# Announcements. CS311H: Discrete Mathematics. First Order Logic, Rules of Inference. Satisfiability, Validity in FOL. Example.

Save this PDF as:

Size: px
Start display at page:

Download "Announcements. CS311H: Discrete Mathematics. First Order Logic, Rules of Inference. Satisfiability, Validity in FOL. Example."

## Transcription

1 Announcements CS311H: Discrete Mathematics First Order Logic, Rules of Inference Instructor: Işıl Dillig Homework 1 is due now! Homework 2 is handed out today Homework 2 is due next Wednesday Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 1/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 2/38 Satisfiability, Validity in FOL An FOL formula F is satisfiable if there exists some domain and some interpretation such that F evaluates to true Example: Prove that x.(p(x) Q(x)) is satisfiable. D = { }, P( ) = true, Q( ) = true Example Is the following formula valid, unsat, or contingent? Prove your answer. (() ( x.q(x))) ( x.(p(x) Q(x))) An FOL formula F is valid if, for all domains and all interpretations, F evaluates to true Prove that x.(p(x) Q(x)) is not valid. D = { }, P( ) = true, Q( ) = false Formulas that are satisfiable, but not valid are contingent, e.g., x.(p(x) Q(x)) Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 3/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 4/38 Equivalence Two formulas F 1 and F 2 are equivalent if F 1 F 2 is valid In PL, we could prove equivalence using truth tables, but not possible in FOL However, we can still use known equivalences to rewrite one formula as the other Example: Prove that ( x. (P(x) Q(x))) and x. (P(x) Q(x)) are equivalent. Example: Prove that x. y.p(x, y) and x. y. P(x, y) are equivalent. Rules of Inference We can prove validity in FOL by using proof rules Proof rules are written as rules of inference: An example inference rule: Hypothesis1 Hypothesis2... Conclusion All men are mortal Socrates is a man Socrates is mortal We ll learn about more general inference rules that will allow constructing formal proofs Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 5/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 6/38 1

2 Modus Ponens Example Uses of Modus Ponens Most basic inference rule is modus ponens: Modus ponens applicable to both propositional logic and first-order logic Application of modus ponens in propositional logic: p q (p q) r r Application of modus ponens in first-order logic: P(a) P(a) Q(b) Q(b) Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 7/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 8/38 Modus Tollens Example Uses of Modus Tollens Application of modus tollens in propositional logic: Second imporant inference rule is modus tollens: p (q r) (q r) p Application of modus tollens in first-order logic: Q(a) P(a) Q(a) P(a) Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 9/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 10/38 Hypothetical Syllogism (HS) Or Introduction and Elimination φ 3 φ 3 Basically says implication is transitive Example: P(a) Q(b) Q(b) R(c) P(a) R(c) Or introduction: Example application: Socrates is a man. Therefore, either Socrates is a man or there are red elephants on the moon. Or elimination: Example application: It is either a dog or a cat. It is not a dog. Therefore, it must be a cat. Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 11/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 12/38 2

3 And Introduction and Elimination And introduction: Example application: It is Tuesday. It s the afternoon. Therefore, it s Tuesday afternoon. And elimination: Example application: It is Tuesday afternoon. Therefore, it is Tuesday. Resolution Final inference rule: resolution φ 3 φ 3 To see why this is correct, observe is either true or false. Suppose is true. Then, is false. Therefore, by second hypothesis, φ 3 must be true. Suppose is false. Then, by 1st hypothesis, must be true. In any case, either or φ 3 must be true; φ 3 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 13/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 14/38 Resolution Example Example 1: Example 2: P(a) Q(b) Q(b) R(c) P(a) R(c) p q q p q Summary Name Rule of Inference Modus ponens Modus tollens Hypothetical syllogism φ 3 φ 3 Or introduction Or elimination And introduction And elimination Resolution φ 3 φ 3 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 15/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 16/38 Using the Rules of Inference Encoding in Logic Assume the following hypotheses: 1. It is not sunny today and it is colder than yesterday. First, encode hypotheses and conclusion as logical formulas. To do this, identify propositions used in the argument: 2. We will go to the lake only if it is sunny. 3. If we do not go to the lake, then we will go hiking. 4. If we go hiking, then we will be back by sunset. Show these lead to the conclusion: We will be back by sunset. s = It is sunny today c= It is colder than yesterday l = We ll go to the lake h = We ll go hiking b= We ll be back by sunset Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 17/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 18/38 3

