Logic and probabilistic systems |
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Authors: | Franco Montagna Giulia Simi Andrea Sorbi |
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Institution: | (1) Department of Mathematics, University of Siena, Via del Capitano 15, I-53100 Siena, Italy (e-mail: sorbi@unisi.it) , IT |
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Abstract: | Following some ideas of Roberto Magari, we propose trial and error
probabilistic functions, i.e. probability measures on the sentences of
arithmetic that evolve in time
by trial and error. The set
of the sentences that get
limit probability 1 is a
theory, in fact
can be a
complete set.
We prove incompleteness results for this setting, by
showing for instance that for every
there are true
sentences
that get limit probability less than
. No set
as above
can contain the set of all true
sentences,
although we exhibit some
containing all the true
sentences.
We also consider an approach based on the notions of inner probability
and outer probability, and we compare this approach with the one based
on trial and error probabilistic functions. Although the
two approaches are shown to be different, we single out an important
case in which they are equivalent.
Received March 20, 1995 |
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Keywords: | |
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