Hardegree, Set Theory; Rules of Derivation 1 of 8 A =========== A & ~B
|
|
- Anthony Horton
- 5 years ago
- Views:
Transcription
1 Hardegree, Set Theory; Rules of Derivation 1 of 8 1. Sentential Logic Henceforth,,, C, D are closed formulas. 1. Inference Rules &I &O &I/O & & & ~(&) =========== ~ vi vo vi/o ~( ) =========== & ~ I O I/O ~( ) =========== & ~ I O I/O ~( ) =========== I O DN ~ ===== Note: The ~O and ~I rules are combined, using a long equals sign ===. Henceforth, any rule that is displayed with === is a bi-directional rule, which can be used both as an in-rule and as an out-rule.
2 Hardegree, Set Theory; Rules of Derivation 2 of 8 2. Strategic Rules Direct Derivation (DD) : DD Indirect Derivation (ID) : ID ~ s : Conditional Derivation (CD) Tilde Indirect Derivation ( D) : : CD s : : D s mpersand Derivation (&D) iconditional Derivation ( D) : & : &D : : D : : Wedge Indirect Derivation ( ID) Separation of Cases (SC) : D 1 D 2... D k ID ~D 1 s D 2 s D k : D 1 D 2... D k : C SC c1: D 1 s : C c2: D 2 s : C... ck: D k : C s
3 Hardegree, Set Theory; Rules of Derivation 3 of 8 2. Quantifier Logic (Free Logic Version) 1. Introduction Classical first-order logic is based on the following two presuppositions: (C1) (C2) The domain (universe) of discourse is not empty; accordingly, the sentence there is something is logically true, even though it is not necessarily true. Every singular term, no matter how silly, denotes an existing object (i.e., element of the domain). In contrast to classical logic, there is Free Logic, of which there are two variants. The more radical version (Universally-Free Logic) denies both (C1) and (C2). The less radical version of Free Logic denies (C2), but accepts (C1). In what follows, we pursue the more radical variant. 2. Constants In intro logic, the distinction between unquantified variables ("constants") ( a, b, c, etc.) and proper nouns ( Jay, Kay, the U.S., etc.) is not important. y contrast, in free logic, the distinction is very important. In particular, whereas constants always denote existing objects (in the domain of quantification), proper nouns need not denote anything. In doing derivations in free logic, one treats constants as purely intra-derivational symbols. In particular, we have the following definition. constant is an atomic singular-term that is introduced by UD or O. Ordinarily, any atomic singular-term that occurs in the premises or conclusion is regarded as a proper noun, not a constant. However, in mathematics (including set theory), one often does derivations in which universal quantification is taken for granted, so constants are allowed in premises and show-lines. These are understood as having been introduced by UD. constant counts as old precisely when it occurs in a line that is neither boxed nor cancelled; otherwise, it counts as new. (as before)
4 Hardegree, Set Theory; Rules of Derivation 4 of 8 3. Quantifier Rules In what follows, Φ is a formula, in which ν is the only variable (if any) that occurs free, and Φ[ε/ν] is the formula that results when ε replaces every occurrence of ν that is free in Φ. n expression is closed iff it contains no free occurrence of any variable. n occurrence of a variable ν is free in expression ℇ iff that occurrence does not lie within the scope of an operator binding ν i.e., ν, ν, or ν. Universal-Out ( O) νφ Φ[o/ν] Existential-In ( I) Φ[o/ν] νφ o is any old constant. Universal-Derivation (UD) : νφ : Φ[n/ν] Existential-Out ( O) νφ Φ[n/ν] n is any new constant. Quantifier-Negation (QN) ~ νφ ν~φ ~ νφ ν~φ Tilde-Universal-Out ( O) O = QN+ O ~ νφ Φ[n/ν] n is any new constant. Tilde-Existential-Out ( O) O = QN+ O ~ νφ Φ[o/ν] o is any old constant.