4 Encoding in Logic, cont. Formal Proof Using Inference Rules It s not sunny today and colder than yesterday. s c We will go to the lake only if it is sunny l s If we do not go to the lake, then we will go hiking. l h If we go hiking, then we will be back by sunset. h b Conclusion: We ll be back by sunset b 1. s c Hypothesis 2. l s Hypothesis 3. l h Hypothesis 4. h b Hypothesis 5. s And Elim (1) 6. l Modus tollens (2,5) 7. l b Hypothetical syllogism (3,4) 8. b Modus ponens (6,7) Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 19/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 20/38 Another Example Encoding in Logic Assume the following hypotheses: 1. It is not raining or Kate has her umbrella 2. Kate does not have her umbrella or she does not get wet First, encode hypotheses and conclusion as logical formulas. To do this, identify propositions used in the argument: r = It is raining 3. It is raining or Kate does not get wet 4. Kate is grumpy only if she is wet Show these lead to the conclusion: Kate is not grumpy. u= Kate has her umbrella w = Kate is wet g = Kate is grumpy Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 21/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 22/38 Encoding in Logic, cont. Formal Proof Using Inference Rules It is not raining or Kate has her umbrella. r u Kate does not have her umbrella or she does not get wet u w It is raining or Kate does not get wet. r w Kate is grumpy only if she is wet. g w Conclusion: Kate is not grumpy. g 1. r u Hypothesis 2. u w Hypothesis 3. r w Hypothesis 4. g w Hypothesis 5. r w Resolution 1,2 6. w w Resolution 3,5 7. w Idempotence 8. g Modus tollens 4,7 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 23/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 24/38 4

5 Additional Inference Rules for Quantified Formulas Universal Instantiation Inference rules we learned so far are sufficient for reasoning about quantifier-free statements Four more inference rules for making deductions from quantified formulas These come in pairs for each quantifier (universal/existential) One is called generalization, the other one called instantiation If we know something is true for all members of a group, we can conclude it is also true for a specific member of this group This idea is formally called universal instantiation: (for any c) If we know All CS classes at UT are hard, universal instantiation allows us to conclude CS311 is hard! Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 25/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 26/38 Example Universal Generalization Consider predicates man(x) and mortal(x) and the hypotheses: 1. All men are mortal: x.(man(x) mortal(x)) 2. Socrates is a man: man(socrates) Using rules of inference, prove mortal(socrates) 3. man(socrates) mortal(socrates) -inst. (1) 4. mortal(socrates) Modus ponens (1), (3) Suppose we can prove a claim for an arbitrary element in the domain. Since we ve made no assumptions about this element, proof should apply to all elements in the domain. This correct reasoning is captured by universal generalization for arbitrary c Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 27/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 28/38 Example Caveat About Universal Generalization Prove x.q(x) from the hypotheses: 1. x. (P(x) Q(x)) Hypothesis 2. x. P(x) Hypothesis 3. Q(c) -inst (1) 4. -inst (2) When using universal generalization, need to ensure that c is truly arbitrary! If you prove something about a specific person Mary, you cannot make generalizations about all people 5. Q(c) Modus ponens (3), (4) 6. x.q(x) -gen (5) Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 29/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 30/38 5

6 Existential Instantiation Example Using Existential Instantiation Consider formula. We know there is some element, say c, in the domain for which is true. This is called existential instantiation: (for unused c) Here, c is a fresh name (i.e., not used before in proof). Otherwise, can prove non-sensical things such as: There exists some animal that can fly. Thus, rabbits can fly! Consider the hypotheses and x. P(x). Prove that we can derive a contradiction (i.e., false) from these hypotheses. 1. Hypothesis 2. x. P(x) Hypothesis 3. instantiation 1, c fresh 4. instantiation intro 6. false Negation law Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 31/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 32/38 Existential Generalization Example Using Existential Generalization Suppose we know is true for some constant c Then, there exists an element for which P is true Thus, we can conlude This inference rule called existential generalization: Consider the hypotheses atut (George) and smart(george). Prove x. (atut (x) smart(x)) 1. atut (George) Hypothesis 2. smart(george) Hypothesis 3. atut (George) smart(george) intro, 1,2 4. x.(atut (x) smart(x)) generalization, 3 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 33/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 34/38 Summary of Inference Rules for Quantifiers Example I Prove that these hypotheses imply x.(p(x) B(x)): Name Universal Instantiation Universal Generalization Existential Instantiation Existential Generalization Rule of Inference (anyc) (for arbitraryc) for fresh c 1. x. (C (x) B(x)) (Hypothesis) 2. x. (C (x) P(x)) (Hypothesis) 3. C (a) B(a) ( -inst, 1) 4. C (a) ( -elim, 3 ) 5. B(a) ( -elim, 3 ) 6. C (a) P(a) ( -inst, 2) 7. P(a) (Modus ponens, 4, 6) 8. P(a) B(a) ( -intro, 5,7) 9. x.(p(x) B(x)) ( -gen, 8) Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 35/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 36/38 6

