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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>
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>
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 <math>{}^{\backprime\backprime} L {}^{\prime\prime}</math> is used equivocally in a statement like <math>L = L^{-1}(1),\!</math> where a sensible reading requires it to denote the relational set <math>L \subseteq \textstyle\prod_i X_i</math> on the first appearance and the propositional function <math>L : \textstyle\prod_i X_i \to \mathbb{B}</math> on the second appearance.
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A "triadic relation" is a relation on an ordered triple of nonempty sets. Thus, R is a triadic relation on <X, Y, Z> if and only if R c XxYxZ. Exercising a proper degree of flexibility with notation, one can use the name of a triadic relation R c XxYxZ to refer to a logical predicate or a propositional function, of the type XxYxZ >B, or any one of the derived binary operations, of the types XxY >Pow(Z), XxZ >Pow(Y), YxZ >Pow(X).
A "triadic relation" is a relation on an ordered triple of nonempty sets. Thus, R is a triadic relation on <X, Y, Z> if and only if R c XxYxZ. Exercising a proper degree of flexibility with notation, one can use the name of a triadic relation R c XxYxZ to refer to a logical predicate or a propositional function, of the type XxYxZ >B, or any one of the derived binary operations, of the types XxY >Pow(Z), XxZ >Pow(Y), YxZ >Pow(X).
