(ÀB Ä (A Â C)) (A Ä ÀC) Á B. This is our sample argument. Formal Proofs

Similar documents
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.

For a horseshoe statement, having the matching p (left side) gives you the q (right side) by itself. It does NOT work with matching q s.

Today s Lecture 2/25/10. Truth Tables Continued Introduction to Proofs (the implicational rules of inference)

Manual of Logical Style (fresh version 2018)

2. The Logic of Compound Statements Summary. Aaron Tan August 2017

Lecture 2. Logic Compound Statements Conditional Statements Valid & Invalid Arguments Digital Logic Circuits. Reading (Epp s textbook)

3 The Semantics of the Propositional Calculus

In this chapter, we specify a deductive apparatus for PL.

Manual of Logical Style

Section 1.1 Propositions

Formal Logic. Critical Thinking

THE LOGIC OF COMPOUND STATEMENTS

Propositional Logic: Part II - Syntax & Proofs 0-0

Logical Form 5 Famous Valid Forms. Today s Lecture 1/26/10

PHI Propositional Logic Lecture 2. Truth Tables

Proofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction

CHAPTER 1 - LOGIC OF COMPOUND STATEMENTS

Section 1.2: Propositional Logic

Chapter 1, Logic and Proofs (3) 1.6. Rules of Inference

Natural Deduction for Propositional Logic

Propositional Logic. Chrysippos (3 rd Head of Stoic Academy). Main early logician. AKA Modern Logic AKA Symbolic Logic. AKA Boolean Logic.

PHIL012. SYMBOLIC LOGIC PROPOSITIONAL LOGIC DERIVATIONS

Discrete Structures of Computer Science Propositional Logic III Rules of Inference

Proof Tactics, Strategies and Derived Rules. CS 270 Math Foundations of CS Jeremy Johnson

A Quick Lesson on Negation

i.e. The conclusion to the following argument says If you had an A, then you d have a ~(B v Z).

DERIVATIONS AND TRUTH TABLES

Arguments and Proofs. 1. A set of sentences (the premises) 2. A sentence (the conclusion)

15414/614 Optional Lecture 1: Propositional Logic

Supplementary Logic Notes CSE 321 Winter 2009

PROPOSITIONAL CALCULUS

Propositional Logic: Syntax

A. Propositional Logic

1.1 Statements and Compound Statements

Proof Worksheet 2, Math 187 Fall 2017 (with solutions)

Warm-Up Problem. Write a Resolution Proof for. Res 1/32

Packet #1: Logic & Proofs. Applied Discrete Mathematics

3.0. OBJECTIVES 3.1.INTRODUCTION

Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014

Propositional Logic. Argument Forms. Ioan Despi. University of New England. July 19, 2013

Proofs. Introduction II. Notes. Notes. Notes. Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry. Fall 2007

PL: Truth Trees. Handout Truth Trees: The Setup

The Logic of Compound Statements cont.

Chapter 1 Elementary Logic

Deductive Systems. Lecture - 3

Collins' notes on Lemmon's Logic

Introduction Logic Inference. Discrete Mathematics Andrei Bulatov

Language of Propositional Logic

First-Degree Entailment

CHAPTER 6 - THINKING ABOUT AND PRACTICING PROPOSITIONAL LOGIC

Proof strategies, or, a manual of logical style

Formal (natural) deduction in propositional logic

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.

Propositional Logic. Spring Propositional Logic Spring / 32

COMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University

Natural deduction for truth-functional logic

4 Derivations in the Propositional Calculus

Proofs. Example of an axiom in this system: Given two distinct points, there is exactly one line that contains them.

