Predicate Logic 1
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Discrete Mathematics
Discrete Mathematics1 http://lcs.ios.ac.cn/~znj/DM2017 Naijun Zhan March 19, 2017 1Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course. Contents 1. The Foundations: Logic and Proofs 2. Basic Structures: Sets, Functions, Sequences, Sum- s, and Matrices 3. Algorithms 4. Number Theory and Cryptography 5. Induction and Recursion 6. Counting 7. Discrete Probability 8. Advanced Counting Techniques 9. Relations 10. Graphs 11. Trees 12. Boolean Algebra 13. Modeling Computation 1 Chapter 1 The Foundations: Logic and Proofs Logic in Computer Science During the past fifty years there has been extensive, continuous, and growing interaction between logic and computer science. In many respects, logic provides computer science with both a u- nifying foundational framework and a tool for modeling compu- tational systems. In fact, logic has been called the calculus of computer science. The argument is that logic plays a fundamen- tal role in computer science, similar to that played by calculus in the physical sciences and traditional engineering disciplines. Indeed, logic plays an important role in areas of computer sci- ence as disparate as machine architecture, computer-aided de- sign, programming languages, databases, artificial intelligence, algorithms, and computability and complexity. Moshe Vardi 2 • The origins of logic can be dated back to Aristotle's time. • The birth of mathematical logic: { Leibnitz's idea { Russell paradox { Hilbert's plan { Three schools of modern logic: logicism (Frege, Russell, Whitehead) formalism (Hilbert) intuitionism (Brouwer) • One of the central problem for logicians is that: \why is this proof correct/incorrect?" • Boolean algebra owes to George Boole. -
Terms First-Order Logic: Syntax - Formulas
First-order logic FOL Query Evaluation Giuseppe De Giacomo • First-order logic (FOL) is the logic to speak about object, which are the domain of discourse or universe. Universita` di Roma “La Sapienza” • FOL is concerned about Properties of these objects and Relations over objects (resp. unary and n-ary Predicates) • FOL also has Functions including Constants that denote objects. Corso di Seminari di Ingegneria del Software: Data and Service Integration Laurea Specialistica in Ingegneria Informatica Universita` degli Studi di Roma “La Sapienza” A.A. 2005-06 G. De Giacomo FOL queries 1 First-order logic: syntax - terms First-order logic: syntax - formulas Ter ms : defined inductively as follows Formulas: defined inductively as follows k • Vars: A set {x1,...,xn} of individual variables (variables that denote • if t1,...,tk ∈ Ter ms and P is a k-ary predicate, then k single objects) P (t1,...,tk) ∈ Formulas (atomic formulas) • Function symbols (including constants: a set of functions symbols of given • φ ∈ Formulas and ψ ∈ Formulas then arity > 0. Functions of arity 0 are called constants. – ¬φ ∈ Formulas • Vars ⊆ Ter ms – φ ∧ ψ ∈ Formulas k – φ ∨ ψ ∈ Formulas • if t1,...,tk ∈ Ter ms and f is a k-ary function, then k ⊃ ∈ f (t1,...,tk) ∈ Ter ms – φ ψ Formulas • nothing else is in Ter ms . • φ ∈ Formulas and x ∈ Vars then – ∃x.φ ∈ Formulas – ∀x.φ ∈ Formulas G. De Giacomo FOL queries 2 G. De Giacomo FOL queries 3 • nothing else is in Formulas. First-order logic: Semantics - interpretations Note: if a predicate is of arity Pi , then it is a proposition of propositional logic. -
Sets, Propositional Logic, Predicates, and Quantifiers
COMP 182 Algorithmic Thinking Sets, Propositional Logic, Luay Nakhleh Computer Science Predicates, and Quantifiers Rice University !1 Reading Material ❖ Chapter 1, Sections 1, 4, 5 ❖ Chapter 2, Sections 1, 2 !2 ❖ Mathematics is about statements that are either true or false. ❖ Such statements are called propositions. ❖ We use logic to describe them, and proof techniques to prove whether they are true or false. !3 Propositions ❖ 5>7 ❖ The square root of 2 is irrational. ❖ A graph is bipartite if and only if it doesn’t have a cycle of odd length. ❖ For n>1, the sum of the numbers 1,2,3,…,n is n2. !4 Propositions? ❖ E=mc2 ❖ The sun rises from the East every day. ❖ All species on Earth evolved from a common ancestor. ❖ God does not exist. ❖ Everyone eventually dies. !5 ❖ And some of you might already be wondering: “If I wanted to study mathematics, I would have majored in Math. I came here to study computer science.” !6 ❖ Computer Science is mathematics, but we almost exclusively focus on aspects of mathematics that relate to computation (that can be implemented in software and/or hardware). !7 ❖Logic is the language of computer science and, mathematics is the computer scientist’s most essential toolbox. !8 Examples of “CS-relevant” Math ❖ Algorithm A correctly solves problem P. ❖ Algorithm A has a worst-case running time of O(n3). ❖ Problem P has no solution. ❖ Using comparison between two elements as the basic operation, we cannot sort a list of n elements in less than O(n log n) time. ❖ Problem A is NP-Complete. -
