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===Commentary Note 12.3 : Older Notes===

===Commentary on Selection 12 : Old Notes===

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In other words, <math>(\mathfrak{S}^\mathfrak{L})_{ab} = 0</math> if and only if there exists an <math>x \in X</math> such that <math>\mathfrak{S}_{ax} = 0</math> and <math>\mathfrak{L}_{xb} = 1.</math>

In other words, <math>(\mathfrak{S}^\mathfrak{L})_{ab} = 0</math> if and only if there exists an <math>x \in X</math> such that <math>\mathfrak{S}_{ax} = 0</math> and <math>\mathfrak{L}_{xb} = 1.</math>

===Commentary on Selection 12===

===Commentary on Selection 12 : Older Notes===

The logic of terms is something of a lost art these days, when the current thinking in logic tends to treat the complete proposition as the quantum of discourse, ''ne plus infra''.  With absolute terms, or monadic relatives, and the simpler operations on dyadic relatives, the necessary translations between propositions and terms are obvious enough, but now that we've reached the threshold of higher adic relatives and operations as complex as exponentiation, it is useful to stop and consider the links between these two languages.

The logic of terms is something of a lost art these days, when the current thinking in logic tends to treat the complete proposition as the quantum of discourse, ''ne plus infra''.  With absolute terms, or monadic relatives, and the simpler operations on dyadic relatives, the necessary translations between propositions and terms are obvious enough, but now that we've reached the threshold of higher adic relatives and operations as complex as exponentiation, it is useful to stop and consider the links between these two languages.

In the special case where <math>Y = \mathbb{B} = \{ 0, 1 \},</math> the function space <math>\mathbb{B}^X</math> is the set of functions <math>\{ f : X \to \mathbb{B} \}.</math>  If the elements <math>0, 1 \in \mathbb{B}</math> are interpreted as the logical values <math>\operatorname{false}, \operatorname{true},</math> respectively, then a function of the type <math>X \to \mathbb{B}</math> may be interpreted as a ''proposition'' about the elements in <math>X.\!</math>

In the special case where <math>Y = \mathbb{B} = \{ 0, 1 \},</math> the function space <math>\mathbb{B}^X</math> is the set of functions <math>\{ f : X \to \mathbb{B} \}.</math>  If the elements <math>0, 1 \in \mathbb{B}</math> are interpreted as the logical values <math>\operatorname{false}, \operatorname{true},</math> respectively, then a function of the type <math>X \to \mathbb{B}</math> may be interpreted as a ''proposition'' about the elements in <math>X.\!</math>

===Old Commentary Notes===

===Really Old Commentary Notes===

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