Logic Overview, I. and T T T T F F F T F F F F

Chapter 1: The Logic of Compound Statements. January 7, 2008

Axiomatic systems. Revisiting the rules of inference. Example: A theorem and its proof in an abstract axiomatic system:

MACM 101 Discrete Mathematics I. Exercises on Propositional Logic. Due: Tuesday, September 29th (at the beginning of the class)

Inference in Propositional Logic

Logic. Logic is a discipline that studies the principles and methods used in correct reasoning. It includes:

Four Basic Logical Connectives & Symbolization

Logic, Sets, and Proofs

CS 2336 Discrete Mathematics

10/5/2012. Logic? What is logic? Propositional Logic. Propositional Logic (Rosen, Chapter ) Logic is a truth-preserving system of inference

Review The Conditional Logical symbols Argument forms. Logic 5: Material Implication and Argument Forms Jan. 28, 2014

8. Reductio ad absurdum

Resolution (14A) Young W. Lim 8/15/14

Semantics and Pragmatics of NLP

Propositional Logic Review

Propositional Logic. Jason Filippou UMCP. ason Filippou UMCP) Propositional Logic / 38

Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012

FORMAL PROOFS DONU ARAPURA

Topic 1: Propositional logic

Computation and Logic Definitions

Deductive and Inductive Logic

The Importance of Being Formal. Martin Henz. February 5, Propositional Logic

CSE 20 DISCRETE MATH. Winter

Artificial Intelligence: Knowledge Representation and Reasoning Week 2 Assessment 1 - Answers

CSC Discrete Math I, Spring Propositional Logic

What is Logic? Introduction to Logic. Simple Statements. Which one is statement?

Notes on Propositional and First-Order Logic (CPSC 229 Class Notes, January )

Deduction by Daniel Bonevac. Chapter 3 Truth Trees

Analyzing Arguments with Truth Tables

CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)

Propositional Calculus

Discrete Structures & Algorithms. Propositional Logic EECE 320 // UBC

Packet #2: Set Theory & Predicate Calculus. Applied Discrete Mathematics

LING 106. Knowledge of Meaning Lecture 3-1 Yimei Xiang Feb 6, Propositional logic

CSE 20: Discrete Mathematics

CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter p. 1/33

Mat2345 Week 2. Chap 1.5, 1.6. Fall Mat2345 Week 2. Chap 1.5, 1.6. Week2. Negation. 1.5 Inference. Modus Ponens. Modus Tollens. Rules.

What is the decimal (base 10) representation of the binary number ? Show your work and place your final answer in the box.

Mathacle. PSet ---- Algebra, Logic. Level Number Name: Date: I. BASICS OF PROPOSITIONAL LOGIC

Intermediate Logic. Natural Deduction for TFL

Outline. Rules of Inferences Discrete Mathematics I MATH/COSC 1056E. Example: Existence of Superman. Outline

2/2/2018. CS 103 Discrete Structures. Chapter 1. Propositional Logic. Chapter 1.1. Propositional Logic

Transcription:

(ÀB Ä (A Â C)) (A Ä ÀC) Á B This is our sample argument. Formal Proofs From now on, formal proofs will be our main way to test arguments. We ll begin with easier proofs. Our initial strategy for constructing proofs has three steps. LogiCola G (EV) Pages 146 52

1 (ÀB Ä (A Â C)) 2 (A Ä ÀC) [ Á B 3 asm: ÀB [ Block off conclusion Assume the opposite Step 1: START Block off the conclusion and add asm: followed by the conclusion s simpler contradictory. LogiCola G (EV) Pages 146 52

1 (ÀB Ä (A Â C)) 2 (A Ä ÀC) [ Á B 3 asm: ÀB Here the complex wffs are 1 and 2, both IF-THENs. You can infer from these if you have the first part true or the second false. Step 2: S&I Begin the S&I step by glancing at the complex wffs and noticing their forms. You can simplify AND, NOR, and NIF and you can infer with NOT-BOTH, OR, and IF-THEN if certain other wffs are available. LogiCola G (EV) Pages 146 52

* 1 (ÀB Ä (A Â C)) 2 (A Ä ÀC) [ Á B 3 asm: ÀB 4 Á (A Â C) {from 1 and 3} IF-THEN rule: (ÀB Ä (A Â C)) ÀB (A Â C) Step 2: S&I Go through the complex wffs that aren t starred or blocked off and use these to derive new wffs using S- and I-rules. Star any wff you simplify using an S-rule, or the longer wff used in an I-rule inference. LogiCola G (EV) Pages 146 52