5 Hardegree, Set Theory; Rules of Derivation 5 of 8 3. Identity Logic Reflexivity (R=) Symmetry (S=) Transitivity (T=) LL σ = σ σ = τ τ = σ ρ = σ σ = τ ρ = τ σ, τ, and ρ are any closed singular-terms. σ = τ Φ[σ/ν] Φ[τ/ν] σ = τ Φ[τ/ν] Φ[σ/ν] 4. Description Logic Iota-Out ( O) c = νφ ν(φ ν=c) Iota-In ( I) ν(φ ν=c) c = νφ c must be a constant. 5. Short-Cut Rules 1. The Immediate Show-Cancel Rule If a show-line follows from available lines (earlier or later!) by a rule, then it can be cancelled by that rule. nnotation: cite the line number(s) and the rule. 2. The Conjunction Rule ny available conjunctive line (with any number of conjuncts) can be treated as the appropriate number of separate lines, numbered (e.g.) 7a, 7b, 7c. nd conversely, any number of available lines can be treated as the corresponding conjunction. 3. Rule-Multiplication ny one-place rule can be multiplied, provided the particular rule also applies to the intermediate line. For example, O+ O = O2; UD+UD = U2D; ~ O+~ O = ~ O2.
6 Hardegree, Set Theory; Rules of Derivation 6 of 8 4. Further Rule Combinations O, O2, etc. can be combined with O to produce O, 2 O, etc., and UD, UD2, etc., can be combined with CD to produce UCD, U2CD, etc. Similarly, UD, UD2, etc., can be combined with D to produce UD, U2D, etc. 5. Contraposition Rule For every genuine one-place rule, there is an associated contrapositive rule ( ), which is obtained by reversing and negating and the premise and conclusion. NOTE CREFULLY: O is not a genuine inference rule, but is rather an assumption rule. 6. lphabetic Variance (V) Φ[u] Φ[v] Here, u, v are variables, Φ[u] is a formula in which v does not occur, Φ[v] is a formula in which u does not occur, and Φ[v] results when every occurrence of u in Φ[u] is replaced by an occurrence of v. 7. The Rule SL If a line can be derived from available lines using only SL rules, then it may be written down by the rule SL. This rule may be used in place of any combination of SL rules, including inference rules and show rules. 8. The Rule QL If a line can be derived from previous available lines using only quantifier rules, but it cannot be derived using just SL rules, then it may be written down by the rule QL. This rule may be used in place of any QL inference rule, as well as any combination of QL rules, including inference rules and show rules. NOTE CREFULLY: The rule O, and hence ~ O, are not genuine inference rules, but assumption rules. No instance of O is valid by QL! So when you cite O or ~ O, do not cite it as QL. 9. The Rule IL If a line can be derived from previous available lines using only identity rules, but it cannot be derived using just QL rules, then it may be written down by the rule IL. This rule may be used in place of any IL inference rule, as well as any combination of IL rules, including inference rules and show rules.
7 Hardegree, Set Theory; Rules of Derivation 7 of 8 6. Theories (including Set Theory) 1. ppealing to a Definition Sentential definitions may be used like rules; each such definition serves as three different rules; out-rule, in-rule, show-rule. The annotation is the same for all three uses Def, where is filled by the particular defined symbol; see examples below. sentential definition is one in which the definiens and definiendum are both sentences (formulas). General form of rule: Given an instance of a definition Φ 1 Φ 2, where Φ 1 and Φ 2 are formulas, Out-Rule: In-Rule: Show-Rule: Examples from Set Theory: If one has a line Φ 1, then one may infer Φ 2. If one has a line Φ 2, then one may infer Φ 1. If one has a show-line show: Φ 1, then one can resolve this show-line to the show-line show: Φ 2. Note: there is a single associated box for both show-lines. Def : x(x x ) [note: and are schematic.] Instance: C x(x x C) [note: and are constants.] out: 1. C given 2. x(x x C) 1, Def in: 1. x(x x C) given 2. C 1, Def show: 1. C Def 2. x(x x C) 2. ppealing to an xiom or Previously Proved Theorem xioms and previously proved theorems can be treated as (unstated) premises. The annotation cites the axiom/theorem and line numbers (optional when line is directly below). Example from Set Theory: show: 1. = E (axiom of extentionality) 2. x(x x )
