6,386 research outputs found
Boolean Dependence Logic and Partially-Ordered Connectives
We introduce a new variant of dependence logic called Boolean dependence
logic. In Boolean dependence logic dependence atoms are of the type
=(x_1,...,x_n,\alpha), where \alpha is a Boolean variable. Intuitively, with
Boolean dependence atoms one can express quantification of relations, while
standard dependence atoms express quantification over functions.
We compare the expressive power of Boolean dependence logic to dependence
logic and first-order logic enriched by partially-ordered connectives. We show
that the expressive power of Boolean dependence logic and dependence logic
coincide. We define natural syntactic fragments of Boolean dependence logic and
show that they coincide with the corresponding fragments of first-order logic
enriched by partially-ordered connectives with respect to expressive power. We
then show that the fragments form a strict hierarchy.Comment: 41 page
Generic filters in partially ordered sets
The concept of partial-order valued models and that of D-generic filters play a central role in the present day development of Set Theory;In this dissertation, we consider questions related to both of the above concepts;In Section 2, we prove the existence of a model (M,(ELEM)) for the unrestricted Comprehension Scheme;((FOR ALL)x)((x (epsilon) M) (---\u3e) ((THERE EXISTS)s)((s(ELEM)M) (WEDGE) ((VBAR)(VBAR)x (ELEM) s(VBAR)(VBAR) = (VBAR)(VBAR)F(x)(VBAR)(VBAR))));in a certain n-valued logic (L(,n), N, D) with connectives N and D defined by appropriate truth tables;In Section 3, we construct partial-order valued models where (VBAR)(VBAR)x (epsilon) y(VBAR)(VBAR) is a subset of a partially ordered set P with (VBAR)(VBAR)(IL-PERP)F(VBAR)(VBAR) defined as z (VBAR)z (epsilon) P and z is incompatible with every element of (VBAR)(VBAR)F(VBAR)(VBAR) and with the other connectives defined in their usual set-theoretical interpretations. These models are reduced to two valued models via the notion of a generic filter of P. In Section 4, we introduce the notion of a molecule m of P by requiring that every two elements compatible with m be themselves compatible. Then, we prove the equivalence of the existence of a generic filter to that of a molecule in a partially ordered set. In Section 5, we introduce some P-lattice algebras and prove the existence of a D-complete ultrafilter in a Boolean algebra for the denumerable case;In Section 6, we introduce the notion of k-inducive partially ordered sets as partially ordered sets in which every inversely well-ordered subset of cardinality less then k of nonzero elements has a nonzero lower bound. Based on k-inducive partially ordered set, we prove the existence of some E-complete filters of partially ordered sets with the condition E(\u27 )\u3c(\u27 )k imposed on cardinality of E;In Sections 7 to 10, we prove some equivalent forms of Martin\u27s Axiom in connection with Boolean algebras and also we give some consequences of Martin\u27s Axiom pertaining to the cardinal exponentiation;Finally, in Section 11 we introduce the notion of a receding sequence (S(,i))(,
On noncommutative extensions of linear logic
Pomset logic introduced by Retor\'e is an extension of linear logic with a
self-dual noncommutative connective. The logic is defined by means of
proof-nets, rather than a sequent calculus. Later a deep inference system BV
was developed with an eye to capturing Pomset logic, but equivalence of system
has not been proven up to now. As for a sequent calculus formulation, it has
not been known for either of these logics, and there are convincing arguments
that such a sequent calculus in the usual sense simply does not exist for them.
In an on-going work on semantics we discovered a system similar to Pomset
logic, where a noncommutative connective is no longer self-dual. Pomset logic
appears as a degeneration, when the class of models is restricted. Motivated by
these semantic considerations, we define in the current work a semicommutative
multiplicative linear logic}, which is multiplicative linear logic extended
with two nonisomorphic noncommutative connectives (not to be confused with very
different Abrusci-Ruet noncommutative logic). We develop a syntax of proof-nets
and show how this logic degenerates to Pomset logic. However, a more
interesting problem than just finding yet another noncommutative logic is to
find a sequent calculus for this logic. We introduce decorated sequents, which
are sequents equipped with an extra structure of a binary relation of
reachability on formulas. We define a decorated sequent calculus for
semicommutative logic and prove that it is cut-free, sound and complete. This
is adapted to "degenerate" variations, including Pomset logic. Thus, in
particular, we give a variant of sequent calculus formulation for Pomset logic,
which is one of the key results of the paper
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
Stone-type representations and dualities for varieties of bisemilattices
In this article we will focus our attention on the variety of distributive
bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and
involutive bisemilattices. After extending Balbes' representation theorem to
bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn
duality and introduce the categories of 2spaces and 2spaces. The
categories of 2spaces and 2spaces will play with respect to the
categories of distributive bisemilattices and De Morgan bisemilattices,
respectively, a role analogous to the category of Stone spaces with respect to
the category of Boolean algebras. Actually, the aim of this work is to show
that these categories are, in fact, dually equivalent
Physical propositions and quantum languages
The word \textit{proposition} is used in physics with different meanings,
which must be distinguished to avoid interpretational problems. We construct
two languages and with classical
set-theoretical semantics which allow us to illustrate those meanings and to
show that the non-Boolean lattice of propositions of quantum logic (QL) can be
obtained by selecting a subset of \textit{p-testable} propositions within the
Boolean lattice of all propositions associated with sentences of
. Yet, the aforesaid semantics is incompatible with the
standard interpretation of quantum mechanics (QM) because of known no-go
theorems. But if one accepts our criticism of these theorems and the ensuing SR
(semantic realism) interpretation of QM, the incompatibility disappears, and
the classical and quantum notions of truth can coexist, since they refer to
different metalinguistic concepts (\textit{truth} and \textit{verifiability
according to QM}, respectively). Moreover one can construct a quantum language
whose Lindenbaum-Tarski algebra is isomorphic to QL, the
sentences of which state (testable) properties of individual samples of
physical systems, while standard QL does not bear this interpretation.Comment: 15 pages, no figure, standard Late
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