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- LOGIC For, the study of philosophy is not undertaken to know what men feel, but to know what is the truth of things" -Aquinas
- Why Study Logic and learn Critical Thinking? In republican nation, whose citizens are to be led by reason and persuasion and not by force, the art of reasoning becomes of the first importance – Thomas Jefferson • Logic allows us to improve the quality of the argument we use • It improves the our ability to evaluate the arguments of others • It improves the our ability to communicate more clearly and efficiently • It teaches us how reasoning can go wrong and what is needed for reasoning to be structured • It teaches us to develop emotional and intellectual distance between yourself and idea – whether your own or others – in order to better evaluate their truth, validity, and reasonableness
- What is Logic? • Logic may be defined as the organized body of knowledge, or science, that evaluates arguments • It is the study of methods and principles used to distinguish good reasoning from bad reasoning • We all encounter arguments in life: reading books & newspapers, hear them on television, and when communicating with friends, science & religion, ethics & law, • Hence the aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own
- • Whatever the subject or content of an argument, the logician is interested in its form and quality • i.e. does the argument do what it claims to do? Does its conclusion logically infer from its premises? • Let us now look at What is an argument and its components:
- Arguments, Propositions, Premises, & Conclusions • An argument does not mean a verbal fight with someone but rather: • An argument is a group of statements (or propositions), one or more of which (the premises) are claimed to provide support for, or reasons to believe, one of the others (the conclusion). • E.g. All film stars are celebrities. (Premise) Halle Berry is a film star. (Premise) Therefore, Halle Berry is a celebrity. (Conclusion) • All arguments fall into two groups: • Good Arguments: Those in which the premises really do support the conclusion • Bad Arguments: Those in which the premises do not support the conclusion though they claim to e.g. Some film stars are men. (Premise) Cameron Diaz is a film star. (Premise) Therefore, Cameron Diaz is a man (Conclusion)
- • The purpose of logic, as the science that evaluates arguments, is thus to develop methods and techniques that allow us to distinguish good arguments from bad. • To understand arguments fully we must first understand its basic components that are propositions (statements), inference, premises, and conclusions:
- Basic concepts to note about Arguments • Arguments are made up of statements (propositions) which are the building the building blocks of arguments • A Statement (proposition) is declarative sentence that is either true or false • E.g. Chocolate truffles are loaded with calories (TRUE) • Political candidates always tell the complete truth (FALSE) • Tiger Woods plays golf and Maria Sharapova plays tennis (Since both parts are true hence the whole statement is TRUE) • Therefore a statement asserts that something is (or is not) the case • Any proposition can be affirmed or denied
- • Truth Value: The Truth or Falsity of a statement is known as the Truth Value of that proposition • Thus if a proposition is true its Truth Value is ‘True’ and if a proposition is false its truth Value is ‘False’ • The Truth or Falsity of a proposition may not be known but it must either be true or false • E.g. the Truth Value of the statement that ‘There is life on some other planet in our galaxy’ is not known but it must either be true or false • Unlike statements many sentence cannot be said to be either true or false • Where is Kalam? (question) • Let’s go to a movie tonight. (proposal) • I suggest you get contact lenses (suggestion) • Turn off the TV right now (command) • Fantastic! (exclamation)
- • Sentences are part of some language but propositions are not part of any language: • Today is Monday (English) • Aaj Peer hai (Urdu) • Nan Peer dai (Pushto) All the three sentences above mean the same thing (proposition) • The same sentence may make different statements if the context changes • E.g. The statement ‘The largest country in the world is the third most populous country in the world’ was true for USSR but is now not true for Russia
- • Simple Proposition: that makes only one assertion • E.g. It is raining outside • Compound Proposition: A proposition containing two or more simple propositions • E.g. • Conjunctive Proposition … if true all of its components must be true • Pakistan is located in South-East Asia and it is an Islamic State • Disjunctive (or Alternate) Proposition … if true at least one of its components must be true • Either Imran Khan will win the national elections or he will not win Part 1 Part 2 Part 1 Part 2
- • Hypothetical or Conditional (If Then) Proposition … False only when the antecedent is true consequent is false • It takes the form of: If ________________ Then _________________ • E.g. If God did not exist, it would be necessary to invent him (Voltaire) • More on Hypothetical conditions and their role in arguments latter on … Antecedent Consequent Antecedent Consequent
- • Inference: In arguments when we link propositions by affirming (or reaching) one proposition (conclusion) on the basis of one or more other propositions (premises) we say an inference has been drawn • The statements that make up an argument are divided into one or more premises and one and only one conclusion. • Example of an argument consisting of one premise: No one was present when life first appeared on earth (P). Therefore any statement about origins of life on earth should be considered a theory, not a fact. (Conclusion) This is the simplest form of an argument containing one premise and one conclusion
- Premises Conclusion Claimed Evidence What is claimed to follow from the evidence Inference drawn Argument
- • Sometimes both the premise and the conclusion may be stated in the same sentence: • Since it turns out that all humans are descendants from a small number of African ancestors in our recent evolutionary past, believing in profound differences between humans is as ridiculous as believing in a flat earth • Sometimes the conclusion of an argument may precede its premise: • Smoking should be banned in public places. After all passive smoking can cause cancer. • You can’t separate peace from freedom because no one can be at peace if they don’t have freedom • These are all simple propositions. Most arguments are complicated as they contain compound propositions with several intricately related components. But whether simple or complex, arguments contain a group of propositions with one conclusion and the other are premises to support the conclusion.
- • Consider the hypothetical proposition: • If life evolved on Mars in the early period of its history when it had an atmosphere and climate similar to earth, then it is likely that life evolved on countless other planets scientists believe to exist in our galaxy There is no argument here. No inference is drawn in this passage. The proposition asserts only that if the former is true then the latter must be true. But both could very well be false. • On the other hand the following passage containing IF-Then statement is an argument If country A is developing nuclear weapons, then country A is a threat to world peace. Country A is developing nuclear weapons. Therefore, country A is a threat to world peace. Antecedent Consequent
- If our borders are porous, then terrorists can enter the country at will. If terrorists can enter the country at will, then all of us are less secure. Therefore, if our borders are porous, then all of us are less secure. • The relation between conditional statements and arguments may now be summarized as follows: 1. A single conditional statement is not an argument. 2. A conditional statement may serve as either the premise or the conclusion (or both) of an argument. • Conditional statements are especially important in logic because they express the relationship between necessary and sufficient conditions. • A is said to be a sufficient condition for B whenever the occurrence of A is all that is needed for the occurrence of B. For example, being a dog is a sufficient condition for being an animal. • On the other hand, B is said to be a necessary condition for A whenever A cannot occur without the occurrence of B. Thus, being an animal is a necessary condition for being a dog.
- If X isa dog,then X isan animal. (beinga dogis sufficient for beingananimal) If X isan animal,then X may or may not be a dog (not sufficient further evidence isneededto determine) If X isnot an animal,then X isnot a dog. (beingan animalis necessaryfor beinga dog) • Dog in the box example: Dog Animal Animal Dog? Cat? Racoon?
