Part II: How to Evaluate Deductive Arguments


 Miles Julius Ball
 9 months ago
 Views:
Transcription
1 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 if they are valid. By comparison: inductive arguments are presented as being strong (not necc. valid) and are successful to the extent they are strong. Remember this example from 3.1: Mary has a child who is pregnant. Only daughters can become pregnant. Mary has at least one daughter Other arguments may use the same form but differ in subject matter: Terry has a job in which she arrests people. Only police officers arrest people. At least one of Terry s jobs is as a police officer. Both arguments are valid it s impossible to imagine that both premises are true and the conclusion false. Forms of valid argument Quantifiers: only, at least, some, all, none, etc. we ll go into these in next week s categorical logic lessons Propositional connectives (partic. truth functional connectives), which we ll discuss this week. Lecture 4.2: Propositions and propositional connectives propositional logic the rules that det. the validity of an argument based on the propositional connectives used within it. What is a proposition? Something that can be either true or false, and that can be the premise or conclusion of an argument. Proposition: The binoculars are in my hand. Not propositions: Binoculars, a hand What is a propositional connective? Something that takes propositions and makes new propositions out of them. Example: and I am holding the binoculars and looking through them.
2 Lecture 4.3: And and truthfunctional connectives And doesn t always function as a propositional connective. Jack and Jill finally talked and could serve three functions, giving sentence 3 meanings: Joins two names to make a complex name ( The fast food company Jack and Jill finally talked. as in, its spokesman finally held a news conference.) Propositional connective that joins two propositions to make a complex proposition made of just those two propositions. ( Each of two contestants, Jack and Jill, finally talked. they re in a silence contest with a third party and I m relaying results: Jack finally talked and Jill finally talked.) Expresses a proposition that involves something over and above Jack s finally talking and JIll s finally talking not just a propositional connective. ( Two people, Jack and Jill, finally talked to each other. ) Here, we ll focus on and as propositional connective we ll look at other words used that way such as or, not, but, only, if, etc. later on. And can be used in diff. ways, even as a propositional connective. I took a shower and got dressed conveys sense of temporal order; propositional connective. I am holding the binoculars and looking through them does not convey sense of temporal order; here, it s not just a PC but also a truth functional connective. Truth functional connective propositional connective that makes a new proposition whose truth or falsity depends solely on the truth or falsity of its propositional ingredients. conjunction a truth functional connective expressed with the use of and Examples of truth functional connectives: As long as it s true that Jack finally talked and Jill finally talked, it s true that Jack and Jill finally talked. Jack finally talked Jill finally talked Jack and Jill finally talked True True True True False False False True False False False False Some arguments are valid because of the truth functional connectives they use. Lecture 4.4: Using truth tables to show validity What is a truth table?
3 Device that can be used to explain how a truth functional connector or a truth functional operator works. (For example, at the end of the last lesson.) Conjunction introduction inference introduces a conjunction that wasn t present in either of the inference s two premises: P Q P & Q Truth table tells you that in any situation in which both premises are true, the conclusion must be true. That means conjunction introduction inferences are all going to be valid. P Q P & Q True True True True False False False True False False False False Conjunction elimination conclusion of argument eliminates a conjunction in its premise P&Q P&Q P Q Truth table for conjunctions (above) still applies therefore, all conjunction elimination arguments are valid, regardless of what they re about. Lecture 4.5: Rules, variables and generality So, why do we use variables when we state rules? Rules are general Designed to apply to many different possible cases RULE: You should never under any circumstances hit another person. Use of pronoun you and never under any circumstances indicate this rule applies to a multitude of possible situations. RULE: You should never hit your brother. NOT A RULE: Stop hitting your brother right now! Use of right now indicates that this command applies to a particular situation. Rule could get even more general never hit another person never... do violence to another person never... do something unkind to another person Another example: Walter should not force his dog to kill his cat some generality to it