8 Hardegree, Set Theory; Rules of Derivation 8 of 8 7. xioms a1. x y[ z(z x z y) x=y] [Extensionality] a2. x y[y x] [Empty Set] a3. x y z[z y (z x & )] [[y nor free in ] [Separation] a4. x y z w(w z [w x w y]) [Simple Unions] a5. x y z(z y z=x) [Singletons] a6. x y z[z y w(w x & z w)] [General Unions] a7. x y z[z y w(w z w x)] [Power Sets] 8. Definitions d0.1 a b [a = b] [negation] d0.2 a b [a b] [negation] d1.!ν x ν( ν=x) [unique existence] d2. Σν!S ν(ν S ) [legitimacy] d3. x(x x ) [inclusion] d4. & [proper inclusion] d5. [converse inclusion] d6. [converse proper inclusion] d7. x(x & x ) [exclusion] d8. {ν: } S ν(ν S ) [set-abstract] d9.1 {a} {x : x=a} [singleton] d9.2 {a,b} {x : x=a x=b} [doubleton] d9.3 {a,b,c} {x : x=a x=b x=c} [tripleton] etc. d10. {x : x=x} [universal set] d11. {x : x x} [empty set] d12. {x : x & x } [simple intersection] d13. {x : x & x } [set-difference] d14. {x : x x } [simple union] d15. + ( ) ( ) [oolean sum] d16. C {x : y(y C & x y)} [general union] d17. C {x : y(y C x y)} [gen intersection] d18. () {X : X } [power set] 9. Definitions (Sentential Versions) cd8. a {ν: } [a/ν] cd9.1 a {b} a=b cd9.2 a {b,c} a=b a=c cd9.3 a {b,c,d} a=b a=c a=d etc. cd11. a a a cd12. a a & a cd13. a a & a cd14. a a a cd16. a C X(X C & a X) cd17. a C X(X C a X) [provided C ] cd18. ()
Deductive Characterization of Logic
6 The Deductive Characterization of Logic 1. Derivations...2 2. Deductive Systems...3 3. Axioms in Deductive Systems...4 4. Axiomatic Systems...5 5. Validity and Entailment in the Deductive Context...6
More information1. The Semantic Enterprise. 2. Semantic Values Intensions and Extensions. 3. Situations
Hardegree, Formal Semantics, Handout, 2015-02-03 1 of 8 1. The Semantic Enterprise The semantic-analysis of a phrase φ consists in the following. (1) providing a semantic-value for φ, and each of its component
More informationSecond-Order Modal Logic
Hardegree, Modal Logic, 11: Second-Order Modal Logic 16 XI-1 11 Second-Order Modal Logic 1. Introduction...2 2. Second-Order Considerations...2 3. Monadic Plurals...4 4. Syntax for Modal Logic with Plurals
More informationClassical Sentential Logic
II-1 2 Classical Sentential Logic 1. Introduction...2 2. Sentences and Connectives...2 3. Declarative Sentences and Truth-Values...3 4. Truth-Functional Connectives...4 5. Sentence Forms and rgument Forms...5
More informationAxiomatic set theory. Chapter Why axiomatic set theory?
Chapter 1 Axiomatic set theory 1.1 Why axiomatic set theory? Essentially all mathematical theories deal with sets in one way or another. In most cases, however, the use of set theory is limited to its
More information3. The Logic of Quantified Statements Summary. Aaron Tan August 2017
3. The Logic of Quantified Statements Summary Aaron Tan 28 31 August 2017 1 3. The Logic of Quantified Statements 3.1 Predicates and Quantified Statements I Predicate; domain; truth set Universal quantifier,
More informationMathematics 114L Spring 2018 D.A. Martin. Mathematical Logic
Mathematics 114L Spring 2018 D.A. Martin Mathematical Logic 1 First-Order Languages. Symbols. All first-order languages we consider will have the following symbols: (i) variables v 1, v 2, v 3,... ; (ii)
More informationSecond-Order Modal Logic
Hardegree, Modal Logic, Chapter 11: Second-Order Modal Logic 1 of 17 11 Second-Order Modal Logic 1. Introduction...2 2. Second-Order Considerations...2 3. Monadic Plurals...3 4. Syntax for Modal Logic
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More information7 Classical Quantified Logic
7 Classical Quantified Logic efore examining quantified modal logic, we review a bare-bones system for first-order predicate or quantificational logic, i.e., that involving the quantifiers x... and x...,
More information6. THE OFFICIAL INFERENCE RULES
154 Hardegree, Symbolic Logic 6. THE OFFICIAL INFERENCE RULES So far, we have discussed only four inference rules: modus ponens, modus tollens, and the two forms of modus tollendo ponens. In the present
More informationTwo-Dimensional Modal Logic
Hardegree, Modal Logic, Chapter 10: Two-Dimensional Modal Logic 1 of 12 10 Two-Dimensional Modal Logic 1. Introduction...2 2. Scoped-Actuality...2 3. System 2D(1)...2 4. Examples of Derivations in 2D(1)...3
More informationTwo-Dimensional Modal Logic
Hardegree, Modal Logic, 10: Two-Dimensional Modal Logic 13 X-1 10 Two-Dimensional Modal Logic 1. Introduction...2 2. Scoped-Actuality...2 3. System 2D(1)...3 4. Examples of Derivations in 2D(1)...4 5.
