An Axiomatic Approach to Physical Systems
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On the Boundary Between Mereology and Topology
On the Boundary Between Mereology and Topology Achille C. Varzi Istituto per la Ricerca Scientifica e Tecnologica, I-38100 Trento, Italy [email protected] (Final version published in Roberto Casati, Barry Smith, and Graham White (eds.), Philosophy and the Cognitive Sciences. Proceedings of the 16th International Wittgenstein Symposium, Vienna, Hölder-Pichler-Tempsky, 1994, pp. 423–442.) 1. Introduction Much recent work aimed at providing a formal ontology for the common-sense world has emphasized the need for a mereological account to be supplemented with topological concepts and principles. There are at least two reasons under- lying this view. The first is truly metaphysical and relates to the task of charac- terizing individual integrity or organic unity: since the notion of connectedness runs afoul of plain mereology, a theory of parts and wholes really needs to in- corporate a topological machinery of some sort. The second reason has been stressed mainly in connection with applications to certain areas of artificial in- telligence, most notably naive physics and qualitative reasoning about space and time: here mereology proves useful to account for certain basic relation- ships among things or events; but one needs topology to account for the fact that, say, two events can be continuous with each other, or that something can be inside, outside, abutting, or surrounding something else. These motivations (at times combined with others, e.g., semantic transpar- ency or computational efficiency) have led to the development of theories in which both mereological and topological notions play a pivotal role. How ex- actly these notions are related, however, and how the underlying principles should interact with one another, is still a rather unexplored issue. -
A Translation Approach to Portable Ontology Specifications
Knowledge Systems Laboratory September 1992 Technical Report KSL 92-71 Revised April 1993 A Translation Approach to Portable Ontology Specifications by Thomas R. Gruber Appeared in Knowledge Acquisition, 5(2):199-220, 1993. KNOWLEDGE SYSTEMS LABORATORY Computer Science Department Stanford University Stanford, California 94305 A Translation Approach to Portable Ontology Specifications Thomas R. Gruber Knowledge System Laboratory Stanford University 701 Welch Road, Building C Palo Alto, CA 94304 [email protected] Abstract To support the sharing and reuse of formally represented knowledge among AI systems, it is useful to define the common vocabulary in which shared knowledge is represented. A specification of a representational vocabulary for a shared domain of discourse — definitions of classes, relations, functions, and other objects — is called an ontology. This paper describes a mechanism for defining ontologies that are portable over representation systems. Definitions written in a standard format for predicate calculus are translated by a system called Ontolingua into specialized representations, including frame-based systems as well as relational languages. This allows researchers to share and reuse ontologies, while retaining the computational benefits of specialized implementations. We discuss how the translation approach to portability addresses several technical problems. One problem is how to accommodate the stylistic and organizational differences among representations while preserving declarative content. Another is how -
Functional Declarative Language Design and Predicate Calculus: a Practical Approach
Functional Declarative Language Design and Predicate Calculus: A Practical Approach RAYMOND BOUTE INTEC, Ghent University, Belgium In programming language and software engineering, the main mathematical tool is de facto some form of predicate logic. Yet, as elsewhere in applied mathematics, it is used mostly far below its potential, due to its traditional formulation as just a topic in logic instead of a calculus for everyday practical use. The proposed alternative combines a language of utmost simplicity (four constructs only) that is devoid of the defects of common mathematical conventions, with a set of convenient calculation rules that is sufficiently comprehensive to make it practical for everyday use in most (if not all) domains of interest. Its main elements are a functional predicate calculus and concrete generic functionals. The first supports formal calculation with quantifiers with the same fluency as with derivatives and integrals in classical applied mathematics and engineering. The second achieves the same for calculating with functionals, including smooth transition between pointwise and point-free expression. The extensive collection of examples pertains mainly to software specification, language seman- tics and its mathematical basis, program calculation etc., but occasionally shows wider applicability throughout applied mathematics and engineering. Often it illustrates how formal reasoning guided by the shape of the expressions is an instrument for discovery and expanding intuition, or highlights design opportunities in declarative -
