Overview. I Review of natural deduction. I Soundness and completeness. I Semantics of propositional formulas. I Soundness proof. I Completeness proof.
|
|
- Dennis Roberts
- 5 years ago
- Views:
Transcription
1 Overview I Review of natural deduction. I Soundness and completeness. I Semantics of propositional formulas. I Soundness proof. I Completeness proof.
2 Propositional formulas Grammar: ::= p j (:) j ( ^ ) j ( _ ) j (! ) Precedence rules: : > _; ^ >! Example: p ^ (q _ :r! q) (p ^ ((q _ (:r))! q))
3 Natural deduction 1 ; : : : ; n ` Formula can be proved from formulas 1 ; : : : ; n. Rules for reasoning about formulas. I Introduction rules: Connective in the conclusion. Construct a complex formula..!!-i I Elimination rules: Connective in a premise. Extract information from proved formulas. xxx!!-e
4 Basic rules ^ Introduction xxx ^ ^-i ^ Elimination ^-e 1 ^ ^-e2 _-i 1 -i 2 _ xxx... xxx... _-e!...!-i xxx!!-e! :...? : :-i xxx:? :-e? no rule??-e :: :: ::-e
5 Proof p! q; r! a ` (p _ r)! (q _ s) 1: p _ r assumption 2: p assumption 3: p! q premise 4: q!-e : 2; 3 5: q _ s _-i 1 : 4 6: r assumption 7: r! s premise 8: s!-e : 6; 7 9: q _ s _-i 2 : 8 10: q _ s _-e : 2 9
6 Proof p! q; r! a ` (p _ r)! (q _ s) 1: p _ r assumption 2: p assumption 3: p! q premise 4: q!-e : 2; 3 5: q _ s _-i 1 : 4 6: r assumption 7: r! s premise 8: s!-e : 6; 7 9: q _ s _-i 2 : 8 10: q _ s _-e : 2 9
7 Reasoning vs. reality Reasoning: argument based on observations and derivational rules. Reality: reality. How does reasoning relate to reality? I Soundness: reasoning derives only true statements. Trivial solution: reasoning that derives nothing. I Completeness: reasoning derives all true statements. Trivial solution: reasoning that derives everything.
8 Semantics What is a true formula? A formula is either true T or false F. The meaning depends on the meaning of the subterms. Examples: I [[p]] = T and [[q]] = F implies [[p ^ q]] = F I [[p]] = T implies [[:p]] = F I [[p]] = T and [[q]] = F implies [[p ^ :q]] = T
9 Truth tables The value of a formula for all possible inputs. Basic connectives: > T xxx? F xxx : F T T F xxx ^ T T T T F F F T F F F F _ T T T T F T F T T F F F xxx xxx! T T T T F F F T T F F T Other tables dene other functions.
10 Meaning of a formula Begin with the meanings of atoms. Compute the value bottom-up. Example: (p! q) ^ (p! r) p q r p! q p! r (p! q) ^ (p! r) T T T T T F T F T T F F F T T F T F F F T F F F I Tautology: Always true. I Contradiction: always false.
11 Semantic entailment true when 1 ; : : : n are true. 1 ; : : : n j= 1. Compute truth tables of the 1 ; : : : n. 2. Collect lines where 1 ; : : : n are all true. 3. Evaluate in these cases.
12 Example q ^ r j= (p! q) ^ (p! r) p q r q ^ r p! q p! r (p! q) ^ (p! r) T T T T T T F F T F T F T F F F F T T T F T F F F F T F F F F F Thus, q ^ r j= (p! q) ^ (p! r)
13 Example q ^ r j= (p! q) ^ (p! r) p q r q ^ r p! q p! r (p! q) ^ (p! r) T T T T T T T T T F F T F T F T F F F F T T T T T T F T F F F F T F F F F F Thus, q ^ r j= (p! q) ^ (p! r)
14 Soundness and completeness Relate provability to semantic entailment. I Soundness: 1 ; : : : ; n ` implies that 1 ; : : : ; n j=. I Completeness: 1 ; : : : ; n j= implies that 1 ; : : : ; n `.
15 Soundness Theorem: Let 1 ; : : : ; n and be propositional logic formulas. Then, if 1 ; : : : ; n `, then 1 ; : : : ; n j=. Proof idea: each proof step is justied by truth tables.