- Recognizing Arguments • Every argument is a structured cluster of propositions but, not every structured cluster of propositions is an argument • E.g. • Leonardo da Vinci understood described and illustrated the principles of hydrodynamics, gross anatomy, physics and astronomy, He invented the helicopter, the armoured tank, and the submarine. He painted like an angel and despite being phobic about deadline wrote often and well. In addition, according to Vasari, he was drop dead gorgeous. And he generated all the near magical accomplishments from a behind a curtain of personal discretion so dense and insulating that no historian or psychologist has been able to pull it aside to reveal the person behind the personage. • There is NO Argument here! • Therefore for a structured cluster of propositions to be an argument two conditions must be fulfilled: 1. At least one of the statements must claim to present evidence or reasons … the premise 2. There must be a claim that the alleged evidence supports or implies something—that is, a claim that something follows from the alleged evidence or reasons …. the conclusion
- • The first condition expresses a factual claim, and deciding whether it is fulfilled often falls outside the domain of logic. • This second condition expresses what is called an inferential claim which says that something supports or implies something, or that something follows from something. • E.g. Since Imran Khan is the president of Pakistan Therefore he can bring the desired change in the country Factual claim (we are not concerned whether it is fulfilled or not) Inferential claim (we are concerned with this)
- Explicit & Implicit Inferential Claims • Inferential claim can be Explicit or Implicit • Explicit Inferential claims are usually asserted by premise indicator and conclusion indicators: • Premise Indicators: • Conclusion Indicators: Seeing that tortured prisoners will say anything just to relieve the pain. Consequently, torture is not a reliable method of interrogation. since in that seeing that as indicated by may be inferred from for the reason that because as in as much as for given that owing to therefore accordingly entails that wherefore we may conclude hence thus it must be that it follows that consequently for this reason implies that we may infer so as a result
- • Implicit Inferential Claims: Some arguments contain no indicators. With these, the reader/listener must ask such questions as: What single statement is claimed (implicitly) to follow from the others? What is the arguer trying to prove? What is the main point in the passage? The answers to these questions should point to the conclusion. • The space program deserves increased expenditures in the years ahead (c). Not only does the national defence depend on it (p1), but the program will more than pay for itself in terms of technological spinoff s (p2). Furthermore, at current funding levels the program cannot fulfil its anticipated potential (p3) No premise or conclusion indicators are used in the above passage but it is a valid argument in which the inferential claims are made implicitly
- Not every structured proposition is an argument! • Simple Non-inferential passages: Those that lack a claim that anything is being proved e.g.: • A Warning • A piece of advice • A statement of belief or opinion • A report: to provide information on something • Illustrations: to show what something means or how something is done • Conditional Statements: as discussed before • An Explanation: that aims to shed light on some event or phenomenon
- Explanations and Arguments Premises Conclusion Claimed Evidence What is claimed to follow from the evidence Inference drawn Explanations Explanandum Claimed to shed light on Accepted Fact To distinguish explanations from arguments, identify the statement that is either the explanandum or the conclusion (usually this is the statement that precedes the word “because”). If this statement describes an accepted matter of fact, and if the remaining statements purport to shed light on this statement, then the passage is an explanation. E.g. The sky appears blue from the earth’s surface because light rays from the sun are scattered by particles in the atmosphere.
- Exercise I. Each of the following passages contains a single argument. Using the letters “P” and “C,” identify the premises and conclusion of each argument, writing premises first and conclusion last: 1. Titanium combines readily with oxygen, nitrogen, and hydrogen, all of which have an adverse effect on its mechanical properties. As a result, titanium must be processed in their absence. 4. When individuals voluntarily abandon property, they forfeit any expectation of privacy in it that they might have had. Therefore, a warrantless search or seizure of abandoned property is not unreasonable under the Fourth Amendment. 7. It really does matter if you get enough sleep. We need sleep to think clearly, react quickly, and create memories. Studies show that people who are taught mentally challenging tasks do better after a good night’s sleep. Other research suggests that sleep is needed for creative problem solving. 10. Punishment, when speedy and specific, may suppress undesirable behaviour, but it cannot teach or encourage desirable alternatives. Therefore, it is crucial to use positive techniques to model and reinforce appropriate behaviour that the person can use in place of the unacceptable response that has to be suppressed. 16. Radioactive fallout isn’t the only concern in the aftermath of nuclear explosions. The nations of planet Earth have acquired nuclear weapons with an explosive power equal to more than a million Hiroshima bombs. Studies suggest that explosion of only half these weapons would produce enough soot, smoke, and dust to blanket the Earth, block out the sun, and bring on a nuclear winter that would threaten the survival of the human race.
- 1. P: Titanium combines readily with oxygen, nitrogen, and hydrogen, all of which have an adverse effect on its mechanical properties. C: Titanium must be processed in their absence. 4. P: When individuals voluntarily abandon property, they forfeit any expectation of privacy in it that they might have had. C: A warrantless search and seizure of abandoned property is not unreasonable under the Fourth Amendment. 7. P1: We need sleep to think clearly, react quickly, and create memories. P2: Studies show that people who are taught mentally challenging tasks do better after a good night’s sleep. P3: Other research suggests that sleep is needed for creative problem solving. C: It really does matter if you get enough sleep. 10. P1: Punishment, when speedy and specific, may suppress undesirable behaviour. P2: Punishment cannot teach or encourage desirable alternatives. C: It is crucial to use positive techniques to model and reinforce appropriate behaviour that the person can use in place of the unacceptable response that has to be suppressed. 16. P1: The nations of planet earth have acquired nuclear weapons with an explosive power equal to more than a million Hiroshima bombs. P2: Studies suggest that explosion of only half these weapons would produce enough soot, smoke, and dust to blanket the earth, block out the sun, and bring on a nuclear winter that would threaten the survival of the human race. C: Radioactive fallout isn’t the only concern in the aftermath of nuclear explosions.
- Deductive and Inductive Arguments • Every argument makes an inferential claim … the claim that its premises provide grounds for the truth of its conclusion. • But there are two very different ways in which the conclusion is supported by its premises: • Hence we have two different classes of arguments: Deductive arguments and Inductive Arguments • Whether an argument is Deductive or Inductive depends upon the strength of the inferential claim that the argument makes
- Deductive Arguments • Deductive Arguments make a claim that its conclusion is supported by its premises conclusively. • Which means a deductive argument is an argument incorporating the claim that it is impossible for the conclusion to be false given that the premises are true. • Deductive arguments are those that involve necessary reasoning. • E.g. Socrates is a man All men are mortal Therefore, it necessarily follows that Socrates is mortal
- Inductive Arguments • In inductive arguments no claim of conclusiveness is made • Which means even if the premises of inductive arguments are true they do not support its conclusions with certainty • That is an inductive argument is an argument incorporating the claim that it is improbable that the conclusion be false given that the premises are true. • E.g. Most actors are celebrities Xyz is an actress Therefore xyz is a celebrity
- Validity, Truth, Soundness, Strength, Cogency • Remember we talked about the Truth value of a proposition which means a proposition is either true or false • Similarly a deductive argument is either Valid or Invalid. • Valid deductive argument: is an argument in which it is impossible for the conclusion to be false given that the premises are true • That is if a conclusion of a deductive argument does follow from its premises with strict necessity then that deductive argument is valid • E.g. All television networks are media companies. NBC is a television network. Therefore, NBC is a media company.
- Media Company Television Company NBC Valid
- • Invalid deductive argument: is a deductive argument in which it is possible for the conclusion to be false given that the premises are true. • In these arguments the conclusion does not follow with strict necessity from the premises, even though it is claimed to. • E.g. All banks are financial institutions. Wells Fargo is a financial institution. Therefore, Wells Fargo is a bank.
