math> to yield a related proposition <math>f : \mathbb{B}^n \to \mathbb{B},\!</math> one that speaks directly about coordinate <math>n\!</math>-tuples but indirectly about points of <math>X.\!</math> Composing maps on the right, the mapping <math>f\!</math> is defined by the equation <math>F = T \circ f.\!</math> For all practical purposes served by the representation <math>T,\!</math> the proposition <math>f\!</math> can be taken as a proxy for the proposition <math>F,\!</math> saying things about the points of <math>X\!</math> by means of <math>X\!</math>'s encoding to <math>\mathbb{B}^n.\!</math>
Working under the functional perspective, the formal system known as ''propositional calculus'' is introduced as a general system of notations for referring to boolean functions. Typically, one takes a space <math>X\!</math> and a coordinate representation <math>T : X \to \mathbb{B}^n\!</math> as parameters of a particular system and speaks of the propositional calculus on a finite set of variables <math>\{ \underline{\underline{x_i}} \}.\!</math> In objective terms, this constitutes the ''domain of propositions'' on the basis <math>\{ \underline{\underline{x_i}} \},\!</math> notated as <math>\operatorname{DOP}\{ \underline{\underline{x_i}} \}.\!</math> Ideally, one does not want to become too fixed on a particular set of logical features or to let the momentary dimensions of the space be cast in stone. In practice, this means that the formalism and its computational implementation should allow for the automatic embedding of <math>\operatorname{DOP}(\underline{\underline{X}})\!</math> into <math>\operatorname{DOP}(\underline{\underline{Y}})\!</math> whenever <math>\underline{\underline{X}} \subseteq \underline{\underline{Y}}.\!</math>
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The rest of this section presents the elements of a particular calculus for propositional logic. First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.
The rest of this section presents the elements of a particular calculus for propositional logic. First, I establish the basic notations and summarize the axiomatic presentation of the calculus, and then I give special attention to its functional and geometric interpretations.