16 Proof: inductive denition I : premise is a proof of `, where 2. I Let: i prove ` i j prove ; j ` j For any rule R: 1 : : : m xxx 1. 1 : : :. n n Then 1 ; : : : ; m ; 1 ; : : : ; n ; R proves `.
17 Base case Given: ` Show: j= Proof of ` : : premise Show: j= Evaluate for truth table lines where are all T. is a premise, so 2. Thus is T.
18 Induction case Given: ` Show: j= ` must have a proof 1 ; : : : ; m ; 1 ; : : : ; n ; R, for R: 1 : : : m xxx 1. 1 : : :. n n Induction hypothesis: I i proves ` i implies j= i. I j proves ; j ` j implies ; j j= j : Proceed by cases on the possible rules.
19 Rule: ^-i Given: ` ^, show: j= ^. Proof of ` ^ : 1 ; 2 ; ^-i xxx ^ ^-i 1 proves `, 2 proves ` By induction: j=, j= Evaluate ^ when [[]] = [[ ]] = T: Thus, j= ^. ^ T T T
20 Rule: ^-i Given: ` ^, show: j= ^. Proof of ` ^ : 1 ; 2 ; ^-i xxx ^ ^-i 1 proves `, 2 proves ` By induction: j=, j= Evaluate ^ when [[]] = [[ ]] = T: Thus, j= ^. ^ T T T
21 Rule: ^-i Given: ` ^, show: j= ^. Proof of ` ^ : 1 ; 2 ; ^-i xxx ^ ^-i 1 proves `, 2 proves ` By induction: j=, j= Evaluate ^ when [[]] = [[ ]] = T: Thus, j= ^. ^ T T T
22 Rule: :-e Given: `?, show: j=?. Proof of `?: 1 ; 2 ; :-e xxx:? :-e 1 proves `, 2 proves ` : By induction: j=, j= : Truth table for : : F T T F never all T. Thus, trivially: j=? NB: Truth table for?:? F
23 Rule: :-e Given: `?, show: j=?. Proof of `?: 1 ; 2 ; :-e xxx:? :-e 1 proves `, 2 proves ` : By induction: j=, j= : Truth table for : : F T T F never all T. Thus, trivially: j=? NB: Truth table for?:? F
24 Rule: :-e Given: `?, show: j=?. Proof of `?: 1 ; 2 ; :-e xxx:? :-e 1 proves `, 2 proves ` : By induction: j=, j= : Truth table for : : F T T F never all T. Thus, trivially: j=? NB: Truth table for?:? F
25 Rule:?-e Given: `, show: j=. Proof of ` : ; :-e??-e proves `?. By induction: j=?. Truth table for?: never all T.? F Thus, trivially: j=
26 Rule:?-e Given: `, show: j=. Proof of ` : ; :-e??-e proves `?. By induction: j=?. Truth table for?: never all T.? F Thus, trivially: j=
27 Rule:?-e Given: `, show: j=. Proof of ` : ; :-e??-e proves `?. By induction: j=?. Truth table for?: never all T.? F Thus, trivially: j=
28 Rule!-i Given: `!, show: j=!. Proof of `! : ;!-i xxx.!-i proves ; `! By induction: ; j=. Show: when all true, so is!.! T T T T F F xxx! F T T F F T Potential problem when [[]] = T and [[ ]] = F. But, by induction, if all true and [[]] = T, then [[ ]] = T. Thus, j=!.
29 Rule!-i Given: `!, show: j=!. Proof of `! : ;!-i xxx.!-i proves ; `! By induction: ; j=. Show: when all true, so is!.! T T T T F F xxx! F T T F F T Potential problem when [[]] = T and [[ ]] = F. But, by induction, if all true and [[]] = T, then [[ ]] = T. Thus, j=!.
30 Rule!-i Given: `!, show: j=!. Proof of `! : ;!-i xxx.!-i proves ; `! By induction: ; j=. Show: when all true, so is!.! T T T T F F xxx! F T T F F T Potential problem when [[]] = T and [[ ]] = F. But, by induction, if all true and [[]] = T, then [[ ]] = T. Thus, j=!.
31 Completeness Theorem: Let 1 ; : : : ; n and be propositional logic formulas. Then, if 1 ; : : : ; n j=, then 1 ; : : : ; n `. Proof idea: construct a proof from a truth table.
32 Completeness proof structure 1. Eliminate premises: 1 ; 2 ; : : : ; n j= implies j= 1! ( 2! : : :! ( n! )). 2. Show provability: j= 1! ( 2! : : :! ( n! )) implies ` 1! ( 2! : : :! ( n! )). 3. Reintroduce premises: ` 1! ( 2! : : :! ( n! )) implies 1 ; 2 ; : : : ; n `.
