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With the appropriate correspondences between these three domains in mind, the general term ''proposition'' can be interpreted in a flexible manner to cover logical, functional, and geometric types of objects.  Thus, a locution like <math>{}^{\backprime\backprime} \text{the proposition}~ F {}^{\prime\prime}\!</math> can be interpreted in three ways:  (1) literally, to denote a logical proposition, (2) functionally, to denote a mapping from a space <math>X\!</math> of propertied or proposed objects to the domain <math>\mathbb{B} = \{ 0, 1 \}\!</math> of truth values, and (3) geometrically, to denote the so-called ''fiber of truth'' <math>F^{-1}(1)\!</math> as a region or a subset of <math>X.\!</math>  For all of these reasons, it is desirable to set up a suitably flexible interpretive framework for propositional logic, where an object introduced as a logical proposition <math>F\!</math> can be recast as a boolean function <math>F : X \to \mathbb{B},\!</math> and understood to indicate the region of the space <math>X\!</math> that is ruled by <math>F.\!</math>

With the appropriate correspondences between these three domains in mind, the general term ''proposition'' can be interpreted in a flexible manner to cover logical, functional, and geometric types of objects.  Thus, a locution like <math>{}^{\backprime\backprime} \text{the proposition}~ F {}^{\prime\prime}\!</math> can be interpreted in three ways:  (1) literally, to denote a logical proposition, (2) functionally, to denote a mapping from a space <math>X\!</math> of propertied or proposed objects to the domain <math>\mathbb{B} = \{ 0, 1 \}\!</math> of truth values, and (3) geometrically, to denote the so-called ''fiber of truth'' <math>F^{-1}(1)\!</math> as a region or a subset of <math>X.\!</math>  For all of these reasons, it is desirable to set up a suitably flexible interpretive framework for propositional logic, where an object introduced as a logical proposition <math>F\!</math> can be recast as a boolean function <math>F : X \to \mathbb{B},\!</math> and understood to indicate the region of the space <math>X\!</math> that is ruled by <math>F.\!</math>

Generally speaking, it does not seem possible to disentangle these three domains from each other or to determine which one is more fundamental.  In practice, due to its concern with the computational implementations of every concept it uses, the present work is biased toward the functional interpretation of propositions.  From this point of view, the abstract intention of a logical proposition <math>F\!</math> is regarded as being realized only when a program is found that computes the function <math>F : X \to \mathbb{B}.\!</math>

Generally speaking, it does not seem possible to disentangle these three domains from each other or to determine which one is more fundamental.  In practice, due to its concern with the computational implementations of every concept it uses, the present work is biased toward the functional interpretation of propositions.  From this point of view, the abstract intention of a logical proposition F is regarded as being realized only when a program is found that computes the function F : X >B.

The functional interpretation of propositional calculus goes hand in hand with an approach to logical reasoning that incorporates "semantic" or "model theoretic" methods, as distinguished from the purely "syntactic" or "proof theoretic" option.  Indeed, the functional conception of a proposition is model theoretic in a double sense, not only because its notations denote functions as their semantic objects, but also because the domains of these functions are spaces of logical interpretations for the propositions, with the points of the domain that lie in the inverse image of truth under the function being the "models" of the proposition.

The functional interpretation of propositional calculus goes hand in hand with an approach to logical reasoning that incorporates ''semantic'' or ''model-theoretic'' methods, as distinguished from the purely ''syntactic'' or ''proof-theoretic'' option.  Indeed, the functional conception of a proposition is model-theoretic in a double sense, not only because its notations denote functions as their semantic objects, but also because the domains of these functions are spaces of logical interpretations for the propositions, with the points of the domain that lie in the inverse image of truth under the function being the ''models'' of the proposition.

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.

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.

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.

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.