- Financial Institution Banks Wells Fargo Dotted line indicates that it may or may not be the case Invalid
- • Important point to note about validity of an argument: validity is something that is determined by the relationship between premises and conclusion. The question is not whether the premises and conclusion are true or false, but whether the premises support the conclusion. Therefore a deductive argument that has false premises and/or a false conclusion can still be valid: All automakers are computer manufacturers. United Airlines is an automaker. Therefore, United Airlines is a computer manufacturer. Similarly a deductive argument with all true premises and conclusion can be invalid All banks are financial institutions. Wells Fargo is a financial institution. Therefore, Wells Fargo is a bank. False False False VALID Argument True True True INVALID Argument
- Deductive Args Valid Invalid True Premises True Conclusion All Primates have hearts All humans are primates Therefore all humans have hearts If I owned all the currency in the reserve bank of Pakistan, then I would be wealthy I do not own all the currency in the reserve bank of Pakistan Therefore I am not wealthy True Premises False Conclusion If Imran Khan owned all the currency in the state bank of Pakistan, then he would be wealthy Imran Khan does not own all the currency in the state bank of Pakistan Therefore Imran Khan is not wealthy False Premises True Conclusion All insects are primates All humans are insects Therefore all humans are primates All primates have horns All humans have horns Therefore all humans are primates False Premises False Conclusion All bipeds have horns All snakes are bipeds Therefore all snakes have horns All primates have horns All humans have horns Therefore all primates are humans The empty cell shows that any deductive argument having true premises and a false conclusion is must be invalid Sound
- • A sound argument is a deductive argument that is valid and has all true premises • An Unsound argument is a deductive argument that is invalid, has one or more false premises, or both Sound Argument Valid Argument All True Premises
- • Just as deductive arguments are valid or invalid, inductive arguments are strong or weak • Strong Inductive Argument: is an inductive argument in which it is improbable that the conclusion be false given that the premises are true • In such arguments, the conclusion does in fact follow probably from the premise but not with certainty as is the claim made in deductive arguments. No such claim is made in inductive arguments. • E.g. All dinosaur bones discovered to this day have been at least 50 million years old. Therefore, probably the next dinosaur bone to be found will be at least 50 million years old. E.g. 2 All meteorites found to this day have contained salt. Therefore, probably the next meteorite to be found will contain salt.
- • Weak inductive argument is an argument in which the conclusion does not follow probably from the premises, even though it is claimed to • E.g. Mr Ali is from Pakistan and he is dark skinned Mr Imran is also from Pakistan Therefore probably Mr Imran will also be dark skinned Through experience we know that is not true therefore it is a weak inductive argument E.g. During the past fifty years, inflation has consistently reduced the value of the American dollar. Therefore, industrial productivity will probably increase in the years ahead. Discuss why inductive arguments are important in everyday life No relation
- Inductive Args Strong Weak True Premises Probably true Conclusion All previous US presidents were older than 40 Therefore, probably the next US president will be older than 40 A few US presidents were lawyers Therefore, probably the next US president will probably be older than 40 True Premises Probably false Conclusion A few US presidents were unmarried Therefore, probably the next US president will unmarried False Premises Probably true Conclusion All previous US presidents were TV debaters Therefore, probably the next US president will be a TV debater A few US presidents were dentists Therefore, probably the next US president will be a TV debater False Premises Probably false Conclusion All previous US presidents died in office Therefore, probably the next US president will die in office A few US presidents were dentists Therefore, probably the next US president will be a dentist Cogent
- Note on deductive and inductive arguments • A deductive argument is either valid or invalid. There is no middle ground. • No new information added to a deductive argument can make it more valid or more invalid. • The strength or weakness of inductive argument is a matter of degree. To be considered strong, an inductive argument must have a conclusion that is more probable than improbable. Otherwise it’s a weak argument. • But as opposed to deductive arguments the inductive arguments can become more stronger or weaker by introducing new information. • E.g. Most actors are celebrities Anna Das is an actor Therefore, Anna Das is a celebrity Most actors are celebrities Anna Das is a theatre actor Most theatre actors are not celebrities Therefore, Anna Das is a celebrity Strong Inductive argument becomes weak inductive argument By adding new information
- Most actors are celebrities Anna Das is an actor Therefore, Anna Das is a celebrity Most actors are celebrities Anna Das is an actor Anna Das is also a cabinet minister Therefore, Anna Das is a celebrity By adding new information Strong Inductive argument becomes even stronger inductive argument Similarly we an make an argument stronger by adding more information
- • A cogent argument is an inductive argument that is strong and has all true premises. Also, the premises must be true in the sense of meeting the total evidence requirement. • An Uncogent argument is an inductive argument that is weak, has one or more false premises, fails to meet the total evidence requirement, or any combination of these. Cogent Argument Strong Argument All True Premises
- Summary of how to evaluate a deductive and inductive arguments • For both deductive and inductive arguments, two separate questions need to be answered: (1) Do the premises support the conclusion? (2) Are all the premises true? • To answer the first question we begin by assuming the premises to be true. • Then, for deductive arguments we determine whether, in light of this assumption, it necessarily follows that the conclusion is true. If it does, the argument is valid; if not, it is invalid. • For inductive arguments we determine whether it probably follows that the conclusion is true. If it does, the argument is strong; if not, it is weak. • For inductive arguments we keep in mind the requirements that the premises actually support the conclusion and that they not ignore important evidence. • Finally, if the argument is either valid or strong, we turn to the second question and determine whether the premises are actually true. If all the premises are true, the argument is sound (in the case of deduction) or cogent (in the case of induction). • All invalid deductive arguments are unsound, and all weak inductive arguments are uncogent.
- Statement True False Group of Statements Arguments Non-Arguments Deductive Inductive Deductive Arguments Valid Invalid (all are unsound) Sound Unsound Inductive Arguments Strong Weak (all are uncogent) Cogent Uncogent
- Exercise I. The following arguments are deductive or Inductive. Determine whether each is valid or invalid/ strong or weak. Furthermore, determine whether the argument is sound or unsound/ cogent or uncogent. 1. Since Moby Dick was written by Shakespeare, and Moby Dick is a science fiction novel, it follows that Shakespeare wrote a science fiction novel. 4. The longest river in South America is the Amazon, and the Amazon flows through Brazil. Therefore, the longest river in South America flows through Brazil. 10. Every province in Canada has exactly one city as its capital. Therefore, since there are thirty provinces in Canada, there are thirty provincial capitals. 1. The grave marker at Arlington National Cemetery says that John F. Kennedy is buried there. It must be the case that Kennedy really is buried in that cemetery. 7. People have been listening to rock and roll music for over a hundred years. Probably people will still be listening to it a year from now. 10. Coca-Cola is an extremely popular soft drink. Therefore, probably someone, somewhere, is drinking a Coke right this minute.
- 1. Valid, unsound; false premises, false conclusion. 4. Valid, sound; true premises, true conclusion. 10. Valid, unsound; false premise, false conclusion. 1. Strong, cogent; true premise, probably true conclusion. 7. Strong, uncogent; false premise, probably true conclusion. 10. Strong, cogent; true premise, probably true conclusion.
