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A comparative survey of integrated learning systems
This paper presents the duction framework for unifying the three basic forms of inference - deduction, abduction, and induction - by specifying the possible relationships and influences among them in the context of integrated learning. Special assumptive forms of inference are defined that extend the use of these inference methods, and the properties of these forms are explored. A comparison to a related inference-based learning frame work is made. Finally several existing integrated learning programs are examined in the perspective of the duction framework
Deduction as Reduction
Deduction systems and graph rewriting systems are compared within a common
categorical framework. This leads to an improved deduction method in
diagrammatic logics
Grammatical structures and logical deductions
The three essays presented here concern natural connections between grammatical derivations and structures provided by certain standard grammar formalisms, on the one hand, and deductions in logical systems, on the other hand. In the first essay we analyse the adequacy of Polish notation for higher-order languages. The Ajdukiewicz algorithm (Ajdukiewicz 1935) is discussed in terms of generalized MP-deductions. We exhibit a failure in Ajdukiewicz’s original version of the algorithm and give a correct one; we prove that generalized MP-deductions have the frontier property, which is essential for the plausibility of Polish notation. The second essay deals with logical systems corresponding to different grammar formalisms, as e.g. Finite State Acceptors, Context-Free Grammars, Categorial Grammars, and others. We show how can logical methods be used to establish certain linguistically significant properties of formal grammars. The third essay discusses the interplay between Natural Deduction proofs in grammar oriented logics and semantic structures expressible by typed lambda terms and combinators
Dependence Logic with Generalized Quantifiers: Axiomatizations
We prove two completeness results, one for the extension of dependence logic
by a monotone generalized quantifier Q with weak interpretation, weak in the
meaning that the interpretation of Q varies with the structures. The second
result considers the extension of dependence logic where Q is interpreted as
"there exists uncountable many." Both of the axiomatizations are shown to be
sound and complete for FO(Q) consequences.Comment: 17 page
Denotational Semantics of the Simplified Lambda-Mu Calculus and a New Deduction System of Classical Type Theory
Classical (or Boolean) type theory is the type theory that allows the type
inference (the type counterpart of
double-negation elimination), where is any type and is
absurdity type. This paper first presents a denotational semantics for a
simplified version of Parigot's lambda-mu calculus, a premier example of
classical type theory. In this semantics the domain of each type is divided
into infinitely many ranks and contains not only the usual members of the type
at rank 0 but also their negative, conjunctive, and disjunctive shadows in the
higher ranks, which form an infinitely nested Boolean structure. Absurdity type
is identified as the type of truth values. The paper then presents a new
deduction system of classical type theory, a sequent calculus called the
classical type system (CTS), which involves the standard logical operators such
as negation, conjunction, and disjunction and thus reflects the discussed
semantic structure in a more straightforward fashion.Comment: In Proceedings CL&C 2016, arXiv:1606.0582
A Framework for Combining Defeasible Argumentation with Labeled Deduction
In the last years, there has been an increasing demand of a variety of
logical systems, prompted mostly by applications of logic in AI and other
related areas. Labeled Deductive Systems (LDS) were developed as a flexible
methodology to formalize such a kind of complex logical systems. Defeasible
argumentation has proven to be a successful approach to formalizing commonsense
reasoning, encompassing many other alternative formalisms for defeasible
reasoning. Argument-based frameworks share some common notions (such as the
concept of argument, defeater, etc.) along with a number of particular features
which make it difficult to compare them with each other from a logical
viewpoint. This paper introduces LDSar, a LDS for defeasible argumentation in
which many important issues concerning defeasible argumentation are captured
within a unified logical framework. We also discuss some logical properties and
extensions that emerge from the proposed framework.Comment: 15 pages, presented at CMSRA Workshop 2003. Buenos Aires, Argentin
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