33 Eliminating premises Theorem: If 1 ; 2 ; : : : ; n j=, then j= 1! ( 2! : : :! ( n! )). Proof: By induction on n. Base case: n = 0. Clearly j= implies j=. Induction case: We showed ; j= implies j=!. Thus, 1 ; 2 ; : : : ; n j= implies 1 ; : : : n 1 j= n!. By induction, j= 1! ( 2! : : :! ( n! ))
34 Showing provability Theorem: I Let be a formula such that p1; p2; : : : ; p n are its only propositional atoms. I Let ` be any line in 's truth table. I For any atom or formula, let `() be if the truth table entry in line ` for is T, and : if the truth table entry for is F. Then, `(p1); `(p2); : : : ; `(p n ) ` `() is provable.
35 Example (p ^ r)! (q ^ s) A truth table line: p q r s p ^ r q ^ s (p ^ r)! (q ^ s) T F T T T F F Constructed sequent: p; :q; r ; s ` :((p ^ r)! (q ^ s)) Another truth table line: p q r s p ^ r q ^ s (p ^ r)! (q ^ s) F T F T F T T Constructed sequent: :p; q; :r ; s ` (p ^ r)! (q ^ s)
36 Proof Induction on the structure of. Base case: p. Truth table: p T F p T F `(p) ` `(p) is p ` p, or :p ` :p: I Proof of p ` p: 1: p premise 2: p 1 I Proof of :p ` :p: 1: :p premise 2: :p 1 Thus, `(p) ` `(p).
37 Negation : Possible truth table lines: F T : T F If [[ ]] = F and [[: ]] = T: xxxshow `(p1); `(p2); : : : ; `(p n ) ` : By induction, since [[ ]] = F: `(p1); `(p2); : : : ; `(p n ) ` :
38 Negation, continued If [[ ]] = T and [[: ]] = F: xxxshow `(p1); `(p2); : : : ; `(p n ) ` :: By induction, since [[ ]] = T: `(p1); `(p2); : : : ; `(p n ) ` To prove `(p1); `(p2); : : : ; `(p n ) ` :: : First prove `(p1); `(p2); : : : ; `(p n ) `. Use :-i.
39 Implication 1! 2 Proof strategy: I Consider possible truth values of 1! 2. I Use induction to nd proofs of `( 1 ) and `( 2 ). I Construct a proof of `( 1! 2 ). Four lines, so four cases ! 2 T T T T F F F T T F F T
40 Use of the induction hypothesis Consider 1, 2 : 1 2 1! 2 T T T T F F F T T F F T I Propositional atoms of 1 : a1; : : : ; a x. I Propositional atoms of 2 : b1; : : : ; b y. Atoms of 1! 2 = fa1; : : : ; a x g [ fb1; : : : ; b y g. Thus, a truth table line for 1! 2 also gives meaning to 1 and 2.
41 Using the induction hypothesis For a truth table line ` for 1! 2, by induction: `(a1); : : : ; `(a x ) ` `( 1 ) `(b1); : : : ; `(b y ) ` `( 2 ) Our goal: `(a1); : : : ; `(a x ); `(b1); : : : ; `(b y ) ` `( 1! 2 ) Proof structure: I Proof of `(a1); : : : ; `(a x ) ` `( 1 ). I Proof of `(b1); : : : ; `(b y ) ` `( 2 ). I `( 1 ) ^ `( 2 ) : ^-i I Derive `( 1! 2 ) from `( 1 ) ^ `( 2 ).
42 Cases I 1 is T, 2 is T, and is T: Show 1 ^ 2 implies 1! 2. I 1 is T, 2 is F, and is F: Show 1 ^ : 2 implies :( 1! 2 ). I 1 is F, 2 is T, and is T: Show : 1 ^ 2 implies 1! 2. I 1 is F, 2 is F, and is T: Show : 1 ^ : 2 implies 1! 2. Conjunction and disjunction similar.
43 Combining sequents We have a \complete" collection of sequents: `(p1); `(p2); : : : ; `(p n ) ` `() Note: j= implies 8`:`() =. We want to prove: ` Proof idea: Use the Law of the Excluded Middle.
