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‘The madman is not the man who has lost his reason. The madman is the man who has lost everything but his reason.’. Reasoning lesson 1. Some logic problems to get you thinking!. How good are you at reasoning?. Uncritical inference test. Knights, knaves and normals.
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‘The madman is not the man who has lost his reason. The madman is the man who has lost everything but his reason.’
Reasoning lesson 1 • Some logic problems to get you thinking!
How good are you at reasoning? • Uncritical inference test
Knights, knaves and normals • On a distant island live three types of humans - Knights, Knaves and Normals. The Knights always tell the truth, the Knaves always lie, and the Normals sometimes lie and sometimes tell the truth.
Knights, knaves and normals • Detectives questioned three inhabitants of the island - Al, Bob, and Clark - as part of the investigation of a terrible crime. The investigators knew that one of the three committed the crime, but did not at first know which one. They also knew that the criminal was a Knight, and that the other two were not. How they knew these things is not important for the solution.
Knights, knaves and normals • Additionally, the investigators made a transcript of the statements made by each of the three men. What follows is that transcript: • Al: I am innocent. • Bob: That is true. • Clark: Bob is not a Normal.
Knights, knaves and normals • After carefully and logically analyzing their information, the investigators positively identified the guilty man. • Was it Al, Bob or Clark?
Possible solutions • Al is the Knight/criminal. • Bob is the Knight/criminal. • Clark is the Knight/criminal.
Fact list • Knights always tell the truth. • Knaves always lie. • Normals sometimes lie and sometimes tell the truth. • Only one of the three men is a Knight. • The guilty man is that Knight. • Al says, "I am innocent." • Bob says, "That is true." • Clark says, "Bob is not a Normal."
Evaluation of possible solutions • If Al is the guilty Knight, his statement is a lie. Since we know it is impossible for Knights to lie, Al cannot be guilty. Therefore, he must be telling the truth. Since there is only 1 Knight, we now know it cannot be Al.
Evaluation of possible solutions • Al is a truth-telling Normal. • If Bob is lying, Al must be guilty. Since we know from evaluating solution #1 that this cannot be, Bob must be telling the truth. Therefore, Bob is either a truth-telling Normal or the guilty Knight.
Evaluation of possible solutions • Bob is either a truth-telling Normal or the guilty Knight. • If Bob is a Normal, then Clark is lying. If Clark is lying, he is either a Knave or a Normal. In either case, nobody would be a Knight. Since we know one of them must be a Knight, Bob cannot be a Normal. Therefore Clark is a truth-telling Normal, and Bob is the guilty Knight
Parking! • At a small company, parking spaces are reserved for the top executives: CEO, president, vice president, secretary, and treasurer with the spaces lined up in that order. The parking lot guard can tell at a glance if the cars are parked correctly by looking at the color of the cars. The cars are yellow, green, purple, red, and blue, and the executives names are Alice, Bert, Cheryl, David, and Enid. • * The car in the first space is red.* A blue car is parked between the red car and the green car.* The car in the last space is purple.* The secretary drives a yellow car.* Alice's car is parked next to David's.* Enid drives a green car.* Bert's car is parked between Cheryl's and Enid's.* David's car is parked in the last space. • 1. Who is the secretary? • 2. Who is the CEO? • 3. What colour is the vice-president’s car?
Lesson 2 • Deductive reasoning - syllogisms
Reasoning This are all examples of gaining knowledge by reasoning. Can you discuss in your groups the benefits of gaining knowledge by this way?
Rationalism Rationalists believe that reason is the most important source of knowledge because it seems to give us some certainty.
The curious incident An expensive racehorse has been stolen. A policeman talks to Sherlock Holmes. I’ve been stolen, that’s why I’ve got such a long face!
The curious incident Does any aspect of the crime strike you as significent Mr Holmes?
The curious incident Does any aspect of the crime strike you as significent Mr Holmes? Yes constable, the curious incident of the dog in the night.
The curious incident The dog did nothing in the night Sir.
The curious incident The dog did nothing in the night Sir. That is the curious incident!
Holme’s reasoning The solution to the crime hinges on the fact that the guard dog did not bark in the night, and from this Holmes deduces that the thief must have been known to the dog. I know him!
Holmes’ reasoning Holmes’ reasoning can be laid out as follows • Guard dogs bark at strangers • The guard dog did not bark at the thief • Therefore the thief was not a stranger
Syllogisms Holmes reasoning is an example of a syllogism.
The Socrates Syllogism • All human beings are mortal • Socrates is a human being • Therefore Socrates is mortal premises Rationalism – A branch of philosophy which takes reason as the most important source of knowledge conclusion
Syllogisms contain: • Two premises and a conclusion • Three terms, each must occur twice (“Socrates”, “human”, “mortal”.) • Quantifiers, such as “all”, or “some” or “no” which tell us of the quantity being referred to “TOK for the IB Diploma”, Richard van de Lagemaat, Cambridge
Another example • All boys like to fart • Chris is a boy • Chris likes to fart!
Truth and validity An argument is valid if the conclusion follows logically from the premises. All hippopotamuses eat cockroaches Mr Porter is a hippopotamus Therefore Mr Porter eats cockroaches Both premises and conclusion are false, but the argument is valid. Imagine that some strange planet exists where the premises are true
Truth and validity All rats are teachers Mr Porter is a rat Therefore Mr Porter is a teacher Both premises are false and conclusion is true! (but the argument is still valid). =
Deductive reasoning and truth • Just because an argument is valid (Some IB students are from Russia, all Russians are good at drinking vodka, therefore some IB students are good at drinking vodka) does not mean that the conclusion is true.
Deductive reasoning and truth For an argument to be true you must be able to answer “yes” to the following questions: • Are the premises true? • Is the argument valid?
Socrates • Socrates is a man • All men are mortal • Therefore Socrates is mortal The conclusion is only true if the premises are true.
Make up your own VALID syllogisms to illustrate each of the following; • 2 true premises and a true conclusion • 1 true premise, 1 false premise and a true conclusion • 1 true premise, 1 false premise and a false conclusion • 2 false premises and a true conclusion • 2 false premises and a false conclusion
Deciding whether a syllogism is valid Trying to decide if a syllogism is valid is not easy. Venn diagrams can help (at last a good use for Venn diagrams!)
Deciding whether a syllogism is valid Mmmmmmm…Vodka! • Some IB students are from Russia • All Russians are good at drinking vodka • Therefore some IB students are good at drinking vodka Is this a valid argument? IB students from Oslo International School
Using Venn diagrams • Some IB students are from Russia IB Students who are Russian IB Students Russians
Using Venn diagrams • All Russians are good at drinking vodka Good vodka drinkers IB Students Russians
Using Venn diagrams • Therefore some IB students are good at drinking vodka Good vodka drinkers IB Students Russians
Another example • All As are Bs • All Bs are Cs • Therefore all Cs are As
Another example • All As are Bs A B
Another example • All Bs are Cs A B C
Another example • Therefore all Cs are As These Cs are not As A B C The syllogism is not valid
Another example! • All As are Bs • Some As are Cs • Therefore some Bs are Cs
Another example! • All As are Bs A B
Another example! • Some As are Cs A C B
Another example! • Therefore some Bs are Cs A C B The syllogism is valid