What Is an Embedding? : a Problem for Category-Theoretic Structuralism
What is an Embedding? : A Problem for Category-theoretic Structuralism Angere, Staffan Published in: Preprint without journal information 2014 Link to publication Citation for published version (APA): Angere, S. (2014). What is an Embedding? : A Problem for Category-theoretic Structuralism. Unpublished. Total number of authors: 1 General rights Unless other specific re-use rights are stated the following general rights apply: Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Read more about Creative commons licenses: https://creativecommons.org/licenses/ Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. LUND UNIVERSITY PO Box 117 221 00 Lund +46 46-222 00 00 What is an Embedding? A Problem for Category-theoretic Structuralism PREPRINT Staffan Angere April 16, 2014 Abstract This paper concerns the proper definition of embeddings in purely category-theoretical terms. It is argued that plain category theory can- not capture what, in the general case, constitutes an embedding of one structure in another. -
Proposals for Mathematical Extensions for Event-B
Proposals for Mathematical Extensions for Event-B J.-R. Abrial, M. Butler, M Schmalz, S. Hallerstede, L. Voisin 9 January 2009 Mathematical Extensions 1 Introduction In this document we propose an approach to support user-defined extension of the mathematical language and theory of Event-B. The proposal consists of considering three kinds of extension: (1) Extensions of set-theoretic expressions or predicates: example extensions of this kind consist of adding the transitive closure of relations or various ordered relations. (2) Extensions of the library of theorems for predicates and operators. (2) Extensions of the Set Theory itself through the definition of algebraic types such as lists or ordered trees using new set constructors. 2 Brief Overview of Mathematical Language Structure A full definition of the mathematical language of Event-B may be found in [1]. Here we give a very brief overview of the structure of the mathematical language to help motivate the remaining sections. Event-B distinguishes predicates and expressions as separate syntactic categories. Predicates are defined in term of the usual basic predicates (>, ?, A = B, x 2 S, y ≤ z, etc), predicate combinators (:, ^, _, etc) and quantifiers (8, 9). Expressions are defined in terms of constants (0, ?, etc), (logical) variables (x, y, etc) and operators (+, [, etc). Basic predicates have expressions as arguments. For example in the predicate E 2 S, both E and S are expressions. Expression operators may have expressions as arguments. For example, the set union operator has two expressions as arguments, i.e., S [ T . Expression operators may also have predicates as arguments. For example, set comprehension is defined in terms of a predicate P , i.e., f x j P g. -
Semantics of First-Order Logic
CSC 244/444 Lecture Notes Sept. 7, 2021 Semantics of First-Order Logic A notation becomes a \representation" only when we can explain, in a formal way, how the notation can make true or false statements about some do- main; this is what model-theoretic semantics does in logic. We now have a set of syntactic expressions for formalizing knowledge, but we have only talked about the meanings of these expressions in an informal way. We now need to provide a model-theoretic semantics for this syntax, specifying precisely what the possible ways are of using a first-order language to talk about some domain of interest. Why is this important? Because, (1) that is the only way that we can verify that the facts we write down in the representation actually can mean what we intuitively intend them to mean; representations without formal semantics have often turned out to be incoherent when examined from this perspective; and (2) that is the only way we can judge whether proposed methods of making inferences are reasonable { i.e., that they give conclusions that are guaranteed to be true whenever the premises are, or at least are probably true, or at least possibly true. Without such guarantees, we will probably end up with an irrational \intelligent agent"! Models Our goal, then, is to specify precisely how all the terms, predicates, and formulas (sen- tences) of a first-order language can be interpreted so that terms refer to individuals, predicates refer to properties and relations, and formulas are either true or false. So we begin by fixing a domain of interest, called the domain of discourse: D. -