- AnalysingArguments Arguments in everyday life are more complex: • the number of premises may vary • the order of the premises may vary • premises may be repeated using different words • premises may be left out etc. We can use two techniques for analysing arguments: 1. Paraphrasing 2. Diagraming
- 1. Paraphrasing • Consider the following example: Archimedes will be remembered when Aeschylus is forgotten, because languages die but mathematical ideas do not • Now this single sentence can be analysed as follows by bringing forward the unstated premises and inferences: 1. Languages die 2. The plays of Aeschylus are in a language 3. So the work of Aeschylus will eventually die 4. Mathematical ideas are permanent and therefore never die 5. The work of Archimedes was with mathematical ideas 6. So the work of Archimedes will not die Therefore Archimedes will be remembered when Aeschylus is forgotten
- • Another example of paraphrasing: Geese are migratory waterfowl, so they fly south for winter This argument has a missing premise: Migratory waterfowl fly south of winter The argument can now be rephrased as: All Geese are migratory waterfowl All migratory waterfowls are birds that fly south for winter Therefore, all geese are birds that fly south for winter
- 2. Diagramming • Logical analysis of arguments found in editorials, essays, letters etc. is very difficult because they involve • Non-arguments: such as explanations, advise, illustrations, statements of opinions etc. • Lengthy arguments involve complex arrangements of sub-arguments that feed into the main argument. • Some argumentative passages involve completely separate strands of argumentation that lead to separate arguments. • Diagraming helps facilitate these extended arguments by assigning numerals to various statements in a passage and use arrows to represent the inferential link
- Consider all these various forms: The contamination of underground aquifers represents a pollution problem of catastrophic proportions. Half the nations dirking water, which comes from these aquifers, is poisoned by chemical wastes dumped into the soil for generations. This arguments is diagrammed as: This diagram says that the statement , the premises, support statement , the conclusion 1 2 1 2 2 1
- The selling of human organs such as heart, kidney, and cornea should be outlawed. Allowing human organs to be sold will inevitably lead to a situation where the only the rich can afford transplants. This is so, because whenever something scarce is bought and sold as a commodity, the price always goes up. The law of supply and demand requires it. The argument is drawn as follows: 1 2 3 4 4 3 2 1 Vertical Pattern
- The selling of human organs such as heart, kidney, and cornea should be outlawed. If this practise is allowed to foothold people in dire financial straits will start selling their own organs to pay for their bills. Alternatively, those will criminal bent will take to killing healthy young people and selling their organs on the market. In the financial analysis, buying and selling human organs comes close to buying and selling of life itself. The diagram says that statement the conclusion, is supported by premises Indipendantly 1 2 3 4 1 2 3 4 Horizontal Pattern 1 2 3 4
- Getting poor people off welfare rolls require that we modify their behaviour pattern. The vast majority of people on welfare are high school dropouts, single parents, or people who abuse alcohol and drugs. These behaviour patterns frustrate any desire poor people may have to get a job and improve their condition in life. Statement 2 and 3 do not support the conclusion 1 independently. But when combined they do support the conclusion. Premises 2 and 3 are dependant upon each other for their support of the conclusion 1. 1 2 3 1 2 3 Conjoint Premises
- Dropping out of school and bearing children outside of marriage are two of the primary causes of poverty in this country. Therefore, to eliminate poverty we must offer incentives for people to get high school diplomas. Also, we must find some way to encourage people to get married before they start having children Statement 1, the premise, supports both statements 2 and 3, the conclusions 1 2 3 1 2 3 Multiple Conclusions
- Complex Arguments Government mandates for zero-emission vehicles won’t work because only electric cars qualify as zero-emission vehicles, and electric cars won’t sell. They are too expensive, their range of operation is too limited, and recharging facilities are not generally available. 1 2 3 4 5 6 1 2 3 4 5 6 Example taken from newspaper editorial
- Rhinos in Kenya are threatened with extinction because poachers are killing them for their horn. Since the rhino has no natural predators, it does not need its horn to survive. Thus there should be an organized program to capture rhinos in the wild and remove their horn. Such a program would eliminate the incentive of the poachers. 1 2 3 4 5 6 1 2 3 4 5 6 Example from a letter to the editor
- Skating is a wonderful form of exercise and relaxation, but today’s rollerbladers are a growing menace and something should be done to control them. Rollerbladers are oblivious to traffic regulations as they breeze through red lights and skim down the wrong way on one-way streets. They pose a threat to pedestrians because a collision can cause serious injury. Rollerbladers are even a hazard to shopkeepers as they zoom through stores and damage merchandise. 1 2 3 4 5 6 7 8 9 10 11 1 23 4 5 610 7 811 9 is merely an introductory sentence Example from magazine article
- We can expect small changes to occur in the length of our calendar year for an indefinite time to come. This is true for two reasons. First, the rotation of the earth exhibits certain irregularities. And why is this so? The rotation of any body is affected by its distribution of mass, and the earth’s mass distribution is continually subject to change. For example, earthquakes alter the location of the tectonic plates. Also, the liquid core of the earth sloshes as the earth turns, and rainfall redistributes water from the oceans. The second reason is that the motion of the tides causes a continual slowing down of earth’s rotation. Tidal motion produces heat, and the loss of this heat removes energy from the system. 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 10 7 8 11 9 12 Are premise indicators and do not add any support
- Exercise The following arguments were abstracted from newspaper articles, editorials, and letters to the editor. Use the method presented in this section to construct argument patterns. If a statement is redundant or plays no role in the argument, do not include it in the pattern. 1 The conditions under which many food animals are raised are unhealthy for humans. 2 To keep these animals alive, large quantities of drugs must be administered. 3 These drugs remain in the animals’ flesh and are passed on to the humans who eat it. 1 When parents become old and destitute, the obligation of caring for them should be imposed on their children. 2 Clearly, children owe a debt to their parents. 3 Their parents brought them into the world and cared for them when they were unable to care for themselves. 4 This debt could be appropriately discharged by having grown children care for their parents. 1 There is a lot of pressure on untenured college teachers to dumb down their courses. 2 Administrators tend to rehire teachers who bring in more money, and 3 teachers who dumb down their classes do precisely this. Why? Because 4 easier classes attract more students, and 5 more students means more money for the school.
- CATEGORICAL PROPOSITIONS • A proposition (statement) that relates two classes or categories is called a categorical proposition • By a class we mean a collection of all objects having some specified characteristics in common • Such as ‘humans’ or ‘doctors’ or ‘junk food’ or ‘school cafeterias’ or ‘happy endings’ or ‘magazine’ or ‘students learning logic’ etc. • Therefore a categorical proposition is one which relates two classes, one denoted by subject term ‘S’ and the other by the predicate term ‘P’ , by asserting that the whole or part of the class ‘S’ is included in or excluded from class ‘P’
- • Therefore there are exactly 4 ways in which categorical propositions relate two classes: 1. All of one class ‘S’ are included all of the other class ‘P’: • E.g. The class of all dogs is included in the class of all mammals 2. All of one class ‘S’ are excluded from all of the other class ‘P’: i.e. they have no members in common • E.g. The class of triangles is excluded from the class of circles 3. Part of the subject class ‘S’ is included in the predicate class ‘P’: • E.g. The class of all chess players is partially included in the class of females 4. Part of the subject class ‘S’ is excluded from the predicate class ‘P’: • E.g. The class of all animals is partially excluded from the class of mammals • Hence we have exactly 4 kinds of categorical propositions:
- The 4 kinds of standard form of categorical propositions • A. - Universal Affirmative • All S are P • All politicians are liars • E. - Universal Negative • No S are P • No politicians are liars • I. - Particular Affirmative • Some S are P • Some politicians are liars • O. - Particular Negative • Some S are not P • Some politicians are not liars
- Further points to note about categorical propositions 1. The four categorical propositions A, E, I, O are the building blocks of the deductive arguments 2. Quantifiers: The words “all,” “no,” and “some” are called quantifiers because they specify how much of the subject class is included in or excluded from the predicate class. 3. In formal deductive logic ‘some’ means ‘at least’ one 4. The letters S and P stand respectively for the subject and predicate terms 5. The words “are” and “are not” are called the copula because they link (or “couple”) the subject term with the predicate term. 6. “subject term” and “predicate term” do not mean the same thing in logic that “ subject” and “predicate” mean in grammar
- • For example consider the following example: All members of the Pakistan Medical Association are people holding degrees from recognized academic institutions. • This standard-form categorical proposition is analysed as follows: quantifier: all subject term: members of the Pakistan Medical Association Copula : are (because they link or couple the subject and predicate) predicate term: people holding degrees from recognized academic institutions Quantifier (subject term) copula (predicate term) All s are p Some s are p No s are p Some s are not p
- Exercise • Some executive pay packages are insults to ordinary workers. • Some preachers who are intolerant of others’ beliefs are not television evangelists. • All oil-based paints are products that contribute to photochemical smog. • No stressful jobs are occupations conducive to a healthy lifestyle.
