[FOM] Compactness of second order propositional logic

Guillermo Badia guillebadia89 at gmail.com
Fri May 8 19:30:38 EDT 2015

Dear Prof. Urquhart,
Sorry for my ignorance, but how does it follows from the simple fact
that second order propositional relevant logic and second order logic
are recursively isomorphic (which is a property referring to
validities only) that second order propositional relevant logic  fails
(in the R-M semantics) to be compact? You suggested adapting the
little argument in the SEP that compactness fails for SOL, but Kremer
doesn't provide a way to do this precisely because he doesn't provide
a way to interpret material implication in the language of second
order propositional relevant logic (in fact, he conjectures this is
downright impossible) with respect to the R-M semantics (to show his
result, he makes a detour through a class of well-behaved R-M models
preserving validity, not satisfiability). The sentence of second order
logic expressing that a model is infinite certainly requires material
implication. Could you help me understand what you mean when you say
that the failure of compactness follows from Kremer's result?
Thanks very much for your help,

On 5/9/15, Alasdair Urquhart <urquhart at cs.toronto.edu> wrote:
> 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
> cases.
> 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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