4 More general: People should not force creatures to kill innocent creatures. Last sentence is unclear: how should creatures be understood? Plural understanding: People should not force a whole bunch of creatures to kill innocent creatures. Distributive understanding: For any creature you pick, people should not force that creature to kill an innocent creature. So we clarify: Where x is any creature, and y is any innocent creature: People should not force x to kill y. Using variables here helps us express the distributive rule precisely and clearly. Variables are equally useful in propositional logic. I m holding my binoculars and looking through them. I m holding my binoculars. P & Q P For any argument where the premise is two propositions, and the conclusion is one of those conjoined propositions, that argument is also going to be valid. Notation Variables are arbitrary we ve used Roman letters, but anything else that s easy to produce and recognize can be used. (Symbols may not be easy to recognize as variables because they re commonly used to express other things.) Notation for truth functions (also arbitrary, but here s what we ll use) Conjunction: & Disjunction: (not entirely unlike a letter V) Negation: or or ~ Lecture 4.6: Disjunction Disjunction truth functional connective usually expressed with the inclusive use of or Either Manchester won it or Barcelona won it. exclusive use; here, they cannot both win. This is breakfast or lunch. inclusive use; here, it could be either or both. Truth table for disjunctions P Q P V Q true true true true false true false true true false false false
5 If P and/or Q is true, then their disjunction (P V Q) is true. If both P and Q are false, then their disjunction is false. Disjunction introduction argument: P Q P V Q P P V Q Any argument of either form is valid. Disjunction elimination argument: P V Q P I m going to tickle you with my right or left hand. I m going to tickle you with my right hand. An argument of this form need not be valid. (For one thing, I could be tickling you with my left hand.) Lecture 4.7: Combining conjunctions and disjunctions They can be used together to string a bunch of propositions into a larger proposition. This example strings together three propositions: I m going to tickle you with hand #1 & I m either going to tickle you with hand #2 or hand #3. (Each hand is a proposition) Truth table for conjoining a disjunction I m going to tickle you with hand #1 I m going to tickle you with hand #2 I m going to tickle you with hand #3 I m going to tickle you with either hand #2 or hand #3 I m going to tickle you with hand #1, and with either hand #2 or hand #3 true true true true true true true false true true true false true true true true false false false false false true true true false false true false true false
6 false false true true false false false false false false The disjunction can only be true if one part is true (either hand #2 or #3). The full proposition can only be true if the hand #1 part is true and the disjunction is true. Conjunction and disjunction are not associative When we use truth functional connectives to build new propositions, the truth table for the new proposition depends on the order in which the connectives are applied. Truth table for (P & Q) V R (below) is different from that for P & (Q V R). Truth table for disjoining a conjunction I m going to tickle you with hand #1 I m going to tickle you with hand #2 I m going to tickle you with hand #3 I m going to tickle you with hand #1 and hand #2 I m going to tickle you with hand #1 and hand #2, or with hand #3 true true true true true true true false true true true false true false true true false false false false false true true false true false true false false false false false true false true false false false false false Lecture 4.8: Negation and truthfunctional operators propositional operator a propositional device that operates on a proposition to convert it to another proposition (rather than connecting two to make a larger one). Ex: I believe that truth functional operator propositional operator that creates a proposition whose truth depends solely on the truth of the proposition to which the operator is applied. I believe that is not a truth functional operator the truth or falsity of I believe that it s raining today doesn t depend solely on the truth or falsity of it s raining today. Negation truth functional operator that takes a proposition and creates a new proposition whose truth value is the opposite of the original. Can be expressed with it is not the case that or simply not.
7 P P true false false true Does not express negation? Sometimes, but not always. Walter has stopped beating his dogs. vs. Walter has not stopped beating his dogs. The former can be false even when the latter is not true. (Walter may still be beating dogs, may have never beaten dogs, may not even have dogs to beat, or flat out may not exist.) But Walter has stopped beating his dogs NOT! is true. Lecture 4.9: Negating conjunctions and disjunctions Negating a conjunction says only that it s not the case that both of the two conjuncts are true one or none could be true. Negating a disjunction says that neither conjunct is the case. Truth table for negating a conjunction I m going to eat the plug I m going to eat the cylinder I m going to eat the plug & the cylinder It s not the case that I m going to eat the plug & the cylinder true true true false true false false true false true false true false false false true Truth table for negating a disjunction I m going to eat the plug I m going to eat the cylinder I m going to eat the plug or the cylinder It s not the case that (I m going to eat the plug or the cylinder) true true true false true false true false false true true false false false false true Lecture 4.10: Commutativity and associativity Two properties we find in conjunction and disjunction, but not in other truth functions. Commutativity A function of two things is commutative when it delivers the same result regardless of the order in which it operates on those things.