- • Quantifier: some; subject term: executive pay packages; copula: are; predicate term: insults to ordinary workers. • Quantifier: some; subject term: preachers who are intolerant of others’ beliefs; copula: are not; predicate term: television evangelists. • Quantifier: All; Subject term: Oil based paints; Copula: are Predicate term: products that contribute to photochemical smog • Quantifier: No; Subject Term: Stressful jobs; Copula: are; Predicate term: occupations conducive to healthy life style
- Quality, Quantity and Distribution • Quality and quantity are attributes of categorical propositions. To understand these concepts consider the following example: Proposition Meaning in class notation All S are P. Every member of the S class is a member of the P class; that is, the S class is included in the P class. No S are P. No member of the S class is a member of the P class; that is, the S class is excluded from the P class. Some S are P. At least one member of the S class is a member of the P class. Some S are not P. At least one member of the S class is not a member of the P class.
- Quality: The quality of a categorical proposition is either affirmative or negative depending on whether it affirms or denies class membership. Accordingly, “All S are P” and “Some S are P” have affirmative quality, and “No S are P” and “Some S are not P” have negative quality. These are called affirmative propositions and negative propositions, respectively. Quantity: The quantity of a categorical proposition is either universal or particular, depending on whether the statement makes a claim about every member or just some member of the class denoted by the subject term. “All S are P” and “No S are P” each assert something about every member of the S class and thus are universal propositions. “Some S are P” and “Some S are not P” assert something about one or more members of the S class and hence are particular propositions. Imp note: ‘All’ and ‘No’ imply universal quantity wile ‘Some’ imply particular quantity. But categorical propositions have no “qualifier”. In universal propositions the quality is determined by the quantifier, and in particular propositions it is determined by the copula.
- Distribution: • Unlike quality and quantity, which are attributes of propositions, distribution is an attribute of the terms (subject and predicate) of propositions. • A proposition distributes a term if it refers to all members of the class designated by that term; otherwise it is undistributed. I.e. If a term is distributed it says something about every member of that term. A: All S are P ……. S is distributed and P is not E: No S are P ……. Both S and P are distributed I: Some S are P …… Neither S nor P is distributed O: Some S are not P ….. P is distributed and S is not P S P S *S P *S P
- Proposition Letter name Quantity Quality Terms distributed All S are P A universal affirmative S No S are P E universal negative S and P Some S are P I particular affirmative none Some S are not P O particular negative P
- Exercise I. For each of the following categorical propositions identify the letter name, quantity, and quality. Then state whether the subject and predicate terms are distributed or undistributed. • No vampire movies are films without blood. • Some Chinese leaders are not thoroughgoing opponents of capitalist economics • Some hospitals are organizations that overcharge the Medicare program. II. Change the quality but not the quantity of the following statements. • All drunk drivers are threats to others on the highway. • Some CIA operatives are not champions of human rights.
- III. Change the quantity but not the quality of the following statements. • All owners of pit bull terriers are people who can expect expensive lawsuits. • Some residents of Manhattan are not people who can afford to live there. IV. Change both the quality and the quantity of the following statements. • All oil spills are events catastrophic to the environment. • Some corporate lawyers are not people with a social conscience.
- I. • E proposition, universal, negative, subject and predicate terms are distributed. • O proposition, particular, negative, subject term undistributed, predicate term distributed. • I proposition, particular, affirmative, subject and predicate terms undistributed. II. • No drunk drivers are threats to others on the highway. • Some CIA operatives are champions of human rights. III. • Some owners of pit bull terriers are people who can expect expensive lawsuits. • No residents of Manhattan are people who can afford to live there. IV. • Some oil spills are not events catastrophic to the environment. • All corporate lawyers are people with a social conscience.
- Drawing Immediate Inferences • Immediate inference is one where the conclusion is drawn directly from only one premise E.g. If it is true that: (A) All horses have 4 legs Therefore, it is true that (I) some horses have 4 legs If it is true that: (A) All horses have 4 legs Therefore, it is false that: (E) no horses have 4 legs • There are a number of relations amongst the 4 propositions that allow us to draw immediate inferences some directly and some through manipulation • These relationships gives us solid grounds for a great deal of reasoning we do in everyday life • Let us have a look at these relationships
- Conversion, Obversion, and Contraposition • Conversion, Obversion, and Contraposition are operations that can be performed on a categorical proposition, resulting in a new statement that may or may not have the same meaning and truth value as the original statement. • They are used for drawing immediate inferences • Venn diagrams are used to determine how the two statements relate to each other. Conversion • It is the simplest of the three operations and is obtained by switching the subject term with the predicate term of a categorical proposition • For example, if the statement “No foxes are hedgehogs” is converted, the resulting statement is “No hedgehogs are foxes.” • This new statement is called the converse of the given statement. • To see how the four types of categorical propositions relate to their converse, compare the following sets of Venn diagrams:
- X X X Given Statement Form Converse . All S are P All P are S No S are P No P are S X Some S are P Some P are S Some P are not SSome S are not P PS PS PS
- S P Switch (Copula)(Quantifier) Conversion We can see from above that Conversion is not valid for A & O statements and applying conversion to them will yield new statements. But Conversion is valid for I & E statements so converting them yields the same truth value as the original statements. Therefore through conversion we can get the following valid immediate inferences: No A are B Therefore, No B are A Some A are B Therefore, some B are A Converting A & O statements, on the other hand, commits Fallacy of Illicit Conversion: All cats are animals (T) Some animals are not dogs (T) Therefore, All animals are cats (F) Therefore, Some dogs are not animals (F)
- Obversion Obversion requires two steps: (1) changing the quality (without changing the quantity), and (2) replacing the predicate with its term complement. Class compliments and Term Compliment Non-Dogs Dogs
- S P(Copula)(Quantifier) Obversion Change Quality Replace with term compliment
- X X X Given Statement Form Obverse . All S are P No S are non-P No S are P All S are non-P Some S are P Some S are not non-P Some S are non-PSome S are not P PS PS PS PS X
- Obverse of all statements is equivalent to their original statements and holds true for all: All A are B. Some A are B. Therefore, no A are non-B Therefore, some A are not non-B. e.g. All horses are animals Some trees are maples No horses are non-animals Some trees are not non-maples No A are B Some A are not B Therefore, all A are non-B Therefore, some A are non-B. e.g. No cats are dogs Some humans are not women All cats are non-dogs Some humans are non-women
- Contraposition Like Obversion, contraposition requires two steps: 1. Switching the subject and predicate terms and 2. Replacing the subject and predicate terms with their term complements. S P Switch & replace with (Copula)(Quantifier) term compliment
- X X X Given Statement Form Contrapositive . All S are P All non-P are non-S No S are P X Some S are P Some S are not P PS PS PS PS PS No non-P are non-S Some non-P are non-S Some non-P are not non-S
- • As seen from the above Venn diagrams of all the 4 propositions only the contrapositives of A and O statements give the same truth value as original statements and hence immediate inferences for them are valid: All dogs are animals Therefore, All non-animals are non-dogs • Whereas the contrapositives of E and I statements are not equivalent to their original statements and therefore commit the fallacy of Illicit Contraposition: E: No dogs are cats. (True) Therefore, no non-cats are non-dogs. (False) I: Some animals are non-cats. (True) Therefore, some cats are non-animals. (False)
- Summary of Conversion, Obversion, & Contraposition Given Statement Converse Truth Value E: No A are B. No B are A. Same truth value as given I: Some A are B. Some B are A. statement A: All A are B. All B are A. Undetermined truth value O: Some A are not B. Some B are not A. Given Statement Obverse Truth Value A: All A are B. No A are non-B. E: No A are B. All A are non-B. Same truth value as given statement I: Some A are B. Some A are not non-B. O: Some A are not B. Some A are non-B. Given Statement Contrapositive Truth Value A: All A are B. All non-B are non- A. Same truth value as given O: Some A are not B. Some non-B are not non- A. statement E: No A are B. No non-B are non- A. Undetermined truth value I: Some A are B. Some non-B are non- A.