Principle of Acquaintance and Axiom of Reducibility
Principle of acquaintance and axiom of reducibility Brice Halimi Université Paris Nanterre 15 mars 2018 Russell-the-epistemologist is the founding father of the concept of acquaintance. In this talk, I would like to show that Russell’s theory of knowledge is not simply the next step following his logic, but that his logic (especially the system of Principia Mathematica) can also be understood as the formal underpinning of a theory of knowledge. So there is a concept of acquaintance for Russell-the-logician as well. Principia Mathematica’s logical types In Principia, Russell gives the following examples of first-order propositional functions of individuals: φx; (x; y); (y) (x; y);::: Then, introducing φ!zb as a variable first-order propositional function of one individual, he gives examples of second-order propositional functions: f (φ!zb); g(φ!zb; !zb); F(φ!zb; x); (x) F(φ!zb; x); (φ) g(φ!zb; !zb); (φ) F(φ!zb; x);::: Then f !(φb!zb) is introduced as a variable second-order propositional function of one first-order propositional function. And so on. A possible value of f !(φb!zb) is . φ!a. (Example given by Russell.) This has to do with the fact that Principia’s schematic letters are variables: It will be seen that “φ!x” is itself a function of two variables, namely φ!zb and x. [. ] (Principia, p. 51) And variables are understood substitutionally (see Kevin Klement, “Russell on Ontological Fundamentality and Existence”, 2017). This explains that the language of Principia does not constitute an autonomous formal language. -
Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 I I 73-26,859
INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus cause a blurred image. You will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at die upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again - beginning below the first row and continuing on until complete. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. -
Chapter 6: Translations in Monadic Predicate Logic 223
TRANSLATIONS IN 6 MONADIC PREDICATE LOGIC 1. Introduction ....................................................................................................222 2. The Subject-Predicate Form of Atomic Statements......................................223 3. Predicates........................................................................................................224 4. Singular Terms ...............................................................................................226 5. Atomic Formulas ............................................................................................228 6. Variables And Pronouns.................................................................................230 7. Compound Formulas ......................................................................................232 8. Quantifiers ......................................................................................................232 9. Combining Quantifiers With Negation ..........................................................236 10. Symbolizing The Statement Forms Of Syllogistic Logic ..............................243 11. Summary of the Basic Quantifier Translation Patterns so far Examined ......248 12. Further Translations Involving Single Quantifiers ........................................251 13. Conjunctive Combinations of Predicates.......................................................255 14. Summary of Basic Translation Patterns from Sections 12 and 13.................262 15. ‘Only’..............................................................................................................263 -
Handout 5 – the Semantics of Predicate Logic LX 502 – Semantics I October 17, 2008
Handout 5 – The Semantics of Predicate Logic LX 502 – Semantics I October 17, 2008 1. The Interpretation Function This handout is a continuation of the previous handout and deals exclusively with the semantics of Predicate Logic. When you feel comfortable with the syntax of Predicate Logic, I urge you to read these notes carefully. The business of semantics, as I have stated in class, is to determine the truth-conditions of a proposition and, in so doing, describe the models in which a proposition is true (and those in which it is false). Although we’ve extended our logical language, our goal remains the same. By adopting a more sophisticated logical language with a lexicon that includes elements below the sentence-level, we now need a more sophisticated semantics, one that derives the meaning of the proposition from the meaning of its constituent individuals, predicates, and variables. But the essentials are still the same. Propositions correspond to states of affairs in the outside world (Correspondence Theory of Truth). For example, the proposition in (1) is true if and only if it correctly describes a state of affairs in the outside world in which the object corresponding to the name Aristotle has the property of being a man. (1) MAN(a) Aristotle is a man The semantics of Predicate Logic does two things. It assigns a meaning to the individuals, predicates, and variables in the syntax. It also systematically determines the meaning of a proposition from the meaning of its constituent parts and the order in which those parts combine (Principle of Compositionality). -
