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A Mechanization of Strong Kleene Logic for Partial Functions
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
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Cited by 28 (11 self)
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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.
A Tableau Calculus for Partial Functions
, 1996
"... . Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
Abstract

Cited by 6 (5 self)
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. Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. Keywords: Partial functions, manyvalued logic, sorted logic, tableau. 1 Introduction Many practical applications of deduction systems in mathematics and computer science rely on the correct and efficient treatment of partial functions. For this purpose...
A Resolution Calculus for Presuppositions
 Proceedings of the 12th ECAI
, 1996
"... . The semantics of everyday language and the semantics of its naive translation into classical firstorder language considerably differ. An important discrepancy that is addressed in this paper is about the implicit assumption what exists. For instance, in the case of universal quantification natura ..."
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Cited by 4 (3 self)
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. The semantics of everyday language and the semantics of its naive translation into classical firstorder language considerably differ. An important discrepancy that is addressed in this paper is about the implicit assumption what exists. For instance, in the case of universal quantification natural language uses restrictions and presupposes that these restrictions are nonempty, while in classical logic it is only assumed that the whole universe is nonempty. On the other hand, all constants mentioned in classical logic are presupposed to exist, while it makes no problems to speak about hypothetical objects in everyday language. These problems have been discussed in philosophical logic and some adequate manyvalued logics were developed to model these phenomena much better than classical firstorder logic can do. An adequate calculus, however, has not yet been given. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. Unfortunately, restricted quantifications are not truthfunctional, hence they do not fit the framework directly. We solve this problem by applying recent methods from sorted logics.
Reasoning without Believing  On the Mechanization of Presuppositions and Partiality
"... . It is wellknown that many relevant aspects of everyday reasoning based on natural language cannot be adequately expressed in classical firstorder logic. In this paper we address two of the problems, firstly that of socalled presuppositions, expressions from which it is possible to draw implicit ..."
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Cited by 1 (1 self)
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. It is wellknown that many relevant aspects of everyday reasoning based on natural language cannot be adequately expressed in classical firstorder logic. In this paper we address two of the problems, firstly that of socalled presuppositions, expressions from which it is possible to draw implicit conclusion, which classical logic normally does not warrant, and secondly the related problem of partiality and the adequate treatment of undefined expressions. In natural language, presuppositions are quite common, they can, however, only insufficiently be modeled in classical firstorder logic. For instance, in the case of universal quantification one normally uses restrictions in natural language and presupposes that these restrictions are nonempty, while in classical logic it is only assumed that the whole universe is nonempty. On the other hand, all constants mentioned in classical logic are presupposed to exist, while it makes no problems to speak about hypothetical objects in every...
HigherOrder OrderSorted Resolution
 FB Informatik, Universitat des Saarlandes
, 1994
"... The introduction of sorts to firstorder automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search space. It is therefore promising to treat sorts in higher order theorem proving as well. In this paper we present a generalizati ..."
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Cited by 1 (1 self)
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The introduction of sorts to firstorder automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search space. It is therefore promising to treat sorts in higher order theorem proving as well. In this paper we present a generalization of Huet's Constrained Resolution to an ordersorted type theory \SigmaT with term declarations. This system builds certain taxonomic axioms into the unification and conducts reasoning with them in a controlled way. We make this notion precise by giving a relativization operator that totally and faithfully encodes \SigmaT into simple type theory.
Semantic Tableaux for a Logic with Preorders and Dynamic Sorts
, 1996
"... . In this paper we present a logic for dealing with preorders, where functions and predicates behave monotonically or antimonotonically in their arguments, and incorporating ordersorted relations into the syntax of the language. For this logic a ground tableaubased deduction method and a free varia ..."
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. In this paper we present a logic for dealing with preorders, where functions and predicates behave monotonically or antimonotonically in their arguments, and incorporating ordersorted relations into the syntax of the language. For this logic a ground tableaubased deduction method and a free variable extension version are proposed, proving their soundness and completeness. Finally, an implementation of the kernel of the logic based on logic programming is outlined. 1 Introduction The study of efficient methods for dealing with equality has been traditionally considered an important workline in different areas of theoretical computer science. However some recent research has revealed the need of extending this study in order to cover transitive relations different from those of equivalence. This is the case, for example, in CLP [JM 94], where constraint solving merges with logic programming. In the field of automated deduction this situation has resulted in the development of provers...
A Many Sorted Natural Deduction
, 1994
"... The goal of this paper is to motivate and define yet another sorted logic, called SND. All the previous sorted logics which can be found in the Artificial Intelligence literature have been designed to be used in (completely) automated deduction. SND has been designed to be used in interactive theor ..."
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The goal of this paper is to motivate and define yet another sorted logic, called SND. All the previous sorted logics which can be found in the Artificial Intelligence literature have been designed to be used in (completely) automated deduction. SND has been designed to be used in interactive theorem proving. Because of this shift of focus, SND has been designed to satisfy three innovative design requirements; that is: it is defined on top of a natural deduction calculus, and in a way to be a definitional extension of such calculus; and it is implemented on top of its implementation. In turn, because of this fact, SND has various innovative technical properties; among them: it allows us to deal with free variables, it has no notion of wellsortedness and of wellsortedness being a prerequisite of wellformedness, its implementation is such that, in the default mode, the system behaves exactly as with the original unsorted calculus. The formal system presented here was originally defin...