864 research outputs found
Model-theoretic characterization of predicate intuitionistic formulas
Notions of asimulation and k-asimulation introduced in [Olkhovikov, 2011] are
extended onto the level of predicate logic. We then prove that a first-order
formula is equivalent to a standard translation of an intuitionistic predicate
formula iff it is invariant with respect to k-asimulations for some k, and then
that a first-order formula is equivalent to a standard translation of an
intuitionistic predicate formula iff it is invariant with respect to
asimulations. Finally, it is proved that a first-order formula is equivalent to
a standard translation of an intuitionistic predicate formula over a class of
intuitionistic models (intuitionistic models with constant domain) iff it is
invariant with respect to asimulations between intuitionistic models
(intuitionistic models with constant domain)
Model-theoretic characterization of intuitionistic propositional formulas
Notions of k-asimulation and asimulation are introduced as asymmetric
counterparts to k-bisimulation and bisimulation, respectively. It is proved
that a first-order formula is equivalent to a standard translation of an
intuitionistic propositional formula iff it is invariant with respect to
k-asimulations for some k, and then that a first-order formula is equivalent to
a standard translation of an intuitionistic propositional formula iff it is
invariant with respect to asimulations. Finally, it is proved that a
first-order formula is intuitionistically equivalent to a standard translation
of an intuitionistic propositional formula iff it is invariant with respect to
asimulations between intuitionistic models.Comment: 16 pages, 0 figures. arXiv admin note: substantial text overlap with
arXiv:1202.119
Failure of interpolation in the intuitionistic logic of constant domains
This paper shows that the interpolation theorem fails in the intuitionistic
logic of constant domains. This result refutes two previously published claims
that the interpolation property holds.Comment: 13 pages, 0 figures. Overlaps with arXiv 1202.1195 removed, the text
thouroughly reworked in terms of notation and style, historical notes as well
as some other minor details adde
Failure of interpolation in the intuitionistic logic of constant domains
This paper shows that the interpolation theorem fails in the intuitionistic
logic of constant domains. This result refutes two previously published claims
that the interpolation property holds.Comment: 13 pages, 0 figures. Overlaps with arXiv 1202.1195 removed, the text
thouroughly reworked in terms of notation and style, historical notes as well
as some other minor details adde
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
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
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