8 Consider arithmetic: Addition is commutative (in general: x + y = y + x) Multiplication is commutative (in general: xy = yx) But subtraction is not commutative (in general: x y y x) Division is also not commutative. Commutativity in propositional logic: Conjunction is commutative: I m standing and waving = I m waving and standing (in general: p & q = q & p) Disjunction is commutative: I m standing or waving = I m waving or standing (in general: p V q = q V p) The conditional (which we ll learn about later this week), is not commutative the order determines the result you get. Associativity A function of three or more things is associative when it delivers the same result regardless of the order in which it operates on those things. In arithmetic: Addition is associative [in general: x + (y + z) = (x + y) + z] Multiplication is associative [in general: x(yz) = (xy)z] Subtraction is not associative [in general: x (y z) (x y) z]. Nor is division. In propositional logic: Conjunction is associative: I m standing and he s sitting and waving = I m standing and he s sitting, and he s also waving [in general: p & (q + r) = (p & q) + r] Disjunction is associative: I m standing or he s sitting and waving = I m standing or he s sitting, or he s waving [in general: p V (q V r) = (p V q) V r] Lecture 4.11: The conditional conditional a truth functional connective that is especially important in understanding the rules by which we assess validity of deductive arguments. If Walter is eating lunch at New Havana, then the private investigator is eating lunch at New Havana. If then... is a propositional connective How to argue that it s a truth functional connective as well? Consider this construction: (P & Q) is true. (negation of the conjunction of P and negation of Q) P & Q is false. If P is true, then Q is false If P is true, then Q is true That is to say: If P, then Q So: If P, then Q follows from (P & Q)
9
10 Truth table for (P & Q) P Q Q P & Q (P & Q) true true false false true true false true true false false true false false true false false true false true Now let s make that argument starting with If P, then Q : If P, then Q. If P is true, then Q is true. If P is true, then Q is false. (P & Q) is not true. (P & Q) is true. So: (P & Q) follows from If P, then Q. The truth table for the conditional If P, then Q is the same as that of (P & Q): P Q If P, then Q true true true true false false false true true false false true Since If P then Q has a truth table, that proves it s a truth functional connective. Some good rules governing our use of the conditional: Modus ponens: From the premises P and If P, then Q, infer the conclusion Q. (In other words: In any situation where P is true and If P, then Q is true, Q has got to be true.) Conditional truth table shows this is a good rule of inference for P to be true, it must be scenario 1 or 2; for If P, then Q to be true, it rules out scenario 2. Modus tollens: From the premises Q and If P, then Q, infer the conclusion P (In any situation where the negation of Q is true and If P, then Q is true, the negation of P has got to be true.) Truth table: Q means it s scenario 2 or 4; If P, then Q rules out 2. Why is the conditional an important truth functional connective? Can use it to express, in the form of a single proposition, the validity of an argument from premises P to conclusion C whatever the argument s about.
11 Lecture 4.12: Conditionals in ordinary language Use of If then often but not always signifies a material conditional or any other truth functional connective: For example, propositions within it cannot be in the subjunctive mood: If I had been 4 feet tall, then I would have been in the Guinness Book of World Records. Both propositions are false, but statement may or may not be true. And phrases other than if then may be used, such as: if : The private investigator is eating lunch at New Havana if Walter is. only if : Walter is eating lunch at New Havana only if the P.I. is. Lecture 4.13: Biconditionals if and only if conjoins conditional if P, then Q and conditional If Q, then P biconditional propositional connective connecting two propositions into a larger proposition, and the larger proposition is true just in case the two propositions that are part of it have the same truth value (both true or both false). P if and only if Q is true just in case P and Q are both true, or both false. Bob was born in the U.S. if, and only if, George was. equivalent to saying both Bob was born in the U.S. if George was. AND Bob was born in the U.S. only if George was. Truth table for biconditional: P Q P = Q true true true true false false false true false false false true Some examples would seem to counter claim that conditional and biconditional are truth functional connectives: If = 4, [then] Pierre is the capital of South Dakota. first proposition after if is antecedent. second proposition after then is consequent. Isn t this a baffling thing to say? Yes, but that doesn t mean it s not true.