44 Example (p ^ q)! q Constructed sequents: p; q ` (p ^ q)! q :p; q ` (p ^ q)! q p; :q ` (p ^ q)! q :p; :q ` (p ^ q)! q Consider p; q ` (p ^ q)! q and p; :q ` (p ^ q)! q. Proof of p ` (p ^ q)! q: 1: q _ :q LEM 2: q assumption 3 proof of p; q ` (p ^ q)! q 4: :q assumption 5 proof of p; :q ` (p ^ q)! q 6: (p ^ q)! q _-e : 1 5
45 Proof idea Goal: Reduce `(p1); `(p2); : : : ; `(p n ) ` to `. Proof by induction on n. Find pairs of sequents that only dier in `(p n ). Use LEM to prove a single sequent without `(p n ) in the premise.
46 Reintroducing premises Theorem: If ` 1! ( 2! : : :! ( n! )), then 1 ; 2 ; : : : ; n `. Proof idea: Convert ` 1! ( 2! : : :! ( n! )) to 1 ` 2! : : :! ( n! ), and proceed by induction. Problem: Theorem not general enough. Restatement: If ` 1! ( 2! : : :! ( n! )), then ; 1 ; 2 ; : : : ; n `.
47 Proof Given a proof of: ` 1! ( 2! : : :! ( n! )) Prove: ; 1 ` 2! : : :! ( n! ) I Proof of ` 1! ( 2! : : :! ( n! )) I 1 : premise I 2! : : :! ( n! ) :!-e By induction, ; 1 ; 2 ; : : : ; n `. Take = ; to prove the original theorem.
48 Summary Soundness: Show that the deduction rules are justied by truth tables. Completeness: 1. Eliminate premises: 1 ; 2 ; : : : ; n j= implies j= 1! ( 2! : : :! ( n! )). 2. Show provability: j= 1! ( 2! : : :! ( n! )) implies ` 1! ( 2! : : :! ( n! )). 3. Reintroduce premises: ` 1! ( 2! : : :! ( n! )) implies 1 ; 2 ; : : : ; n `. Conclusion: can freely use either natural deduction or truth tables.
Natural Deduction. Formal Methods in Verification of Computer Systems Jeremy Johnson
Natural Deduction Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. An example 1. Validity by truth table 2. Validity by proof 2. What s a proof 1. Proof checker 3. Rules of
More informationNatural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
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 information2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
More information15414/614 Optional Lecture 1: Propositional Logic
15414/614 Optional Lecture 1: Propositional Logic Qinsi Wang Logic is the study of information encoded in the form of logical sentences. We use the language of Logic to state observations, to define concepts,
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 informationPropositional Logic: Part II - Syntax & Proofs 0-0
Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems
More informationWarm-Up Problem. Write a Resolution Proof for. Res 1/32
Warm-Up Problem Write a Resolution Proof for Res 1/32 A second Rule Sometimes throughout we need to also make simplifications: You can do this in line without explicitly mentioning it (just pretend you
More informationLecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)
Lecture 2 Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits Reading (Epp s textbook) 2.1-2.4 1 Logic Logic is a system based on statements. A statement (or
More informationChapter 11: Automated Proof Systems (1)
Chapter 11: Automated Proof Systems (1) SYSTEM RS OVERVIEW Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. Automated systems
More informationLogic for Computer Science - Week 5 Natural Deduction
Logic for Computer Science - Week 5 Natural Deduction Ștefan Ciobâcă November 30, 2017 1 An Alternative View of Implication and Double Implication So far, we have understood as a shorthand of However,
More informationThe Importance of Being Formal. Martin Henz. February 5, Propositional Logic
The Importance of Being Formal Martin Henz February 5, 2014 Propositional Logic 1 Motivation In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets
More informationPropositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
More informationANALYSIS EXERCISE 1 SOLUTIONS
ANALYSIS EXERCISE 1 SOLUTIONS 1. (a) Let B The main course will be beef. F The main course will be fish. P The vegetable will be peas. C The vegetable will be corn. The logical form of the argument is
More informationCOMP219: Artificial Intelligence. Lecture 19: Logic for KR
COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof
More informationPropositional logic. Programming and Modal Logic
Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness
More informationCOMP219: Artificial Intelligence. Lecture 19: Logic for KR
COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof
More information2.2: Logical Equivalence: The Laws of Logic
Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q
More informationAdvanced Topics in LP and FP
Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection
More informationPropositional Logic: Review
Propositional Logic: Review Propositional logic Logical constants: true, false Propositional symbols: P, Q, S,... (atomic sentences) Wrapping parentheses: ( ) Sentences are combined by connectives:...and...or
More information03 Propositional Logic II
Martin Henz February 12, 2014 Generated on Wednesday 12 th February, 2014, 09:49 1 Review: Syntax and Semantics of Propositional Logic 2 3 Propositional Atoms and Propositions Semantics of Formulas Validity,
