On extremal conditions, inspired by the implementable set theory thread
Aatu Koskensilta
has, among other things, proposed a completeness axiom, to the effect that
only sets whose existence follows from the axioms of his implementable set
theory, IST, should exist. He's been widely criticised for this, and it has
been pointed it's not at all obvious how to formulate this axiom in, say,
the language of set theory.
The criticism has not been without foundation, but it seems to have escaped
some of the posters in the thread that this sort of axioms, known as
extremal conditions or clauses, are not an invention of Han's. Hilbert's
original axiomatisation of geometry included an extremal clause, a
completeness axiom stating that the domain of objects can not be extended
with new objects without violating the axioms. This stipulation was later
replaced with the more familiar second order form of geometric completeness.
In general, extremal conditions are expressible in the following
framework, which, to the best of my knowledge, has not been systematically,
or at all, studied in the literature. We take a first order language L and
extend it with four modal operators, [], <>, B and F. [] and <> act as
universal and existential quantifiers over structures of the signature of L,
while B and F act as "backward looking" and "forward looking" operators,
respectively. The operator B cancels the effect of the innermost enclosing
modal operator, while F returns it. Thus e.g. []BP is equivalent to P and
[]BFP to []P. In the propositional case the operators B and F do not
increase expressive power, but in conjunction with first order quantifiers,
the resulting logic is extremely expressive (much more expressive than, say,
omega-order logic).
In particular, we can express the condition that the model is a maximal
model of the axioms A by the following formula
~<>(BAx(FEy(y=x)) & ExBAy(x =/= y) & A & Pred-T)
where Pred-T is the conjunction of the formulas
Ax1x2x3....(BP(x1,...,xn) <--> P(x1,...,xn))
for each predicate symbol P in the language A is expressed in. We can
construct similar sentences for expressing that the model is a minimal model
of A.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Herman Jurjus
> In the thread /Implementable Set Theory and Consistency of ZFC/ Han de Brujn
> has, among other things, proposed a completeness axiom, to the effect that
> only sets whose existence follows from the axioms of his implementable set
> theory, IST, should exist. He's been widely criticised for this, and it has
> been pointed it's not at all obvious how to formulate this axiom in, say,
> the language of set theory.
>
> The criticism has not been without foundation, but it seems to have escaped
> some of the posters in the thread that this sort of axioms, known as
> extremal conditions or clauses, are not an invention of Han's. Hilbert's
> original axiomatisation of geometry included an extremal clause, a
> completeness axiom stating that the domain of objects can not be extended
> with new objects without violating the axioms. This stipulation was later
> replaced with the more familiar second order form of geometric completeness.
>
> In general, extremal conditions are expressible in the following
> framework, which, to the best of my knowledge, has not been systematically,
> or at all, studied in the literature.
Perhaps the following counts?
M. Ryan, P.-Y. Schobbens, and O. Rodrigues. Counterfactuals and updates
as inverse modalities. In Y. Shoham, editor, TARK'96: Proc. Theoretical
Aspects of Rationality and Knowledge, pages 163--173. Morgan Kaufmann, 1996
Other things that come to mind: circumscription, non-monotonic logic.
We take a first order language L and
> extend it with four modal operators, [], <>, B and F. [] and <> act as
> universal and existential quantifiers over structures of the signature of L,
> while B and F act as "backward looking" and "forward looking" operators,
> respectively. The operator B cancels the effect of the innermost enclosing
> modal operator, while F returns it. Thus e.g. []BP is equivalent to P and
> []BFP to []P. In the propositional case the operators B and F do not
> increase expressive power, but in conjunction with first order quantifiers,
> the resulting logic is extremely expressive (much more expressive than, say,
> omega-order logic).
>
> In particular, we can express the condition that the model is a maximal
> model of the axioms A by the following formula
>
> ~<>(BAx(FEy(y=x)) & ExBAy(x =/= y) & A & Pred-T)
>
> where Pred-T is the conjunction of the formulas
>
> Ax1x2x3....(BP(x1,...,xn) <--> P(x1,...,xn))
>
> for each predicate symbol P in the language A is expressed in. We can
> construct similar sentences for expressing that the model is a minimal model
> of A.
--
Cheers,
Herman Jurjus
Aatu Koskensilta
> Perhaps the following counts?
Not really. The paper discusses "inverse modalities", but the operators B
and F are not "inverse modalities" nor can they be defined in terms of
such.
> Other things that come to mind: circumscription, non-monotonic logic.
I fail to see the relation of either of these to the extension of
(first-order) modal logic I described. Perhaps you can be more explicit?