- Exercise Part I Exercises 1 through 6 provide a statement, its truth value in parentheses, and an operation to be performed on that statement. Supply the new statement and the truth value of the new statement: Given statement Operation New statement Truth Value 1. No A are non-B. (T) conv. 2. Some A are B. (T) contrap. 3. All A are non-B. (F) obv. 4. All non-A are B. (F) contrap. 5. Some non-A are not B. (T) conv. 6. Some non-A are non-B. (T) obv.
- Key Part I Given statement Operation New statement Truth Value 1. No A are non-B. (T) conv. No non-B are A T 2. Some A are B. (T) contrap. Some non-A are non-B Undermined 3. All A are non-B. (F) obv. No A are B F 4. All non-A are B. (F) contrap. All non-B are A F 5. Some non-A are not B. (T) conv. Some B are not non-A Undermined 6. Some non-A are non-B. (T) obv. Some non-A are not B T
- Exercise Part II Exercises 7 through 12 provide a statement, its truth value in parentheses, and a new statement. Determine how the new statement was derived from the given statement and supply the truth value of the new statement. Given statement Operation New statement Truth Value 7. No non-A are non-B. (F) No B are A. 8. Some A are not non-B. (T) Some A are B. 9. All A are non-B. (F) All non-B are A. 10. No non-A are B. (F) All non-A are non-B. 11. Some non-A are not B. (T) Some non-B are not A. 12. Some A are non-B. (F) Some non-B are A.
- Key Part II Given statement Operation New statement Truth Value 7. No non-A are non-B. (F) Contrap. No B are A. Undermined 8. Some A are not non-B. (T) Obv. Some A are B. T 9. All A are non-B. (F) Conv. All non-B are A. Undermined 10. No non-A are B. (F) Obv. All non-A are non-B. F 11. Some non-A are not B. (T) Contrap. Some non-B are not A. T 12. Some A are non-B. (F) Conv. Some non-B are A. F
- The Traditional Square of Opposition A E I O T F T F Contrary Subcontrary Subalternation Subalternation
- The four relations in the traditional square of opposition may be characterized as follows: Contradictory = opposite truth value Contrary = at least one is false (not both true) Subcontrary = at least one is true (not both false) Subalternation = truth flows downward, falsity flows upward The Contradictory relation expresses opposition thus: If A is true = O is false If A is false = O is true If O is true = A is false If O is false = A is true Same relations hold between E and I A: All dogs are animals. (T) O: Therefore, it is false that some dogs are not animals. (F) I: Some humans are aliens (F) E: Therefore, it is true that no humans are aliens (T)
- The Contrary relation expresses only partial opposition between A & E propositions …. Both statements cannot be true at the same time: at least one is false If A is true = then E is false If E is true = then A is false But If A is false = then E can either be true or false …. Thus E is undetermined If E is false = then A can either be true or false …. Thus A is undetermined The following statements can both be false but both cannot be true so if one is true the other is false: India will win this match against Pakistan Pakistan will win this match against India A All cats are animals (T) E Therefore, it is false that no cats are animals (F) E No cats are dogs (T) A All cats are dogs (F) A All animals are cats (F) E No animals are cats (F)
- The Subcontrary relation also expresses only partial opposition between I & O propositions …. But here Both statements cannot be false at the same time : at least one is True If I is False = then O is true If O is false = then I is true But If I is true = then O can either be true or false …. Thus O is undetermined If O is true= then I can either be true or false …. Thus I is undetermined The following statements can both be true but both cannot be false so if one is false the other is true: Some diamonds are precious stones Some diamonds are not precious stones I Some cats are dogs (F) O Some cats are not dogs (T) O Some cats are not animals (F) I Some cats are animals (T)
- The Subalternation relation expresses corresponding relation between the A & I as well as between E & O propositions Thus the A proposition: All spiders are 8 legged creatures (T) Has a corresponding I proposition: Some spiders are 8 legged creatures (T) Similarly the E proposition: No whales are fishes (T) Has the following O proposition: Some whales are not fishes (T) Only the Truth flows from top to bottom thus: If A is true = I is true & If E is true = then O is true But if A is false = I can be true or false & If E is false = then O can be true or false Only the Falsity flows from bottom to top thus: If I is False = then A is false & If O is false = then E is false But if I is True = then A can be true or false & If O is True = then E can be true or false
- Formal Fallacies • Fallacy is a type of argument that seems to be correct; but that proves on examination not to be so. • There are two types of fallacies: • Formal fallacies are those that can be determined by the form of a categorical syllogism • Informal fallacies are those that can only be determined by the content of the argument and cannot be determined merely by its form Exercise Cover from Book
- Categorical Syllogisms A Syllogism is a deductive argument consisting of three propositions: of which two are premises and one is conclusion Both the premises and the conclusion are one of the A, E, I, O propositions Eg. No Politicians are professors Some doctors are professors Therefore, Some doctors are not politicians • The above syllogism is called categorical because the propositions are categorical • Which means they relate two categories and the syllogism as a whole relates three categories • These categories are called terms:
- • Each of the three terms in a categorical syllogism has its own name depending on its position in the argument. • The major term, by definition, is the predicate of the conclusion, and • the minor term is the subject of the conclusion. • The middle term, which provides the middle ground between the two premises, is the one that occurs once in each premise and does not occur in the conclusion. • Thus, for the argument just given, the major term is ‘politicians’, the minor term is ‘doctors’, and the middle term is ‘professors’ • The premise that has the major term is called the ‘Major Premise’, the premise that has the minor term is called the ‘Minor Premise’
- A standard-form categorical syllogism is one that meets the following four conditions: 1. All three statements are standard-form categorical propositions i.e. A, E, I, O 2. Each term is used in the same sense throughout the argument. 3. The major premise is listed first, the minor premise second, and the conclusion last. Note: to put a categorical syllogism into standard form: First identify the conclusion From that identify the major term and the minor terms Based on the above identify the relative major and minor premises Then put the major premise first, minor premise second, and the conclusion last in the following format: 1st: Quantifier __________ copula ________ 2nd: Quantifier __________ copula ________ 3rd: Quantifier __________ copula ________ Major premise (containing major term) Minor premise (containing minor term) Conclusion Minor term Major term
- Mood and Figure The mood of a categorical syllogism consists of the letter names of the propositions that make it up. No painters are sculptors. (E) Some sculptors are artists. (I) Therefore, some artists are not painters. (O) The mood of this standard-form categorical proposition is EIO The figure of a categorical syllogism is determined by the location of the two occurrences of the middle term in the premises. Four different arrangements are possible: Figure 1 M P MS PS Figure 2 MP MS PS Figure 3 M P M S PS Figure 4 MP M S PS
- E.g. No painters are sculptors. (E) Some sculptors are artists. (I) Therefore, some artists are not painters. (O) The mood of the above standard-form categorical proposition is EIO and the figure is 4 Hence the form of the syllogism is EIO-4 After the categorical proposition is put into standard form then its validity or invalidity maybe determined by mere inspection of its form. 1 2 3 4
- Since there are 4 kinds of categorical propositions and there are three categorical propositions in a categorical syllogism, there are 64 possible moods (4 × 4 × 4 = 64). And since there are four different figures, there are 256 different forms of categorical syllogisms (4 × 64 = 256). Of these 256 possible categorical syllogism only 15 are (unconditionally) valid and the rest are invalid: Figure 1 Figure 2 Figure 3 Figure 4 AAA EAE IAI AEE EAE AEE AII IAI AII EIO OAO EIO EIO AOO EIO
- Exercises I The following syllogisms are in standard form. Identify the major, minor, and middle terms, as well as the mood and figure of each. 1. All neutron stars are things that produce intense gravity. All neutron stars are extremely dense objects. Therefore, all extremely dense objects are things that produce intense gravity. 2. No insects that eat mosquitoes are insects that should be killed. All dragonflies are insects that eat mosquitoes. Therefore, no dragonflies are insects that should be killed. 3. No environmentally produced diseases are inherited afflictions. Some psychological disorders are not inherited afflictions. Therefore, some psychological disorders are environmentally produced diseases. 4. No people who mix fact with fantasy are good witnesses. Some hypnotized people are people who mix fact with fantasy. Therefore, some hypnotized people are not good witnesses.