More informationAI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic
AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic and Predicate Logic Propositional logic Logical connectives Rules for wffs Truth tables for the connectives Using Truth Tables to evaluate
More informationPart Two: The Basic Components of the SOFL Specification Language
Part Two: The Basic Components of the SOFL Specification Language SOFL logic Module Condition Data Flow Diagrams Process specification Function definition and specification Process decomposition Other
More informationOverview. Knowledge-Based Agents. Introduction. COMP219: Artificial Intelligence. Lecture 19: Logic for KR
COMP219: Artificial Intelligence Lecture 19: Logic for KR Last time Expert Systems and Ontologies oday Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof theory Natural
More information02 Propositional Logic
SE 2F03 Fall 2005 02 Propositional Logic Instructor: W. M. Farmer Revised: 25 September 2005 1 What is Propositional Logic? Propositional logic is the study of the truth or falsehood of propositions or
More informationWarm-Up Problem. Is the following true or false? 1/35
Warm-Up Problem Is the following true or false? 1/35 Propositional Logic: Resolution Carmen Bruni Lecture 6 Based on work by J Buss, A Gao, L Kari, A Lubiw, B Bonakdarpour, D Maftuleac, C Roberts, R Trefler,
More informationComputation and Logic Definitions
Computation and Logic Definitions True and False Also called Boolean truth values, True and False represent the two values or states an atom can assume. We can use any two distinct objects to represent
More informationMathematics for linguists
Mathematics for linguists WS 2009/2010 University of Tübingen January 7, 2010 Gerhard Jäger Mathematics for linguists p. 1 Inferences and truth trees Inferences (with a finite set of premises; from now
More informationPropositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
More informationCOMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
More informationChapter 4: Classical Propositional Semantics
Chapter 4: Classical Propositional Semantics Language : L {,,, }. Classical Semantics assumptions: TWO VALUES: there are only two logical values: truth (T) and false (F), and EXTENSIONALITY: the logical
More informationCOMP2411 Lecture 6: Soundness and Completeness. Reading: Huth and Ryan, Sections 1.4
COMP2411 Lecture 6: Soundness and Completeness Reading: Huth and Ryan, Sections 14 Arithmetic is useful in the world because it is an example of the diagram: symbols - symbolic manipulation -> symbols
More informationSection 1.2: Propositional Logic
Section 1.2: Propositional Logic January 17, 2017 Abstract Now we re going to use the tools of formal logic to reach logical conclusions ( prove theorems ) based on wffs formed by some given statements.
More informationChapter 11: Automated Proof Systems
Chapter 11: Automated Proof Systems SYSTEM RS OVERVIEW Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. Automated systems are
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 informationCSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter p. 1/33
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter 4.1-4.8 p. 1/33 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer
More informationPropositional Logic. CS 3234: Logic and Formal Systems. Martin Henz and Aquinas Hobor. August 26, Generated on Tuesday 31 August, 2010, 16:54
Propositional Logic CS 3234: Logic and Formal Systems Martin Henz and Aquinas Hobor August 26, 2010 Generated on Tuesday 31 August, 2010, 16:54 1 Motivation In traditional logic, terms represent sets,
More informationPropositional natural deduction
Propositional natural deduction COMP2600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2016 Major proof techniques 1 / 25 Three major styles of proof in logic and mathematics Model
More informationPropositional Logic Arguments (5A) Young W. Lim 10/11/16
Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationLogic for Computer Science - Week 4 Natural Deduction
Logic for Computer Science - Week 4 Natural Deduction 1 Introduction In the previous lecture we have discussed some important notions about the semantics of propositional logic. 1. the truth value of a
More informationCS206 Lecture 03. Propositional Logic Proofs. Plan for Lecture 03. Axioms. Normal Forms
CS206 Lecture 03 Propositional Logic Proofs G. Sivakumar Computer Science Department IIT Bombay siva@iitb.ac.in http://www.cse.iitb.ac.in/ siva Page 1 of 12 Fri, Jan 03, 2003 Plan for Lecture 03 Axioms
More informationPropositional Calculus - Soundness & Completeness of H
Propositional Calculus - Soundness & Completeness of H Moonzoo Kim CS Dept. KAIST moonzoo@cs.kaist.ac.kr 1 Review Goal of logic To check whether given a formula Á is valid To prove a given formula Á `
More informationLogic. Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another.
