[FOM] Compactness of second order propositional logic

Alasdair Urquhart urquhart at cs.toronto.edu
Fri May 8 15:47:56 EDT 2015

Philip Kremer (JSL Volume 58 (1993), pp. 334-349) proves that
the second order formulations of quantified relevant propositional
logic are equivalent to full second order classical logic.
This is with respect to both the semilattice semantics of
my 1972 paper, and the Routley-Meyer ternary relational
semantics (where the quantifiers range over increasing subsets
of frames).  The failure of compactness follows in both

Richard Zach called my attention to the fact that Kremer
had already proved the equivalence that I conjectured
in an earlier posting.  I had forgotten this because
the title of Kremer's paper only gives the much
weaker claim of non-axiomatizability.

On Sat, 2 May 2015, Guillermo Badia wrote:

> Dear Prof. Urquhart,
> But what about compactness with respect to the Routley-Meyer
> semantics? I think with respect to your semantics you should be right
> since Kremer gives a way to interpret material implication, which
> makes the claim that second order propositional relevant logic is as
> expressive as full second order logic much more plausible. However, in
> the case of the Routley-Meyer semantics he doesn't do this, in fact he
> conjectures it's impossible.
> Kind regards,
> Guillermo

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