On 28 May 2012, at 18:35, meekerdb wrote:
On 5/28/2012 1:37 AM, Bruno Marchal wrote:
I am mute on the subject of whether p is true in any other
world (unless I can use an axiom like the above).
By the logicians notion of proof, if you prove a proposition, it is
true in all worlds/model/interpretation.
But the 'worlds' are defined by the axioms and rules of inference.
Not as such. the axioms and rules define only the truth common to all
models, when the theory is sound (and vice versa if the theory is
complete). Individual models have their lives of their own. They lives
in other theories or theories models. The models of PA exists in ZF's
model. Models (semantics) are beyond the theory.
So you could change or add axioms and get different 'worlds'. In
this logicians idea of 'world' it is not the case that you only
prove things in the one world you're in.
That was my point. We alway "prove" what is true in all worlds.
Proving is a trans-world notion. G proves <> t -> <> f, makes that
formula true in all worlds of all models based on all finite
irreflexive realist models, for example.
Here the world can be related to the models, but it leads to a very
special model. So different completeness theorem are being used in
that context, and we have to be cautious which one we are talking about.
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