Math 0413 Appendix A.0 Logic Definition [1] A logic is a formal language that comes with rules for deducing the truth of one proposition from the truth of another. This type of logic is called propositional.
More informationPropositional logic. Programming and Modal Logic
Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness
More informationKnowledge base (KB) = set of sentences in a formal language Declarative approach to building an agent (or other system):
Logic Knowledge-based agents Inference engine Knowledge base Domain-independent algorithms Domain-specific content Knowledge base (KB) = set of sentences in a formal language Declarative approach to building
More informationMarie Duží
Marie Duží marie.duzi@vsb.cz 1 Formal systems, Proof calculi A proof calculus (of a theory) is given by: 1. a language 2. a set of axioms 3. a set of deduction rules ad 1. The definition of a language
More informationCSE 20: Discrete Mathematics
Spring 2018 Summary Last time: Today: Logical connectives: not, and, or, implies Using Turth Tables to define logical connectives Logical equivalences, tautologies Some applications Proofs in propositional
More informationPřednáška 12. Důkazové kalkuly Kalkul Hilbertova typu. 11/29/2006 Hilbertův kalkul 1
Přednáška 12 Důkazové kalkuly Kalkul Hilbertova typu 11/29/2006 Hilbertův kalkul 1 Formal systems, Proof calculi A proof calculus (of a theory) is given by: A. a language B. a set of axioms C. a set of
More information1. Propositional Calculus
1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:
More informationNatural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms)
More information1 Propositional Logic
1 Propositional Logic Required reading: Foundations of Computation. Sections 1.1 and 1.2. 1. Introduction to Logic a. Logical consequences. If you know all humans are mortal, and you know that you are
More informationDeductive Systems. Lecture - 3
Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL AS is based on the set of only three axioms and one rule of deduction. It is minimal in structure but as powerful as the truth
More informationPropositional Logic Arguments (5A) Young W. Lim 11/8/16
Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationPropositional Logic Arguments (5A) Young W. Lim 11/30/16
Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationIntroduction to Intuitionistic Logic
Introduction to Intuitionistic Logic August 31, 2016 We deal exclusively with propositional intuitionistic logic. The language is defined as follows. φ := p φ ψ φ ψ φ ψ φ := φ and φ ψ := (φ ψ) (ψ φ). A
More informationInference in Propositional Logic
Inference in Propositional Logic Deepak Kumar November 2017 Propositional Logic A language for symbolic reasoning Proposition a statement that is either True or False. E.g. Bryn Mawr College is located
More informationCOT 2104 Homework Assignment 1 (Answers)
1) Classify true or false COT 2104 Homework Assignment 1 (Answers) a) 4 2 + 2 and 7 < 50. False because one of the two statements is false. b) 4 = 2 + 2 7 < 50. True because both statements are true. c)
More informationPHIL12A Section answers, 28 Feb 2011
PHIL12A Section answers, 28 Feb 2011 Julian Jonker 1 How much do you know? Give formal proofs for the following arguments. 1. (Ex 6.18) 1 A B 2 A B 1 A B 2 A 3 A B Elim: 2 4 B 5 B 6 Intro: 4,5 7 B Intro:
More informationPropositional logic (revision) & semantic entailment. p. 1/34
Propositional logic (revision) & semantic entailment p. 1/34 Reading The background reading for propositional logic is Chapter 1 of Huth/Ryan. (This will cover approximately the first three lectures.)
