[FOM] quantum logic

[Vaughan Pratt]

> Arnold Neumaier<Arnold.Neumaier at univie.ac.at>u>
> Sent: Sunday, April 25, 2010 9:37 PM wrote:
>
> No serious mathematics can be deduced from quantum logic.
>
> The latter has no useful form of implication that would cooperate
> with conjunction in a way that lets one build up a theory in the
> usual stepwise fashion.

Like Antonino Drago I feel this needs more calibration.  Maria Dalla 
Chiara surveyed a number of candidates for a quantum logic implication 
in her Handbook of Philosophical Logic article (427-469, ed. Gabbay, 
Guenthner, Reidel, 1986) and found all of them wanting in some way or other.

I summarized these in 1992 at the start of section 2.4 of

   http://boole.stanford.edu/pub/ql.pdf

and suggested that a solution might lie in dropping the usual 
restriction that the model be a poset, this being the usual approach to 
modeling linear logic, namely as a category understood as a poset 
admitting multiple comparisons between two elements.  These comparisons 
admit various interpretations, including primitive proofs, of a rather 
more abstract kind however than envisaged by Harvey just now in his 
fleshing out of "having a scientific theory of mathematical proof."  See 
the paper for more details.

Vaughan Pratt