First-Order Logic Without Bound Variables: Compositional Semantics
First-order Logic without bound variables: Compositional Semantics W. W. Tait This paper was written in honor of Dag Prawitz for the occasion, now alas long past, of his 70th birthday. An attractive format for semantics is that in which composite expressions are built up from atomic ones by means of the operation of concatenation and the concatenation XY expresses the application of a function denoted by X to an argument denoted by Y . The use of relative pronouns presents an obstacle to this form of compositional semantics, since the reference of a relative pronoun in one component may occur in another. In the standard notation of first-order predicate logic this occurs in the form of variable- binding operations of quantification: in the sentence 8xφ(x), the reference of x in 8x is in φ(x) and neither component has an independent meaning. Frege, in the interests of compositional semantics, was led by this to declare that the open formula φ(x) is semantically significant: it simply denotes an `incomplete object'. We won't discuss here the many reasons for rejecting this very ugly idea, but reject it we will. So the demands of compositional semantics require that we formalize first-order predicate logic without using bound variables. Of course the use of bound variables is very natural and the means that we use to eliminate them can result in quite complex expressions. Our purpose, therefore, is not the practical one of finding the most readable notation: it is the theoretical one of obtaining a compositional semantics. On the other hand, we shouldn't be too humble: although the notation φ(v) with free variable v and instances φ(t) where t is a term is quite intuitive, the substi- tutions involved in actual cases, in substituting t for the possibly multiple occurrences of v in φ(v), can create long expressions. -
The Foundations: Logic and Proof, Sets, and Functions
The Foundations: Logic and Proof, Sets, and Functions his chapter reviews the foundations of discrete mathematics. Three important topics are covered: logic, sets, and functions. The rules of logic specify the mean- ing of mathematical statements For instance, these rules help us understand and reason with statements such as "There exists an integer that is not the sum of two squares," and "For every positive integer n the sum of the positive integers not exceeding n is n(n + 1)/2." Logic is the basis of all mathematical reasoning, and it has practical applications to the design of computing machines, to system specifications, to artificial intelligence, to computer programming, to programming languages, and to other areas of computer science, as well as to many other fields of study. To understand mathematics, we must understand what makes up a correct mathemat- ical argument, that is, a proof. Moreover, to learn mathematics, a person needs to actively construct mathematical arguments and not just read exposition. In this chapter, we ex- plain what makes up a correct mathematical argument and introduce tools to construct these arguments. Proofs are important not only in mathematics, but also in many parts of computer science, including program verification, algorithm correctness, and system security. Furthermore, automated reasoning systems have been constructed that allow computers to construct their own proofs. Much of discrete mathematics is devoted to the study of discrete structures, which are used to represent discrete objects. Many important discrete structures are built using sets, which are collections of objects. Among the discrete structures built from sets are combinations, which are unordered collections of objects used extensively in counting; relations, which are sets of ordered pairs that represent relationships between objects; graphs, which consist of sets of vertices and of edges that connect vertices: and finite state machines, which are used to model computing machines. -
Logic Via Foundational Algorithms