- II II. Put the following syllogisms into standard form, using letters to represent the terms, and name the mood and figure. 1. No Republicans are Democrats, so no Republicans are big spenders, since all big spenders are Democrats. 4. Some insects that feed on milkweed are not foods suitable for birds, inasmuch as no monarch butterflies are foods suitable for birds and all monarch butterflies are insects that feed on milkweed. 5. No illegal aliens are people who have a right to welfare payments, and some migrant workers are illegal aliens. Thus, some people who have a right to welfare payments are migrant workers. 6. Some African nations are not countries deserving military aid, because some African nations are not upholders of human rights, and all countries deserving military aid are upholders of human rights. 7. All pranksters are exasperating individuals, consequently some leprechauns are exasperating individuals, since all leprechauns are pranksters.
- III Reconstruct the syllogistic forms from the following combinations of mood and figure. OAE-3 EIA-4 AII-3 IAE-1 AOO-2 AAA-1 OEA-4 IV. Construct the following syllogisms. 1. An EIO-2 syllogism with these terms: major: dogmatists; minor: theologians; middle: scholars who encourage free thinking. 2. An unconditionally valid syllogism in the first figure with a particular affirmative conclusion and these terms: major: people incapable of objectivity; minor: Supreme Court justices; middle: lockstep ideologues. 3. An unconditionally valid syllogism in the fourth fi gure having two universal premises and these terms: major: teenage suicides; minor: heroic episodes; middle: tragic occurrences. 4. A valid syllogism having mood OAO and these terms: major: things capable of replicating by themselves; minor: structures that invade cells; middle: viruses. 5. A valid syllogism in the fi rst fi gure having a universal negative conclusion and these terms: major: guarantees of marital happiness; minor: prenuptial agreements; middle: legally enforceable documents.
- Explain the whole process of proving validity or invalidity of a standard form categorical syllogism through Venn-Diagrams 1 2 3 4 5 6 7
- Pointers for using Venn Diagrams 1. Marks (shading or placing an X) are entered only for the premises. No marks are made for the conclusion. 2. If the argument contains one universal premise, this premise should be entered first in the diagram. If there are two universal premises, either one can be done first. 3. When entering the information contained in a premise, one should concentrate on the circles corresponding to the two terms in the statement. While the third circle cannot be ignored altogether, it should be given only minimal attention. 4. When inspecting a completed diagram to see whether it supports a particular conclusion, one should remember that particular statements assert two things. “Some S are P” means “At least one S exists and that S is a P”; “Some S are not P” means “At least one S exists and that S is not a P.” 5. When shading an area, one must be careful to shade all of the area in question. Examples:
- 6. The area where an X goes is always initially divided into two parts. If one of these parts has already been shaded, the X goes in the non-shaded part. Examples: --------------------------- If one of the two parts is not shaded, the X goes on the line separating the two parts. Examples: ----------------------------- This means that the X may be in either (or both) of the two areas—but it is not known which one. 7. An X should never be placed in such a way that it dangles outside of the diagram, and it should never be placed on the intersection of two lines.
- Illustrations • EAE-2 (valid) • AEE-1 (invalid) • IAI-4 (valid) • AOO-2 (valid) • IAI-1 (invalid) • AOO-1 (invalid) • AAA-1 (valid) • OIO-1 (invalid)
- Syllogistic Rules and Formal Fallacies • Valid syllogisms confirm to certain rules. • If any of these rules are violated a specific fallacy is committed and the syllogism is invalid. • Since the fallacies is due to the form of the syllogism therefore it is called formal fallacies (as opposed to informal fallacies discussed latter) • These rules can be used to cross-check with Venn-diagram technique discussed above. • Of the five rules presented in this section, the first two depend on the concept of distribution, the second two on the concept of quality, and the last on the concept of quantity.
- Statement Type Terms Distributed A Subject E Subject, predicate I None O Predicate Rule 1: The middle term must be distributed at least once. Fallacy: Undistributed middle. Example: All sharks are fish. All salmon are fish. All salmon are sharks. Rule 2: If a term is distributed in the conclusion, then it must be distributed in a premise. Fallacies: Illicit major; illicit minor. Examples: All horses are animals. Some dogs are not horses. Some dogs are not animals. All tigers are mammals. All mammals are animals. All animals are tigers.
- Rule 3: Two negative premises are not allowed. Fallacy: Exclusive premises. Example: No fish are mammals. Some dogs are not fish. Some dogs are not mammals. Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise. Fallacy: Drawing an affirmative conclusion from a negative premise. or Drawing a negative conclusion from affirmative premises. Examples: All crows are birds. Some wolves are not crows. Some wolves are birds. All triangles are three-angled polygons. All three-angled polygons are three-sided polygons. Some three-sided polygons are not triangles.