12 Week 5: Categorical Logic and Syllogisms Lecture 5.1: Categorical logic This week: looking at arguments that are valid for a reason other than because of the propositional connectives they use some of these have no propositional connectives. Propositional logic: 1. Jill is riding her bicycle if, and only if, John is walking to the park 2. John is walking to the park if, and only if, premise 1 is true. 3. Jill is riding her bicycle We have a tricky deductive argument whose validity is not obvious it takes a while to check: Jill is riding her bike. John is walking to the park. Jill is riding her bike if, and only if, John is walking to the park. John is walking to the park if, and only if, the statement to the left is true. true true true true true false false true false true false false false false true false We don t just use truth tables to determine when or if arguments are valid we can also use truth tables to explain why they re valid, even in cases where it s obvious (as the ones we looked at last week were). This week: We can t use truth tables on inferences/arguments that don t use truth functional connectives, so we need to find another method to discover whether and why an argument is valid. No fish have wings All birds have wings All animals with gills are fish No birds have gills This is a valid argument we ll learn how to suss it out via the central method of categorical logic. Mary has a child who is pregnant Only daughters can become pregnant. Therefore, Mary has at least one daughter. What is it about the use of the terms only and at least that makes this valid? The concept of the quantifier is our central concept this week. Lecture 5.2: Categories and quantifiers Categories in arguments
13 Brazilians speak Portuguese. Portuguese speakers understand Spanish. Brazilians understand Spanish This argument brings 3 categories into relation with each other: Brazilians, speakers of Portuguese, and people who understand Spanish. Using some as quantifier on premises and conclusion = not a valid argument: Possible scenario: all the Portuguese speakers who understand Spanish live outside Brazil. Using most as quantifier = also not valid. Possible scenario: most Portuguese speakers live outside Brazil, and only those outside Brazil understand Spanish. Using all as quantifier = valid, but not sound. It s almost certainly not the case that both premises are true. So we started with 3 categories, then saw that by use of certain modifiers, we could make the original argument more precise and thus test its validity. To test arguments that use categories and quantifiers, we use a Venn diagram. The one at right covers just one premise; to cover all the premises and the conclusion, we d use one like the diagram below, for the some argument: At the right is one representing the all argument. The unshaded portion shows
14 Brazilians who ve got to understand Spanish; the Venn diagram shows the argument to be valid. Lecture 5.3: How quantifiers modify categories Types of quantifiers (F and G represent any category at all.) A: All Fs are Gs. All things in the first category also fall into the second category. E: No Fs are Gs. No things in the first category also fall into the second. I: Some Fs are Gs. Some things in the first category also fall into the second. O: Some Fs are not Gs. Some things in the first category don t fall into the second. As these Venn diagrams show, propositions of the A and O forms are negations of one another; the same is true between forms E and I. Lecture 5.4: Immediate categorical inference Immediate categorical inference: an inference with just one premise, in which both the premise and the conclusion are of form A, E, I or O. (They don t both have to be the same form.) Subject term and predicate term Each proposition of form A, E, I or O has a subject term and a predicate term. subject term the one directly modified by the quantifier (in our examples, labeled F) predicate term the other term, not directly modified by quantifier (labeled G here)
15 Conversion Inference in which the conclusion switches the subject and predicate term that appear in the premise. The most common example of immediate categorical inference. No Fs are Gs. All Fs are Gs. No Gs are Fs. All Gs are Fs. The first example is plausibly valid; the second is not. Conversion inferences are valid for E and I propositions (No Fs are Gs/Some Fs are Gs), but not A and O propositions. Venn diagrams illustrate why. A All Fs are Gs/All Gs are Fs : Non overlapping F circle is excluded, which could mean all the Gs there are are in overlapping section but it leaves open that there are plenty of Gs in the rest of that circle E No Fs are Gs/No Fs are Gs : Overlapping section is excluded, showing that there is the possibility of Fs and Gs, but not both at the same time. I Some Fs are Gs/Some Gs are Fs : X in overlapping section can be read as something being in both circles. O Some Fs are not Gs/Some Gs are not Fs : X in F circle outside the G circle. Does not imply that some Gs are not Fs there may be no Gs at all, or they may all be in overlapping section. Lecture 5.5: Syllogisms Syllogism: an argument with two premises and a conclusion, where all three propositions are of the A, E, I or O form. subject term of the syllogism = subject term of the conclusion (modified by quantifier). One of the premises the minor premise must also contain the subject term. predicate term of the syllogism = predicate term of the conclusion (not modified). One of the premises the major premise must also contain the predicate term. Example 1: Example 2: All Duke students are humans. Some Duke students are humans. All humans are animals. All humans are animals. All Duke students are animals. Some Duke students are animals.
16 Example 3: No Duke students are humans. All humans are animals. No Duke students are animals. Lecture 5.6: Categories, individuals and language How ordinary language can mislead us: Many statements that seem to be about individuals really are about categories. Example: Mary owns a Ferrari. Doesn t appear to involve categories, does it? A particular person owns a particular car. But compare that to: Some of Mary s possessions are Ferrari cars. It s a statement of the I form ( Some Fs are Gs. ) The two statements seem to amount to the same thing. Mary owns a Ferrari is true if and only if some of Mary s possessions are Ferrari cars. Thus, our categorical logic and Venn diagrams can be used to study the validity of such arguments. Not every statement is of the A, E, I or O forms, but lots of ordinary statements we make are. Lecture 5.7: Venn diagrams and validity Let s try to diagram some of our deductive arguments from last week: Mary has a child who is pregnant Only daughters can become pregnant. Mary has at least one daughter Terry has a job in which she arrests people. Only police officers arrest people. At least one of Terry s jobs is as a police officer.
