Dear Bruno, From http://www.andrew.cmu.edu/user/awodey/preprints/fold.pdf First-Order Logical Duality we read:

`"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`

`'algebraically`

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`

`presentation,`

`and two theories are equivalent if they present 'the same' -- i.e.`

`isomorphic --`

algebras."

## Advertising

The construction of the Lindenbaum-Tarski algebra is implemented by 1) identification of provably equivalent formulas and 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`

`<http://commons.wikimedia.org/wiki/Logic_diagram#Implication_diagrams> ,`

`so how is bisimulation between them defined????? There has to be a`

`transformation that generates a diffeomorphism between them.`

-- Onward! Stephen "Nature, to be commanded, must be obeyed." ~ Francis Bacon -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.