- • Note: As a result of the interaction of these first four rules, it turns out that no valid syllogism can have two particular premises. Rule 5: If both premises are universal, the conclusion cannot be particular. Fallacy: Existential fallacy. Example: All mammals are animals. All tigers are mammals. Some tigers are animals. • Before deciding if a syllogism breaks Rule 5 make sure it does not break other 4 rules. • Discuss Aristotelian vs. Boolean standpoints
- Propositional Logic • In Categorical Propositions and Categorical Syllogism letters represented terms • The fundamental elements in Propositional Logic are whole statements (propositions) • And those Statements are represented by letters, and these letters are then combined by means of the operators to form more-complex symbolic representations. • Remember Simple statement is one that does not contain any other statement as a component • Whereas Compound statement is one that contains at least one simple statement as a component. E.g.: • It is not the case that Al Qaeda is a humanitarian organization. It is not the case that A • Dianne Reeves sings jazz, and Christina Aguilera sings pop. D and C • Either people get serious about conservation or energy prices will skyrocket. Either P or E • If nations spurn international law, then future wars are guaranteed. If N then F • The Broncos will win if and only if they run the ball. B if and only if R
- Operator Name Logical function Used to translate ∼ tilde negation not, it is not the case that • dot conjunction and, also, moreover ∨ wedge disjunction or, unless ⊃ horseshoe implication if . . . then . . . , only if ≡ triple bar equivalence if and only if It is not the case that A. ∼A D and C. D • C Either P or E. P ∨ E If N then F. N ⊃ F B if and only if R. B ≡ R Disjuncts, conjuncts, Conditional statement representing material implication, Antecedent, consequent, Bi-conditional statement representing material equivalence,
- Negation (The Tilde) • the tilde is always placed in front of the proposition it negates. All of the other operators are placed between two propositions. • Also, unlike the other operators, the tilde cannot be used to connect two propositions. Thus, G ∼ H is not a proper expression. • But the tilde is the only operator that can immediately follow another operator. Thus, it would be proper to write G • ∼H. • The tilde is used to negate a simple proposition, but it can also be used to negate a compound proposition—for example ∼(G • F). • In this case the tilde negates the entire expression inside the parentheses. • These statements are all negations. The main operator is a tilde. • ∼ B • ∼(G ⊃ H) • ∼[ (A ≡ F) • (C ≡ G) ]
- • The main operator is the operator that has as its scope everything else in the statement. H • (J ∨ K), ∼(K • M), K ⊃ ∼(L • M), • The dot symbol is used to translate such conjunctions as “and”, “also”, “but”, “however”, “yet”, “still”, “moreover”, “although”, “nevertheless”, and “both”: • Tiffany sells jewellery, and Gucci sells cologne. T • G • Tiffany sells jewellery, but Gucci sells cologne. T • G • Tiffany sells jewellery; however, Gucci sells cologne. T • G • Tiffany and Ben Bridge sell jewellery. T • B • These statements are all conjunctions. The main operator is a dot. • K • L • (E v F) • (G v H ) • [ (R ⊃ T ) v (S ⊃ U) ] • [ (W ≡ X ) v (Y ≡ Z ) ]
- • The wedge symbol is used to translate “or”, “unless”, and “either” • Aspen allows snowboards or Telluride does. A ∨ T • Either Aspen allows snowboards or Telluride does. A ∨ T • Aspen allows snowboards unless Telluride does. A ∨ T • Unless Aspen allows snowboards, Telluride does. A ∨ T • These statements are all disjunctions. The main operator is a wedge. • C v D • (F • H) v (K • L) • [S • (T ⊃ U) ] v [X • (Y ≡ Z ) ] • Note that A ∨ T is logically equivalent to T ∨ A. • Also T • G is logically equivalent to G • T.
- • The horseshoe symbol is used to translate “if . . . then . . . ,”, “only if”, “in case”, “provided that”, “given that”, “on condition that”, and “implies that” • The function of “only if” is, in a sense, just the reverse of “if”. For example, the statement “You will catch a fish only if your hook is baited” does not mean “If your hook is baited, then you will catch a fish.” • If it meant this, then everyone with a baited hook would catch a fish. Rather, the statement means “If your hook is not baited, then you will not catch a fish” which is logically equivalent to “If you catch a fish, then your hook was baited”. • To avoid mistakes in translating “if” and “only if” remember this rule: The statement that follows “if” is always the antecedent, and the statement that follows “only if” is always the consequent. Thus, “C only if H” is translated C ⊃ H, whereas “C if H” is translated H ⊃ C. • If Purdue raises tuition, then so does Notre Dame. P ⊃ N • Notre Dame raises tuition if Purdue does. P ⊃ N • Purdue raises tuition only if Notre Dame does. P ⊃ N • Cornell cuts enrolment provided that Brown does. B ⊃ C • Cornell cuts enrolment on condition that Brown does. B ⊃ C • Brown’s cutting enrolment implies that Cornell does. B ⊃ C
- • In translating conditional statements, it is essential not to confuse antecedent with consequent. The statement A ⊃ B is not logically equivalent to B ⊃ A. • H ⊃ J • (A v C ) ⊃ (D • E ) • [K v (S • T ) ] ⊃ [F v (M • O) ] Necessary and Sufficient Conditions • Event A is said to be a sufficient condition for event B whenever the occurrence of A is all that is required for the occurrence of B. • On the other hand, event A is said to be a necessary condition for event B whenever B cannot occur without the occurrence of A. • E.G.: Flu = Feeling miserable (sufficient) and air = alive (necessary) • SUN mnemonic S ⊃ N • Hilton’s opening a new hotel is a sufficient condition for Marriott’s doing so. H ⊃ M • Hilton’s opening a new hotel is a necessary condition for Marriott’s doing so. M ⊃ H
- • The triple bar symbol is used to translate the expressions “if and only if” and “is a sufficient and necessary condition for”: • JFK tightens security if and only if O’Hare does. J ≡ O • JFK’s tightening security is a sufficient and necessary condition for O’Hare’s doing so. J ≡ O • Analysis of the first statement reveals that J ≡ O is logically equivalent to (J ⊃ O) • (O ⊃ J). • The statement “JFK tightens security only if O’Hare does” is translated J ⊃ O, and “JFK tightens security if O’Hare does” is translated O ⊃ J. Combining the two English statements, we have (J ⊃ O) • (O ⊃ J), which is just a longer way of writing J ≡ O. • These statements are all bi-conditionals (material equivalences). The main operator is a triple bar. • M ≡ T • (B v D) ≡ (A • C ) • [K v (F ⊃ I ) ] ≡ [L • (G v H) ]
- Whenever more than two letters appear in a translated statement, we must use parentheses, brackets, or braces to indicate the proper range of the operators. Prozac relieves depression and Allegra combats allergies, or Zocor lowers cholesterol. (P • A) ∨ Z Prozac relieves depression, and Allegra combats allergies or Zocor lowers cholesterol. P • (A ∨ Z) Either Prozac relieves depression and Allegra combats allergies or Zocor lowers cholesterol. (P • A) ∨ Z Prozac relieves depression and either Allegra combats allergies or Zocor lowers cholesterol. P • (A ∨ Z) Prozac relieves depression or both Allegra combats allergies and Zocor lowers cholesterol. P ∨ (A • Z) Prozac relieves depression and Allegra or Zocor lowers cholesterol. P • (A ∨ Z) If Merck changes its logo, then if Pfizer increases sales, then Lilly will reorganize. M ⊃ (P ⊃ L) If Merck’s changing its logo implies that Pfizer increases sales, then Lilly will reorganize. (M ⊃ P) ⊃ L If Schering and Pfizer lower prices or Novartis downsizes, then Warner will expand production. [(S • P) ∨ N] ⊃ W
- • Do not confuse these three statement forms: A if B B ⊃ A A only if B A ⊃ B A if and only if B A ≡ B Further Examples: It is not the case that K or M ∼K ∨ M Not both S and T ∼(S • T) By De Morgan’s rule ∼(S • T) is equivalent to ∼S ∨ ∼T Similarly: Not either S or T ∼(S ∨ T) Which by De Morgan’s rule is equivalent to ∼S • ∼T.
- Further illustrations: • Megan is not a winner, but Kathy is ∼ M • K • Not both Megan and Kathy are winners ∼ (M • K) • Either Megan or Kathy is not a winner ∼ M ∨ ∼ K • Both Megan and Kathy are not winners ∼ M • ∼ K • Not either Megan or Kathy is a winner ∼ (M ∨ K) • Neither Megan nor Kathy is a winner ∼ (M ∨ K) Notice the function of “either” and “both”: • Not either A or B. ∼( A v B ) • Either not A or not B. ∼A v ∼B • Not both A and B. ∼( A • B ) • Both not A and not B. ∼A • ∼B

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