* 1 (ÀB Ä (A Â C)) 2 (A Ä ÀC) [ Á B 3 asm: ÀB * 4 Á (A Â C) {from 1 and 3} 5 Á A {from 4} 6 Á C {from 4} AND rule: (A Â C) A, C Step 2: S&I Go through the complex wffs that aren t starred or blocked off and use these to derive new wffs using S- and I-rules. Star any wff you simplify using an S-rule, or the longer wff used in an I-rule inference. LogiCola G (EV) Pages 146 52

* 1 (ÀB Ä (A Â C)) * 2 (A Ä ÀC) [ Á B 3 asm: ÀB * 4 Á (A Â C) {from 1 and 3} 5 Á A {from 4} 6 Á C {from 4} 7 Á ÀC {from 2 and 5} IF-THEN rule: (A Ä ÀC) A ÀC Step 2: S&I Go through the complex wffs that aren t starred or blocked off and use these to derive new wffs using S- and I-rules. Star any wff you simplify using an S-rule, or the longer wff used in an I-rule inference. LogiCola G (EV) Pages 146 52

* 1 (ÀB Ä (A Â C)) * 2 (A Ä ÀC) [ Á B 3 1 asm: ÀB * 4 2 Á (A Â C) {from 1 and 3} 5 2 Á A {from 4} 6 2 Á C {from 4} 7 3 Á ÀC {from 2 and 5} 8 Á B {from 3; 6 contradicts 7} 1 2 2 2 3 Valid block off from the assumption on down derive conclusion Step 3: RAA When some pair of not-blocked-off lines contradicts, apply RAA and derive the original conclusion. Your proof is done! LogiCola G (EV) Pages 146 52

S- and I-Rules AND (P Â Q) P, Q NOR À(P Ã Q) ÀP, ÀQ NIF À(P Ä Q) P, ÀQ NOT-BOTH OR IF-THEN À(P Â Q) P ÀQ À(P Â Q) Q ÀP (P Ã Q) ÀP Q (P Ã Q) ÀQ P (P Ä Q) P Q (P Ä Q) ÀQ ÀP affirm one part deny one part affirm 1 st or deny 2 nd LogiCola G (EV) Pages 146 52

S- and I-Rules AND (P Â Q) P, Q NOR À(P Ã Q) ÀP, ÀQ NIF À(P Ä Q) P, ÀQ NN ÀÀP P IFF (P Å Q) (P Ä Q), (Q Ä P) NIFF À(P Å Q) (P Ã Q), À(P Â Q) NOT-BOTH OR IF-THEN À(P Â Q) P ÀQ À(P Â Q) Q ÀP (P Ã Q) ÀP Q (P Ã Q) ÀQ P (P Ä Q) P Q (P Ä Q) ÀQ ÀP RAA: Suppose that some pair of not-blocked-off lines has contradictory wffs. Then block off all the lines from the last not-blocked-off assumption on down and infer a line consisting in Á followed by a contradictory of that assumption. LogiCola G (EV) and F (IE & IH) Pages 146 52

* 1 (ÀB Ä (A Â C)) * 2 (A Ä ÀC) [ Á B 3 1 asm: ÀB * 4 2 Á (A Â C) {from 1 and 3} 5 2 Á A {from 4} 6 2 Á C {from 4} 7 3 Á ÀC {from 2 and 5} 8 Á B {from 3; 6 contradicts 7} premises (no asm or Á ) assumption ( asm ) derived lines ( Á ) A formal proof is a vertical sequence of zero or more premises followed by one or more assumptions or derived lines, where each derived line follows from previously not-blocked-off lines by one of the S- and I-rules listed above or by RAA, and each assumption is blocked off using RAA. Two wffs are contradictories if they are exactly alike except that one starts with an additional À. A simple wff is a letter or its negation; any other wff is complex. LogiCola G (EV) and F (IE & IH) Pages 146 52

