Part II: How to Evaluate Deductive Arguments


 Miles Julius Ball
 2 years 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.
Study Guides. Chapter 1  Basic Training
Study Guides Chapter 1  Basic Training Argument: A group of propositions is an argument when one or more of the propositions in the group is/are used to give evidence (or if you like, reasons, or grounds)
More informationLogic Book Part 1! by Skylar Ruloff!
Logic Book Part 1 by Skylar Ruloff Contents Introduction 3 I Validity and Soundness 4 II Argument Forms 10 III Counterexamples and Categorical Statements 15 IV Strength and Cogency 21 2 Introduction This
More informationA. 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 informationPHI 1500: Major Issues in Philosophy
PHI 1500: Major Issues in Philosophy Session 3 September 9 th, 2015 All About Arguments (Part II) 1 A common theme linking many fallacies is that they make unwarranted assumptions. An assumption is a claim
More information1 Clarion Logic Notes Chapter 4
1 Clarion Logic Notes Chapter 4 Summary Notes These are summary notes so that you can really listen in class and not spend the entire time copying notes. These notes will not substitute for reading the
More informationLecture 3 Arguments Jim Pryor What is an Argument? Jim Pryor Vocabulary Describing Arguments
Lecture 3 Arguments Jim Pryor What is an Argument? Jim Pryor Vocabulary Describing Arguments 1 Agenda 1. What is an Argument? 2. Evaluating Arguments 3. Validity 4. Soundness 5. Persuasive Arguments 6.
More informationLogic: 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
More informationPHI Introduction Lecture 4. An Overview of the Two Branches of Logic
PHI 103  Introduction Lecture 4 An Overview of the wo Branches of Logic he wo Branches of Logic Argument  at least two statements where one provides logical support for the other. I. Deduction  a conclusion
More informationLogic Appendix: More detailed instruction in deductive logic
Logic Appendix: More detailed instruction in deductive logic Standardizing and Diagramming In Reason and the Balance we have taken the approach of using a simple outline to standardize short arguments,
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 informationArtificial Intelligence: Valid Arguments and Proof Systems. Prof. Deepak Khemani. Department of Computer Science and Engineering
Artificial Intelligence: Valid Arguments and Proof Systems Prof. Deepak Khemani Department of Computer Science and Engineering Indian Institute of Technology, Madras Module 02 Lecture  03 So in the last
More informationLOGIC ANTHONY KAPOLKA FYF 1019/3/2010
LOGIC ANTHONY KAPOLKA FYF 1019/3/2010 LIBERALLY EDUCATED PEOPLE......RESPECT RIGOR NOT SO MUCH FOR ITS OWN SAKE BUT AS A WAY OF SEEKING TRUTH. LOGIC PUZZLE COOPER IS MURDERED. 3 SUSPECTS: SMITH, JONES,
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 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 informationA 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
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 informationSemantic Entailment and Natural Deduction
Semantic Entailment and Natural Deduction Alice Gao Lecture 6, September 26, 2017 Entailment 1/55 Learning goals Semantic entailment Define semantic entailment. Explain subtleties of semantic entailment.
More informationPhilosophy 1100: Ethics
Philosophy 1100: Ethics Topic 1  Course Introduction: 1. What is Philosophy? 2. What is Ethics? 3. Logic a. Truth b. Arguments c. Validity d. Soundness What is Philosophy? The Three Fundamental Questions
More informationSYLLOGISTIC LOGIC CATEGORICAL PROPOSITIONS
Prof. C. Byrne Dept. of Philosophy SYLLOGISTIC LOGIC Syllogistic logic is the original form in which formal logic was developed; hence it is sometimes also referred to as Aristotelian logic after Aristotle,
More information4.7 Constructing Categorical Propositions
4.7 Constructing Categorical Propositions We have spent the last couple of weeks studying categorical propositions. Unfortunately, in the real world, the statements that people make seldom have that form.
