797 research outputs found
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
Double-Negation Elimination in Some Propositional Logics
This article answers two questions (posed in the literature), each concerning
the guaranteed existence of proofs free of double negation. A proof is free of
double negation if none of its deduced steps contains a term of the form
n(n(t)) for some term t, where n denotes negation. The first question asks for
conditions on the hypotheses that, if satisfied, guarantee the existence of a
double-negation-free proof when the conclusion is free of double negation. The
second question asks about the existence of an axiom system for classical
propositional calculus whose use, for theorems with a conclusion free of double
negation, guarantees the existence of a double-negation-free proof. After
giving conditions that answer the first question, we answer the second question
by focusing on the Lukasiewicz three-axiom system. We then extend our studies
to infinite-valued sentential calculus and to intuitionistic logic and
generalize the notion of being double-negation free. The double-negation proofs
of interest rely exclusively on the inference rule condensed detachment, a rule
that combines modus ponens with an appropriately general rule of substitution.
The automated reasoning program OTTER played an indispensable role in this
study.Comment: 32 pages, no figure
A simple sequent calculus for partial functions
AbstractUsually, the extension of classical logic to a three-level valued logic results in a complicated calculus, with side-conditions on the rules of logic in order to ensure consistency. One reason for the necessity of side-conditions is the presence of nonmonotonic operators. Another reason is the choice of consequence relation. Side-conditions severely violate the symmetry of the logic. By limiting the extension to monotonic cases and by choosing an appropriate consequence relation, a simple calculus for three-valued logic arises. The logic has strong correspondences to ordinary classical logic and, in particular, the symmetry of the Genzen sequent calculus (LK) is preserved, leading to a simple proof for cut elimination
Deductive Systems in Traditional and Modern Logic
The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic
Classical BI: Its Semantics and Proof Theory
We present Classical BI (CBI), a new addition to the family of bunched logics
which originates in O'Hearn and Pym's logic of bunched implications BI. CBI
differs from existing bunched logics in that its multiplicative connectives
behave classically rather than intuitionistically (including in particular a
multiplicative version of classical negation). At the semantic level,
CBI-formulas have the normal bunched logic reading as declarative statements
about resources, but its resource models necessarily feature more structure
than those for other bunched logics; principally, they satisfy the requirement
that every resource has a unique dual. At the proof-theoretic level, a very
natural formalism for CBI is provided by a display calculus \`a la Belnap,
which can be seen as a generalisation of the bunched sequent calculus for BI.
In this paper we formulate the aforementioned model theory and proof theory for
CBI, and prove some fundamental results about the logic, most notably
completeness of the proof theory with respect to the semantics.Comment: 42 pages, 8 figure
From Quantum Metalanguage to the Logic of Qubits
The main aim of this thesis is to look for a logical deductive calculus (we
will adopt sequent calculus, originally introduced in Gentzen, 1935), which
could describe quantum information and its properties. More precisely, we
intended to describe in logical terms the formation of the qubit (the unit of
quantum information) which is a particular linear superposition of the two
classical bits 0 and 1. To do so, we had to introduce the new connective
"quantum superposition", in the logic of one qubit, Lq, as the classical
conjunction cannot describe this quantum link.Comment: 138 pages, PhD thesis in Mathematic
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
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