More informationDiscrete Mathematical Structures: Theory and Applications
Chapter 1: Foundations: Sets, Logic, and Algorithms Discrete Mathematical Structures: Theory and Applications Learning Objectives Learn about sets Explore various operations on sets Become familiar with
More informationLogic and Proof. On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes!
Logic and Proof On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes! 26-Aug-2011 MA 341 001 2 Requirements for Proof 1. Mutual understanding
More informationGeneric Size Theory Gary Hardegree Department of Philosophy University of Massachusetts Amherst, MA 01003
Generic Size Theory Gary Hardegree Department of Philosophy University of Massachusetts Amherst, MA 01003 1. Introduction The Euclidian Paradigm...1 2. A Simple Example A Generic Theory of Size...1 1.
More informationLecture 3. Logic Predicates and Quantified Statements Statements with Multiple Quantifiers. Introduction to Proofs. Reading (Epp s textbook)
Lecture 3 Logic Predicates and Quantified Statements Statements with Multiple Quantifiers Reading (Epp s textbook) 3.1-3.3 Introduction to Proofs Reading (Epp s textbook) 4.1-4.2 1 Propositional Functions
More informationOn my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes! 26-Aug-2011 MA
Logic and Proof On my first day of school my parents dropped me off at the wrong nursery. There I was...surrounded by trees and bushes! 26-Aug-2011 MA 341 001 2 Requirements for Proof 1. Mutual understanding
More informationCSC384: Intro to Artificial Intelligence Knowledge Representation II. Required Readings: 9.1, 9.2, and 9.5 Announcements:
CSC384: Intro to Artificial Intelligence Knowledge Representation II Required Readings: 9.1, 9.2, and 9.5 Announcements: 1 Models Examples. Environment A Language (Syntax) Constants: a,b,c,e Functions:
More informationAxiom Systems For Classical Sentential Logic
8 Axiom Systems For Classical Sentential Logic 1. Introduction...2 2. Axiom System AS1...2 3. Examples of Derivations in System AS1...3 4. Other Axiom Systems for CSL...6 2 Hardegree, MetaLogic 1. Introduction
More informationTopics in Logic, Set Theory and Computability
Topics in Logic, Set Theory and Computability Homework Set #3 Due Friday 4/6 at 3pm (by email or in person at 08-3234) Exercises from Handouts 7-C-2 7-E-6 7-E-7(a) 8-A-4 8-A-9(a) 8-B-2 8-C-2(a,b,c) 8-D-4(a)
More informationHANDOUT AND SET THEORY. Ariyadi Wijaya
HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics
More informationModal Predicate Logic
Hardegree, Modal Logic, Chapter 06: Modal Predicate Logic 1 of 26 6 Modal Predicate Logic 1. Overview...2 A. Ordinary Predicate Logic...2 2. Introduction...2 3. Noun Phrases...2 4. Two Key Simplifications
More informationIntroduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
More informationPART II QUANTIFICATIONAL LOGIC
Page 1 PART II QUANTIFICATIONAL LOGIC The language m of part I was built from sentence letters, symbols that stand in for sentences. The logical truth of a sentence or the logical validity of an argument,
More informationCLASSICAL EXTENSIONAL MEREOLOGY. Mereology
1 Mereology Core axioms and concepts parthood sum Higher order properties: cumulativity divisivity (aka divisiveness) atomicity 2 Mereology is the theory of parthood derived from the Greek µέρος (meros)
More informationLogic, Sets, and Proofs
Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be assigned a value either true or false.