More informationPropositional Logic: Deductive Proof & Natural Deduction Part 1
Propositional Logic: Deductive Proof & Natural Deduction Part 1 CS402, Spring 2016 Shin Yoo Deductive Proof In propositional logic, a valid formula is a tautology. So far, we could show the validity of
More informationDeduction by Daniel Bonevac. Chapter 3 Truth Trees
Deduction by Daniel Bonevac Chapter 3 Truth Trees Truth trees Truth trees provide an alternate decision procedure for assessing validity, logical equivalence, satisfiability and other logical properties
More informationKnowledge based Agents
Knowledge based Agents Shobhanjana Kalita Dept. of Computer Science & Engineering Tezpur University Slides prepared from Artificial Intelligence A Modern approach by Russell & Norvig Knowledge Based Agents
More informationPropositional Logics and their Algebraic Equivalents
Propositional Logics and their Algebraic Equivalents Kyle Brooks April 18, 2012 Contents 1 Introduction 1 2 Formal Logic Systems 1 2.1 Consequence Relations......................... 2 3 Propositional Logic
More informationINTRODUCTION TO LOGIC. Propositional Logic. Examples of syntactic claims
Introduction INTRODUCTION TO LOGIC 2 Syntax and Semantics of Propositional Logic Volker Halbach In what follows I look at some formal languages that are much simpler than English and define validity of
More informationWhat is Logic? Introduction to Logic. Simple Statements. Which one is statement?
What is Logic? Introduction to Logic Peter Lo Logic is the study of reasoning It is specifically concerned with whether reasoning is correct Logic is also known as Propositional Calculus CS218 Peter Lo
More informationType Systems. Lecture 9: Classical Logic. Neel Krishnaswami University of Cambridge
Type Systems Lecture 9: Classical Logic Neel Krishnaswami University of Cambridge Where We Are We have seen the Curry Howard correspondence: Intuitionistic propositional logic Simply-typed lambda calculus
More informationCOMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning
COMP9414, Monday 26 March, 2012 Propositional Logic 2 COMP9414: Artificial Intelligence Propositional Logic: Automated Reasoning Overview Proof systems (including soundness and completeness) Normal Forms
More informationFormal (natural) deduction in propositional logic
Formal (natural) deduction in propositional logic Lila Kari University of Waterloo Formal (natural) deduction in propositional logic CS245, Logic and Computation 1 / 67 I know what you re thinking about,
More informationArtificial Intelligence
Artificial Intelligence Propositional Logic [1] Boolean algebras by examples U X U U = {a} U = {a, b} U = {a, b, c} {a} {b} {a, b} {a, c} {b, c}... {a} {b} {c} {a, b} {a} The arrows represents proper inclusion
More informationSemantics and Pragmatics of NLP
Semantics and Pragmatics of NLP Alex Ewan School of Informatics University of Edinburgh 28 January 2008 1 2 3 Taking Stock We have: Introduced syntax and semantics for FOL plus lambdas. Represented FOL
More informationLanguage of Propositional Logic
Logic A logic has: 1. An alphabet that contains all the symbols of the language of the logic. 2. A syntax giving the rules that define the well formed expressions of the language of the logic (often called
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 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 informationEquivalents of Mingle and Positive Paradox
Eric Schechter Equivalents of Mingle and Positive Paradox Abstract. Relevant logic is a proper subset of classical logic. It does not include among itstheoremsanyof positive paradox A (B A) mingle A (A
More information1. Propositional Calculus
1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:
More informationCSCE 222 Discrete Structures for Computing. Propositional Logic. Dr. Hyunyoung Lee. !!!!!! Based on slides by Andreas Klappenecker
CSCE 222 Discrete Structures for Computing Propositional Logic Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Propositions A proposition is a declarative sentence that is either true or false
More informationPropositional Logic Part 1
Propositional Logic Part 1 Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [Based on slides from Louis Oliphant, Andrew Moore, Jerry Zhu] slide 1 5 is even
More informationNotes on Inference and Deduction
Notes on Inference and Deduction Consider the following argument 1 Assumptions: If the races are fixed or the gambling houses are crooked, then the tourist trade will decline. If the tourist trade declines
More information1.3 Propositional Equivalences
1 1.3 Propositional Equivalences The replacement of a statement with another statement with the same truth is an important step often used in Mathematical arguments. Due to this methods that produce propositions
More informationFundamentals of Logic
Fundamentals of Logic No.5 Soundness and Completeness Tatsuya Hagino Faculty of Environment and Information Studies Keio University 2015/5/18 Tatsuya Hagino (Faculty of Environment and InformationFundamentals
More informationA Weak Post s Theorem and the Deduction Theorem Retold
Chapter I A Weak Post s Theorem and the Deduction Theorem Retold This note retells (1) A weak form of Post s theorem: If Γ is finite and Γ = taut A, then Γ A and derives as a corollary the Deduction Theorem:
