Formal Proofs and Quantifiers

Slides:



Advertisements
Similar presentations
Chapter 1: The Foundations: Logic and Proofs 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Predicates and Quantifiers 1.4 Nested Quantifiers.
Advertisements

Methods of Proof for Quantifiers Chapter 12 Language, Proof and Logic.
Predicate Calculus Formal Methods in Verification of Computer Systems Jeremy Johnson.
Predicate Logic.
EE1J2 - Slide 1 EE1J2 – Discrete Maths Lecture 6 Limitations of propositional logic Introduction to predicate logic Symbols, terms and formulae, Parse.
Formal Logic Mathematical Structures for Computer Science Chapter 1 Copyright © 2006 W.H. Freeman & Co.MSCS SlidesFormal Logic.
Section 1.3: Predicates and Quantifiers
Methods of Proof & Proof Strategies
Chapter 1: The Foundations: Logic and Proofs
INTRODUCTION TO LOGIC FALL 2009 Quiz Game. ConceptsTrue/FalseTranslations Informal Proofs Formal Proofs
CSCI 115 Chapter 2 Logic. CSCI 115 §2.1 Propositions and Logical Operations.
Formal Logic Mathematical Structures for Computer Science Chapter Copyright © 2006 W.H. Freeman & Co.MSCS SlidesFormal Logic.
Advanced Topics in FOL Chapter 18 Language, Proof and Logic.
1 Chapter 7 Propositional and Predicate Logic. 2 Chapter 7 Contents (1) l What is Logic? l Logical Operators l Translating between English and Logic l.
Discrete Mathematics CS 2610 August 24, Agenda Last class Introduction to predicates and quantifiers This class Nested quantifiers Proofs.
1 Section 7.2 Equivalent Formulas Two wffs A and B are equivalent, written A  B, if they have the same truth value for every interpretation. Property:
Logic CL4 Episode 16 0 The language of CL4 The rules of CL4 CL4 as a conservative extension of classical logic The soundness and completeness of CL4 The.
Introduction to Quantification Chapter 9 Language, Proof and Logic.
The Logic of Quantifiers Chapter 10 Language, Proof and Logic.
2.3Logical Implication: Rules of Inference From the notion of a valid argument, we begin a formal study of what we shall mean by an argument and when such.
ARTIFICIAL INTELLIGENCE [INTELLIGENT AGENTS PARADIGM] Professor Janis Grundspenkis Riga Technical University Faculty of Computer Science and Information.
Mathematical Induction Chapter 16 Language, Proof and Logic.
1 Predicate (Relational) Logic 1. Introduction The propositional logic is not powerful enough to express certain types of relationship between propositions.
Scope, free variable, closed wff §In  X(A) or  X(A), where A is a wff : X is called the variable quantified over; A is said to be (within) the scope.
1 Introduction to Abstract Mathematics Chapter 2: The Logic of Quantified Statements. Predicate Calculus Instructor: Hayk Melikya 2.3.
1 Introduction to Abstract Mathematics Proofs in Predicate Logic , , ~, ,  Instructor: Hayk Melikya Purpose of Section: The theorems.
0 Propositional logic versus first-order (predicate) logic The universe of discourse Constants, variables, terms and valuations Predicates as generalized.
1 Introduction to Abstract Mathematics Predicate Logic Instructor: Hayk Melikya Purpose of Section: To introduce predicate logic (or.
CSS342: Quantifiers1 Professor: Munehiro Fukuda. CSS342: Quantifiers2 Review of Propositions Proposition: a statement that is either true or false, but.
1 Section 9.1 Automatic Reasoning Recall that a wff W is valid iff ¬ W is unsatisfiable. Resolution is an inference rule used to prove unsatisfiability.
Methods of Proof for Boolean Logic Chapter 5 Language, Proof and Logic.
Formal Proofs and Boolean Logic Chapter 6 Language, Proof and Logic.
The Logic of Atomic Sentences Chapter 2 Language, Proof and Logic.
1 Outline Quantifiers and predicates Translation of English sentences Predicate formulas with single variable Predicate formulas involving multiple variables.
1 CS1502 Formal Methods in Computer Science Notes 15 Problem Sessions.
Chapter Nine Predicate Logic Proofs. 1. Proving Validity The eighteen valid argument forms plus CP and IP that are the proof machinery of sentential logic.
Chapter Ten Relational Predicate Logic. 1. Relational Predicates We now broaden our coverage of predicate logic to include relational predicates. This.
The Logic of Conditionals Chapter 8 Language, Proof and Logic.
More Proofs. REVIEW The Rule of Assumption: A Assumption is the easiest rule to learn. It says at any stage in the derivation, we may write down any.
1 Section 7.3 Formal Proofs in Predicate Calculus All proof rules for propositional calculus extend to predicate calculus. Example. … k.  x p(x) P k+1.
1 Section 7.1 First-Order Predicate Calculus Predicate calculus studies the internal structure of sentences where subjects are applied to predicates existentially.
Foundations of Computing I CSE 311 Fall Announcements Homework #2 due today – Solutions available (paper format) in front – HW #3 will be posted.
Lecture 1-3: Quantifiers and Predicates. Variables –A variable is a symbol that stands for an individual in a collection or set. –Example, a variable.
رياضيات متقطعة لعلوم الحاسب MATH 226. Chapter 1 Predicates and Quantifiers 1.4.
By P. S. Suryateja Asst. Professor, CSE Vishnu Institute of Technology
Introduction to Logic for Artificial Intelligence Lecture 2
Logical Inference 2 Rule-based reasoning
CSNB 143 Discrete Mathematical Structures
Rationale Behind the Precise Formulation of the Four Quantifier Rules
2.1 Propositions and Logical Operations
Predicate Calculus Discussion #14 Chapter 2, Section 1 1/20.
Discussion #14 Predicate Calculus
Chapter 1: The Foundations: Logic and Proofs
Logical Inference 2 Rule-based reasoning
Artificial Intelligence
Relational Proofs Formal proofs in Relational Logic are analogous to formal proofs in Propositional Logic. The major difference is that there are additional.
Today’s Topics Universes of Discourse
CS 1502 Formal Methods in Computer Science
Predicate Calculus Discussion #14 Chapter 2, Section 1.
Mathematical Structures for Computer Science Chapter 1
Chapter 1 Logic and Proof.
First Order Logic Rosen Lecture 3: Sept 11, 12.
Relational Proofs Computational Logic Lecture 8
MA/CSSE 474 More Math Review Theory of Computation
Metatheorems Computational Logic Lecture 8
Relational Proofs Computational Logic Lecture 7
Formal Methods in software development
Relational Proofs Computational Logic Lecture 7
Relational Proofs Computational Logic Lecture 7
Herbrand Semantics Computational Logic Lecture 15
Presentation transcript:

Formal Proofs and Quantifiers Language, Proof and Logic Formal Proofs and Quantifiers Chapter 13

Universal quantifier rules 13.1  Elim: x --- variable t --- constant term (variable-free term) c --- constant which does not occur outside the subproof where it is introduced P(x), Q(x) --- any wffs only containing x free P(c), Q(c) --- the result of replacing in P(x), Q(x) all free occurrences of x by c P(t) --- the result of replacing in P(x) all free occurrences of x by t xP(x) … P(t)  Intro (General Cond. Proof):  Intro (Universal introduct.): c P(c) … Q(c) x[P(x)Q(x)] c … Q(c) xQ(x) You try it, pp. 353, 354 [Universal 1-2]

Existential quantifier rules 13.2  Intro:  Elim: P(t) … xP(x) xP(x) … c P(c) Q x --- variable t --- constant term (variable-free term) c --- constant which does not occur outside the subproof where it is introduced P(x) --- any wff only containing x free P(c) --- the result of replacing in P(x) all free occurrences of x by c P(t) --- the result of replacing in P(x) all free occurrences of x by t You try it, pp. 358 [Existential 1]

1. Always be clear about the meaning of the sentences you are using. Strategy and tactics 13.3.a General tips: 1. Always be clear about the meaning of the sentences you are using. Practically zero chance to succeed without that! 2. A good strategy is to find an informal proof and then try to formalize it. 3. Working backwards can be very useful in proving universal claims. You typically use  Intro in these cases. 4. Working backwards ( Intro) is not useful in proving an existential claim xS(x) unless you can think of a particular instance S(c) of the claim that follows from the premises. 5. If you get stuck, consider using proof by contradiction.

x[Small(x)LeftOf(x,b)] xLeftOf(x,b) Strategy and tactics 13.3.b x[Tet(x)Small(x)] x[Small(x)LeftOf(x,b)] xLeftOf(x,b) Informal proof: Look, Bozo, we are told that there is a small tetrahedron. So we know that it is small, right? But we’re also told that anything that’s small is left of b. So if it’s small, it’s got to be left of b, too. So, something is left of b, namely, the small tetrahedron. Formal proof: 1. x[Tet(x)  Small(x)] 2. x[Small(x)  LeftOf(x,b)] 3. c Tet(c)  Small(c) 4. Small(c)  Elim: 3 5. Small(c)  LeftOf(c,b)]  Elim: 2 6. LeftOf(c,b)  Elim: 4,5 7. xLeftOf(x,b)  Inro: 6 8. xLeftOf(x,b)  Elim: 1, 3-7 You try it, p.366 [Quantifier Strategy 1]

Soundness and completeness 13.4 As in the propositional case, we have: Q is provable in Fitch from premises P1,…, Pn if (completeness) and only if (soundness) Q is a FO consequence of P1,…, Pn