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 informationHANDBOOK. IV. Argument Construction Determine the Ultimate Conclusion Construct the Chain of Reasoning Communicate the Argument 13
1 HANDBOOK TABLE OF CONTENTS I. Argument Recognition 2 II. Argument Analysis 3 1. Identify Important Ideas 3 2. Identify Argumentative Role of These Ideas 4 3. Identify Inferences 5 4. Reconstruct the
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 informationPART III  Symbolic Logic Chapter 7  Sentential Propositions
Logic: A Brief Introduction Ronald L. Hall, Stetson University 7.1 Introduction PART III  Symbolic Logic Chapter 7  Sentential Propositions What has been made abundantly clear in the previous discussion
More informationDeduction. Of all the modes of reasoning, deductive arguments have the strongest relationship between the premises
Deduction Deductive arguments, deduction, deductive logic all means the same thing. They are different ways of referring to the same style of reasoning Deduction is just one mode of reasoning, but it is
More informationAnnouncements. CS243: Discrete Structures. First Order Logic, Rules of Inference. Review of Last Lecture. Translating English into FirstOrder 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
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 informationAppendix: The Logic Behind the Inferential Test
Appendix: The Logic Behind the Inferential Test In the Introduction, I stated that the basic underlying problem with forensic doctors is so easy to understand that even a twelveyearold could understand
More informationSHORT 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
More informationILLOCUTIONARY ORIGINS OF FAMILIAR LOGICAL OPERATORS
ILLOCUTIONARY ORIGINS OF FAMILIAR LOGICAL OPERATORS 1. ACTS OF USING LANGUAGE Illocutionary logic is the logic of speech acts, or language acts. Systems of illocutionary logic have both an ontological,
More informationAn Introduction to. Formal Logic. Second edition. Peter Smith, February 27, 2019
An Introduction to Formal Logic Second edition Peter Smith February 27, 2019 Peter Smith 2018. Not for reposting or recirculation. Comments and corrections please to ps218 at cam dot ac dot uk 1 What
More informationAlso, in Argument #1 (Lecture 11, Slide 11), the inference from steps 2 and 3 to 4 is stated as:
by SALVATORE  5 September 2009, 10:44 PM I`m having difficulty understanding what steps to take in applying valid argument forms to do a proof. What determines which given premises one should select to
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 informationArtificial Intelligence Prof. P. Dasgupta Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur
Artificial Intelligence Prof. P. Dasgupta Department of Computer Science & Engineering Indian Institute of Technology, Kharagpur Lecture 9 First Order Logic In the last class, we had seen we have studied
More informationPart 2 Module 4: Categorical Syllogisms
Part 2 Module 4: Categorical Syllogisms Consider Argument 1 and Argument 2, and select the option that correctly identifies the valid argument(s), if any. Argument 1 All bears are omnivores. All omnivores
More informationA R G U M E N T S I N A C T I O N
ARGUMENTS IN ACTION Descriptions: creates a textual/verbal account of what something is, was, or could be (shape, size, colour, etc.) Used to give you or your audience a mental picture of the world around
More informationCritical Thinking 5.7 Validity in inductive, conductive, and abductive arguments
5.7 Validity in inductive, conductive, and abductive arguments REMEMBER as explained in an earlier section formal language is used for expressing relations in abstract form, based on clear and unambiguous
More informationPhilosophy 1100: Introduction to Ethics. Critical Thinking Lecture 1. Background Material for the Exercise on Validity
Philosophy 1100: Introduction to Ethics Critical Thinking Lecture 1 Background Material for the Exercise on Validity Reasons, Arguments, and the Concept of Validity 1. The Concept of Validity Consider
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 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 informationRevisiting 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
More informationRecall. Validity: If the premises are true the conclusion must be true. Soundness. Valid; and. Premises are true
Recall Validity: If the premises are true the conclusion must be true Soundness Valid; and Premises are true Validity In order to determine if an argument is valid, we must evaluate all of the sets of
More informationChapter 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 truthfunctional arguments.
More informationThere are two common forms of deductively valid conditional argument: modus ponens and modus tollens.
INTRODUCTION TO LOGICAL THINKING Lecture 6: Two types of argument and their role in science: Deduction and induction 1. Deductive arguments Arguments that claim to provide logically conclusive grounds
More information5.3 The Four Kinds of Categorical Propositions
M05_COI1396_13_E_C05.QXD 11/13/07 8:39 AM age 182 182 CHATER 5 Categorical ropositions Categorical propositions are the fundamental elements, the building blocks of argument, in the classical account of
More informationChapter 3: More Deductive Reasoning (Symbolic Logic)
Chapter 3: More Deductive Reasoning (Symbolic Logic) There's no easy way to say this, the material you're about to learn in this chapter can be pretty hard for some students. Other students, on the other
More informationCriticizing Arguments
Kareem Khalifa Criticizing Arguments 1 Criticizing Arguments Kareem Khalifa Department of Philosophy Middlebury College Written August, 2012 Table of Contents Introduction... 1 Step 1: Initial Evaluation
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 informationLGCS 199DR: Independent Study in Pragmatics
LGCS 99DR: Independent Study in Pragmatics Jesse Harris & Meredith Landman September 0, 203 Last class, we discussed the difference between semantics and pragmatics: Semantics The study of the literal
More information1.2. What is said: propositions
1.2. What is said: propositions 1.2.0. Overview In 1.1.5, we saw the close relation between two properties of a deductive inference: (i) it is a transition from premises to conclusion that is free of any
More informationCHAPTER THREE Philosophical Argument
CHAPTER THREE Philosophical Argument General Overview: As our students often attest, we all live in a complex world filled with demanding issues and bewildering challenges. In order to determine those
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 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 informationTransition to Quantified Predicate Logic
Transition to Quantified Predicate Logic Predicates You may remember (but of course you do!) during the first class period, I introduced the notion of validity with an argument much like (with the same
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 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 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 informationThe way we convince people is generally to refer to sufficiently many things that they already know are correct.
