1,574 research outputs found
Real Analysis in Paraconsistent Logic
This paper begins an analysis of the real line using an inconsistency-
tolerant (paraconsistent) logic. We show that basic field and compactness
properties hold, by way of novel proofs that make no use of consistency-
reliant inferences; some techniques from constructive analysis are used
instead. While no inconsistencies are found in the algebraic operations on
the real number field, prospects for other non-trivializing contradictions
are left open
Towards a Paraconsistent Quantum Set Theory
In this paper, we will attempt to establish a connection between quantum set
theory, as developed by Ozawa, Takeuti and Titani, and topos quantum theory, as
developed by Isham, Butterfield and Doring, amongst others. Towards this end,
we will study algebraic valued set-theoretic structures whose truth values
correspond to the clopen subobjects of the spectral presheaf of an orthomodular
lattice of projections onto a given Hilbert space. In particular, we will
attempt to recreate, in these new structures, Takeuti's original isomorphism
between the set of all Dedekind real numbers in a suitably constructed model of
set theory and the set of all self adjoint operators on a chosen Hilbert space.Comment: In Proceedings QPL 2015, arXiv:1511.0118
Potentiality and Contradiction in Quantum Mechanics
Following J.-Y.B\'eziau in his pioneer work on non-standard interpretations
of the traditional square of opposition, we have applied the abstract structure
of the square to study the relation of opposition between states in
superposition in orthodox quantum mechanics in \cite{are14}. Our conclusion was
that such states are \ita{contraries} (\ita{i.e.} both can be false, but both
cannot be true), contradicting previous analyzes that have led to different
results, such as those claiming that those states represent \ita{contradictory}
properties (\ita{i. e.} they must have opposite truth values). In this chapter
we bring the issue once again into the center of the stage, but now discussing
the metaphysical presuppositions which underlie each kind of analysis and which
lead to each kind of result, discussing in particular the idea that
superpositions represent potential contradictions. We shall argue that the
analysis according to which states in superposition are contrary rather than
contradictory is still more plausible
A Paraconsistent Higher Order Logic
Classical logic predicts that everything (thus nothing useful at all) follows
from inconsistency. A paraconsistent logic is a logic where an inconsistency
does not lead to such an explosion, and since in practice consistency is
difficult to achieve there are many potential applications of paraconsistent
logics in knowledge-based systems, logical semantics of natural language, etc.
Higher order logics have the advantages of being expressive and with several
automated theorem provers available. Also the type system can be helpful. We
present a concise description of a paraconsistent higher order logic with
countable infinite indeterminacy, where each basic formula can get its own
indeterminate truth value (or as we prefer: truth code). The meaning of the
logical operators is new and rather different from traditional many-valued
logics as well as from logics based on bilattices. The adequacy of the logic is
examined by a case study in the domain of medicine. Thus we try to build a
bridge between the HOL and MVL communities. A sequent calculus is proposed
based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker,
Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte
Inconsistent boundaries
Research on this paper was supported by a grant from the Marsden Fund, Royal Society of New Zealand.Mereotopology is a theory of connected parts. The existence of boundaries, as parts of everyday objects, is basic to any such theory; but in classical mereotopology, there is a problem: if boundaries exist, then either distinct entities cannot be in contact, or else space is not topologically connected (Varzi in Noûs 31:26–58, 1997). In this paper we urge that this problem can be met with a paraconsistent mereotopology, and sketch the details of one such approach. The resulting theory focuses attention on the role of empty parts, in delivering a balanced and bounded metaphysics of naive space.PostprintPeer reviewe
The Paraconsistent Approach to Quantum Superpositions Reloaded: Formalizing Contradictory Powers in the Potential Realm
In [7] the authors of this paper argued in favor of the possibility to
consider a Paraconsistent Approach to Quantum Superpositions (PAQS). We claimed
that, even though most interpretations of quantum mechanics (QM) attempt to
escape contradictions, there are many hints -coming from present technical and
experimental developments in QM- that indicate it could be worth while to
engage in a research of this kind. Recently, Arenhart and Krause have raised
several arguments against the PAQS [1, 2, 3]. In [11, 12] it was argued that
their reasoning presupposes a metaphysical stance according to which the
physical representation of reality must be exclusively considered in terms of
the equation: Actuality = Reality. However, from a different metaphysical
standpoint their problems disappear. It was also argued that, if we accept the
idea that quantum superpositions exist in a (contradictory) potential realm, it
makes perfect sense to develop QM in terms of a paraconsistent approach and
claim that quantum superpositions are contradictory, contextual existents.
Following these ideas, and taking as a standpoint an interpretation in terms of
the physical notions of power and potentia put forward in [10, 12, 15], we
present a paraconsistent formalization of quantum superpositions that attempts
to capture the main features of QM.Comment: 26 pages, no figures. arXiv admin note: substantial text overlap with
arXiv:1502.05081, arXiv:1404.5186, arXiv:1506.0737
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