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Revision as of 14:27, 15 April 2013
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One way to extend the generic brand of partiality among relations in a non-trivial direction can be charted as follows. If the name or formula of a relation is a PIR to elementary relations, that is, if a sign or expression of an <math>n\!</math>-place relation is interpreted as a proposition about <math>n\!</math>-tuples, then a PIR to relations …
One way to extend the generic brand of partiality among relations in a non-trivial direction can be charted as follows. If the name or formula of a relation is a PIR to elementary relations, that is, if a sign or expression of an <math>n\!</math>-place relation is interpreted as a proposition about <math>n\!</math>-tuples, then a PIR to relations …
===6.37. Propositional Types===
Consider a relation <math>L\!</math> of the following type.
{| align="center" cellspacing="8" width="90%"
| <math>L : \texttt{(} S \texttt{(} T \texttt{))}\!</math>
|}
<pre>
Finally, the set of triples of dyadic relations, with pairwise cartesian products chosen in a pre arranged order from a collection of three sets {X, Y, Z}, is called the "dyadic explosion" of {X, Y, Z}. This object is denoted as "Explo (X, Y, Z; 2)", read as the "explosion of XxYxZ by 2s" or simply as "X, Y, Z, choose 2", and is defined as follows:
Explo (X, Y, Z; 2) = Pow (XxY) x Pow (XxZ) x Pow (YxZ).
This domain is defined well enough for now to serve the immediate purposes of this section, but later it will be necessary to examine its construction more closely.
Just to provide a hint of what's at stake, consider the suggestive identity,
2XY x 2XZ x 2YZ = 2(XY + XZ + YZ),
and ask what sense would have to be found for the sums on the right in order to interpret this equation as a set theoretic isomorphism. Answering this question requires the concept of a "co product", roughly speaking, a "disjointed union" of sets. By the time this discussion has detailed the forms of indexing necessary to maintain these constructions, it should have become patently obvious that the forms of analysis and synthesis that are called on to achieve the putative "reductions to" and "reconstructions from" dyadic relations in actual fact never really leave the realm of genuinely triadic relations, but merely reshuffle its contents in various convenient fashions.
</pre>
==Scrap Area==
==Scrap Area==
