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===6.37. Propositional Types===
===6.37. Propositional Types===
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In this section, I describe a formal system of "type expressions" that are analogous to formulas of propositional logic, and I discuss their use as a calculus of predicates for classifying, analyzing, and drawing typical inferences about n place relations, in particular, for reasoning about the results of operations indicated or performed on relations and about the properties of their transformations and combinations.
Definition. Given a cartesian product XxY, an ordered pair <x, y> C XxY has the type S.T, written <x, y> : S.T, iff x C S c X and y C T c Y. Notice that an ordered pair can have many types.
Definition. A relation R c XxY has type S.T, written R : S.T, iff every <x, y> C R has type S.T, that is, iff R c SxT for some S c X and T c Y.
Notation. "Barred parentheses", like "(" and ")", will be used in pairs to indicate the negations of propositions and the complements of sets. When an n place relation R is initially given relative to the domains X1, ... , Xn and a set S is being mentioned as a subset of one of them, say S c Xi, then the "relevant complement" of S in such a context is the one taken relative to Xi, that is:
(S) = S = (Xi S).
When there is occasion for ambiguities that are not resolved by context then one must resort to indices on the bars, as "(S)i", or revert to writing out the intended term in full, as "(Xi S)".
R : (S(T)).
Fragments
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 2's" 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.
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===6.38. Considering the Source===
===6.38. Considering the Source===