7 Example II Prove the below hypotheses are contradictory by deriving false 1. x.(p(x) (Q(x) S(x))) (Hypothesis) 2. x.(p(x) R(x)) (Hypothesis) 3. x.( R(x) S(x)) (Hypothesis) 4. R(a) S(a) ( -inst, 3) 5. P(a) R(a) ( -inst, 2) Example III Prove x. father(x, Evan) from the following premises: 1. x. y. ((parent(x, y) male(x)) father(x, y)) 2. parent(tom, Evan) 3. male(tom) 6. P(a) ( -elim, 5) 7. R(a) ( -elim, 5) 8. S(a) (Resolution, 4, 7) 9. P(a) Q(a) S(a) ( -inst, 1) 10. Q(a) S(a) (Modus ponens, 6, 9) 11. S(a) -elim, S(a) S(a) false ( -intro, 8, 11; double negation) Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 37/38 Instructor: Işıl Dillig, CS311H: Discrete Mathematics First Order Logic, Rules of Inference 38/38 7

### Announcements. CS243: Discrete Structures. First Order Logic, Rules of Inference. Review of Last Lecture. Translating English into First-Order Logic

Announcements CS243: Discrete Structures First Order Logic, Rules of Inference Işıl Dillig Homework 1 is due now Homework 2 is handed out today Homework 2 is due next Tuesday Işıl Dillig, CS243: Discrete

### Revisiting the Socrates Example

Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments Rules of Inference for Quantified Statements Building Arguments for Quantified

### Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing February, 2018

Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing February, 2018 Alice E. Fischer... 1/28 1 Examples and Varieties Order of Quantifiers and Negations 2 3 Universal Existential 4 Universal Modus

### Selections from Aristotle s Prior Analytics 41a21 41b5

Lesson Seventeen The Conditional Syllogism Selections from Aristotle s Prior Analytics 41a21 41b5 It is clear then that the ostensive syllogisms are effected by means of the aforesaid figures; these considerations

### HOW TO ANALYZE AN ARGUMENT

What does it mean to provide an argument for a statement? To provide an argument for a statement is an activity we carry out both in our everyday lives and within the sciences. We provide arguments for

### In this section you will learn three basic aspects of logic. When you are done, you will understand the following:

Basic Principles of Deductive Logic Part One: In this section you will learn three basic aspects of logic. When you are done, you will understand the following: Mental Act Simple Apprehension Judgment

### 10.3 Universal and Existential Quantifiers

M10_COPI1396_13_SE_C10.QXD 10/22/07 8:42 AM Page 441 10.3 Universal and Existential Quantifiers 441 and Wx, and so on. We call these propositional functions simple predicates, to distinguish them from

### Foundations of Non-Monotonic Reasoning

Foundations of Non-Monotonic Reasoning Notation S A - from a set of premisses S we can derive a conclusion A. Example S: All men are mortal Socrates is a man. A: Socrates is mortal. x.man(x) mortal(x)

### Exercise Sets. KS Philosophical Logic: Modality, Conditionals Vagueness. Dirk Kindermann University of Graz July 2014

Exercise Sets KS Philosophical Logic: Modality, Conditionals Vagueness Dirk Kindermann University of Graz July 2014 1 Exercise Set 1 Propositional and Predicate Logic 1. Use Definition 1.1 (Handout I Propositional

### Logic: A Brief Introduction. Ronald L. Hall, Stetson University

Logic: A Brief Introduction Ronald L. Hall, Stetson University 2012 CONTENTS Part I Critical Thinking Chapter 1 Basic Training 1.1 Introduction 1.2 Logic, Propositions and Arguments 1.3 Deduction and Induction

### Chapter 1. Introduction. 1.1 Deductive and Plausible Reasoning Strong Syllogism

Contents 1 Introduction 3 1.1 Deductive and Plausible Reasoning................... 3 1.1.1 Strong Syllogism......................... 3 1.1.2 Weak Syllogism.......................... 4 1.1.3 Transitivity

### A BRIEF INTRODUCTION TO LOGIC FOR METAPHYSICIANS

A BRIEF INTRODUCTION TO LOGIC FOR METAPHYSICIANS 0. Logic, Probability, and Formal Structure Logic is often divided into two distinct areas, inductive logic and deductive logic. Inductive logic is concerned

### Essential Logic Ronald C. Pine

Essential Logic Ronald C. Pine Chapter 11: Other Logical Tools Syllogisms and Quantification Introduction A persistent theme of this book has been the interpretation of logic as a set of practical tools.