Solutions to Inclass Problems Week 2, Wed
Massachusetts Institute of Technology 6.042J/18.062J, Fall ’05: Mathematics for Computer Science September 14 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised September 13, 2005, 1279 minutes Solutions to InClass Problems Week 2, Wed. Problem 1. For each of the logical formulas, indicate whether or not it is true when the domain of discourse is N (the natural numbers 0, 1, 2, . ), Z (the integers), Q (the rationals), R (the real numbers), and C (the complex numbers). ∃x (x2 =2) ∀x ∃y (x2 =y) ∀y ∃x (x2 =y) ∀x�= 0∃y (xy =1) ∃x ∃y (x+ 2y=2)∧ (2x+ 4y=5) Solution. Statement N Z Q R √ C ∃x(x2= 2) f f f t(x= 2) t ∀x∃y(x2=y) t t t t(y=x2) t ∀y∃x(x2=y) f f f f(take y <0)t ∀x�=∃ 0y(xy= 1) f f t t(y=/x 1 ) t ∃x∃y(x+ 2y= 2)∧ (2x+ 4y=5)f f f f f � Problem 2. The goal of this problem is to translate some assertions about binary strings into logic notation. The domain of discourse is the set of all finitelength binary strings: λ, 0, 1, 00, 01, 10, 11, 000, 001, . (Here λdenotes the empty string.) In your translations, you may use all the ordinary logic symbols (including =), variables, and the binary symbols 0, 1 denoting 0, 1. A string like 01x0yof binary symbols and variables denotes the concatenation of the symbols and the binary strings represented by the variables. For example, if the value of xis 011 and the value of yis 1111, then the value of 01x0yis the binary string 0101101111. -
Discrete Mathematics for Computer Science I
University of Hawaii ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., University of Hawaii Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross Provided by McGraw-Hill ICS 141: Discrete Mathematics I – Fall 2011 4-1 University of Hawaii Lecture 4 Chapter 1. The Foundations 1.3 Predicates and Quantifiers ICS 141: Discrete Mathematics I – Fall 2011 4-2 Topic #3 – Predicate Logic Previously… University of Hawaii In predicate logic, a predicate is modeled as a proposional function P(·) from subjects to propositions. P(x): “x is a prime number” (x: any subject) P(3): “3 is a prime number.” (proposition!) Propositional functions of any number of arguments, each of which may take any grammatical role that a noun can take P(x,y,z): “x gave y the grade z” P(Mike,Mary,A): “Mike gave Mary the grade A.” ICS 141: Discrete Mathematics I – Fall 2011 4-3 Topic #3 – Predicate Logic Universe of Discourse (U.D.) University of Hawaii The power of distinguishing subjects from predicates is that it lets you state things about many objects at once. e.g., let P(x) = “x + 1 x”. We can then say, “For any number x, P(x) is true” instead of (0 + 1 0) (1 + 1 1) (2 + 1 2) ... The collection of values that a variable x can take is called x’s universe of discourse or the domain of discourse (often just referred to as the domain). ICS 141: Discrete Mathematics I – Fall 2011 4-4 Topic #3 – Predicate Logic Quantifier Expressions University of Hawaii Quantifiers provide a notation that allows us to quantify (count) how many objects in the universe of discourse satisfy the given predicate. -
Leibniz on Providence, Foreknowledge and Freedom
University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations 1896 - February 2014 1-1-1994 Leibniz on providence, foreknowledge and freedom. Jack D. Davidson University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_1 Recommended Citation Davidson, Jack D., "Leibniz on providence, foreknowledge and freedom." (1994). Doctoral Dissertations 1896 - February 2014. 2221. https://scholarworks.umass.edu/dissertations_1/2221 This Open Access Dissertation is brought to you for free and open access by ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations 1896 - February 2014 by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. LEIBNIZ ON PROVIDENCE, FOREKNOWLEDGE AND FREEDOM A Dissertation Presented by JACK D. DAVIDSON Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY September 1994 Department of Philosophy Copyright by Jack D. Davidson 1994 All Rights Reserved ^ LEIBNIZ ON PROVIDENCE, FOREKNOWLEDGE AND FREEDOM A Dissertation Presented by JACK D. DAVIDSON Approved as to style and content by: \J J2SU2- Vere Chappell, Member Vincent Cleary, Membie r Member oe Jobln Robison, Department Head Department of Philosophy ACKNOWLEDGEMENTS I want to thank Vere Chappell for encouraging my work in early modern philosophy. From him I have learned a great deal about the intricate business of exegetical history. His careful work is, and will remain, an inspiration. I also thank Gary Matthews for providing me with an example of a philosopher who integrates his work with his life. Much of my interest in ancient and medieval philosophy has been inspired by his work. -
DRUM–II: Efficient Model–Based Diagnosis of Technical Systems
