Topic
Deduction theorem
About: Deduction theorem is a research topic. Over the lifetime, 392 publications have been published within this topic receiving 7910 citations.
Papers published on a yearly basis
1 / 2
Papers
15 Jul 1992
TL;DR: This paper presents a way of extending the paradigm "proofs as programs" to classical proofs, which can be seen as a simple extension of intuitionistic natural deduction, whose algorithmic interpretation is very well known.
Abstract: 1 INTRODUCTION This paper presents a way of extending the paradigm "proofs as programs" to classical proofs. The system we use is derived from the general Free Deduction system presented in [31. Usually when considering proofs as programs, one has only in mind some kind of intuitionistic proofs. There is an obvious reason for that restriction: only intuitionistic proofs are contructive, in the sense that from the proof of an existential statement, one can get a witness of this existential statement. But from the programming point of view, constructivity is only needed for E~-statements, for which classical and intuitionistic provability coincide. This means that, classical proofs are also candidates for being programs. In order to use them as programs, one has two tasks to achieve: (i) to find a system in which one can extract directly a program from a classical proof (and not by means of a translation to intuitionistic logic), and (ii) to understand the algorithmic meaning of classical constructions. The system we will consider is a natural deduction system with multiple conclusions, we will call it Classical Natural Deduction (the one with the absurdity rule being called Usual Natural Deduction). It is a particular subsystem of Free Deduction (FD) with inputs fixed to the left, chosen for its simplicity: it can be seen as a simple extension of intuitionistic natural deduction, whose algorithmic interpretation is very well known. In this context, the contribution of classical constructs to programming appears clearly: they correspond to control operators added to functional languages, like call/ce in Scheme. In both contexts, the role of the classical constructs is the same: they allow to take shorter routes in the construction of a proof~program. The link between control operators and classical constructs has first been made by T. Griffin in [1], where he proposes to type the C operator of Felleisen, with the type-~'-,A-* A. The system he obtains is not satisfactory from the logical point of view: the reduction is in fact a reduction strategy and the type assigned to C doesn't fit in general the reduction rule for C. C. Murthy further analysed the connections t91 between control operators, classical constructs and translations from classical logic to intuitionistic logic (see [4]). The difficulties met in trying to use-~-A ~ A (or the classical absurdity rule) as a type for control operators is not really due to classical logic, but much nore to …
691 citations
TL;DR: A logical language which extends the syntax of positive Horn clauses by permitting implications in goals and in the bodies of clauses is presented and it is shown how to build a Kripke-like model for programs by a fixed-point construction and shown that the operational meaning of implication is sound and complete for intuitionistic logic.
Abstract: We present a logical language which extends the syntax of positive Horn clauses by permitting implications in goals and in the bodies of clauses. The operational meaning of a goal which is an implication is given by the deduction theorem: a goal D ⊃ G is provable from a program P if the goal G is provable from the larger program P ∪ {D}. This paper explores the qualitative nature of this extension to logic programming. For example, if the formula D is the conjunction of universally quantified clauses, we interpret the goal D ⊃ G as a request to load the code in D prior to attempting G and then unload that code after G succeeds or fails. This extended use of implication provides a logical explanation of parametric modules, some uses of PROLOG's assert predicate, and aspects of abstract datatypes. Both a model theory and proof theory are presented for this logical language. In particular, we show how to build a Kripke-like model for programs by a fixed-point construction and show that the operational meaning of implication mentioned above is sound and complete for intuitionistic logic. We also examine a weak notion of negation which is easily implemented in this language and show how database constraints can be represented by it.
309 citations
TL;DR: A rule based approach to defining multivalued implications is suggested which allows a user to specify an implication to met their customized performance requirements.
246 citations
Book•
5 Dec 2010
TL;DR: In this paper, the authors present a general framework for automated reasoning in many-valued logic systems based on the Lattice-Valued Propositional Logic (LVPL).
