Handbook Of Logic Research Articles

Bipartite graphs as polynomials and polynomials as bipartite graphs

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial [Formula: see text], and any directed finite bipartite graph can be considered as a polynomial [Formula: see text], and vise verse. We also show that the multiplication in the semirings [Formula: see text], [Formula: see text] corresponds to an operation of the corresponding graphs. This operation is exactly the product of Petri nets in the sense of Winskel [G. Winskel and M. Nielsen, Models of concurrency, in Handbook of Logic in Computer Science, Vol. 4, eds. Abamsky, Gabbay and Maibaum (Oxford University Press, 1995), pp. 1–148]. As an application, we give an approach to dividing in the semirings [Formula: see text], [Formula: see text], and a criteria for parallalization of Petri nets. Finally, we endow the set of all bipartite graphs with the Zariski topology.

Context-Sensitivity in Jain Philosophy: A Dialogical Study of Siddharṣigaṇi’s Commentary on the Handbook of Logic

In classical India, Jain philosophers developed a theory of viewpoints (naya-v¯ ada) according to which any statement is always performed within and dependent upon a given epistemic perspective or viewpoint. The Jainas furnished this epistemology with an (epistemic) theory of disputation that takes into account the viewpoint in which the main thesis has been stated. The main aim of our paper is to delve into the Jain notion of viewpoint-contextualisation and to develop the elements of a suitable logical system that should offer a reconstruction of the Jainas' epistemic theory of disputation. A crucial step of our project is to approach the Jain theory of disputa-tion with the help of a theory of meaning for logical constants based on argumentative practices called dialogical logic. Since in the dialogical framework the meaning of the logical constants is given by the norms or rules for their use in a debate, it provides a meaning theory closer to the Jain context-sensitive disputation theory than the mainstream formal model-theoretic semantics.

Designing and Implementing an Ontology for Logic and Linguistics

This article reports on the design and implementation of a domain-specific, manually built ontology, as a scaffolding for the electronic publication of and access to scientific handbooks. The ontology will provide access to the case study handbook (the Handbook of Logic and Language, van Benthem, J. and ter Meulen, A. (1997)) by means of semi-automatically detected links. The ontology's structure is given by a set of hierarchical relations plus two non-hierarchical relations introduced for navigational purposes. The main expected advantage of this way of providing access to the text consists in an enhanced browsing system, providing the user with an explicit map of the contents of the handbook. Moreover, by manually designing and populating the hierarchy, we expect a coherent representation of the domain and a good quality result. When designing and implementing the ontology, we put into practice some of the ideas and technologies already available from the Semantic Web. In the course of this study we describe the design and implementation of the ontology, and draw some preliminary conclusions about the feasibility of such an enterprise and its suitability to support reading in an electronic environment. Here we do not treat the issue of automatically generating links from the ontology to the text, which is currently a work in progress.

A NEW FINITE DIFFERENCE METHOD FOR THE HELMHOLTZ EQUATION USING SYMBOLIC COMPUTATION

A new finite difference method for the Helmholtz equation is presented. The method involves replacing the standard "weights" in the central difference quotients (Secs. 2.1, 2.2, and 2.3) by weights that are optimal in a sense that will be explained in the sections just mentioned. The calculation of the optimal weights involves some complicated and error-prone manipulations of integral formulas that is best done using computer-aided symbolic computation (SC). In addition, we discuss the important problem of interpolation involving meshes that have been refined in certain subregions. Analytic formulae are derived using SC for these interpolation schemes. Our results are discussed in Sec. 5. Some hints about the computer methods we used to accomplish these results are given in the Appendix. More information is available and access to that information is referenced. While we do not want to make SC the focus of this work, we also do not want to underestimate its value. Armed with robust and efficient SC librar...

Handbook of logic in artificial intelligence and logic programming, Volume 3, Nonmonotonic reasoning and uncertain reasoning, edited by Dov M. Gabbay, C. J. Hogger, and J. A. Robinson, with D. Nute, Handbooks of logic in computer science and artificial intelligence and logic programming, Clarendon Press, Oxford University Press, Oxford, New York, etc., 1994, xix + 529 pp.–

Handbook of logic in artificial intelligence and logic programming, Volume 3, Nonmonotonic reasoning and uncertain reasoning, edited by Dov M. Gabbay, C. J. Hogger, and J. A. Robinson, with D. Nute, Handbooks of logic in computer science and artificial intelligence and logic programming, Clarendon Press, Oxford University Press, Oxford, New York, etc., 1994, xix + 529 pp.– - Volume 6 Issue 4

Dov M. Gabbay, C.J. Hogger, J.A. Robinson (Eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 2: Deduction Methodologies

Dov M. Gabbay, C.J. Hogger, J.A. Robinson (Eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 2: Deduction Methodologies

