2 Aristotle Logic


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Uses and abuses of Logic with Ian Kluge

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  • Thank you very much for your summary. It has given me a very helpful overview of the topic -- something I've been struggling to get for about six months! (I teach logic to 13 to 15-year-olds, so understanding the big picture is crucial, to say the least.)

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2 Aristotle Logic

  1. 1. The Legacy of Aristotle Part 2 Logic
  2. 2. LOGIC
  3. 3. <ul><ul><li>LOGIC </li></ul></ul><ul><li>Aristotle was the first to systematically study and catalogue the rules of correct logical reasoning </li></ul><ul><li>His logic is important because it dominated all western thought, including scientific thought, until the 19 th century CE; it also had enormous influence on the development of Jewish, Christian and Muslim philosophy. It is still influential today. </li></ul><ul><li>Although other types of logical systems exist, Aristotelian logic is still a powerful tool used to teach reasoning skills in numerous academic disciplines. </li></ul>
  4. 4. <ul><li>In his logic, Aristotle explicitly established three laws of logical thought. </li></ul><ul><ul><li>Law # 1: law of identity : “each thing is inseparable from itself and its being one just meant this” ( Metaphysics , 7, 17). A thing is just itself and not something else: e.g. a soccer ball is a soccer ball and not a kitchen stove. </li></ul></ul><ul><ul><li>* Sometimes this is expressed as A = A. </li></ul></ul><ul><ul><ul><li>Note: the fact that we can use a book for a doorstop does not mean it is not a book. Its use does not contradict the law of identity. What a thing is and how it is used are two different issues. </li></ul></ul></ul>
  5. 5. <ul><ul><li>Law # 2: the law of contradiction : “the same attribute cannot at the same time belong and not belong to the same subject and in the same respect ” ( Metaphysics , 4, 3). E.g. my cup cannot be blue and not-blue at the same time </li></ul></ul><ul><ul><li>A cannot be A and not-A at the same time in the same way/respect. </li></ul></ul><ul><ul><li>Note: things may have and not have the same attributes in different ways: e.g. man is the most intelligent creature compared to animals but he is not intelligent compared to God. So man is both intelligent (compared to animals) and not intelligent (compared to God). There is no contradiction because ‘intelligent’ is being used in different ways. </li></ul></ul>
  6. 6. <ul><ul><li>Law # 3: the law of the excluded middle or excluded third : “there cannot be an intermediate between two contradictories, but of one subject we must either affirm or deny any one predicate [statement]” ( Metaphysics 4, 7). </li></ul></ul><ul><ul><li>A statement about a topic must either be true or false. It cannot be both, i.e. there is no middle between them. It cannot be neither true nor false. </li></ul></ul><ul><ul><li>Note: It is either true that Socrates is mortal or it is not true that he is not mortal. He is not both. Nor can he be neither mortal nor immortal. </li></ul></ul><ul><ul><li>Another example: It is either true that there is a rubber duck in my bath tub or it is not true. Nor can we say neither of these choices is true. </li></ul></ul>
  7. 7. <ul><li>Reflections on the Rules of Logic </li></ul><ul><li>According to Aristotle, the laws of logic are not man-made prescriptions – but are ‘rules of reality,’ i.e. if we violate them we will reach false conclusions about the real world. </li></ul><ul><li>They are ‘necessary’ (unchangeable, non-arbitrary) and not ‘normative’ (changeable and arbitrary) rules. </li></ul><ul><li>A traffic law is a ‘normative ‘rule, i.e. the government can change it. The law of non-contradiction is ‘necessary,’ i.e. if you try to violate it you get into trouble with reality; no government can change it. E.g., it is true that either there is or there is not a car coming towards you as you cross the street; it can’t be both are true or neither. </li></ul>
