[Inquiry] Re: Futures Of Logical Graphs

[Jon Awbrey]

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FOLG.  Note 64

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Arisbe List, Cybernetics List,

Viewing relations of distinction and equivalence as special cases
of order relations, a measure of reflection with respect to these
types of relation is a way of turning some of the statements that
we might make about difference or equality of elements in a given
ordered system into elements residing within that very same order.

Ordinary propositional calculus, in whatever brand of syntax is adequate
to its tasks, will enjoy this type of reflection, since statements about
equivalence or inequivalence, taking the shapes "p <=> q" or "p <=/=> q",
respectively, will themselves be statements that fall within the purview
of its propositional order.

In the light of this reflection on distinction and equivalence, however,
we have already observed that some styles of syntactic calculi are more
direct, efficient, flexible, and succinct than others in the expression
of logical differences and equations, and it's a curious fact that both
Peirce's alpha graphs and Spencer-Brown's primary algebra, in which the
roles of distinctions and equational inferences are so paramount, don't
afford us better expressions of logical difference and logical equality.

This is yet another one of those deficiencies that the cactus language,
which arose after all from the application of reflective operations to
forms of expression, namely, the use of operator variables to discover
additional layers of lawfulness in formal expressions, seems unusually
well suited to supply.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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