Handbook of logic and language

From the Publisher: The combined study of logic and language goes back at least as far as the Middle Ages. In the last twenty-five years it has gained momentum with the formulation of Montague semantics and Generative Syntax, and the subsequent diversification of research programs. The Handbook of Logic and Language is the first comprehensive survey of the field. The twenty chapters show both sides of the interaction between logic and language: how logical systems are designed and modified in response to linguistic needs, and how mathematical theory arises out of this process and affects subsequent linguistic theory. Contributors: N. Asher, D. Beaver, W. Buszkowski, D. de Jongh, J. E. Fenstad, J. Groenendijk, H. Hendriks, J. Higginbotham, J. Hintikka, T. M. V. Janssen, H. Kamp, E. J. Keenan, J. T. Lonning, E. Martin, M. J. Moortgat, L. S. Moss, R. Muskens, D. Osherson, B. H. Partee, F. J. Pelletier, W. C. Rounds, G. Sandu, J. Seligman, M. Steedman, M. Stokhof, R. H. Thomason, R. Turner, J. van Benthem, J. van Eijck, A. Visser, D. Westerstahl

Comparing cubes of typed and type assignment systems

We study the cube of type assignment systems, as introduced in Giannini et al. (Fund. Inform. 19 (1993) 87–126), and confront it with Barendregt's typed gl-cube (Barendregt, in: Handbook of Logic in Computer Science, Vol. 2, Clarenden Press, Oxford, 1992). The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property holds only for systems without polymorphism. The type assignment systems we consider satisfy the properties ‘subject reduction’ and ‘strong normalization’. Moreover, we define a new type assignment cube that is isomorphic to the typed one.

Book review: Handbook of Logic in Artificial Intelligence and Logic Programming. Volume 1. Logical Foundations. Edited by Dov M. Gabbay, C. J. Hogger, and J. A. Robinson (Claredon Press, Oxford, 1993 )

One of the main goals of Artificial Intelligence is to design programs (automatic systems) capable of reasoning. It is therefore natural that many of these systems use logic, i.e., a well-formalized part of human reasoning. There are different opinions in the AI community on the importance of logic: John McCarthy pioneered the approach in which logic is the main tool, while some other researchers put more hope into domain-specific heuristics, expert's knowledge, simulating physiology (like neural networks), etc. However, most AI researchers, even those who do not consider logic the main basis for AI, agree that logic is important for AI. This book is the first volume of a series that presents logic to AI researchers, with an emphasis on those parts of logic that are applicable and useful to AI.

Handbook of logic in artificial intelligence and logic programming: v.1: Logical foundations; v.2: Deduction methodologies; v.3: Nonmonotonic reasoning and uncertain reasoning

Handbook of logic in artificial intelligence and logic programming: v.1: Logical foundations; v.2: Deduction methodologies; v.3: Nonmonotonic reasoning and uncertain reasoning