More informationPropositions and Proofs
Chapter 2 Propositions and Proofs The goal of this chapter is to develop the two principal notions of logic, namely propositions and proofs There is no universal agreement about the proper foundations
More informationPacket #1: Logic & Proofs. Applied Discrete Mathematics
Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should
More informationArguments and Proofs. 1. A set of sentences (the premises) 2. A sentence (the conclusion)
Arguments and Proofs For the next section of this course, we will study PROOFS. A proof can be thought of as the formal representation of a process of reasoning. Proofs are comparable to arguments, since
More informationPropositional Logic Not Enough
Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks
More information2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic
CS 103 Discrete Structures Chapter 1 Propositional Logic Chapter 1.1 Propositional Logic 1 1.1 Propositional Logic Definition: A proposition :is a declarative sentence (that is, a sentence that declares
More informationRelative Modal Logic System K
Hardegree, Modal Logic, Chapter 04: Relative Modal Logic 1 of 28 4 Relative Modal Logic System K A. System K...2 1. Absolute versus Relative Modalities...2 2. Index Points in Relative Modal Logic...2 3.
More information13. APPENDIX 1: THE SYNTAX OF PREDICATE LOGIC
394 Hardegree, Symbolic Logic 13. APPENDIX 1: THE SYNTAX OF PREDICATE LOGIC In this appendix, we review the syntactic features of predicate logic that are crucial to understanding derivations in predicate
More informationPacket #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics
CSC 224/226 Notes Packet #2: Set Theory & Predicate Calculus Barnes Packet #2: Set Theory & Predicate Calculus Applied Discrete Mathematics Table of Contents Full Adder Information Page 1 Predicate Calculus
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationSoundness Theorem for System AS1
10 The Soundness Theorem for System AS1 1. Introduction...2 2. Soundness, Completeness, and Mutual Consistency...2 3. The Weak Soundness Theorem...4 4. The Strong Soundness Theorem...6 5. Appendix: Supporting
More information6. COMPLEX PREDICATES
318 Hardegree, Symbolic Logic (1) x... y...rxy (1a) x(sx & y(py Rxy)) (1b) x(px & y(sy Rxy)) (2) x... y...ryx (2a) x(sx & y(py Ryx)) (2b) x(px & y(sy Ryx)) (3) x... y...rxy (3a) x(sx y(py & Rxy)) (3b)
More informationDiscrete Mathematics. Instructor: Sourav Chakraborty. Lecture 4: Propositional Logic and Predicate Lo
gic Instructor: Sourav Chakraborty Propositional logic and Predicate Logic Propositional logic and Predicate Logic Every statement (or proposition) is either TRUE or FALSE. Propositional logic and Predicate
More informationNotes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.
Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3
More informationLearning Goals of CS245 Logic and Computation
Learning Goals of CS245 Logic and Computation Alice Gao April 27, 2018 Contents 1 Propositional Logic 2 2 Predicate Logic 4 3 Program Verification 6 4 Undecidability 7 1 1 Propositional Logic Introduction
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationMat 243 Exam 1 Review
OBJECTIVES (Review problems: on next page) 1.1 Distinguish between propositions and non-propositions. Know the truth tables (i.e., the definitions) of the logical operators,,,, and Write truth tables for
More informationarxiv: v1 [cs.lo] 1 Sep 2017
A DECISION PROCEDURE FOR HERBRAND FORMULAE WITHOUT SKOLEMIZATION arxiv:1709.00191v1 [cs.lo] 1 Sep 2017 TIMM LAMPERT Humboldt University Berlin, Unter den Linden 6, D-10099 Berlin e-mail address: lampertt@staff.hu-berlin.de
More information2.3 Exercises. (a) F P(A). (Solution)
2.3 Exercises 1. Analyze the logical forms of the following statements. You may use the symbols, /, =,,,,,,, and in your answers, but not,, P,,, {, }, or. (Thus, you must write out the definitions of some
More informationBasic Propositional Logic. Inductive Theory of the Natural Numbers. Conjunction. Equivalence. Negation and Inequivalence. Implication.