  8. 8. <ul><ul><li>Aristotle’s systemization of thought laid the basis for scientific progress which requires careful, systematic, step-by-step habits of thinking, i.e. of inquiring, investigating, evaluating, comparing and contrasting and drawing conclusions. </li></ul></ul><ul><ul><li>It also improved society in general because logic teaches people to regard their own and other people’s thinking critically in an objective and systematic way. They become more self-critical, i.e. learn to evaluate the validity of their own thoughts. </li></ul></ul><ul><ul><li>Aristotle’s logic taught people that orderly critical, and objective thought can give us knowledge about reality, that correct human thought processes are ‘adapted’ to the real world and capable of discovering truth. Reality and truth are directly available to us – if we use these logical tools correctly. </li></ul></ul>
  9. 9. <ul><li>The Syllogism </li></ul><ul><li>To help us reason correctly, Aristotle invented the syllogism. A syllogism is a three-part reasoning process beginning with 2 premises and ending with 1 conclusion . </li></ul><ul><ul><li>Aristotle works mainly with categorical syllogisms which affirm or deny something. There are other types of syllogisms but these will not concern us here. </li></ul></ul><ul><ul><li>The value of studying and learning to work with syllogisms is that we learn to break our ideas down into simple parts and by putting them into syllogistic form we can make sure we are developing a logical argument. Here is an example of a syllogism: </li></ul></ul><ul><li>(1) No reptiles have fur; </li></ul><ul><li>(2) All snakes are reptiles; </li></ul><ul><li>(3) Therefore, no snakes have fur. </li></ul>
  10. 10. <ul><ul><li>Here is another example – the most famous syllogism of all time: </li></ul></ul><ul><ul><li>(1) All men are mortal; </li></ul></ul><ul><ul><li>(2) Socrates is a man; </li></ul></ul><ul><ul><li>(3) Therefore, Socrates is mortal. </li></ul></ul><ul><ul><li>Here we see the essential requirements of a syllogism : </li></ul></ul><ul><ul><li>3 statements: 2 premises (1) and (2) and a conclusion (3) </li></ul></ul><ul><ul><li>3 terms; each is used twice: men/man, mortal, Socrates </li></ul></ul>
  11. 11. <ul><ul><li>each statement has a subject (what the statement is about) e.g. “All men” in the statement “All men are mortal.” </li></ul></ul><ul><ul><li>and a predicate (what is being said about the subject) e.g. “are mortal” in the statement “All men are mortal.” </li></ul></ul><ul><ul><li>(e) the subject of the conclusion is the minor term , e.g. “Socrates” in the statement “Therefore Socrates is mortal.” </li></ul></ul><ul><li>(f) the predicate of the conclusion is the major term , e.g. “mortal” in the statement “Therefore Socrates is mortal.” </li></ul><ul><li>(g) the term that appears in premises (1) and (2) but not in the conclusion is the middle term . </li></ul><ul><li>(h) the premise with the major term is the major premise </li></ul><ul><li>(i) the premise with the minor term is the minor premise . </li></ul>
  12. 12. <ul><li>Analysing a syllogism </li></ul><ul><li>To analyze a syllogism follow the steps in this order: </li></ul><ul><li>Step 1: Identify the conclusion </li></ul><ul><li>Step 2: Identify the minor and major terms in the conclusion </li></ul><ul><li>Step 3: Identify the major and minor premises </li></ul><ul><li>Step 4: Identify the middle term (it is not in the conclusion but is the same in the first two premises.) </li></ul>
  13. 13. <ul><li>To identify these parts we use: </li></ul><ul><li>*S for the minor term; </li></ul><ul><li>*P for the major term; and </li></ul><ul><li>* M for the middle term (repeated in both premises) </li></ul><ul><li>For example: </li></ul><ul><li>(1) All men (M) are mortal (P); </li></ul><ul><li>(2) Socrates (S) is a man (M); </li></ul><ul><li>(3) Therefore, Socrates (S) is mortal (P). </li></ul><ul><li>The major premise is “All men are mortal”. </li></ul><ul><li>The minor premise is “Socrates is a man.” </li></ul><ul><li>NOTE: we could have switched the places of the first two premises: </li></ul>
  14. 14. <ul><li>(1) Socrates (S) is a man (M); </li></ul><ul><li>(2) All men (M) are mortal (P); </li></ul><ul><li>(3) Therefore, Socrates (S) is mortal (P). </li></ul><ul><li>Now (2) is the major premise and (1) is the minor premise. </li></ul><ul><li>Practice exercise # 1: Analyze the following syllogisms for S,M, P and the major and minor premise. </li></ul><ul><li>1) Lions are meat-eaters; </li></ul><ul><li>(2) Leo is a lion; </li></ul><ul><li>(3) Therefore Leo is a meat-eater. </li></ul><ul><li>1) No computer is alive; </li></ul><ul><li>(2) Humans are alive; </li></ul><ul><li>(3) Therefore, no humans are computers </li></ul>
