FURTHER ON ARGUMENTS

1. FURTHER ON ARGUMENTS Varieties of Arguments

2. Varieties of arguments • Two kinds of argument • Deductive - Premises are taken to provide complete, watertight support for the conclusion (may or may not be successful) • Inductive - Premises are taken to provide probable support for the conclusion, but not watertight support • (may or may not be successful)

3. Varieties of arguments • Example of deductive argument • 1. If I file my taxes I will get a refund. • 2. I will file my taxes. • 3. I will get a refund.

4. Varieties of arguments • Example of inductive argument • 1. Southpark has always been on Wednesday at 10 pm. • 2. It is now Wednesday at 10 pm. • 3. Therefore, Southpark is [probably] on now.

5. Bad or Good arguments • Difference between deductive and inductive arguments is not a matter of how good the arguments are. • There are good and bad inductive arguments, and good and bad deductive arguments.

6. Bad or Good arguments • There are two distinct measures of an argument’s goodness: • 1. The ‘inferential relationship’ between the premises and the conclusion • 2. The truth of the premises

7. Good or Bad arguments • The ‘inferential relationship’ between the premises and the conclusion • If the premises were true, would the conclusion necessarily (deductive), or probably (inductive), be true? • Note: This can be assessed even if the premises are in fact false.

8. Good or Bad arguments • What is INFERENCE? • a conclusion reached on the basis of evidence and reasoning. • a belief or opinion that you develop from the information that you know. • a guess that you make or an opinion that you form based on the information that you have

9. Good or Bad arguments • If a deductive argument has a good inferential relationship between the premises and conclusion, then it is valid. • An argument (deductive) is valid if: • -If all the premises of the argument were true, the conclusion would have to be true • -If the conclusion is false, then one or more of the premises must be false

10. Good or Bad arguments • A valid deductive argument with false premises and a false conclusion: • 1. If I am President of the US, then I get all the free BMWs I want. • 2. I am President of the US. • ∴ 3. I get all the free BMWs I want.

11. Good or Bad arguments • An invalid deductive argument with true premises and a true conclusion: 1. If I have a mass greater than 0 kg, then I am subject to gravitational forces. 2. I like pizza. ∴ 3. Paris is in France.

12. Good or Bad arguments • If an inductive argument has a good inferential relationship between the premises and conclusion, then it is strong. • An argument (inductive) is strong if: • -If all the premises of the argument were true, the conclusion would probably be true • -If the conclusion is false, then one or more of the premises is probably false

13. Good or Bad arguments • Example of inductive argument • 1. The last time I watched Southpark, the first commercial was for Old Navy. • ∴ 2. The first commercial on tonight’s episode of Southpark will [probably] be for Old Navy. • In this argument, there is not a good inferential relation between premises and conclusion.

14. Good or Bad arguments • First kind of goodness that an argument can have: an appropriate inferential relation between the premises and the conclusion. • If a deductive argument has it, it is valid. • If an inductive argument has it, it is strong.

15. Good or Bad arguments • Second kind of goodness an argument can have is the truth of its premises • If a deductive argument (i) is valid, and (ii) has all true premises, then it is sound. • If an inductive argument (i) is strong, and (ii) has all true premises, then it is cogent.

16. Arguments Sound and Cogent Deductive Inductive Valid Invalid Strong Weak • ARGUMENT- RECAP Sound Cogent

17. Symbolic Logic Identifying Statement of Forms

18. Statements • This section we will study symbolic logic which was developed in the late 17th century. • All logical reasoning is based on statements. • A statement is a sentence that is either true or false.

19. Statements • Definition: A proposition or statement is a sentence which is either true or false. • Definition: If a proposition is true, then we say its truth value is true, and if a proposition is false, we say its truth value is false.

20. Statements • Definition: A proposition is a declarative sentence that is either -true (denoted either T or 1) or -false (denoted either F or 0).

21. Which of the following are statements? • The 2004 Summer Olympic Games were in Athens, Greece.(Statement - true) • Osofo Dadzie was the best TV comedy of all time. (Not a Statement – opinion) • Did you watch The Godfather? (Not a statement – a question) • The Philadelphia Eagles won Super XV. (statement – false) • I am telling a lie. (not a statement – paradox)

22. Propositional Logic • There are six primary forms of Propositional Logic: • Simple • Conjunction • Disjunction • Conditional • Biconditional • Negation

23. Simple • Simple statements express one idea • Example: • The plant need watering • Accra is the capital of Ghana

24. Compound Statements • A compound statement is a statement that contains one or more simpler statements. • Compound statements can be formed by • inserting the word NOT, • joining two or more statements with connective words such as AND, OR, IF…THEN, ONLY IF, IF AND ONLY IF.

25. Examples • Steve did not do his homework. • This is formed from the simpler statement, Steve did his homework. • Mr. Doe wrote the Logic notes and listened to a Elder Mireku CD. • This statement is formed from the simper statements: Mr. Doe wrote the Logic notes. Mr. Doe listened to Elder Mireku CD.