17 Robert has a pet who is canine. Only mammals are canine. At least one of Robert s pets is mammal. Labels aside, the diagrams for all three of these arguments look the same, which tells us the arguments have the same form, and are valid for the same reason. Lecture 5.8: Other ways of expressing A, E, I or O propositions Propositions of the form not all : Not all Fs are Gs. (= Some Fs are not Gs O form) Not all geniuses take Coursera courses. Not everything that Pat does is intended to annoy Chris. Propositions of the form All Fs are not Gs (=No Fs are Gs E form) All Nobel Prize winners are not alcoholics. Everything that Pat does is not designed to achieve victory. Propositions of the form Some Fs are both Gs and Hs : Not the same as Some Fs are Gs and some Fs are Hs. Some philosophers are both robotic and monotone. Some of the things that Pat does are both intended to amuse and to provoke. Week 6: Representing Information Lecture 6.1: Reasoning from Venn diagrams or truth tables alone All the arguments we ve looked at over the last few weeks are given in ordinary language we can understand. But not all arguments are like that they might be given in technical jargon or in a foreign language. So, could you predict and explain those arguments validity based just on diagrams or truth tables? Yes. Example 1: You re an anthropologist studying another culture and trying to translate its untranslated language the word SPooG has eluded you. By observing their behavior, you devise a hypothesis: SPooG is used as a truth functional connective with this truth table:
18 P Q P SPooG Q True True True True False True False True False False False True You overhear an argument you translate as: John is riding his bicycle SPooG Jill is walking to the park. Jill is walking to the park. Therefore, John is riding his bicycle. Is this argument valid? Use the truth table (with headings John is riding his bicycle / JIll is walking to the park / John is riding his bike SPooG Jill is walking to the park ). The only scenario that works is the first one, in which all three are true. Example 2: John is riding his bicycle SPooG (Jill is walking to the park or Frank is sick) Frank is not sick. John is not riding his bicycle. Therefore, Jill is walking to the park. John is riding Jill is walking Frank is sick Jill is walking or Frank is sick John is riding SPooG (Jill is walking or Frank is sick) True True True True True True True False True True True False True True True True False False False True False True True True False False True False True False False False True True False False False False False True Premise 3 (John is not riding) narrows it down to the last four scenarios. Premise 2 (Frank is not sick) narrows it down to scenarios 1, 3, 5 and 7. Premise 1 (John is riding SPooG (Jill is walking or Frank is sick)) narrows it to first 4 scenarios or #8
19 This leaves us with scenario #8 but that states Jill is not walking to the park, which doesn t match the conclusion that she is. Thus, this argument is not valid. Quantifiers Another word in this language Jid! is always uttered in a high pitched, excited tone. Your hypothesis is that this word is a quantifier which works like this: Jid! (F or G) = There are no Fs outside G, but there are Gs outside F. (See Venn diagram at right.) You hear this argument: Jid! giraffes are herbivores. Jid! herbivores are mammals. There are some giraffes, and all of them are mammals. From premise 1, we rule out all of giraffes that s not under herbivores. We know there s an X in the overlap, but not where yet. From premise 2, we rule out all of herbivores that doesn t overlap with mammals, and an X in the overlap somewhere. We wouldn t know where if it weren t for premise 1. So it goes in the central overlap. Our diagram matches the conclusion, so the argument is valid. How might you translate Jid! into English? All Fs are Gs, but not all Gs are Fs. Lecture 6.2: Different ways of representing information What s the point of using truth tables or Venn diagrams to represent information? Unlike sentence representations, they allow us to clearly see which deductive inferences involving that information are valid. We can see relations of deductive validity that we can t with sentences.
A. Problem set #3 it has been posted and is due Tuesday, 15 November
Lecture 9: Propositional Logic I Philosophy 130 1 & 3 November 2016 O Rourke & Gibson I. Administrative A. Problem set #3 it has been posted and is due Tuesday, 15 November B. I am working on the group
More informationHANDBOOK (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
More informationINTERMEDIATE 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
More informationChapter 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 truthvalue of a given truthfunctional compound proposition depends
More informationPHILOSOPHY 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
More informationUnit 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,
More informationChapter 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
More informationLogic: 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
More informationMCQ 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 .
More informationExercise 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
More informationA 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
More informationIn 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
More informationAlice 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
More informationUnit. 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
More informationResponses to the sorites paradox
Responses to the sorites paradox phil 20229 Jeff Speaks April 21, 2008 1 Rejecting the initial premise: nihilism....................... 1 2 Rejecting one or more of the other premises....................
More informationA star (*) indicates that there are exercises covering this section and previous unmarked sections.
1 An Introduction To Reasoning Some Everyday Reasoning 1 Introduction 2 Reasoning Based On Properties 3 PartWhole Relationships 4 Reasoning With Relations 5 The Tricky Verb 'To Be' 6 Reasoning With Categorical
More informationWhat are TruthTables and What Are They For?