* 1 (ÀB Ä (A Â C)) * 2 (A Ä ÀC) [ Á B 3 1 asm: ÀB * 4 2 Á (A Â C) {from 1 and 3} 5 2 Á A {from 4} 6 2 Á C {from 4} 7 3 Á ÀC {from 2 and 5} 8 Á B {from 3; 6 contradicts 7} Valid Proof Strategy 1 START: Assume the opposite of the conclusion. 2 S&I: Derive whatever you can using S- and I-rules, until you get a contradiction. 3 RAA: Apply RAA and derive the original conclusion. LogiCola G (EV) and F (IE & IH) Pages 146 52

1 (A Ä B) [ Á (B Ä A) * 2 asm: À(B Ä A) 3 Á B {from 2} 4 Á ÀA {from 2} We can derive nothing further. Proof strategy to include invalid arguments: 1 START: Assume the opposite of the conclusion. 2 S&I: Derive whatever you can using S- and I-rules. 3 RAA: If you get a contradiction, apply RAA and derive the original conclusion. 4 REFUTE: If you don t get a contradiction, construct a refutation box. LogiCola F (IE & IH) Pages 154 57

1 (A 0 Ä B 1 ) = 1 [ Á (B 1 Ä A 0 ) = 0 * 2 asm: À(B Ä A) 3 Á B {from 2} 4 Á ÀA {from 2} Invalid B, ÀA Step 4 REFUTE: If you can t get a contradiction, then: draw a box containing any simple wffs (letters or their negation) that aren t blocked off; in the original argument, mark each letter 1 or 0 or? depending on whether you have the letter or its negation or neither in the box; if these truth conditions make the premises all true and conclusion false, then this shows the argument to be invalid. LogiCola F (IE & IH) Pages 154 57

* 1 (ÀB Ä (A Â C)) Valid * 2 (A Ä ÀC) [ Á B 3 1 asm: ÀB * 4 2 Á (A Â C) {from 1 and 3} 5 2 Á A {from 4} 6 2 Á C {from 4} 7 3 Á ÀC {from 2 and 5} 8 Á B {from 3; 6 contradicts 7} 1 (A 0 Ä B 1 ) = 1 [ Á (B 1 Ä A 0 ) = 0 * 2 asm: À(B Ä A) 3 Á B {from 2} 4 Á ÀA {from 2} Invalid B, ÀA 1 START: Assume the opposite of the conclusion. 2 S&I: Derive whatever you can using S- and I-rules. 3 RAA: If you get a contradiction, apply RAA and derive the original conclusion. 4 REFUTE: If you don t get a contradiction, construct a refutation box. LogiCola F (IE & IH) Pages 154 57

1 (B Ã A) 2 (B Ä A) [ Á À(A Ä ÀA) 3 asm: (A Ä ÀA) We re stuck! Here we get stuck using our old strategy so we need to make another assumption. 1 START: Assume the opposite of the conclusion. 2 S&I: Derive whatever you can using S- and I-rules. 3 RAA: If you get a contradiction, apply RAA and derive the original conclusion. 4 REFUTE: If you don t get a contradiction, construct a refutation box. LogiCola G (HV) Pages 161 68

1 (B Ã A) 2 (B Ä A) [ Á À(A Ä ÀA) 3 asm: (A Ä ÀA) We re stuck! We re stuck when: We can t apply S- or I-rules further. And we can t prove the argument VALID (since we have no contradiction) or INVALID (since we don t have enough simple wffs for a refutation). LogiCola G (HV) Pages 161 68

1 (B Ã A) 2 (B Ä A) [ Á À(A Ä ÀA) 3 asm: (A Ä ÀA) 4 asm: B {break up 1} When you re stuck, try to make another assumption. ASSUME: Look for a complex wff that isn t starred or blocked off or broken. This wff will have one of these forms: NOT-BOTH À(A Â B) OR (A Ã B) IF-THEN (A Ä B) Assume one side or its negation and then return to step 2 (S&I). LogiCola G (HV) Pages 161 68

