584 research outputs found
Neutrality and Many-Valued Logics
In this book, we consider various many-valued logics: standard, linear,
hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We
survey also results which show the tree different proof-theoretic frameworks
for many-valued logics, e.g. frameworks of the following deductive calculi:
Hilbert's style, sequent, and hypersequent. We present a general way that
allows to construct systematically analytic calculi for a large family of
non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and
p-adic valued logics characterized by a special format of semantics with an
appropriate rejection of Archimedes' axiom. These logics are built as different
extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's,
Product, and Post's logics). The informal sense of Archimedes' axiom is that
anything can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and it
is not well-founded and well-ordered. On the base of non-Archimedean valued
logics, we construct non-Archimedean valued interval neutrosophic logic INL by
which we can describe neutrality phenomena.Comment: 119 page
Achieving while maintaining:A logic of knowing how with intermediate constraints
In this paper, we propose a ternary knowing how operator to express that the
agent knows how to achieve given while maintaining
in-between. It generalizes the logic of goal-directed knowing how proposed by
Yanjing Wang 2015 'A logic of knowing how'. We give a sound and complete
axiomatization of this logic.Comment: appear in Proceedings of ICLA 201
Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic
Hilbertâs choice operators Ï and Δ, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbertâs operators in relation to his evolving program in the foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummettâs program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+Δ without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummettâs contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummettâs framework is improved by these results as they show that questions of realism and anti-realism are not an âall or nothingâ matter, but that there are plausibly metaphysical stances between the poles of anti-realism (corresponding to acceptance just of intutionistic logic) and realism (corresponding to acceptance of classical logic), because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as âis smart,â which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong (i.e., classical) logic
Hilbert's epsilon as an Operator of Indefinite Committed Choice
Paul Bernays and David Hilbert carefully avoided overspecification of
Hilbert's epsilon-operator and axiomatized only what was relevant for their
proof-theoretic investigations. Semantically, this left the epsilon-operator
underspecified. In the meanwhile, there have been several suggestions for
semantics of the epsilon as a choice operator. After reviewing the literature
on semantics of Hilbert's epsilon operator, we propose a new semantics with the
following features: We avoid overspecification (such as right-uniqueness), but
admit indefinite choice, committed choice, and classical logics. Moreover, our
semantics for the epsilon supports proof search optimally and is natural in the
sense that it does not only mirror some cases of referential interpretation of
indefinite articles in natural language, but may also contribute to philosophy
of language. Finally, we ask the question whether our epsilon within our
free-variable framework can serve as a paradigm useful in the specification and
computation of semantics of discourses in natural language.Comment: ii + 73 pages. arXiv admin note: substantial text overlap with
arXiv:1104.244
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