### Chapter 8 - Sentential Truth Tables and Argument Forms

Logic: A Brief Introduction Ronald L. Hall Stetson University Chapter 8 - Sentential ruth ables and Argument orms 8.1 Introduction he truth-value of a given truth-functional compound proposition depends

### Lecture 17:Inference Michael Fourman

Lecture 17:Inference Michael Fourman 2 Is this a valid argument? Assumptions: If the races are fixed or the gambling houses are crooked, then the tourist trade will decline. If the tourist trade declines

### What is the Nature of Logic? Judy Pelham Philosophy, York University, Canada July 16, 2013 Pan-Hellenic Logic Symposium Athens, Greece

What is the Nature of Logic? Judy Pelham Philosophy, York University, Canada July 16, 2013 Pan-Hellenic Logic Symposium Athens, Greece Outline of this Talk 1. What is the nature of logic? Some history

### 4.1 A problem with semantic demonstrations of validity

4. Proofs 4.1 A problem with semantic demonstrations of validity Given that we can test an argument for validity, it might seem that we have a fully developed system to study arguments. However, there

### PHILOSOPHY 102 INTRODUCTION TO LOGIC PRACTICE EXAM 1. W# Section (10 or 11) 4. T F The statements that compose a disjunction are called conjuncts.

PHILOSOPHY 102 INTRODUCTION TO LOGIC PRACTICE EXAM 1 W# Section (10 or 11) 1. True or False (5 points) Directions: Circle the letter next to the best answer. 1. T F All true statements are valid. 2. T

### Part II: How to Evaluate Deductive Arguments

Part II: How to Evaluate Deductive Arguments Week 4: Propositional Logic and Truth Tables Lecture 4.1: Introduction to deductive logic Deductive arguments = presented as being valid, and successful only

### MCQ IN TRADITIONAL LOGIC. 1. Logic is the science of A) Thought. B) Beauty. C) Mind. D) Goodness

MCQ IN TRADITIONAL LOGIC FOR PRIVATE REGISTRATION TO BA PHILOSOPHY PROGRAMME 1. Logic is the science of-----------. A) Thought B) Beauty C) Mind D) Goodness 2. Aesthetics is the science of ------------.

### A Judgmental Formulation of Modal Logic

A Judgmental Formulation of Modal Logic Sungwoo Park Pohang University of Science and Technology South Korea Estonian Theory Days Jan 30, 2009 Outline Study of logic Model theory vs Proof theory Classical

### Elements of Science (cont.); Conditional Statements. Phil 12: Logic and Decision Making Fall 2010 UC San Diego 9/29/2010

Elements of Science (cont.); Conditional Statements Phil 12: Logic and Decision Making Fall 2010 UC San Diego 9/29/2010 1 Why cover statements and arguments Decision making (whether in science or elsewhere)

### Tutorial A03: Patterns of Valid Arguments By: Jonathan Chan

A03.1 Introduction Tutorial A03: Patterns of Valid Arguments By: With valid arguments, it is impossible to have a false conclusion if the premises are all true. Obviously valid arguments play a very important

### CRITICAL THINKING (CT) MODEL PART 1 GENERAL CONCEPTS

Fall 2001 ENGLISH 20 Professor Tanaka CRITICAL THINKING (CT) MODEL PART 1 GENERAL CONCEPTS In this first handout, I would like to simply give you the basic outlines of our critical thinking model

### Venn Diagrams and Categorical Syllogisms. Unit 5

Venn Diagrams and Categorical Syllogisms Unit 5 John Venn 1834 1923 English logician and philosopher noted for introducing the Venn diagram Used in set theory, probability, logic, statistics, and computer

What is an argument? PHIL 110 Lecture on Chapter 3 of How to think about weird things An argument is a collection of two or more claims, one of which is the conclusion and the rest of which are the premises.

### Session 10 INDUCTIVE REASONONING IN THE SCIENCES & EVERYDAY LIFE( PART 1)

UGRC 150 CRITICAL THINKING & PRACTICAL REASONING Session 10 INDUCTIVE REASONONING IN THE SCIENCES & EVERYDAY LIFE( PART 1) Lecturer: Dr. Mohammed Majeed, Dept. of Philosophy & Classics, UG Contact Information:

### Chapter 9- Sentential Proofs

Logic: A Brief Introduction Ronald L. Hall, Stetson University Chapter 9- Sentential roofs 9.1 Introduction So far we have introduced three ways of assessing the validity of truth-functional arguments.

