On 27 May 2012, at 12:15, Russell Standish wrote:
On Thu, May 24, 2012 at 03:42:15PM +0200, Bruno Marchal wrote:
But "a => Ba" is a valid rule for all logic having a Kripke
semantics. Why? Because it means that a is supposed to be valid (for
example you have already prove it), so a, like any theorem, will be
true in all worlds, so a will be in particular true in all worlds
accessible from anywhere in the model, so Ba will be true in all
worlds of the model, so Ba is also a theorem.
I still don't follow. If I have proved a is true in some world, why
should I infer that it is true in all worlds? What am I missing?
I realize my previous answer might be too long and miss your question.
Apology if it is the case.
Here is a shorter answer. The idea of proving, is that what is proved
in true in all possible world. If not, a world would exist as a
counter-example, invalidating the argument.
You might want to prove something about your actual world, but this
can only have the form of a conditional like if my world satisfy such
a such propositions then it has to satisfy that or this proposition,
and that conditional has better to be true in all worlds, for we never
really know which world we are in, we can only make theories.
Now, the modal Bp, and proof in math, can be study mathematically, and
that is what I described in the preceding post, and constitutes a bit
of the Arithmetical UDA.
Bruno
http://iridia.ulb.ac.be/~marchal/
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