1,357 research outputs found
Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free
By operations on models we show how to relate completeness with respect to
permissive-nominal models to completeness with respect to nominal models with
finite support. Models with finite support are a special case of
permissive-nominal models, so the construction hinges on generating from an
instance of the latter, some instance of the former in which sufficiently many
inequalities are preserved between elements. We do this using an infinite
generalisation of nominal atoms-abstraction.
The results are of interest in their own right, but also, we factor the
mathematics so as to maximise the chances that it could be used off-the-shelf
for other nominal reasoning systems too. Models with infinite support can be
easier to work with, so it is useful to have a semi-automatic theorem to
transfer results from classes of infinitely-supported nominal models to the
more restricted class of models with finite support.
In conclusion, we consider different permissive-nominal syntaxes and nominal
models and discuss how they relate to the results proved here
First-order Goedel logics
First-order Goedel logics are a family of infinite-valued logics where the
sets of truth values V are closed subsets of [0, 1] containing both 0 and 1.
Different such sets V in general determine different Goedel logics G_V (sets of
those formulas which evaluate to 1 in every interpretation into V). It is shown
that G_V is axiomatizable iff V is finite, V is uncountable with 0 isolated in
V, or every neighborhood of 0 in V is uncountable. Complete axiomatizations for
each of these cases are given. The r.e. prenex, negation-free, and existential
fragments of all first-order Goedel logics are also characterized.Comment: 37 page
An analysis of the logic of Riesz Spaces with strong unit
We study \L ukasiewicz logic enriched with a scalar multiplication with
scalars taken in . Its algebraic models, called {\em Riesz MV-algebras},
are, up to isomorphism, unit intervals of Riesz spaces with a strong unit
endowed with an appropriate structure. When only rational scalars are
considered, one gets the class of {\em DMV-algebras} and a corresponding
logical system. Our research follows two objectives. The first one is to deepen
the connections between functional analysis and the logic of Riesz MV-algebras.
The second one is to study the finitely presented MV-algebras, DMV-algebras and
Riesz MV-algebras, connecting them from logical, algebraic and geometric
perspective
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