Theorem A Theorem is a valid deduction. One of the key activities in higher mathematics is identifying whether or not a deduction is actually a theorem and then trying to convince other people that you
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 informationIntro Viewed from a certain angle, philosophy is about what, if anything, we ought to believe.
Overview Philosophy & logic 1.2 What is philosophy? 1.3 nature of philosophy Why philosophy Rules of engagement Punctuality and regularity is of the essence You should be active in class It is good to
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 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 informationDeduction by Daniel Bonevac. Chapter 1 Basic Concepts of Logic
Deduction by Daniel Bonevac Chapter 1 Basic Concepts of Logic Logic defined Logic is the study of correct reasoning. Informal logic is the attempt to represent correct reasoning using the natural language
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 informationComplications for Categorical Syllogisms. PHIL 121: Methods of Reasoning February 27, 2013 Instructor:Karin Howe Binghamton University
Complications for Categorical Syllogisms PHIL 121: Methods of Reasoning February 27, 2013 Instructor:Karin Howe Binghamton University Overall Plan First, I will present some problematic propositions and
More informationOverview of Today s Lecture
Branden Fitelson Philosophy 12A Notes 1 Overview of Today s Lecture Music: Robin Trower, Daydream (King Biscuit Flower Hour concert, 1977) Administrative Stuff (lots of it) Course Website/Syllabus [i.e.,
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 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 informationChapter 1. What is Philosophy? Thinking Philosophically About Life
Chapter 1 What is Philosophy? Thinking Philosophically About Life Why Study Philosophy? Defining Philosophy Studying philosophy in a serious and reflective way will change you as a person Philosophy Is
More informationScott Soames: Understanding Truth
Philosophy and Phenomenological Research Vol. LXV, No. 2, September 2002 Scott Soames: Understanding Truth MAlTHEW MCGRATH Texas A & M University Scott Soames has written a valuable book. It is unmatched
More informationWhat would count as Ibn Sīnā (11th century Persia) having first order logic?
1 2 What would count as Ibn Sīnā (11th century Persia) having first order logic? Wilfrid Hodges Herons Brook, Sticklepath, Okehampton March 2012 http://wilfridhodges.co.uk Ibn Sina, 980 1037 3 4 Ibn Sīnā
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 informationA Romp through the Foothills of Logic: Session 2
A Romp through the Foothills of Logic: Session 2 You might find it easier to understand this podcast if you first watch the short podcast Introducing Truth Tables. (Slide 2) Right, by the time we finish
More informationWorkbook Unit 3: Symbolizations
Workbook Unit 3: Symbolizations 1. Overview 2 2. Symbolization as an Art and as a Skill 3 3. A Variety of Symbolization Tricks 15 3.1. nplace Conjunctions and Disjunctions 15 3.2. Neither nor, Not both
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 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 information7. Some recent rulings of the Supreme Court were politically motivated decisions that flouted the entire history of U.S. legal practice.
M05_COPI1396_13_SE_C05.QXD 10/12/07 9:00 PM Page 193 5.5 The Traditional Square of Opposition 193 EXERCISES Name the quality and quantity of each of the following propositions, and state whether their
More informationAm I free? Freedom vs. Fate
Am I free? Freedom vs. Fate We ve been discussing the free will defense as a response to the argument from evil. This response assumes something about us: that we have free will. But what does this mean?