1 (B Ã A) ** 2 (B Ä A) [ Á À(A Ä ÀA) ** 3 asm: (A Ä ÀA) 4 asm: B {break up 1} 5 Á A {from 2 and 4} 6 Á ÀA {from 3 and 5} Contradiction! S&I: Go through the complex wffs that aren t starred or blocked off and use these to derive new wffs using S- and I-rules. Star (with one star for each live assumption) any wff you simplify using an S-rule, or the longer wff used in an I-rule inference. LogiCola G (HV) Pages 161 68

1 (B Ã A) 2 (B Ä A) [ Á À(A Ä ÀA) 3 asm: (A Ä ÀA) 4 1 asm: B {break up 1} 5 2 Á A {from 2 and 4} 6 3 Á ÀA {from 3 and 5} 7 Á ÀB {from 4; 5 contradicts 6} Apply RAA. RAA: If you have a contradiction, apply RAA on the last live assumption. If all assumptions are now blocked off, you ve proved the argument valid. Otherwise, erase star strings having more stars than the number of live assumptions and then return to step 2 (S&I). LogiCola G (HV) Pages 161 68

* 1 (B Ã A) 2 (B Ä A) [ Á À(A Ä ÀA) * 3 1 asm: (A Ä ÀA) 4 2 1 asm: B {break up 1} 5 2 2 Á A {from 2 and 4} 6 2 3 Á ÀA {from 3 and 5} 7 2 Á ÀB {from 4; 5 contradicts 6} 8 2 Á A {from 1 and 7} 9 3 Á ÀA {from 3 and 8} 10 Á À(A Ä ÀA) {from 3; 8 contradicts 9} Valid We use ÀB to get a contradiction & finish the proof. LogiCola G (HV) Pages 161 68

* 1 (B Ã A) Valid 2 (B Ä A) [ Á À(A Ä ÀA) * 3 1 asm: (A Ä ÀA) 4 2 1 asm: B {break up 1} 5 2 2 Á A {from 2 and 4} 6 2 3 Á ÀA {from 3 and 5} 7 2 Á ÀB {from 4; 5 contradicts 6} 8 2 Á A {from 1 and 7} 9 3 Á ÀA {from 3 and 8} 10 Á À(A Ä ÀA) {from 3; 8 contradicts 9} Strategy: Start S&I RAA Assume Refute LogiCola G (HV) Pages 161 68

1 À(A Â B) [ Á (ÀA Â ÀB) 2 asm: À(ÀA Â ÀB) Assume opposite. Then we re stuck! We can t apply S- or I-rules or RAA; and we don t have enough simple wffs for a refutation. START: Assume the opposite of the conclusion. LogiCola G (HI & HC) Pages 170 71

1 À(A Â B) [ Á (ÀA Â ÀB) 2 asm: À(ÀA Â ÀB) 3 asm: A {break up 1} When you re stuck, try to make another assumption. ASSUME: Look for a complex wff that isn t starred or blocked off or broken. This wff will have one of these forms: NOT-BOTH À(A Â B) OR (A Ã B) IF-THEN (A Ä B) Assume one side or its negation and then return to step 2 (S&I). LogiCola G (HI & HC) Pages 170 71

** 1 À(A Â B) [ Á (ÀA Â ÀB) 2 asm: À(ÀA Â ÀB) 3 asm: A {break up 1} 4 Á ÀB {from 1 and 3} Derive further lines. We re stuck again! But now all complex wffs are either starred or blocked off or broken. S&I: Go through the complex wffs that aren t starred or blocked off and use these to derive new wffs using S- and I-rules. Star (with one star for each live assumption) any wff you simplify using an S-rule, or the longer wff used in an I-rule inference. LogiCola G (HI & HC) Pages 170 71

** 1 À(A 1 Â B 0 ) = 1 [ Á (ÀA 1 Â ÀB 0 ) = 0 2 asm: À(ÀA Â ÀB) 3 asm: A {break up 1} 4 Á ÀB {from 1 and 3} Invalid A, ÀB REFUTE: Construct a refutation box if you can t apply S- and I-rules or RAA further, and yet all complex wffs are either starred or blocked off or broken. LogiCola G (HI & HC) Pages 170 71

