First-Order Logical Duality
"In the propositional case, one passes from a propositional theory to a
Boolean algebra by
constructing the Lindenbaum-Tarski algebra of the theory, a construction
which identifies provably equivalent formulas (and orders them by provable
implication). Thus any two complete theories, for instance, are
equivalent' in the sense of having isomorphic Lindenbaum-Tarski algebras.
The situation is precisely analogous to a presentation of an algebra
by generators and relations: a logical theory corresponds to such a
and two theories are equivalent if they present 'the same' -- i.e.
The construction of the Lindenbaum-Tarski algebra is implemented by
1) identification of provably equivalent formulas
2) ordering them by provable implication
1) might be equivalent to your sheaf of infinities of computations
(but requires a bisimilarity measure) and 2) seems contrary to the
Universal Dovetailer ordering idea as it implies tight sequential
strings (buttightness <http://en.wikipedia.org/wiki/Sequential_space>
might be recovered by Godel Numbering but not uniquely for infinitely
long strings). But there is a question regarding the /constructability/
of the Lindenbaum-Tarski algebra itself!
Does it require Boolean Satisfiability
<http://en.wikipedia.org/wiki/Boolean_satisfiability_problem> for an
arbitrary propositional theory to allow the construction? It surely
seems to! But is there a unique sieve or filter for the ordering of
implication? How do we define invariance of meaning under
transformations of language? Two propositional theories in different
languages would have differing implication diagrams
so how is bisimulation between them defined????? There has to be a
transformation that generates a diffeomorphism between them.
"Nature, to be commanded, must be obeyed."
~ Francis Bacon
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