### Relevance. Premises are relevant to the conclusion when the truth of the premises provide some evidence that the conclusion is true

Relevance Premises are relevant to the conclusion when the truth of the premises provide some evidence that the conclusion is true Premises are irrelevant when they do not 1 Non Sequitur Latin for it does

### UC Berkeley, Philosophy 142, Spring 2016

Logical Consequence UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Intuitive characterizations of consequence Modal: It is necessary (or apriori) that, if the premises are true, the conclusion

### The basic form of a syllogism By Timo Schmitz, Philosopher

The basic form of a syllogism By Timo Schmitz, Philosopher In my article What is logic? (02 April 2017), I pointed out that an apophantic sentence is always a proposition. To find out whether the formal

### Philosophical Arguments

Philosophical Arguments An introduction to logic and philosophical reasoning. Nathan D. Smith, PhD. Houston Community College Nathan D. Smith. Some rights reserved You are free to copy this book, to distribute

### Basic Concepts and Skills!

Basic Concepts and Skills! Critical Thinking tests rationales,! i.e., reasons connected to conclusions by justifying or explaining principles! Why do CT?! Answer: Opinions without logical or evidential

### Chapter 3: Basic Propositional Logic. Based on Harry Gensler s book For CS2209A/B By Dr. Charles Ling;

Chapter 3: Basic Propositional Logic Based on Harry Gensler s book For CS2209A/B By Dr. Charles Ling; cling@csd.uwo.ca The Ultimate Goals Accepting premises (as true), is the conclusion (always) true?

### Logic and Argument Analysis: An Introduction to Formal Logic and Philosophic Method (REVISED)

Carnegie Mellon University Research Showcase @ CMU Department of Philosophy Dietrich College of Humanities and Social Sciences 1985 Logic and Argument Analysis: An Introduction to Formal Logic and Philosophic

### 1. Introduction Formal deductive logic Overview

1. Introduction 1.1. Formal deductive logic 1.1.0. Overview In this course we will study reasoning, but we will study only certain aspects of reasoning and study them only from one perspective. The special

### Introduction to Logic

University of Notre Dame Fall, 2015 Arguments Philosophy is difficult. If questions are easy to decide, they usually don t end up in philosophy The easiest way to proceed on difficult questions is to formulate

### HANDBOOK (New or substantially modified material appears in boxes.)

1 HANDBOOK (New or substantially modified material appears in boxes.) I. ARGUMENT RECOGNITION Important Concepts An argument is a unit of reasoning that attempts to prove that a certain idea is true by

### Suppressed premises in real life. Philosophy and Logic Section 4.3 & Some Exercises

Suppressed premises in real life Philosophy and Logic Section 4.3 & Some Exercises Analyzing inferences: finale Suppressed premises: from mechanical solutions to elegant ones Practicing on some real-life

### The Problem of Induction and Popper s Deductivism

The Problem of Induction and Popper s Deductivism Issues: I. Problem of Induction II. Popper s rejection of induction III. Salmon s critique of deductivism 2 I. The problem of induction 1. Inductive vs.

### 9 Methods of Deduction

M09_COPI1396_13_SE_C09.QXD 10/19/07 3:46 AM Page 372 9 Methods of Deduction 9.1 Formal Proof of Validity 9.2 The Elementary Valid Argument Forms 9.3 Formal Proofs of Validity Exhibited 9.4 Constructing

### Unit. Categorical Syllogism. What is a syllogism? Types of Syllogism

Unit 8 Categorical yllogism What is a syllogism? Inference or reasoning is the process of passing from one or more propositions to another with some justification. This inference when expressed in language

### b) The meaning of "child" would need to be taken in the sense of age, as most people would find the idea of a young child going to jail as wrong.

Explanation for Question 1 in Quiz 8 by Norva Lo - Tuesday, 18 September 2012, 9:39 AM The following is the solution for Question 1 in Quiz 8: (a) Which term in the argument is being equivocated. (b) What

### Characterizing the distinction between the logical and non-logical

Aporia vol. 27 no. 1 2017 The Nature of Logical Constants Lauren Richardson Characterizing the distinction between the logical and non-logical expressions of a language proves a challenging task, and one

### Inference in Cyc. Copyright 2002 Cycorp

Inference in Cyc Logical Aspects of Inference Incompleteness in Searching Incompleteness from Resource Bounds and Continuable Searches Efficiency through Heuristics Inference Features in Cyc We ll be talking

### What is a logical argument? What is deductive reasoning? Fundamentals of Academic Writing

What is a logical argument? What is deductive reasoning? Fundamentals of Academic Writing Logical relations Deductive logic Claims to provide conclusive support for the truth of a conclusion Inductive

### A short introduction to formal logic

A short introduction to formal logic Dan Hicks v0.3.2, July 20, 2012 Thanks to Tim Pawl and my Fall 2011 Intro to Philosophy students for feedback on earlier versions. My approach to teaching logic has

### Name: Course: CAP 4601 Semester: Summer 2013 Assignment: Assignment 06 Date: 08 JUL Complete the following written problems:

Name: Course: CAP 4601 Semester: Summer 2013 Assignment: Assignment 06 Date: 08 JUL 2013 Complete the following written problems: 1. Alpha-Beta Pruning (40 Points). Consider the following min-max tree.