  15. 15. <ul><li>NOTE: </li></ul><ul><li>(a) P is always the predicate of the conclusion but it is not always the predicate of the major premise. </li></ul><ul><li>(b) S is always the subject of the conclusion but it is not always the subject of the minor premise. </li></ul><ul><li>(c) The middle term M can be anywhere in the first two premises; it can be both subjects, both predicates or one of each. For example: </li></ul><ul><li>(1) All horses (P) have hooves (M); </li></ul><ul><li>(2) No humans have hooves (M); </li></ul><ul><li>(3) Therefore, no humans are horses (P). </li></ul><ul><li>*** In this case P is the predicate of the conclusion (as it always is) but is the subject – not the predicate – of the major premise. </li></ul>
  16. 16. <ul><li>Another example: </li></ul><ul><li>(1) All fruit is nutritious; </li></ul><ul><li>(2) Some nutritious things are tasty (S); </li></ul><ul><li>(3) Therefore, some tasty things (S) are fruit. </li></ul><ul><li>*** In this case the minor term (S) is the subject of the conclusion (as it always is) but is now the predicate of the minor premise. </li></ul><ul><li>The purpose of this explanation is to show that there is some flexibility in the construction of a syllogism. </li></ul>
  17. 17. <ul><li>How To Construct a Syllogism </li></ul><ul><li>There are 3 basic steps to constructing a correct syllogism: </li></ul><ul><li>(1) You must know the conclusion you are trying to prove, and put it into logical form. Doing this will give you your S (minor term) and P (major term). E.g. “Justice is a virtue.” </li></ul><ul><li>(2) Find a good middle term. This is the key to writing successful syllogisms. The middle term joins the S and P. E.g. “giving people their appropriate reward.” </li></ul><ul><li>(3) Set up the first two premises that logically lead to the conclusion. </li></ul>
  18. 18. <ul><li>Practice # 2: Write a syllogism (and label the parts S, P, and M) to reach the following conclusion: </li></ul><ul><li>(1) </li></ul><ul><li>(2) </li></ul><ul><li>(3) Therefore, cake is not healthy. </li></ul><ul><li>Example 2: </li></ul><ul><li>(1) </li></ul><ul><li>(2) </li></ul><ul><li>(3) Therefore, people are imperfect. </li></ul>
  19. 19. <ul><li>Remember: </li></ul><ul><li>A syllogism cannot have two negative premises . We cannot reach any conclusion from “(1) No dogs are cats; (2) No cats are nice; </li></ul><ul><li>If a syllogism has a negative premise, the conclusion must be negative. </li></ul><ul><li>(1) No tree is edible; </li></ul><ul><li>(2) Some trees are green; </li></ul><ul><li>(3) Therefore, some green things are not edible. </li></ul><ul><li>(c) Make sure the middle term (M) is distributed. If it is not, we have a fallacy or logical error. We shall ignore the technical reasons for this name and use a simple test: are the two groups being joined by the middle term (M) separate even though they share a quality? For example: </li></ul>
  20. 20. <ul><li>(1) All teenagers are two legged; </li></ul><ul><li>(2) All ostriches are two-legged; </li></ul><ul><li>(3) Therefore, all teenagers are ostriches. </li></ul><ul><li>Obviously the two groups are still separate even though they share the </li></ul><ul><li>quality of being two-legged. </li></ul><ul><li>In the “Socrates is mortal” syllogism, we can see that mankind and </li></ul><ul><li>Socrates are not separate groups. Here is another example of an </li></ul><ul><li>undistributed middle: </li></ul><ul><li>(1) All penguins are black and white; </li></ul><ul><li>(2) Some old films are black and white; </li></ul><ul><li>(3) Therefore, some old films are penguins. </li></ul><ul><li>This is obviously false. </li></ul>
  21. 21. <ul><li>Four Types of Syllogisms </li></ul><ul><li>There are 4 types of syllogisms depending on if the first premise affirms or denies something, and how much it affirms or denies, i.e. all, some, or none. </li></ul><ul><ul><li>(1) All S are P (universal affirmation)** “All men are mortal” </li></ul></ul><ul><ul><ul><li>Called A </li></ul></ul></ul><ul><ul><li>(2) No S are P (universal negation) “No men are birds” </li></ul></ul><ul><ul><ul><li>Called E </li></ul></ul></ul><ul><ul><li>(3) Some S are P (particular affirmation) “Some dogs chase cats” </li></ul></ul><ul><ul><ul><li>Called I </li></ul></ul></ul><ul><ul><li>(4) Some S are not P (particular negation) “Some dogs do not swim” </li></ul></ul><ul><ul><li>* called O </li></ul></ul>
