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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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, 21:18, 17 April 2012→6.6. Basic Notions of Group Theory: markup
First, <math>L\!</math> can be associated with a logical predicate or a proposition that says something about the space of effects, being true of certain effects and false of all others. In this way, <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from <math>\textstyle\prod_i X_i</math> to the domain of truth values <math>\mathbb{B} = \{ 0, 1 \}.</math> With the appropriate understanding, it is permissible to let the notation <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times X_k \to \mathbb{B} {}^{\prime\prime}</math> indicate this interpretation.
First, <math>L\!</math> can be associated with a logical predicate or a proposition that says something about the space of effects, being true of certain effects and false of all others. In this way, <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from <math>\textstyle\prod_i X_i</math> to the domain of truth values <math>\mathbb{B} = \{ 0, 1 \}.</math> With the appropriate understanding, it is permissible to let the notation <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times X_k \to \mathbb{B} {}^{\prime\prime}</math> indicate this interpretation.
Second, <math>L\!</math> can be associated with a piece of information that allows one to complete various sorts of partial data sets in the space of effects. In particular, if one is given a partial effect or an incomplete <math>k\!</math>-tuple, say, one that is missing a value in the <math>j^\text{th}\!</math> place, as indicated by the notation <math>{}^{\backprime\backprime} (x_1, \ldots, \hat{j}, \ldots, x_k) {}^{\prime\prime},</math> then <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> can be interpreted as naming a function from the cartesian product of the domains at the filled places to the power set of the domain at the missing place. With this in mind, it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to \operatorname{Pow}(X_j) {}^{\prime\prime}</math> indicate this use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math> If the sets in the range of this function are all singletons, then it is permissible to let <math>{}^{\backprime\backprime} L : X_1 \times \ldots \times \hat{j} \times \ldots \times X_k \to X_j {}^{\prime\prime}</math> specify the corresponding use of <math>{}^{\backprime\backprime} L {}^{\prime\prime}.</math>
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In general, the indicated degrees of freedom in the interpretation of relation symbols can be exploited properly only if one understands the consequences of this interpretive liberality and is prepared to deal with the problems that arise from its "polymorphic" practices, from using the same sign in different contexts to refer to different types of objects. For example, one should consider what happens, and what sort of IF is demanded to deal with it, when the name "R" is used equivocally in a statement like "R = R 1(1)", where a sensible reading requires it to denote the relational set R c Xi Xi on the first appearance and the propositional function R : Xi Xi > B on the second appearance.
In general, the indicated degrees of freedom in the interpretation of relation symbols can be exploited properly only if one understands the consequences of this interpretive liberality and is prepared to deal with the problems that arise from its "polymorphic" practices, from using the same sign in different contexts to refer to different types of objects. For example, one should consider what happens, and what sort of IF is demanded to deal with it, when the name "R" is used equivocally in a statement like "R = R 1(1)", where a sensible reading requires it to denote the relational set R c Xi Xi on the first appearance and the propositional function R : Xi Xi > B on the second appearance.