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 informationHomework: read in the book pgs and do "You Try It" (to use Submit); Read for lecture. C. Anthony Anderson
Philosophy 183 Page 1 09 / 26 / 08 Friday, September 26, 2008 9:59 AM Homework: read in the book pgs. 110 and do "You Try It" (to use Submit); Read 1929 for lecture. C. Anthony Anderson (caanders@philosophy.ucsb.edu)
More informationPhilosophy 57 Day 10
Branden Fitelson Philosophy 57 Lecture 1 Philosophy 57 Day 10 Quiz #2 Curve (approximate) 100 (A); 70 80 (B); 50 60 (C); 40 (D); < 40 (F) Quiz #3 is next Tuesday 03/04/03 (on chapter 4 not tnanslation)
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 informationThe antecendent always a expresses a sufficient condition for the consequent
Critical Thinking Lecture Four October 5, 2012 Chapter 3 Deductive Argument Patterns Diagramming Arguments Deductive Argument Patterns  There are some common patterns shared by many deductive arguments
More informationDr. Carlo Alvaro Reasoning and Argumentation Distribution & Opposition DISTRIBUTION
DISTRIBUTION Categorical propositions are statements that describe classes (groups) of objects designate by the subject and the predicate terms. A class is a group of things that have something in common
More information7.1. Unit. Terms and Propositions. Nature of propositions. Types of proposition. Classification of propositions
Unit 7.1 Terms and Propositions Nature of propositions A proposition is a unit of reasoning or logical thinking. Both premises and conclusion of reasoning are propositions. Since propositions are so important,
More information2.1 Review. 2.2 Inference and justifications
Applied Logic Lecture 2: Evidence Semantics for Intuitionistic Propositional Logic Formal logic and evidence CS 4860 Fall 2012 Tuesday, August 28, 2012 2.1 Review The purpose of logic is to make reasoning
More informationBaronett, Logic (4th ed.) Chapter Guide
Chapter 6: Categorical Syllogisms Baronett, Logic (4th ed.) Chapter Guide A. Standardform Categorical Syllogisms A categorical syllogism is an argument containing three categorical propositions: two premises
More informationIntroduction Symbolic Logic
An Introduction to Symbolic Logic Copyright 2006 by Terence Parsons all rights reserved CONTENTS Chapter One Sentential Logic with 'if' and 'not' 1 SYMBOLIC NOTATION 2 MEANINGS OF THE SYMBOLIC NOTATION
More informationCHAPTER 1 A PROPOSITIONAL THEORY OF ASSERTIVE ILLOCUTIONARY ARGUMENTS OCTOBER 2017
CHAPTER 1 A PROPOSITIONAL THEORY OF ASSERTIVE ILLOCUTIONARY ARGUMENTS OCTOBER 2017 Man possesses the capacity of constructing languages, in which every sense can be expressed, without having an idea how
More informationPortfolio Project. Phil 251A Logic Fall Due: Friday, December 7
Portfolio Project Phil 251A Logic Fall 2012 Due: Friday, December 7 1 Overview The portfolio is a semesterlong project that should display your logical prowess applied to realworld arguments. The arguments
More informationIn Defense of The WideScope Instrumental Principle. Simon Rippon
In Defense of The WideScope Instrumental Principle Simon Rippon Suppose that people always have reason to take the means to the ends that they intend. 1 Then it would appear that people s intentions to
More informationReasoning SYLLOGISM. follows.
Reasoning SYLLOGISM RULES FOR DERIVING CONCLUSIONS 1. The Conclusion does not contain the Middle Term (M). Premises : All spoons are plates. Some spoons are cups. Invalid Conclusion : All spoons are cups.
More informationIs the law of excluded middle a law of logic?
Is the law of excluded middle a law of logic? Introduction I will conclude that the intuitionist s attempt to rule out the law of excluded middle as a law of logic fails. They do so by appealing to harmony
More informationPHIL2642 CRITICAL THINKING USYD NOTES PART 1: LECTURE NOTES
PHIL2642 CRITICAL THINKING USYD NOTES PART 1: LECTURE NOTES LECTURE CONTENTS LECTURE 1: CLAIMS, EXPLAINATIONS AND ARGUMENTS LECTURE 2: CONDITIONS AND DEDUCTION LECTURE 3: MORE DEDUCTION LECTURE 4: MEANING
More information6. Truth and Possible Worlds
6. Truth and Possible Worlds We have defined logical entailment, consistency, and the connectives,,, all in terms of belief. In view of the close connection between belief and truth, described in the first
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 informationLOGIC LECTURE #3: DEDUCTION AND INDUCTION. Source: A Concise Introduction to Logic, 11 th Ed. (Patrick Hurley, 2012)
LOGIC LECTURE #3: DEDUCTION AND INDUCTION Source: A Concise Introduction to Logic, 11 th Ed. (Patrick Hurley, 2012) Deductive Vs. Inductive If the conclusion is claimed to follow with strict certainty
More information10. Presuppositions Introduction The Phenomenon Tests for presuppositions
10. Presuppositions 10.1 Introduction 10.1.1 The Phenomenon We have encountered the notion of presupposition when we talked about the semantics of the definite article. According to the famous treatment
More information