### INTERMEDIATE LOGIC Glossary of key terms

1 GLOSSARY INTERMEDIATE LOGIC BY JAMES B. NANCE INTERMEDIATE LOGIC Glossary of key terms This glossary includes terms that are defined in the text in the lesson and on the page noted. It does not include

### Quantificational logic and empty names

Quantificational logic and empty names Andrew Bacon 26th of March 2013 1 A Puzzle For Classical Quantificational Theory Empty Names: Consider the sentence 1. There is something identical to Pegasus On

### TWO VERSIONS OF HUME S LAW

DISCUSSION NOTE BY CAMPBELL BROWN JOURNAL OF ETHICS & SOCIAL PHILOSOPHY DISCUSSION NOTE MAY 2015 URL: WWW.JESP.ORG COPYRIGHT CAMPBELL BROWN 2015 Two Versions of Hume s Law MORAL CONCLUSIONS CANNOT VALIDLY

### An alternative understanding of interpretations: Incompatibility Semantics

An alternative understanding of interpretations: Incompatibility Semantics 1. In traditional (truth-theoretic) semantics, interpretations serve to specify when statements are true and when they are false.

### PHIL 115: Philosophical Anthropology. I. Propositional Forms (in Stoic Logic) Lecture #4: Stoic Logic

HIL 115: hilosophical Anthropology Lecture #4: Stoic Logic Arguments from the Euthyphro: Meletus Argument (according to Socrates) [3a-b] Argument: Socrates is a maker of gods; so, Socrates corrupts the

### Logic Dictionary Keith Burgess-Jackson 12 August 2017

Logic Dictionary Keith Burgess-Jackson 12 August 2017 addition (Add). In propositional logic, a rule of inference (i.e., an elementary valid argument form) in which (1) the conclusion is a disjunction

### PHILOSOPHER S TOOL KIT 1. ARGUMENTS PROFESSOR JULIE YOO 1.1 DEDUCTIVE VS INDUCTIVE ARGUMENTS

PHILOSOPHER S TOOL KIT PROFESSOR JULIE YOO 1. Arguments 1.1 Deductive vs Induction Arguments 1.2 Common Deductive Argument Forms 1.3 Common Inductive Argument Forms 1.4 Deduction: Validity and Soundness

### A Solution to the Gettier Problem Keota Fields. the three traditional conditions for knowledge, have been discussed extensively in the

A Solution to the Gettier Problem Keota Fields Problem cases by Edmund Gettier 1 and others 2, intended to undermine the sufficiency of the three traditional conditions for knowledge, have been discussed

### The Little Logic Book Hardy, Ratzsch, Konyndyk De Young and Mellema The Calvin College Press, 2013

The Little Logic Book Hardy, Ratzsch, Konyndyk De Young and Mellema The Calvin College Press, 2013 Exercises for The Little Logic Book may be downloaded by the instructor as Word documents and then modified

### Artificial Intelligence I

Artificial Intelligence I Matthew Huntbach, Dept of Computer Science, Queen Mary and Westfield College, London, UK E 4NS. Email: mmh@dcs.qmw.ac.uk. Notes may be used with the permission of the author.

### Faith indeed tells what the senses do not tell, but not the contrary of what they see. It is above them and not contrary to them.

19 Chapter 3 19 CHAPTER 3: Logic Faith indeed tells what the senses do not tell, but not the contrary of what they see. It is above them and not contrary to them. The last proceeding of reason is to recognize

### What are Truth-Tables and What Are They For?

PY114: Work Obscenely Hard Week 9 (Meeting 7) 30 November, 2010 What are Truth-Tables and What Are They For? 0. Business Matters: The last marked homework of term will be due on Monday, 6 December, at

### Theories of propositions

Theories of propositions phil 93515 Jeff Speaks January 16, 2007 1 Commitment to propositions.......................... 1 2 A Fregean theory of reference.......................... 2 3 Three theories of

### Class #37 - Translation Using Identity I ( 8.7)

Class #37 - Translation Using Identity I ( 8.7) I. The identity predicate is a special predicate, with a special logic Consider the following logical derivation: 1. Superman can fly. Fs 2. Superman is

