7,223 research outputs found
A QBF-based Formalization of Abstract Argumentation Semantics
Supported by the National Research Fund, Luxembourg (LAAMI project) and by the Engineering and Physical Sciences Research Council (EPSRC, UK), grant ref. EP/J012084/1 (SAsSY project).Peer reviewedPostprin
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
Effects and Propositions
The quantum logical and quantum information-theoretic traditions have exerted
an especially powerful influence on Bub's thinking about the conceptual
foundations of quantum mechanics. This paper discusses both the quantum logical
and information-theoretic traditions from the point of view of their
representational frameworks. I argue that it is at this level, at the level of
its framework, that the quantum logical tradition has retained its centrality
to Bub's thought. It is further argued that there is implicit in the quantum
information-theoretic tradition a set of ideas that mark a genuinely new
alternative to the framework of quantum logic. These ideas are of considerable
interest for the philosophy of quantum mechanics, a claim which I defend with
an extended discussion of their application to our understanding of the
philosophical significance of the no hidden variable theorem of Kochen and
Specker.Comment: Presented to the 2007 conference, New Directions in the Foundations
of Physic
Conditionals and modularity in general logics
In this work in progress, we discuss independence and interpolation and
related topics for classical, modal, and non-monotonic logics
Suszko's Problem: Mixed Consequence and Compositionality
Suszko's problem is the problem of finding the minimal number of truth values
needed to semantically characterize a syntactic consequence relation. Suszko
proved that every Tarskian consequence relation can be characterized using only
two truth values. Malinowski showed that this number can equal three if some of
Tarski's structural constraints are relaxed. By so doing, Malinowski introduced
a case of so-called mixed consequence, allowing the notion of a designated
value to vary between the premises and the conclusions of an argument. In this
paper we give a more systematic perspective on Suszko's problem and on mixed
consequence. First, we prove general representation theorems relating
structural properties of a consequence relation to their semantic
interpretation, uncovering the semantic counterpart of substitution-invariance,
and establishing that (intersective) mixed consequence is fundamentally the
semantic counterpart of the structural property of monotonicity. We use those
to derive maximum-rank results proved recently in a different setting by French
and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various
structural properties (reflexivity, transitivity, none, or both). We strengthen
these results into exact rank results for non-permeable logics (roughly, those
which distinguish the role of premises and conclusions). We discuss the
underlying notion of rank, and the associated reduction proposed independently
by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve
compositionality in general, meaning that the resulting semantics is no longer
truth-functional. We propose a modification of that notion of reduction,
allowing us to prove that over compact logics with what we call regular
connectives, rank results are maintained even if we request the preservation of
truth-functionality and additional semantic properties.Comment: Keywords: Suszko's thesis; truth value; logical consequence; mixed
consequence; compositionality; truth-functionality; many-valued logic;
algebraic logic; substructural logics; regular connective
A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
This paper is the first in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper we discuss two different types of language that can be
attached to a system, S. The first is a propositional language, PL(S); the
second is a higher-order, typed language L(S). Both languages provide deductive
systems with an intuitionistic logic. The reason for introducing PL(S) is that,
as shown in paper II of the series, it is the easiest way of understanding, and
expanding on, the earlier work on topos theory and quantum physics. However,
the main thrust of our programme utilises the more powerful language L(S) and
its representation in an appropriate topos.Comment: 36 pages, no figure
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