Necessary Propositions and the Square of Opposition

NECESSARY PROPOSITIONS AND THE SQUARE OF OPPOSITION MARK ROBERTS University of Rhode Island Kingston, Rhode Island IT IS COMMONPLACE to define contradictory, contrary, and subcontrary propositions in the following way: contradictory propositions cannot both be true and cannot both be false; contrary propositions cannot both be true but can both be false; and subcontrary propositions can both be true but cannot both be false. In his Introduction to Logic 1 Irving Copi raises a problem with two of these definitions which he believes forces him to limit the range of propositions which can be used in the square of opposition. Since contrary propositions can both be false, but the falsity of a necessarily true proposition is not possible , a necessarily true proposition has no contrary. Therefore, only contingent propositions can be contraries. Subcontrary propositions can both be true, but the truth of a necessarily false proposition is not possible.2 This problem is rarely raised in other logic texts, but, it is interesting to note, it is mentioned by a number of neoscholastic philosophers, for example, P. Nicholas Russo,3 Rev. H. Grenier,4 1 Irving M. Copi, Introduction to Logic (New York: Macmillan, 1986), pp. 178-179. Copi first introduced this problem in the fifth edition of his work on the basis of an article by David H. Sanford entitled " Contraries and Subcontraries ,'' Nous 2 (1968) : 95-96. Sanford argues that the most reasonable way to resolve the problem of necessary propositions is to assume that contingent propositions comprise the square of opposition. 2 Copi, pp. 178-179. 8 Nicholas Russo, Summa Philosophica (Boston: Apud Marlier et Socios, 1[;85), p. 23. 4 Henri Grenier, Thomistic Philosophy, Vol. I (Charlottetown, Canada: St. Dunstan's University, 1950), p. 69. 427 428 MARK ROBERTS G. Sanseverino,5 Jacques Maritain,6 F.-X. Maquart,7 P. Coffey,8 and others.9 (It is also mentioned by Bishop Whately.10 ) Each of these philosophers makes necessarily true or false propositions an exception to the definition of contrary and subcontrary propositions . The reason is they and Copi interpret the definitions of contrary and subcontrary propositions to mean that the falsity of each contrary proposition is possible and the truth of each subcontrary proposition is possible. It should be mentioned, however, that the position of the philosophers cited above is not entirely the same as Copi's. Copi reasons that these definitions imply that necessary propositions have no contrary or subcontrary. The other philosophers believe that necessary propositions can be contraries and subcontraries. Thus, in the case of necessary propositions contrary and subcontrary propositions are, instead, defined merely in terms of their quantity and quality, not by whether or not they can both be false or can both be true. In fact, Coffey and some of the other philosophers cited above assert that, when the propositions composing the square are necessary , one can infer the truth of the universal from the truth of 5 Gaietano Sanseverino, Philosophia Christiana, Vol. II (Neapoli: Vincentii Manfredi, 1862), pp. DCCXXXVIII ff. 6 J. Maritain, An Introduction to Logic (London: Sheed & Ward, 1937), p. 135, notes 1 & 2. 7 F.-X. Maquart, Elementa Philosophiae, Vol. I (Parisiis: Andreas Blot, 1937)' p. 126 ff. 8 P. Coffey, The Science of Logic, Vol. 1 (New York: Longmans, Green & Co., 1918), p. 225, 226. 9 Joseph Gredt, Elementa Philosophiae, Vol. I (Friburgi: Herder, 1929), p. 46, notes 1 & 2; Sylvester J. Hartman, A Textbook of Logic (New York: American Book Co., 1936), p. 162 ff; Roland Houde and Jerome J. Fischer, Handbook of Logic (Dubuque, Iowa: Wm. C. Brown Co., 1954), p. 61; Eduard Hugon, Cursus Philosophiae Thomisticae, Vol. I (Parisiis: P. Lethielleux n.d), p. 155; Dennis C. Kane, Logic: The Art of Predication and Inference (Providence: Providence College Press, 1978), p. 98 ff; Francis P. Siegfried, Essentialia Philosophiae (Philadelphia: Dolphin Press, 1927), p. 25; Francis Varvello, Minor Logic, trans. and supplemented Arthur D. Fearon (San Francisco: Univ. of San Francisco Press, 1933), p. 73 ff. 10 Richard Whately, Elements of Logic (Boston: James Munroe & Co., 1854), p. 77 ff. PROPOSITIONS AND THE SQUARE OF OPPOSITION 429 the particular and the falsity of the particular from the falsity of the universal.11 Maritain maintains that in the case...

Automated reasoning and nonclassical logics: Introduction

In the mid-1980s researchers in the computer science and artificial intelligence communities began to take an interest in logics studied by logicians and philosophers since at least Aristotle in the fourth century BC. These are logics such as deontic, epistemic, intuitionistic, modal, paraconsistent, relevant, and temporal logics, all of which are usually collected under the somewhat unsatisfactory name of nonclassical, or nonstandard, logics. During the past several years, the interest in nonclassical logics in these communities has increased enormously, as anyone remotely familiar with them will know. To the reader unfamiliar with this territory we recommend [3] for a masterful historical survey of it from the time of Aristotle. A huge amount of modem work has been done in the field during this century, and the total literature would probably exceed 4000 items. Central research directions in this field have been surveyed in many places, but in our opinion the definitive work is Gabbay and Guenthner's multivolume Handbook of Philosophical Logic, the first volume of which was published in 1983 by Reidel. Recent research done on nonclassical logics in computer science and artificial intelligence is to be the subject of two further multivolume series, currently in preparation by Gabbay and others, called the Handbook of Logic in Computer Science and the Handbook of Logic in Artificial Intelligence and Logic Programming. These will be published over the next few years. Sophisticated automated reasoning systems based on classical logic have been in use for over 20 years. However, similar systems for nonclassical logics have only recently made an appearance. This special issue of the Journal of Automated Reasoning addresses the use of automated reasoning systems to solve problems in a ntmaber of nonclassical logics. Indeed, a common theme to all of the papers in this issue is that they discuss the implementations of various automated reasoning systems. Here we have in mind automated reasoning systems in possibly a broader sense than is usual. Automated reasoning systems tend to be equated with automated theorem-proving systems. At least two of the papers in this special issue, those of Slaney and Pritchard, are concerned with the automation of methods for finding non-theorems.

"Handbook of Logic," by E. R. Emmet; and "Disputed Questions in Philosophy," by J. F. Keleher

"Handbook of Logic," by E. R. Emmet; and "Disputed Questions in Philosophy," by J. F. Keleher

An Elementary Handbook of Logic.

An Elementary Handbook of Logic.