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On Knowledge, Strings, and Paradoxes
, 1998
"... . A powerful syntactic theory as well as expressive modal logics have to deal with selfreferentiality. Selfreferentiality and paradoxes seem to be close neighbours and depending on the logical system, they have devastating consequences, since they introduce contradictions and trivialise the logica ..."
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. A powerful syntactic theory as well as expressive modal logics have to deal with selfreferentiality. Selfreferentiality and paradoxes seem to be close neighbours and depending on the logical system, they have devastating consequences, since they introduce contradictions and trivialise the logical system. There is a large amount of different attempts to tackle these problems. Some of them are compared in this paper, futhermore a simple approach based on a threevalued logic is advocated. In this approach paradoxes may occur and are treated formally. However, it is necessary to be very careful, otherwise a system built on such an attempt trivialises as well. In order to be able to formally deal with such a system, the reason for selfreferential paradoxes is studied in more detail and a semantical condition on the connectives is given such that paradoxes are excluded. Keywords: Knowledge representation, selfreferentiality, paradoxe, Kleene logic Ich habe manche Zeit damit verloren, ...
System Description: Kimba, A Model Generator for ManyValued FirstOrder Logics
 In Proc. of the 16th International Conference on Automated Deduction (CADE16
, 1999
"... aint solver. The set of inequations generated is linear in size to that of the input theory and the solutions produced by the constraint solver are models of this input. The method does not require clause normalisation, an advantage in practice where suitable normalisation routines may not be availa ..."
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aint solver. The set of inequations generated is linear in size to that of the input theory and the solutions produced by the constraint solver are models of this input. The method does not require clause normalisation, an advantage in practice where suitable normalisation routines may not be available for the given logic. It also avoids the combinatorial explosion of tableau branches in certain manyvalued logics where the branching factor of propositional rules can equal the number of truth values. However, Hahnle's method cannot be used for firstorder model generation, and the mixedinteger inequations generated quickly lead to intractable CSPs. The model generator Kimba generalises Hahnle's idea to a constraint tableau system whose propagation mechanism can be tailored to the underlying logic. Kimba accepts higherorder specifications and makes use of Oz's builtin propagators and constraint solving system for integer variables. It translates quan
Proving partial correctness of partial functions
 PROC. CADEWORKSHOP MECHANIZATION OF PARTIAL FUNCTIONS
, 1996
"... We present a method for automated induction proofs about partial functions. This method cannot only be used to verify the partial correctness of functional programs, but it also solves some other challenge problems where reasoning about partial functions is necessary. For a further analysis of part ..."
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We present a method for automated induction proofs about partial functions. This method cannot only be used to verify the partial correctness of functional programs, but it also solves some other challenge problems where reasoning about partial functions is necessary. For a further analysis of partial functions we also developed a method to determine (nontrivial subsets of) their domains automatically.
Classical Logic with Partial Functions Hans
"... Abstract. We introduce a semantics for classical logic with partial functions. We believe that the semantics is natural. When a formula contains a subterm in which a function is applied outside of its domain, our semantics ensures that the formula has no truthvalue, so that it cannot be used for re ..."
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Abstract. We introduce a semantics for classical logic with partial functions. We believe that the semantics is natural. When a formula contains a subterm in which a function is applied outside of its domain, our semantics ensures that the formula has no truthvalue, so that it cannot be used for reasoning. The semantics relies on order of formulas. In this way, it is able to ensure that functions and predicates are properly declared before they are used. We define a sequent calculus for the semantics, and prove that this calculus is sound and complete for the semantics. 1
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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. 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...
Subset Types and Partial Functions
 19th International Conference on Automated Deduction
, 2003
"... Abstract. A classical higherorder logic PFsub of partial functions is defined. The logic extends a version of Farmer’s logic PF by enriching the type system of the logic with subset types and dependent types. Validity in PFsub is then reduced to validity in PF by a translation. 1 ..."
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Abstract. A classical higherorder logic PFsub of partial functions is defined. The logic extends a version of Farmer’s logic PF by enriching the type system of the logic with subset types and dependent types. Validity in PFsub is then reduced to validity in PF by a translation. 1
Isabelle/HOL as a Platform for Partiality
"... In Isabelle, there are several possibilities when one wants to support partial functions. One could put Isabelle to its intended use by directly embedding a logic of partial functions in Isabelle's metalogic. Alternatively, one could investigate partial functions in one of Isabelle's already welld ..."
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In Isabelle, there are several possibilities when one wants to support partial functions. One could put Isabelle to its intended use by directly embedding a logic of partial functions in Isabelle's metalogic. Alternatively, one could investigate partial functions in one of Isabelle's already welldeveloped object logics, e.g., ZF or HOL. In this paper we pursue the second approach and describe miscellaneous support for partiality found in the HOL instantiation of Isabelle. This support ranges from (1) underspecification and lifting in a recursive function definition package, (2) inductive definitions, to (3) HOLCF, an embedding of domain theory. All approaches are illustrated by relatively large examples. 1 Introduction The problem of partiality is one that all users of logics of total functions face, and a menagerie of solutions, hacks, workarounds and other good ideas have emerged to deal with this fundamental problem. In this paper, we will not discuss new logics intended to solve...
Unification in a Sorted λCalculus with Term Declarations and Function Sorts
, 1994
"... The introduction of sorts to firstorder automated deduction... ..."
Automatic Deduction for Theories of Algebraic Data Types
, 2011
"... In this thesis we present formal logical systems, concerned with reasoning about algebraic data types. The first formal system is based on the quantifierfree calculus (outermost universally quantified). This calculus is comprised of state change rules, and computations are performed by successive a ..."
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In this thesis we present formal logical systems, concerned with reasoning about algebraic data types. The first formal system is based on the quantifierfree calculus (outermost universally quantified). This calculus is comprised of state change rules, and computations are performed by successive applications of these rules. Thereby, our calculus gives rise to an abstract decision procedure. This decision procedure determines if a given formula involving algebraic type members is valid. It is shown that this calculus is sound and complete. We also examine how this system performs practically and give experimental results. Our main contribution, as compared to previous work on this subject, is a new and more efficient decision procedure for checking satisfiability of the universal fragment within the theory of algebraic data types. The second formal system, called Term Builder, is the deductive system based on higher order type theory, which subsumes second order and higher order logics. The main purpose of this calculus is to formulate and prove theorems about algebraic or other arbitrary userdefined types. Term Builder supports proof objects and is
Journal of Automated Reasoning manuscript No. (will be inserted by the editor) Classical Logic with Partial Functions
"... the date of receipt and acceptance should be inserted later Abstract We introduce a semantics for classical logic with partial functions, in which illtyped formulas are guaranteed to have no truth value, so that they cannot be used in any form of reasoning. The semantics makes it possible to mix rea ..."
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the date of receipt and acceptance should be inserted later Abstract We introduce a semantics for classical logic with partial functions, in which illtyped formulas are guaranteed to have no truth value, so that they cannot be used in any form of reasoning. The semantics makes it possible to mix reasoning about types and preconditions with reasoning about other properties. This makes it possible to deal with partial functions with preconditions of unlimited complexity. We show that, in spite of its increased complexity, the semantics is still a natural generalization of firstorder logic with simple types. If one does not use the increased expressivity, the type system is not stronger than classical logic with simple types. We will define two sequent calculi for our semantics, and prove that they are sound and complete. The first calculus follows the semantics closely, and hence its completeness proof is fairly straightforward. The second calculus is further away from the semantics, but more suitable for practical use because it has better proof theoretic properties. Its completeness can be shown by proving that proofs from the first calculus can be translated.