[FOM] Compactness of second order propositional logic

[Richard Zach]

Phil actually has a paper where he proved the result for relevance logic:

http://individual.utoronto.ca/philipkremer/onlinepapers/propquanrel.pdf

On 2015-04-30 12:17 PM, Alasdair Urquhart wrote:
> I believe that the compactness theorem fails for second order 
> propositional relevant logic, for the same reason it fails for
> second order intuitionistic propositional logic.  The 
> non-axiomatisability
> result of Philip Kremer can, I think, be strengthened to a proof
> that the system defined by the second order propositional model
> theory for relevant logic is recursively isomorphic to full second 
> order classical logic.
>
> Assuming this, I believe that the classical failure of compactness
> in second order classical logic can be transferred to the second
> order propositional logic.  There is a simple proof of the
> classical failure in the entry on second order logic in
> the online Stanford Encyclopedia of Philosophy.
>
> I haven't checked the details of these claims, but I think they
> are likely correct, because the encoding technique used by
> Kremer for second order intuitionistic propositional logic
> is quite generally applicable.

-- 
Richard Zach ...... http://www.ucalgary.ca/rzach/
Professor,             Department  of  Philosophy
University of Calgary, Calgary AB T2N 1N4, Canada