Logic via Foundational Algorithms James Hook and Tim Sheard March 1, 2011 Contents 1 Overview; Introduction to Propositional Logic 3 1.1 Course Goals and Outline . 3 1.2 Propositional logic . 3 1.2.1 How to understand a logic? . 4 1.2.2 Syntax of Propositional Logic . 4 1.2.3 Natural Deduction Proofs for Propositional Logic . 5 1.3 Tim’s Prover and exercise . 9 2 Propositional logic: truth, validity, satisfiability, tautology, and soundness 10 2.1 Semantics . 10 2.1.1 Valuation . 10 2.1.2 Logical equivalence . 11 2.1.3 Tautology . 11 2.1.4 Satisfiable . 11 2.2 Soundness . 11 2.3 Exercise 2: Backward prover . 14 3 Tableau Proof 15 3.1 Signed Tableau prover . 15 3.1.1 A systematic example . 16 3.1.2 Soundness . 18 3.1.3 A Simple Implementation . 20 3.1.4 Completeness . 22 4 Prop logic: completeness, SAT solvers 22 4.1 Propositional Tableau Proofs, Continued . 23 4.1.1 Uniform notation . 23 4.1.2 Improving the Prover . 23 4.2 Gentzen L-style prover . 24 4.3 Normal Forms . 26 4.4 A Framework for Completeness . 28 1 4.4.1 Propositional Hintikka Sets and the Model Existence The- orem . 29 4.4.2 Application to Tableau completeness . 31 4.4.3 Application to other systems . 32 5 Applications of SAT solvers 32 6 Ideas in SAT Solving 32 6.1 Simple, Incomplete SAT solvers . 33 6.1.1 Conceptual Description . 33 6.1.2 Discussion of complexity . 40 6.1.3 An implementation in Haskell . -
The Early Life of Russell's Notion of a Propositional Function
Baltic International Yearbook of Cognition, Logic and Communication Volume 4 200 YEARS OF ANALYTICAL PHILOSOPHY Article 1 2008 The Early Life Of Russell’s Notion Of A Propositional Function Michael Beaney University of York, UK Follow this and additional works at: https://newprairiepress.org/biyclc This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Beaney, Michael (2008) "The Early Life Of Russell’s Notion Of A Propositional Function," Baltic International Yearbook of Cognition, Logic and Communication: Vol. 4. https://doi.org/10.4148/ biyclc.v4i0.125 This Proceeding of the Symposium for Cognition, Logic and Communication is brought to you for free and open access by the Conferences at New Prairie Press. It has been accepted for inclusion in Baltic International Yearbook of Cognition, Logic and Communication by an authorized administrator of New Prairie Press. For more information, please contact [email protected]. The Early Life of Russell’s Notion of a Propositional Function 2 The Baltic International Yearbook of weeks before the whole manuscript was submitted to Cambridge Uni- Cognition, Logic and Communication versity Press in late May 1902. The stage may have been set for its in- troduction, but given the importance of the notion, the precise nature of August 2009 Volume 4: 200 Years of Analytical Philosophy its appearance deserves elucidation. I offer an account of its emergence pages 1-25 DOI: 10.4148/biyclc.v4i0.125 in section 2. My interest in this is not merely antiquarian, however. The evolution of Russell’s conception of a propositional function plays a central role in the development of his thinking about analysis, and ex- MICHAEL BEANEY plaining this thinking is important not only in understanding Russell’s University of York, United Kingdom philosophy in all its changes but also in appreciating the complex na- ture of the practices of analysis in analytic philosophy today. -
Discrete Mathematics1
Discrete Mathematics1 http://lcs.ios.ac.cn/~znj/DM2017 Naijun Zhan April 5, 2017 1Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course. Contents 1. The Foundations: Logic and Proofs 2. Basic Structures: Sets, Functions, Sequences, Sum- s, and Matrices 3. Algorithms 4. Number Theory and Cryptography 5. Induction and Recursion 6. Counting 7. Discrete Probability 8. Advanced Counting Techniques 9. Relations 10. Graphs 11. Trees 12. Boolean Algebra 13. Modeling Computation 1 Chapter 1 The Foundations: Logic and Proofs Logic in Computer Science During the past fifty years there has been extensive, continuous, and growing interaction between logic and computer science. In many respects, logic provides computer science with both a u- nifying foundational framework and a tool for modeling compu- tational systems. In fact, logic has been called the calculus of computer science. The argument is that logic plays a fundamen- tal role in computer science, similar to that played by calculus in the physical sciences and traditional engineering disciplines. Indeed, logic plays an important role in areas of computer sci- ence as disparate as machine architecture, computer-aided de- sign, programming languages, databases, artificial intelligence, algorithms, and computability and complexity. Moshe Vardi 2 • The origins of logic can be dated back to Aristotle's time. • The birth of mathematical logic: { Leibnitz's idea { Russell paradox { Hilbert's plan { Three schools of modern logic: logicism (Frege, Russell, Whitehead) formalism (Hilbert) intuitionism (Brouwer) • One of the central problem for logicians is that: \why is this proof correct/incorrect?" • Boolean algebra owes to George Boole.