DRUM–II: Efficient Model–based Diagnosis of Technical Systems Vom Fachbereich Elektrotechnik und Informationstechnik der Universitat¨ Hannover zur Erlangung des akademischen Grades Doktor–Ingenieur genehmigte Dissertation von Dipl.-Inform. Peter Frohlich¨ geboren am 12. Februar 1970 in Wurselen¨ 1998 1. Referent: Prof. Dr. techn. Wolfgang Nejdl 2. Referent: Prof. Dr.-Ing. Claus-E. Liedtke Tag der Promotion: 23. April 1998 Abstract Diagnosis is one of the central application areas of artificial intelligence. The compu- tation of diagnoses for complex technical systems which consist of several thousand components and exist in many different configurations is a grand challenge. For such systems it is usually impossible to directly deduce diagnoses from observed symptoms using empirical knowledge. Instead, the model–based approach to diagnosis uses a model of the system to simulate the system behaviour given a set of faulty components. The diagnoses are obtained by comparing the simulation results with the observed be- haviour of the system. Since the second half of the 1980’s several model–based diagnosis systems have been developed. However, the flexibility of current systems is limited, because they are based on restricted diagnosis definitions and they lack support for reasoning tasks related to diagnosis like temporal prediction. Furthermore, current diagnosis engines are often not sufficiently efficient for the diagnosis of complex systems, especially because of their exponential memory requirements. In this thesis we describe the new model–based diagnosis system DRUM–II. This system achieves increased flexibility by embedding diagnosis in a general logical framework. It computes diagnoses efficiently by exploiting the structure of the sys- tem model, so that large systems with complex internal structure can be solved. -
Cse541 LOGIC for Computer Science
cse541 LOGIC for Computer Science Professor Anita Wasilewska LECTURE 4 Chapter 4 GENERAL PROOF SYSTEMS PART 1: Introduction- Intuitive definitions PART 2: Formal Definition of a Proof System PART 3: Formal Proofs and Simple Examples PART 4: Consequence, Soundness and Completeness PART 5: Decidable and Syntactically Decidable Proof Systems PART 1: General Introduction Proof Systems - Intuitive Definition Proof systems are built to prove, it means to construct formal proofs of statements formulated in a given language First component of any proof system is hence its formal language L Proof systems are inference machines with statements called provable statements being their final products Semantical Link The starting points of the inference machine of a proof systemS are called its axioms We distinguish two kinds of axioms: logical axioms LA and specific axioms SA Semantical link: we usually build a proof systems for a given language and its semantics i.e. for a logic defined semantically Semantical Link We always choose as a set of logical axioms LA some subset of tautologies, under a given semantics We will consider here only proof systems with finite sets of logical or specific axioms , i.e we will examine only finitely axiomatizable proof systems Semantical Link We can, and we often do, consider proof systems with languages without yet established semantics In this case the logical axiomsLA serve as description of tautologies under a future semantics yet to be built Logical axiomsLA of a proof systemS are hence not only tautologies under an -
First-Order Logic in a Nutshell Syntax
First-Order Logic in a Nutshell 27 numbers is empty, and hence cannot be a member of itself (otherwise, it would not be empty). Now, call a set x good if x is not a member of itself and let C be the col- lection of all sets which are good. Is C, as a set, good or not? If C is good, then C is not a member of itself, but since C contains all sets which are good, C is a member of C, a contradiction. Otherwise, if C is a member of itself, then C must be good, again a contradiction. In order to avoid this paradox we have to exclude the collec- tion C from being a set, but then, we have to give reasons why certain collections are sets and others are not. The axiomatic way to do this is described by Zermelo as follows: Starting with the historically grown Set Theory, one has to search for the principles required for the foundations of this mathematical discipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to ex- clude all contradictions and, on the other hand, take them sufficiently wide to retain all the features of this theory. The principles, which are called axioms, will tell us how to get new sets from already existing ones. In fact, most of the axioms of Set Theory are constructive to some extent, i.e., they tell us how new sets are constructed from already existing ones and what elements they contain. However, before we state the axioms of Set Theory we would like to introduce informally the formal language in which these axioms will be formulated.