### I Don't Believe in God I Believe in Science

I Don't Believe in God I Believe in Science This seems to be a common world view that many people hold today. It is important that when we look at statements like this we spend a proper amount of time

### From Necessary Truth to Necessary Existence

Prequel for Section 4.2 of Defending the Correspondence Theory Published by PJP VII, 1 From Necessary Truth to Necessary Existence Abstract I introduce new details in an argument for necessarily existing

### Leibniz, Principles, and Truth 1

Leibniz, Principles, and Truth 1 Leibniz was a man of principles. 2 Throughout his writings, one finds repeated assertions that his view is developed according to certain fundamental principles. Attempting

### Logic: Deductive and Inductive by Carveth Read M.A. CHAPTER IX CHAPTER IX FORMAL CONDITIONS OF MEDIATE INFERENCE

CHAPTER IX CHAPTER IX FORMAL CONDITIONS OF MEDIATE INFERENCE Section 1. A Mediate Inference is a proposition that depends for proof upon two or more other propositions, so connected together by one or

### Logic: A Brief Introduction

Logic: A Brief Introduction Ronald L. Hall, Stetson University PART III - Symbolic Logic Chapter 7 - Sentential Propositions 7.1 Introduction What has been made abundantly clear in the previous discussion

### ISSA Proceedings 1998 Wilson On Circular Arguments

ISSA Proceedings 1998 Wilson On Circular Arguments 1. Introduction In his paper Circular Arguments Kent Wilson (1988) argues that any account of the fallacy of begging the question based on epistemic conditions

### Possibility and Necessity

Possibility and Necessity 1. Modality: Modality is the study of possibility and necessity. These concepts are intuitive enough. Possibility: Some things could have been different. For instance, I could

### Ayer and Quine on the a priori

Ayer and Quine on the a priori November 23, 2004 1 The problem of a priori knowledge Ayer s book is a defense of a thoroughgoing empiricism, not only about what is required for a belief to be justified

### Denying the antecedent and conditional perfection again

University of Windsor Scholarship at UWindsor OSSA Conference Archive OSSA 10 May 22nd, 9:00 AM - May 25th, 5:00 PM Denying the antecedent and conditional perfection again Andrei Moldovan University of

### Logical Constants as Punctuation Marks

362 Notre Dame Journal of Formal Logic Volume 30, Number 3, Summer 1989 Logical Constants as Punctuation Marks KOSTA DOSEN* Abstract This paper presents a proof-theoretical approach to the question "What

### Based on the translation by E. M. Edghill, with minor emendations by Daniel Kolak.

On Interpretation By Aristotle Based on the translation by E. M. Edghill, with minor emendations by Daniel Kolak. First we must define the terms 'noun' and 'verb', then the terms 'denial' and 'affirmation',

### Instrumental reasoning* John Broome

Instrumental reasoning* John Broome For: Rationality, Rules and Structure, edited by Julian Nida-Rümelin and Wolfgang Spohn, Kluwer. * This paper was written while I was a visiting fellow at the Swedish

### Confirmation Gary Hardegree Department of Philosophy University of Massachusetts Amherst, MA 01003

Confirmation Gary Hardegree Department of Philosophy University of Massachusetts Amherst, MA 01003 1. Hypothesis Testing...1 2. Hempel s Paradox of Confirmation...5 3. How to Deal with a Paradox...6 1.

### The view can concede that there are principled necessary conditions or principled sufficient conditions, or both; just no principled dichotomy.

Pluralism in Logic Hartry Field New York University Abstract: A number of people have proposed that we should be pluralists about logic, but there are a number of things this can mean. Are there versions

### Negative Facts. Negative Facts Kyle Spoor

54 Kyle Spoor Logical Atomism was a view held by many philosophers; Bertrand Russell among them. This theory held that language consists of logical parts which are simplifiable until they can no longer

### Vagueness and supervaluations

Vagueness and supervaluations UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Supervaluations We saw two problems with the three-valued approach: 1. sharp boundaries 2. counterintuitive consequences

### 3.3. Negations as premises Overview

3.3. Negations as premises 3.3.0. Overview A second group of rules for negation interchanges the roles of an affirmative sentence and its negation. 3.3.1. Indirect proof The basic principles for negation

### 10.7 Asyllogistic Inference

M10_COPI1396_13_SE_C10.QXD 10/22/07 8:42 AM Page 468 468 CHAPTER 10 Quantification Theory 8. None but the brave deserve the fair. Every soldier is brave. Therefore none but soldiers deserve the fair. (Dx:

### Dispositionalism and the Modal Operators

Philosophy and Phenomenological Research Philosophy and Phenomenological Research doi: 10.1111/phpr.12132 2014 Philosophy and Phenomenological Research, LLC Dispositionalism and the Modal Operators DAVID

### 6: DEDUCTIVE LOGIC. Chapter 17: Deductive validity and invalidity Ben Bayer Drafted April 25, 2010 Revised August 23, 2010

6: DEDUCTIVE LOGIC Chapter 17: Deductive validity and invalidity Ben Bayer Drafted April 25, 2010 Revised August 23, 2010 Deduction vs. induction reviewed In chapter 14, we spent a fair amount of time

### Philosophy 12 Study Guide #4 Ch. 2, Sections IV.iii VI

Philosophy 12 Study Guide #4 Ch. 2, Sections IV.iii VI Precising definition Theoretical definition Persuasive definition Syntactic definition Operational definition 1. Are questions about defining a phrase

### A Brief History of Thinking about Thinking Thomas Lombardo

A Brief History of Thinking about Thinking Thomas Lombardo "Education is nothing more nor less than learning to think." Peter Facione In this article I review the historical evolution of principles and

### Deflationary Nominalism s Commitment to Meinongianism

Res Cogitans Volume 7 Issue 1 Article 8 6-24-2016 Deflationary Nominalism s Commitment to Meinongianism Anthony Nguyen Reed College Follow this and additional works at: http://commons.pacificu.edu/rescogitans

### Unit 4. Reason as a way of knowing. Tuesday, March 4, 14

Unit 4 Reason as a way of knowing I. Reasoning At its core, reasoning is using what is known as building blocks to create new knowledge I use the words logic and reasoning interchangeably. Technically,

### Statistical Syllogistic, Part 1

University of Windsor Scholarship at UWindsor OSSA Conference Archive OSSA 4 May 17th, 9:00 AM - May 19th, 5:00 PM Statistical Syllogistic, Part 1 Lawrence H. Powers Follow this and additional works at:

### Practice Test Three Fall True or False True = A, False = B

Practice Test Three Fall 2015 True or False True = A, False = B 1. The inclusive "or" means "A or B or both A and B." 2. The conclusion contains both the major term and the middle term. 3. "If, then" statements

### assertoric, and apodeictic and gives an account of these modalities. It is tempting to

Kant s Modalities of Judgment Jessica Leech Abstract This paper proposes a way to understand Kant's modalities of judgment problematic, assertoric, and apodeictic in terms of the location of a judgment

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Exam Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Draw a Venn diagram for the given sets. In words, explain why you drew one set as a subset of

### Russell: On Denoting

Russell: On Denoting DENOTING PHRASES Russell includes all kinds of quantified subject phrases ( a man, every man, some man etc.) but his main interest is in definite descriptions: the present King of

### FACULTY OF ARTS B.A. Part II Examination,

FACULTY OF ARTS B.A. Part II Examination, 2015-16 8. PHILOSOPHY SCHEME Two Papers Min. pass marks 72 Max. Marks 200 Paper - I 3 hrs duration 100 Marks Paper - II 3 hrs duration 100 Marks PAPER - I: HISTORY

### Pragmatic Considerations in the Interpretation of Denying the Antecedent

University of Windsor Scholarship at UWindsor OSSA Conference Archive OSSA 8 Jun 3rd, 9:00 AM - Jun 6th, 5:00 PM Pragmatic Considerations in the Interpretation of Denying the Antecedent Andrei Moldovan

### Ethical Consistency and the Logic of Ought

Ethical Consistency and the Logic of Ought Mathieu Beirlaen Ghent University In Ethical Consistency, Bernard Williams vindicated the possibility of moral conflicts; he proposed to consistently allow for

### 1/6. The Resolution of the Antinomies

1/6 The Resolution of the Antinomies Kant provides us with the resolutions of the antinomies in order, starting with the first and ending with the fourth. The first antinomy, as we recall, concerned the

### The Philosopher s World Cup

The Philosopher s World Cup Monty Python & the Flying Circus http://www.youtube.com/watch?v=92vv3qgagck&feature=related What is an argument? http://www.youtube.com/watch?v=kqfkti6gn9y What is an argument?

### 1. To arrive at the truth we have to reason correctly. 2. Logic is the study of correct reasoning. B. DEDUCTIVE AND INDUCTIVE ARGUMENTS

I. LOGIC AND ARGUMENTATION 1 A. LOGIC 1. To arrive at the truth we have to reason correctly. 2. Logic is the study of correct reasoning. 3. It doesn t attempt to determine how people in fact reason. 4.