PY114: Work Obscenely Hard Week 9 (Meeting 7) 30 November, 2010 What are TruthTables and What Are They For? 0. Business Matters: The last marked homework of term will be due on Monday, 6 December, at
More information9 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
More informationToday s Lecture 1/28/10
Chapter 7.1! Symbolizing English Arguments! 5 Important Logical Operators!The Main Logical Operator Today s Lecture 1/28/10 Quiz State from memory (closed book and notes) the five famous valid forms and
More informationVenn 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
More informationChapter 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?
More informationSelections 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
More informationBasic 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
More informationHOW 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
More information1. 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.
More informationAnnouncements. CS311H: Discrete Mathematics. First Order Logic, Rules of Inference. Satisfiability, Validity in FOL. Example.
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:
More informationWhat is an argument? PHIL 110. Is this an argument? Is this an argument? What about this? And what about this?
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.
More informationWhat is the Nature of Logic? Judy Pelham Philosophy, York University, Canada July 16, 2013 PanHellenic Logic Symposium Athens, Greece
What is the Nature of Logic? Judy Pelham Philosophy, York University, Canada July 16, 2013 PanHellenic Logic Symposium Athens, Greece Outline of this Talk 1. What is the nature of logic? Some history
More informationPHILOSOPHER 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
More informationWhat 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
More informationDigital Logic Lecture 5 Boolean Algebra and Logic Gates Part I
Digital Logic Lecture 5 Boolean Algebra and Logic Gates Part I By Ghada AlMashaqbeh The Hashemite University Computer Engineering Department Outline Introduction. Boolean variables and truth tables. Fundamental
More informationConditionals, Predicates and Probability
Conditionals, Predicates and Probability Abstract Ernest Adams has claimed that a probabilistic account of validity gives the best account of our intuitive judgements about the validity of arguments. In
More informationTest Item File. Full file at
Test Item File 107 CHAPTER 1 Chapter 1: Basic Logical Concepts Multiple Choice 1. In which of the following subjects is reasoning outside the concern of logicians? A) science and medicine B) ethics C)
More informationExposition of Symbolic Logic with KalishMontague derivations
An Exposition of Symbolic Logic with KalishMontague derivations Copyright 200613 by Terence Parsons all rights reserved Aug 2013 Preface The system of logic used here is essentially that of Kalish &
More informationPRACTICE EXAM The state of Israel was in a state of mourning today because of the assassination of Yztzak Rabin.
PRACTICE EXAM 1 I. Decide which of the following are arguments. For those that are, identify the premises and conclusions in them by CIRCLING them and labeling them with a P for the premises or a C for
More informationPHIL 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) [3ab] Argument: Socrates is a maker of gods; so, Socrates corrupts the
More informationRelevance. 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
More informationHandout 1: Arguments  the basics because, since, given that, for because Given that Since for Because
Handout 1: Arguments  the basics It is useful to think of an argument as a list of sentences.[1] The last sentence is the conclusion, and the other sentences are the premises. Thus: (1) No professors
More informationSection 3.5. Symbolic Arguments. Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 3.5 Symbolic Arguments What You Will Learn Symbolic arguments Standard forms of arguments 3.52 Symbolic Arguments A symbolic argument consists of a set of premises and a conclusion. It is called
More informationSyllogism. Exam Importance Exam Importance. CAT Very Important IBPS/Bank PO Very Important. XAT Very Important BANK Clerk Very Important
1 About Disha publication One of the leading publishers in India, Disha Publication provides books and study materials for schools and various competitive exams being continuously held across the country.
More information10.3 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
More informationWhat is the Frege/Russell Analysis of Quantification? Scott Soames
What is the Frege/Russell Analysis of Quantification? Scott Soames The FregeRussell analysis of quantification was a fundamental advance in semantics and philosophical logic. Abstracting away from details
More information4.1 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
More informationInstructor s Manual 1
Instructor s Manual 1 PREFACE This instructor s manual will help instructors prepare to teach logic using the 14th edition of Irving M. Copi, Carl Cohen, and Kenneth McMahon s Introduction to Logic. The
More informationA Brief Introduction to Key Terms
1 A Brief Introduction to Key Terms 5 A Brief Introduction to Key Terms 1.1 Arguments Arguments crop up in conversations, political debates, lectures, editorials, comic strips, novels, television programs,
More informationConditionals IV: Is Modus Ponens Valid?