  22. 22. <ul><li>Here is an A type syllogism, i.e. a universal affirmation . It has 2 universal premises; a “universal premise” applies to all members of a kind or class. </li></ul><ul><ul><li>1) All cats hunt mice; </li></ul></ul><ul><ul><li>2) All Manx’s are cats; </li></ul></ul><ul><ul><li>3) Therefore, all Manx’s hunt mice. </li></ul></ul><ul><ul><li>Example # 2 of a universal affirmative with a singular affirmative statement: </li></ul></ul><ul><ul><li>(1) All educated people can read and write; </li></ul></ul><ul><ul><li>(2) Sam is an educated man; [‘Sam’ is singular, i.e. one] </li></ul></ul><ul><ul><li>(3) Therefore, Sam can read and write. </li></ul></ul>
  23. 23. <ul><ul><li>Note: the 2 universal premises require a universal conclusion; the affirmative conclusion also requires 2 affirmative premises . </li></ul></ul><ul><li>Here is an E type syllogism, a universal negation : t he premise is a universal negative: </li></ul><ul><ul><li>1) No members of the dog family have wings; </li></ul></ul><ul><ul><li>2) Wolves are members of the dog family; </li></ul></ul><ul><ul><li>3) Therefore, no wolves have wings. </li></ul></ul><ul><ul><ul><ul><li>Note: a negative premise requires a negative conclusion. </li></ul></ul></ul></ul><ul><ul><ul><ul><li>A syllogism cannot have 2 negative premises. If one premise is negative, the conclusion must be negative. </li></ul></ul></ul></ul>
  24. 24. <ul><li>Here is a I type syllogism: it is a particular affirmation : </li></ul><ul><ul><li>(1) Some vases are beautiful; </li></ul></ul><ul><ul><li>(2) All vases are useful; </li></ul></ul><ul><ul><li>(3) Therefore, some useful things are beautiful. </li></ul></ul><ul><ul><li>(1) Some computers are out-of-date; </li></ul></ul><ul><ul><li>(2) All out-of-date things should be replaced; </li></ul></ul><ul><ul><li>(3) Therefore, some things that should be replaced are computers. </li></ul></ul><ul><ul><li>Note: the particular premise requires a particular conclusion. We cannot have two particular premises. </li></ul></ul>
  25. 25. <ul><li>Here is an O type syllogism with a negative particular premise : </li></ul><ul><ul><li>1) Some buildings are not tall; </li></ul></ul><ul><ul><li>2) All houses are buildings; </li></ul></ul><ul><ul><li>3) Therefore, some houses are not tall. </li></ul></ul><ul><ul><li>(1) Some cats have no tails; </li></ul></ul><ul><ul><li>(2) All cats are mammals; </li></ul></ul><ul><ul><li>(3) Therefore, some mammals have no tails. </li></ul></ul><ul><ul><li>Note: the negative conclusion requires a negative premise; a particular premise needs a particular conclusion . </li></ul></ul>
  26. 26. <ul><li>Practice # 3: Write an example of an A, E, I, O type syllogism and label the parts S, M, P and the minor and major premise. </li></ul><ul><li>Practical Uses of Syllogisms </li></ul><ul><li>Syllogisms can be used to construct valid arguments. If you think you have a valid argument to make about something, you can try putting it into syllogism form to see whether it is valid. </li></ul><ul><li>Syllogisms do not always have to be about very simple topics, although we always try to keep them as simple as possible. </li></ul><ul><li>(1) Making people work without pay is morally wrong; </li></ul><ul><li>(2) Slavery is making people work without pay; </li></ul><ul><li>(3) Therefore, slavery is morally wrong. </li></ul>
  27. 27. <ul><li>(1) Giving people their proper reward is a virtue; </li></ul><ul><li>(2) Justice gives people their proper reward; </li></ul><ul><li>(3) Therefore, justice is a virtue. </li></ul><ul><li>We shall now look at two examples of complex syllogisms: </li></ul><ul><li>(1) All created things (M) receive divine bounties that should be developed (P); </li></ul><ul><li>(2) We human beings (S) are created things (M); </li></ul><ul><li>(3) Therefore, human beings (S) must develop their divine bounties (P) </li></ul>
  28. 28. <ul><li>Here is another example: </li></ul><ul><li>(1) All living creations (M) undergo outward changes without changing their inner essence (P); </li></ul><ul><li>(2) Human beings (S) are living creations (M); </li></ul><ul><li>(3) Therefore human beings (S) undergo outward changes without changing their inner essence (P). </li></ul><ul><li>Practice # 4: Complete the syllogism: </li></ul><ul><li>1. All fragile things are breakable things. Some glasses are fragile things. Therefore 2. All mammals are warm-blooded animals. All whales are mammals. Therefore </li></ul>
  29. 29. <ul><li>3. All books are things with pages. Some books are mysteries. Therefore </li></ul><ul><li>4. All flowers are pretty objects. All pansies are flowers. Therefore </li></ul><ul><li>  5. No animals are plants. All sheep are animals. Therefore </li></ul>