Abstract: I Introduction.- 1 Introduction.- 1.1 Major Methodologies in Artificial Intelligence.- 1.2 Basic Academic Ideas.- 1.3 Some Related Concepts.- 1.4 Many-Valued Logic and Lattice-Valued Logic.- 1.5 Uncertainty Inference.- 1.5.1 Probability-Based Uncertainty Reasoning.- 1.5.2 Fuzzy Set Based Uncertainty Reasoning.- 1.5.3 Non-Monotonic Logic Based Uncertainty Reasoning.- 1.6 Automated Reasoning in Many-Valued Logic.- II Lattice Implication Algebras.- 2 Concepts and Properties.- 2.1 Lattice Implication Algebras.- 2.1.1 Concepts and Examples.- 2.1.2 Basic Properties.- 2.2 Lattice H Implication Algebras.- 2.3 Lattice Properties.- 2.4 Homomorphisms.- 3 Filters.- 3.1 Filters and Implicative Filters.- 3.2 Generated Filters.- 3.3 Positive Implicative Filters and Associative Filters.- 3.4 Prime Filters and Ultra-Filters.- 3.5 I-Filters, Involution Filters and Obstinate Filters.- 3.6 Fuzzy Filters.- 4 LI-Ideals.- 4.1 LI-Ideals.- 4.2 Fuzzy LI-Ideals.- 4.3 Normal Fuzzy LI-Ideals.- 4.4 Intuitionistic Fuzzy LI-Ideals.- 5 Homomorphisms and Representations.- 5.1 Congruence Relations.- 5.1.1 Congruence Relations Induced by Filters.- 5.1.2 Congruences Relations Induced by LI-ideals.- 5.1.3 Congruence Relations Induced by Fuzzy Filters.- 5.1.4 Congruence Relations Induced by Fuzzy LI-ideals.- 5.2 Proper Lattice Implication Algebras.- 5.3 Representations.- 6 Topological Structure of Filter Spaces.- 6.1 Filter Spaces.- 6.1.1 Basic Concepts.- 6.1.2 Topological Properties.- 6.2 Product Topology and Quotient Topology.- 6.3 Lattice Topology.- 6.4 Prime Spaces.- 7 Connections with Related Algebras.- 7.1 Lattice Implication Algebras and BCK-Algebras.- 7.2 Lattice Implication Algebras and MV-Algebras.- 7.3 Lattice Implication Algebras and Related Algebras.- 8 Related Issues.- 8.1 Category of Lattice Implication Algebras.- 8.2 Category of Fuzzy Lattice Implication Algebras.- 8.3 Fuzzy Power Sets.- 8.4 Adjoint Semigroups.- 8.5 Logical Properties.- III Lattice-Valued Logic Systems.- 9 Lattice-Valued Propositional Logics.- 9.1 Lattice-Valued Propositional Logic LP(X).- 9.1.1 Language.- 9.1.2 Semantics.- 9.1.3 Syntax.- 9.1.4 Examples.- 9.2 Gradational Lattice-Valued Propositional Logic Lvpl.- 9.2.1 Language.- 9.2.2 Rules of Inference.- 9.2.3 Semantics.- 9.2.4 Syntax.- 9.2.5 Satisfiability and Consistency.- 9.2.6 Deduction Theorem.- 9.2.7 Compactness.- 9.2.8 Examples.- 10 Lattice-Valued First-Order Logics.- 10.1 Lattice-Valued First-Order Logic LF(X).- 10.1.1 Language.- 10.1.2 Interpretation.- 10.1.3 Semantics.- 10.1.4 Syntax.- 10.1.5 Properties of Model Theory.- 10.2 Gradational Lattice-Valued First-Order Logic Lvfl.- 10.2.1 Language.- 10.2.2 Interpretation.- 10.2.3 Semantics.- 10.2.4 Standardization of Formulae.- 10.2.5 Syntax.- 10.2.6 Soundness and Completeness.- 10.2.7 Satisfiability and Consistency.- 10.2.8 Deduction Theorem.- 10.2.9 Compactness.- 10.2.10Examples.- 11 Uncertainty and Automated Reasoning.- 11.1 Uncertainty Reasoning Based on LP(X).- 11.2 Uncertainty Reasoning Based on Lvpl.- 11.2.1 Another Kind of Interpretation of X ? Y.- 11.2.2 Basic Theory.- 11.2.3 Examples.- 11.2.4 Multi-Dimensional and Multiple Uncertainty Reasoning.- Models and Methods.- Semantical Interpretation and Syntactical Proof.- 11.3 ?-Resolution Principle Based on LP(X).- 11.3.1 ?-Resolution Principle.- 11.3.2 Soundness and Completeness.- 11.4 ?-Resolution Principle Based on LF(X).- 11.4.1 Interpretation of Formulae.- 11.4.2 ?-Resolution Principle.- References.
228 citations
TL;DR: Strong completeness is demonstrated for the Aristotelian system in the sense that every valid argument formable in the language of the system is demonstrable by means of a formal deduction in the system.
Abstract: In previous articles ([4], [5]) it has been shown that the deductive system developed by Aristotle in his “second logic” (cf. Bochenski [2, p. 43]) is a natural deduction system and not an axiomatic system as previously had been thought [6]. It was also pointed out that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument formable in the language of the system is demonstrable by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system.
172 citations
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| 2016 | 4 |