* 1 (B Ã A) Valid 2 (B Ä A) [ Á À(A Ä ÀA) * 3 1 asm: (A Ä ÀA) 4 2 1 asm: B {break up 1} 5 2 2 Á A {from 2 and 4} 6 2 3 Á ÀA {from 3 and 5} 7 2 Á ÀB {from 4; 5 contradicts 6} 8 2 Á A {from 1 and 7} 9 3 Á ÀA {from 3 and 8} 10 Á À(A Ä ÀA) {from 3; 8 contradicts 9} ** 1 À(A 1 Â B 0 ) = 1 Invalid [ Á (ÀA 1 Â ÀB 0 ) = 0 2 asm: À(ÀA Â ÀB) 3 asm: A {break up 1} 4 Á ÀB {from 1 and 3} A, ÀB Start S&I RAA Assume Refute LogiCola G (HI & HC) Pages 170 71

Traditional Copi proofs use eight inference rules and ten replacement rules. Here are the inference rules: AD Addition CJ Conjunction DI Dilemma DS Disjunctive Syllogism P (P Ã Q) P Q (P Â Q) ((P Ä Q) Â (R Ä S)) (P Ã R) (Q Ã S) (P Ã Q) ÀP Q HS Hypothetical Syllogism MP Modus Ponens MT Modus Tollens SP Simplification (P Ä Q) (Q Ä R) (P Ä R) (P Ä Q) P Q (P Ä Q) ÀQ ÀP (P Â Q) P LogiCola G (EO, HO, & MO) Pages 174 78

Here are the ten Copi replacement rules: AS CM DB DM Association Commutation Distribution De Morgan (P Ã (Q Ã R)) = ((P Ã Q) Ã R) (P Â (Q Â R)) = ((P Â Q) Â R) (P Ã Q) = (Q Ã P) (P Â Q) = (Q Â P) (P Â (Q Ã R)) = ((P Â Q) Ã (P Â R)) (P Ã (Q Â R)) = ((P Ã Q) Â (P Ã R)) À(P Â Q) = (ÀP Ã ÀQ)) À(P Ã Q) = (ÀP Â ÀQ) DN Double Negation P = ÀÀP EQ Equivalence (P Å Q) = ((P Ä Q) Â (Q Ä P)) (P Å Q) = ((P Â Q) Ã (ÀP Â ÀQ)) EX Exportation ((P Â Q) Ä R) = (P Ä (Q Ä R)) IM Implication (P Ä Q) = (ÀP Ã Q) RP Repetition P = (P Ã P) P = (P Â P) TR Transposition (P Ä Q) = (ÀQ Ä ÀP) LogiCola G (EO, HO, & MO) Pages 174 78

Conclusion: B 1 T 2 (T Ä (B Ã M)) 3 (M Ä H) 4 ÀH 5 (B Ã M) {MP 1+2} 6 ÀM {MT 3+4} 7 (M Ã B) {CM 5} 8 B {DS 6+7} Many Copi proofs directly derive the conclusion from the premises. Copi also provides for conditional and indirect (RAA) proofs. CP Conditional Proof RA Reductio ad Absurdum If you assume P and later derive Q, then you can star all the lines from P to Q [showing that you aren t to use them to derive further steps] and then derive (P Ä Q). If you assume P and later derive (Q Â ÀQ), then you can star all the lines from P to (Q Â ÀQ) [showing that you aren t to use them to derive further steps] and then derive ÀP. LogiCola G (EO, HO, & MO) Pages 174 78

Truth trees break formulas into the cases that make them true. Here s a truth tree for (A Â ÀB), (B Ã C) Á C : Conclusion: C (A Â ÀB) * (B Ã C) * ÀC A ÀB Valid: every branch closes B * C * An argument is valid if and only if every branch eventually closes (has a self-contradiction). LogiCola G (EZ, HZ, & MZ) Pages 178 81

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback