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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)

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One of the reasons for pursuing a pragmatic hybrid of semantic and syntactic approaches, rather than keeping to the purely syntactic ways of manipulating meaningless tokens according to abstract rules of proof, is that the model theoretic strategy preserves the form of connection that exists between an agent's concrete particular experiences and the abstract propositions and general properties that it uses to describe its experience. This makes it more likely that a hybrid approach will serve in the realistic pursuits of inquiry, since these efforts involve the integration of deductive, inductive, and abductive sources of knowledge.

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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>

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In this approach to propositional logic, with a view toward computational realization, one begins with a space X, 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 X are effectively known only in terms of their computable features, one can assume that there is a finite set of computable coordinate projections xi : X >B, for i = 1 to n, for some n, that can serve to describe the points of X. This means that there is a computable coordinate representation for X, in other words, a computable map T : X >Bn that describes the points of X insofar as they are known. Thus, each proposition F : X >B can be factored through the coordinate representation T : X >Bn to yield a related proposition f : Bn >B, one that speaks directly about coordinate n tuples but indirectly about points of X. Composing maps on the right, the mapping f is defined by the equation F = T o f. For all practical purposes served by the representation T, the proposition f can be taken as a proxy for the proposition F, saying things about the points of X by means of X's encoding to Bn.

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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 X and a coordinate representation T : X >Bn as parameters of a particular system and speaks of the propositional calculus on a finite set of variables {xi}. In objective terms, this constitutes the "domain of propositions" on the basis {xi}, notated as "DOP{xi}". 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 enmbedding of DOP(X) into DOP(Y) whenever X c Y.

Jon Awbrey

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