Conditionals IV: Is Modus Ponens Valid? UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 The intuitive counterexamples McGee [2] offers these intuitive counterexamples to Modus Ponens: 1. (a)
More informationRussell on Descriptions
Russell on Descriptions Bertrand Russell s analysis of descriptions is certainly one of the most famous (perhaps the most famous) theories in philosophy not just philosophy of language over the last century.
More informationPossibility 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
More informationPhilosophical 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
More information15. Russell on definite descriptions
15. Russell on definite descriptions Martín Abreu Zavaleta July 30, 2015 Russell was another top logician and philosopher of his time. Like Frege, Russell got interested in denotational expressions as
More informationNecessity. Oxford: Oxford University Press. Pp. iix, 379. ISBN $35.00.
Appeared in Linguistics and Philosophy 26 (2003), pp. 367379. Scott Soames. 2002. Beyond Rigidity: The Unfinished Semantic Agenda of Naming and Necessity. Oxford: Oxford University Press. Pp. iix, 379.
More informationb) 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
More informationA 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
More informationDenying 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
More information2016 Philosophy. Higher. Finalised Marking Instructions
National Qualifications 06 06 Philosophy Higher Finalised Marking Instructions Scottish Qualifications Authority 06 The information in this publication may be reproduced to support SQA qualifications only
More informationUnit 4. Reason as a way of knowing
Unit 4 Reason as a way of knowing Zendo The Master will present two Koans  one that follows the rule and one that does not. Teams will take turns presenting their own koans to the master to see if they
More informationFaith 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
More informationStatements, Arguments, Validity. Philosophy and Logic Unit 1, Sections 1.1, 1.2
Statements, Arguments, Validity Philosophy and Logic Unit 1, Sections 1.1, 1.2 Mayor Willy Brown on proposition 209: There is still rank discrimination in this country. If there is rank discrimination,
More informationThe 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.
More informationLogic 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
More informationIn view of the fact that IN CLASS LOGIC EXERCISES
IN CLASS LOGIC EXERCISES Instructions: Determine whether the following are propositions. If some are not propositions, see if they can be rewritten as propositions. (1) I have a very refined sense of smell.
More informationLogic Dictionary Keith BurgessJackson 12 August 2017
Logic Dictionary Keith BurgessJackson 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
More information1.5. Argument Forms: Proving Invalidity
18. If inflation heats up, then interest rates will rise. If interest rates rise, then bond prices will decline. Therefore, if inflation heats up, then bond prices will decline. 19. Statistics reveal that
More informationAn alternative understanding of interpretations: Incompatibility Semantics
An alternative understanding of interpretations: Incompatibility Semantics 1. In traditional (truththeoretic) semantics, interpretations serve to specify when statements are true and when they are false.
More informationTWO 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
More informationIntroducing truth tables. Hello, I m Marianne Talbot and this is the first video in the series supplementing the Formal Logic podcasts.
Introducing truth tables Marianne: Hello, I m Marianne Talbot and this is the first video in the series supplementing the Formal Logic podcasts. Okay, introducing truth tables. (Slide 2) This video supplements
More informationThree Kinds of Arguments
Chapter 27 Three Kinds of Arguments Arguments in general We ve been focusing on Moleculananalyzable arguments for several chapters, but now we want to take a step back and look at the big picture, at
More informationPragmatic 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
More information1. 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
More informationBASIC CONCEPTS OF LOGIC
BASIC CONCEPTS OF LOGIC 1. What is Logic?...2 2. Inferences and Arguments...2 3. Deductive Logic versus Inductive Logic...5 4. Statements versus Propositions...6 5. Form versus Content...7 6. Preliminary
More informationIntro. FirstOrder Necessity and Validity. First Order Attention. First Order Attention
Intro Mark Criley IWU 10/23/2015 We have added some new pieces to our language: Quantifiers and variables. These new pieces are going to add a new layer of NPEC: Necessity, Possibility, Equivalence, Consequence.
More informationhigherorder attitudes, frege s abyss, and the truth in propositions
Mark Schroeder University of Southern California November 28, 2011 higherorder attitudes, frege s abyss, and the truth in propositions In nearly forty years of work, Simon Blackburn has done more than
More informationReductio ad Absurdum, Modulation, and Logical Forms. Miguel LópezAstorga 1
International Journal of Philosophy and Theology June 25, Vol. 3, No., pp. 5965 ISSN: 2333575 (Print), 23335769 (Online) Copyright The Author(s). All Rights Reserved. Published by American Research
More informationPlato s Allegory of the Cave
Logic Plato s Allegory of the Cave The First Word of the Day is Troglodyte From the Greek word for cave (trōglē). The Troglodytae (Τρωγλοδῦται) or Troglodyti (literally cave goers ) are those who live
More informationSession 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:
More informationLogicola Truth Evaluation Exercises
Logicola Truth Evaluation Exercises The Logicola exercises for Ch. 6.3 concern truth evaluations, and in 6.4 this complicated to include unknown evaluations. I wanted to say a couple of things for those
More informationIntersubstitutivity Principles and the Generalization Function of Truth. Anil Gupta University of Pittsburgh. Shawn Standefer University of Melbourne
Intersubstitutivity Principles and the Generalization Function of Truth Anil Gupta University of Pittsburgh Shawn Standefer University of Melbourne Abstract We offer a defense of one aspect of Paul Horwich
More information1. True or False: The terms argument and disagreement mean the same thing. 2. True or False: No arguments have more than two premises.