More informationMAI0203 Lecture 7: Inference and Predicate Calculus
MAI0203 Lecture 7: Inference and Predicate Calculus Methods of Artificial Intelligence WS 2002/2003 Part II: Inference and Knowledge Representation II.7 Inference and Predicate Calculus MAI0203 Lecture
More informationPropositional Logic: Gentzen System, G
CS402, Spring 2017 Quiz on Thursday, 6th April: 15 minutes, two questions. Sequent Calculus in G In Natural Deduction, each line in the proof consists of exactly one proposition. That is, A 1, A 2,...,
More informationManual of Logical Style
Manual of Logical Style Dr. Holmes January 9, 2015 Contents 1 Introduction 2 2 Conjunction 3 2.1 Proving a conjunction...................... 3 2.2 Using a conjunction........................ 3 3 Implication
More informationPropositional Language - Semantics
Propositional Language - Semantics Lila Kari University of Waterloo Propositional Language - Semantics CS245, Logic and Computation 1 / 41 Syntax and semantics Syntax Semantics analyzes Form analyzes Meaning
More informationLogic, Human Logic, and Propositional Logic. Human Logic. Fragments of Information. Conclusions. Foundations of Semantics LING 130 James Pustejovsky
Logic, Human Logic, and Propositional Logic Foundations of Semantics LING 3 James Pustejovsky Human Logic Thanks to Michael Genesereth of Stanford for use of some slides Fragments of Information Conclusions
More informationProofs. Joe Patten August 10, 2018
Proofs Joe Patten August 10, 2018 1 Statements and Open Sentences 1.1 Statements A statement is a declarative sentence or assertion that is either true or false. They are often labelled with a capital
More informationA statement is a sentence that is definitely either true or false but not both.
5 Logic In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial
More informationComputational Logic. Davide Martinenghi. Spring Free University of Bozen-Bolzano. Computational Logic Davide Martinenghi (1/30)
Computational Logic Davide Martinenghi Free University of Bozen-Bolzano Spring 2010 Computational Logic Davide Martinenghi (1/30) Propositional Logic - sequent calculus To overcome the problems of natural
More informationAnnouncements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication
Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before
More informationArtificial Intelligence. Propositional logic
Artificial Intelligence Propositional logic Propositional Logic: Syntax Syntax of propositional logic defines allowable sentences Atomic sentences consists of a single proposition symbol Each symbol stands
More informationThe semantics of propositional logic
The semantics of propositional logic Readings: Sections 1.3 and 1.4 of Huth and Ryan. In this module, we will nail down the formal definition of a logical formula, and describe the semantics of propositional
More informationLogical Agents. Knowledge based agents. Knowledge based agents. Knowledge based agents. The Wumpus World. Knowledge Bases 10/20/14
0/0/4 Knowledge based agents Logical Agents Agents need to be able to: Store information about their environment Update and reason about that information Russell and Norvig, chapter 7 Knowledge based agents
More informationPropositional Logic Basics Propositional Equivalences Normal forms Boolean functions and digital circuits. Propositional Logic.
Propositional Logic Winter 2012 Propositional Logic: Section 1.1 Proposition A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?
More informationUNIT-I: Propositional Logic
1. Introduction to Logic: UNIT-I: Propositional Logic Logic: logic comprises a (formal) language for making statements about objects and reasoning about properties of these objects. Statements in a logical
More informationCompleteness for FOL
Completeness for FOL Overview Adding Witnessing Constants The Henkin Theory The Elimination Theorem The Henkin Construction Lemma 12 This lemma assures us that our construction of M h works for the atomic
More informationOutline. Overview. Syntax Semantics. Introduction Hilbert Calculus Natural Deduction. 1 Introduction. 2 Language: Syntax and Semantics
Introduction Arnd Poetzsch-Heffter Software Technology Group Fachbereich Informatik Technische Universität Kaiserslautern Sommersemester 2010 Arnd Poetzsch-Heffter ( Software Technology Group Fachbereich
More informationApplied Logic. Lecture 1 - Propositional logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw
Applied Logic Lecture 1 - Propositional logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018
More informationIntelligent Agents. First Order Logic. Ute Schmid. Cognitive Systems, Applied Computer Science, Bamberg University. last change: 19.
Intelligent Agents First Order Logic Ute Schmid Cognitive Systems, Applied Computer Science, Bamberg University last change: 19. Mai 2015 U. Schmid (CogSys) Intelligent Agents last change: 19. Mai 2015
More informationPropositional Logic (2A) Young W. Lim 11/8/15
Propositional Logic (2A) Young W. Lim Copyright (c) 2014 2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU ree Documentation License, Version
More information