Two formalisms of extended possibilistic logic programming with context-dependent fuzzy unification A comparative description

AbstractPossibilistic logic is a logic of uncertainty where a certainty degree between 0 and 1, interpreted as a lower bound of a necessity measure, is attached to each classical formula. In this paper we present a comparative description of two models extending first order possibilistic logic so as to allow for fuzzy unification. The first formalism, called PLFC, is a general extension that allows clauses with fuzzy constants and fuzzily restricted quantifiers. The second formalism is an implication-based extension defined on top of Gödel infinitely-valued logic, capable of dealing with fuzzy constants. In this paper we compare these approaches, mainly their Horn-clause fragments, discussing their basic differences, specially in what regards their unification and automated deduction mechanisms

Ramified higher-order unification

While unification in the simple theory of types (a.k.a.\ higher-order logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. But the pure ramified theory of types cannot express even the simplest formulas of logic. The impure ramified type theory has an undecidable unification problem even at order 2. However, the decidability result for the pure subsystem indicates that unification should fail to terminate less often than general higher-order unification. We present applications to two expressive subsystems of second-order Peano arithmetic, \mbox{ACA}_0 and \Pi^1_{k}\mbox{-CA}_0

Parametricity and Dependent Types

Reynolds' abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We (in second order predicate logic) about inhabitants of the type. We obtain a similar result for a single lambda calculus (a pure type system), in which terms, types and their relations are expressed. Working within a single system dispenses with the need for an interpretation layer, allowing for an unusually simple presentation. While the unification puts some constraints on the type system (which we spell out), the result applies to many interesting cases, including dependently-typed ones

Topological Models of Columnar Vagueness

This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness

Nominal Unification from a Higher-Order Perspective

Nominal Logic is a version of first-order logic with equality, name-binding, renaming via name-swapping and freshness of names. Contrarily to higher-order logic, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of lambda-calculus. Despite these differences, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higher-order unification problems: Higher-Order Pattern Unification. This reduction proves that nominal unification can be decided in quadratic deterministic time, using the linear algorithm for Higher-Order Pattern Unification. We also prove that the translation preserves most generality of unifiers

Carnap: an Open Framework for Formal Reasoning in the Browser

This paper presents an overview of Carnap, a free and open framework for the development of formal reasoning applications. Carnap’s design emphasizes flexibility, extensibility, and rapid prototyping. Carnap-based applications are written in Haskell, but can be compiled to JavaScript to run in standard web browsers. This combination of features makes Carnap ideally suited for educational applications, where ease-of-use is crucial for students and adaptability to different teaching strategies and classroom needs is crucial for instructors. The paper describes Carnap’s implementation, along with its current and projected pedagogical applications