Logic Chapter 1 Practice Test: True / False: Mark each of the following statements as True or False. 1. True or False: The terms argument and disagreement mean the same thing. 2. True or False: No arguments
More informationA 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
More informationI. What is an Argument?
I. What is an Argument? In philosophy, an argument is not a dispute or debate, but rather a structured defense of a claim (statement, assertion) about some topic. When making an argument, one does not
More informationEssential 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.
More informationFundamentals of Philosophy
Logic Logic is a comprehensive introduction to the major concepts and techniques involved in the study of logic. It explores both formal and philosophical logic and examines the ways in which we can achieve
More information6.5 Exposition of the Fifteen Valid Forms of the Categorical Syllogism
M06_COPI1396_13_SE_C06.QXD 10/16/07 9:17 PM Page 255 6.5 Exposition of the Fifteen Valid Forms of the Categorical Syllogism 255 7. All supporters of popular government are democrats, so all supporters
More informationIntroduction 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
More informationWhat is reason? The power of the mind to think, understand, and form judgments by a process of logic
WoK 3 Reason What is reason? Webster s Dictionary defines reason as: The power of the mind to think, understand, and form judgments by a process of logic and logic as: reasoning conducted or assessed according
More informationContradictions and Counterfactuals: Generating Belief Revisions in Conditional Inference
Contradictions and Counterfactuals: Generating Belief Revisions in Conditional Inference Ruth M.J. Byrne (rmbyrne@tcd.ie) Psychology Department, University of Dublin, Trinity College, Dublin, Ireland Clare
More informationLecture 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
More informationSensitivity hasn t got a Heterogeneity Problem  a Reply to Melchior
DOI 10.1007/s114060169782z Sensitivity hasn t got a Heterogeneity Problem  a Reply to Melchior Kevin Wallbridge 1 Received: 3 May 2016 / Revised: 7 September 2016 / Accepted: 17 October 2016 # The
More informationElements 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)
More informationtempered expressivism for Oxford Studies in Metaethics, volume 8
Mark Schroeder University of Southern California December 1, 2011 tempered expressivism for Oxford Studies in Metaethics, volume 8 This paper has two main goals. Its overarching goal, like that of some
More informationReason and Judgment, A Primer Istvan S. N. Berkeley, Ph.D. ( istvan at louisiana dot edu) 2008, All Rights Reserved.
Reason and Judgment, A Primer Istvan S. N. Berkeley, Ph.D. (Email: istvan at louisiana dot edu) 2008, All Rights Reserved. I. Introduction Aristotle said that our human capacity to reason is one of the
More information6: 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
More informationThe 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?
More information5.6.1 Formal validity in categorical deductive arguments
Deductive arguments are commonly used in various kinds of academic writing. In order to be able to perform a critique of deductive arguments, we will need to understand their basic structure. As will be
More informationDenying the Antecedent as a Legitimate Argumentative Strategy: A Dialectical Model
Denying the Antecedent as a Legitimate Argumentative Strategy 219 Denying the Antecedent as a Legitimate Argumentative Strategy: A Dialectical Model DAVID M. GODDEN DOUGLAS WALTON University of Windsor
More informationEvaluating Arguments
Govier: A Practical Study of Argument 1 Evaluating Arguments Chapter 4 begins an important discussion on how to evaluate arguments. The basics on how to evaluate arguments are presented in this chapter
More informationRussellianism and Explanation. David Braun. University of Rochester
Forthcoming in Philosophical Perspectives 15 (2001) Russellianism and Explanation David Braun University of Rochester Russellianism is a semantic theory that entails that sentences (1) and (2) express
More informationTable of Contents. What This Book Teaches... iii Four Myths About Critical Thinking... iv Pretest...v
Table of Contents Table of Contents What This Book Teaches... iii Four Myths About Critical Thinking... iv Pretest...v 1. What Is Critical Thinking?...1 2. Decisions and Conclusions...4 3. Beliefs and
More informationLogic: 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
More information