McMaster University COMPSCI&SFWRENG 2DM3 Dept. of Computing and Software Theorem List 4 Dr. W. Kahl 2017-12-09 The names listed here are precisely the names used in the preloaded material you are already
More informationLogic Overview, I. and T T T T F F F T F F F F
Logic Overview, I DEFINITIONS A statement (proposition) is a declarative sentence that can be assigned a truth value T or F, but not both. Statements are denoted by letters p, q, r, s,... The 5 basic logical
More informationPart I: Propositional Calculus
Logic Part I: Propositional Calculus Statements Undefined Terms True, T, #t, 1 False, F, #f, 0 Statement, Proposition Statement/Proposition -- Informal Definition Statement = anything that can meaningfully
More informationMathematical Preliminaries. Sipser pages 1-28
Mathematical Preliminaries Sipser pages 1-28 Mathematical Preliminaries This course is about the fundamental capabilities and limitations of computers. It has 3 parts 1. Automata Models of computation
More informationDiscrete Mathematical Structures. Chapter 1 The Foundation: Logic
Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction
More informationFor all For every For each For any There exists at least one There exists There is Some
Section 1.3 Predicates and Quantifiers Assume universe of discourse is all the people who are participating in this course. Also let us assume that we know each person in the course. Consider the following
More informationSection 1.1 Propositions
Set Theory & Logic Section 1.1 Propositions Fall, 2009 Section 1.1 Propositions In Chapter 1, our main goals are to prove sentences about numbers, equations or functions and to write the proofs. Definition.
More informationSome Review Problems for Exam 1: Solutions
Math 3355 Fall 2018 Some Review Problems for Exam 1: Solutions Here is my quick review of proof techniques. I will focus exclusively on propositions of the form p q, or more properly, x P (x) Q(x) or x
More informationQuantifiers. P. Danziger
- 2 Quantifiers P. Danziger 1 Elementary Quantifiers (2.1) We wish to be able to use variables, such as x or n in logical statements. We do this by using the two quantifiers: 1. - There Exists 2. - For
More informationSYMBOLIC LOGIC UNIT 10: SINGULAR SENTENCES
SYMBOLIC LOGIC UNIT 10: SINGULAR SENTENCES Singular Sentences name Paris is beautiful (monadic) predicate (monadic) predicate letter Bp individual constant Singular Sentences Bp These are our new simple
More informationDERIVATIONS AND TRUTH TABLES
DERIVATIONS AND TRUTH TABLES Tomoya Sato Department of Philosophy University of California, San Diego Phil120: Symbolic Logic Summer 2014 TOMOYA SATO LECTURE 3: DERIVATIONS AND TRUTH TABLES 1 / 65 WHAT
More informationDiscrete Structures Lecture 5
Introduction EXAMPLE 1 Express xx yy(xx + yy = 0) without the existential quantifier. Solution: xx yy(xx + yy = 0) is the same as xxxx(xx) where QQ(xx) is yyyy(xx, yy) and PP(xx, yy) = xx + yy = 0 EXAMPLE
More informationDeductive reasoning is the process of reasoning from accepted facts to a conclusion. if a = b and c = d, c 0, then a/c = b/d
Chapter 2 Reasoning Suppose you know the following two statements are true. 1. Every board member read their back-up material 2. Tom is a board member You can conclude: 3. Tom read his back-up material.
More informationSummary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016
8. For any two events E and F, P (E) = P (E F ) + P (E F c ). Summary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016 Sample space. A sample space consists of a underlying
More informationBefore you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here.
