723 research outputs found
The ubiquity of conservative translations
We study the notion of conservative translation between logics introduced by
Feitosa and D'Ottaviano. We show that classical propositional logic (CPC) is
universal in the sense that every finitary consequence relation over a
countable set of formulas can be conservatively translated into CPC. The
translation is computable if the consequence relation is decidable. More
generally, we show that one can take instead of CPC a broad class of logics
(extensions of a certain fragment of full Lambek calculus FL) including most
nonclassical logics studied in the literature, hence in a sense, (almost) any
two reasonable deductive systems can be conservatively translated into each
other. We also provide some counterexamples, in particular the paraconsistent
logic LP is not universal.Comment: 15 pages; to appear in Review of Symbolic Logi
Scaled Boolean Algebras
Scaled Boolean algebras are a category of mathematical objects that arose
from attempts to understand why the conventional rules of probability should
hold when probabilities are construed, not as frequencies or proportions or the
like, but rather as degrees of belief in uncertain propositions. This paper
separates the study of these objects from that not-entirely-mathematical
problem that motivated them. That motivating problem is explicated in the first
section, and the application of scaled Boolean algebras to it is explained in
the last section. The intermediate sections deal only with the mathematics. It
is hoped that this isolation of the mathematics from the motivating problem
makes the mathematics clearer.Comment: 53 pages, 8 Postscript figures, Uses ajour.sty from Academic Press,
To appear in Advances in Applied Mathematic
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