Chapter 2 Mathematics and Logic Before you get started, make sure you ve read Chapter 1, which sets the tone for the work we will begin doing here. 2.1 A Taste of Number Theory In this section, we will
More informationPredicate in English. Predicates and Quantifiers. Predicate in Logic. Propositional Functions: Prelude. Propositional Function
Predicates and Quantifiers Chuck Cusack Predicate in English In English, a sentence has 2 parts: the subject and the predicate. The predicate is the part of the sentence that states something about the
More informationChapter 1 Elementary Logic
2017-2018 Chapter 1 Elementary Logic The study of logic is the study of the principles and methods used in distinguishing valid arguments from those that are not valid. The aim of this chapter is to help
More information1.1 Language and Logic
c Oksana Shatalov, Spring 2018 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,
More information1. Propositions: Contrapositives and Converses
Preliminaries 1 1. Propositions: Contrapositives and Converses Given two propositions P and Q, the statement If P, then Q is interpreted as the statement that if the proposition P is true, then the statement
More informationLogic. Logic is a discipline that studies the principles and methods used in correct reasoning. It includes:
Logic Logic is a discipline that studies the principles and methods used in correct reasoning It includes: A formal language for expressing statements. An inference mechanism (a collection of rules) to
More informationECOM Discrete Mathematics
ECOM 2311- Discrete Mathematics Chapter # 1 : The Foundations: Logic and Proofs Fall, 2013/2014 ECOM 2311- Discrete Mathematics - Ch.1 Dr. Musbah Shaat 1 / 85 Outline 1 Propositional Logic 2 Propositional
More informationCHAPTER 1 SETS AND EVENTS
CHPTER 1 SETS ND EVENTS 1.1 Universal Set and Subsets DEFINITION: set is a well-defined collection of distinct elements in the universal set. This is denoted by capital latin letters, B, C, If an element
More informationCOMP 409: Logic Homework 5
COMP 409: Logic Homework 5 Note: The pages below refer to the text from the book by Enderton. 1. Exercises 1-6 on p. 78. 1. Translate into this language the English sentences listed below. If the English
More informationRelative Modal Logic System K
IV-1 4 Relative Modal Logic System K A. System K...2 1. Absolute versus Relative Modalities...2 2. Index Points in Relative Modal Logic...2 3. A Potential Notational Problem...3 4. The Official Technical
More informationMath 535: Topology Homework 1. Mueen Nawaz
Math 535: Topology Homework 1 Mueen Nawaz Mueen Nawaz Math 535 Topology Homework 1 Problem 1 Problem 1 Find all topologies on the set X = {0, 1, 2}. In the list below, a, b, c X and it is assumed that
More informationA Guide to Proof-Writing
A Guide to Proof-Writing 437 A Guide to Proof-Writing by Ron Morash, University of Michigan Dearborn Toward the end of Section 1.5, the text states that there is no algorithm for proving theorems.... Such
More informationAnswers to the Exercises -- Chapter 1
Answers to the Exercises -- Chapter 1 SECTION 1 1. a Sentence, official notation ~~~P ~~P ~P P Sentence, informal notation ~Q ~R /\ ~Q ~R Q R c d e Not a sentence; it is impossile to construct "~ " Not
More informationTwo Posts to Fill On School Board
Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83
More informationTheorem. For every positive integer n, the sum of the positive integers from 1 to n is n(n+1)
Week 1: Logic Lecture 1, 8/1 (Sections 1.1 and 1.3) Examples of theorems and proofs Theorem (Pythagoras). Let ABC be a right triangle, with legs of lengths a and b, and hypotenuse of length c. Then a +
More informationLogic and Proof. Aiichiro Nakano
Logic and Proof Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Department of Computer Science Department of Physics & Astronomy Department of Chemical Engineering & Materials Science
More informationChapter 5 Vocabulary:
Geometry Week 11 ch. 5 review sec. 6.3 ch. 5 review Chapter 5 Vocabulary: biconditional conclusion conditional conjunction connective contrapositive converse deductive reasoning disjunction existential
More informationCSC Discrete Math I, Spring Propositional Logic
CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)
More informationICS141: Discrete Mathematics for Computer Science I
ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Originals slides by Dr. Baek and Dr. Still, adapted by J. Stelovsky Based on slides Dr. M. P. Frank and Dr. J.L. Gross
More informationChapter 1 : The language of mathematics.
MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :
More informationMathematical Reasoning. The Foundation of Algorithmics
Mathematical Reasoning The Foundation of Algorithmics The Nature of Truth In mathematics, we deal with statements that are True or False This is known as The Law of the Excluded Middle Despite the fact
More informationIntroduction to Metalogic 1
Philosophy 135 Spring 2012 Tony Martin Introduction to Metalogic 1 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: (i) sentence letters p 0, p 1, p 2,... (ii) connectives,
More informationSection 1.3. Let I be a set. When I is used in the following context,
Section 1.3. Let I be a set. When I is used in the following context, {B i } i I, we call I the index set. The set {B i } i I is the family of sets of the form B i where i I. One could also use set builder
More informationThe Lambek-Grishin calculus for unary connectives
The Lambek-Grishin calculus for unary connectives Anna Chernilovskaya Utrecht Institute of Linguistics OTS, Utrecht University, the Netherlands anna.chernilovskaya@let.uu.nl Introduction In traditional
More informationToday s topics. Introduction to Set Theory ( 1.6) Naïve set theory. Basic notations for sets
Today s topics Introduction to Set Theory ( 1.6) Sets Definitions Operations Proving Set Identities Reading: Sections 1.6-1.7 Upcoming Functions A set is a new type of structure, representing an unordered
More informationPredicate Logic: Sematics Part 1
Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with
More informationCHAPTER 0: BACKGROUND (SPRING 2009 DRAFT)
CHAPTER 0: BACKGROUND (SPRING 2009 DRAFT) MATH 378, CSUSM. SPRING 2009. AITKEN This chapter reviews some of the background concepts needed for Math 378. This chapter is new to the course (added Spring
More informationAbsolute Modal Logic System L
Hardegree, Modal Logic, Chapter 03: Absolute Modal Logic 1 of 30 3 Absolute Modal Logic System L A. Leibnizian World Theory...2 1. Introduction...2 2. Direct and Indirect Quotation...2 3. Sentences and
More informationarxiv: v2 [cs.lo] 22 Nov 2017
A DECISION PROCEDURE FOR HERBRAND FORMULAE WITHOUT SKOLEMIZATION arxiv:1709.00191v2 [cs.lo] 22 Nov 2017 TIMM LAMPERT Humboldt University Berlin, Unter den Linden 6, D-10099 Berlin e-mail address: lampertt@staff.hu-berlin.de
More informationFirst order Logic ( Predicate Logic) and Methods of Proof
First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating
More informationInformal Statement Calculus
FOUNDATIONS OF MATHEMATICS Branches of Logic 1. Theory of Computations (i.e. Recursion Theory). 2. Proof Theory. 3. Model Theory. 4. Set Theory. Informal Statement Calculus STATEMENTS AND CONNECTIVES Example
More informationLecture Notes 1 Basic Concepts of Mathematics MATH 352
Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,
More informationResolution for mixed Post logic
Resolution for mixed Post logic Vladimir Komendantsky Institute of Philosophy of Russian Academy of Science, Volkhonka 14, 119992 Moscow, Russia vycom@pochtamt.ru Abstract. In this paper we present a resolution
More informationDERIVATIONS IN SENTENTIAL LOGIC
5 DERIVATIONS IN SENTENTIAL LOGIC 1. Introduction... 142 2. The Basic Idea... 143 3. Argument Forms and Substitution Instances... 145 4. Simple Inference Rules... 147 5. Simple Derivations... 151 6. The
More informationRecall that the expression x > 3 is not a proposition. Why?
Predicates and Quantifiers Predicates and Quantifiers 1 Recall that the expression x > 3 is not a proposition. Why? Notation: We will use the propositional function notation to denote the expression "
More informationThree Profound Theorems about Mathematical Logic
Power & Limits of Logic Three Profound Theorems about Mathematical Logic Gödel's Completeness Theorem Thm 1, good news: only need to know* a few axioms & rules, to prove all validities. *Theoretically
More informationPHIL 50 INTRODUCTION TO LOGIC 1 FREE AND BOUND VARIABLES MARCELLO DI BELLO STANFORD UNIVERSITY DERIVATIONS IN PREDICATE LOGIC WEEK #8
PHIL 50 INTRODUCTION TO LOGIC MARCELLO DI BELLO STANFORD UNIVERSITY DERIVATIONS IN PREDICATE LOGIC WEEK #8 1 FREE AND BOUND VARIABLES Before discussing the derivation rules for predicate logic, we should
More informationsoftware design & management Gachon University Chulyun Kim
Gachon University Chulyun Kim 2 Outline Propositional Logic Propositional Equivalences Predicates and Quantifiers Nested Quantifiers Rules of Inference Introduction to Proofs 3 1.1 Propositional Logic
More informationCompound Propositions
Discrete Structures Compound Propositions Producing new propositions from existing propositions. Logical Operators or Connectives 1. Not 2. And 3. Or 4. Exclusive or 5. Implication 6. Biconditional Truth
More informationDiscrete Mathematics and Its Applications
Discrete Mathematics and Its Applications Lecture 1: The Foundations: Logic and Proofs (1.3-1.5) MING GAO DASE @ ECNU (for course related communications) mgao@dase.ecnu.edu.cn Sep. 19, 2017 Outline 1 Logical
More informationSemantics of Classical First- Order Logic
14 The Semantics of Classical First- Order Logic 1. Introduction...2 2. Semantic Evaluations...2 3. Semantic Items and their Categories...2 4. Official versus Conventional Identifications of Semantic Items...3
More information