26. Examples of Propositions • There are five types of compound propositions. • They are, Negation, Conjunction, Disjunction, Conditional statement and Bi-Conditional statement.

27. Conjunction • Conjunctions are compound statements made up of two or more statements (which maybe either simple or compound) connected with the word ‘’AND’’ or BUT, YET, ALTHOUGH, ALSO….. • Example: “An NPP candidate is in the Jubilee House and the NDC control the parliament. • The two statements in the example above are: “An NPP candidate is in the Jubilee House “and” the NDC control the parliament.

28. Conjunction • The components of the conjunction are called the conjuncts. • Each conjunct may be either simple or compound

29. CONJUNCTION p ^ q • A conjunction is a compound statement that consists of 2 or more statements connected by the word and. • And is represented by the symbol ^. • p ^ q represents “p and q”. • Example: p: Frimpong Manso is a comedian. q: Frimpong Manso is a millionaire. Express the following in symbolic form: i. Frimpong Manso is a comedian and he is a millionaire. ii. Frimpong Manso is a comedian and he is not a millionaire.

30. Conjunction • Using the symbolic representations p: The lyrics are controversial. q: The performance is banned. Express the following in symbolic form: a. “The lyrics are controversial and the performance is banned.” b. “The lyrics are not controversial and the performance is not banned.” Answers: a. p ^ q b. ~p ^ ~q

31. Disjunction • Disjunctions are compound statements made up of two or more statements (simple or compound) connected with words as “either…or”, “or”, “unless”. • Example: Either Hearts or Kotoko will win the trophy • The two simple statements in this example are: “Hearts will win the trophy” and “Kotoko will win the trophy”

32. DISJUNCTION p v q • When you connect statements with the word or you form a disjunction. • Or is represented by the symbol v. • p v q is read as “p or q”. • Using the p and q, write out in words p v q, and p v ~q. Example: p v q is “the lyrics are controversial or the performance is banned.” • p v ~q is “the lyrics are controversial or the performance is not banned.”

33. Conditional • Conditional Statements are made up of two or more statements (simple or compound) connected by such hypothetical terms as “if….then”, “implies that”, “provided that”, “only if”, “is implied by”. • Example: If tuition goes up, then I’ll have to get another job. • The two simple statements in the example are “tuition will go up” and “I’ll have to get another job”.

34. Conditional • The components of the conditional are called the antecedent and consequent.

35. CONDITIONAL p  q • A conditional is of the form “if p then q”. This is also known as an implication. p is the hypothesis (or premise), and q is the. conclusion. • The representation of “if p then q” is p q. • Again use the p and q from the previous 2 slides. • “If the lyrics are not controversial, the performance is not banned.” • ~p  ~q

36. Biconditional • “Biconditionals are compound statements made up of two components (simple or compound), where each is said to imply the other. • Example: The final exam will include Module 10 if, and only if, we cover it in class. - The two components that make up the statement above are: “The final exam will include Module 10” and “we cover Module 10 in class”. • The example above could be expanded to read “If we cover module 10 in class, then it will be on the final exam and module 10 will be on the final exam only if we cover it in class.

37. Biconditional • The example above could be expanded to read “If we cover module 10 in class, then it will be on the final exam and module 10 will be on the final exam only if we cover it in class. • The expanded version is a conjunction of two conditionals, which is why this statement form is called biconditional.

38. Biconditional • Other logical words/phrases include “implies and is implied by”, “is a necessary and sufficient condition”, “just in case that”, “entails”. • The components of biconditionals donot have special names

39. Negation • A negation is any statement denying that another statement is true. • Simple statements can be denied/negated just as compound statements can be denied. • Example: It is false that money is the root of all evils.

40. NEGATION ~p • The negation of a statement is the denial of that statement. The symbolic representation is a tilde ~. • Negation of a simple statement is formed by inserting not. • Example: The Pastor is a Presbytarian. The negation is: The pastor is not a Presbytarian.

41. Negation • Example : Lisa and James won’t go to the movies. • What the statement is actually saying, in other words, is that it is false that Lisa and James will both go to the movies. • There are many ways to deny/negate a statement: “it is falls that”, “it is not the case that”, “won’t”, “can’t”, “unsuccessful”.

42. Negation • “All of Mr. Doe’s students are Phobia fans.” • The negation is: “Some of Mr. Doe’s students are not Phobia fans.” • To negate the first statement, we don’t need to have all the students to be not Phobia fans, we just need only one student not to be an Phobia fan. Hence the usage of some.

43. Negation • “No students are Hospitality majors.” • To deny this statement, we need at least one instance in which a student does major in Hospitality. • “Some students are Hospitality majors.”

44. Negation • To summarize negation: • All p are q is negated by Some p are not q • No p are q is negated by Some p are q

45. SUMMARY

46. Compound Statements • Types of truth-functionally compound statements • Negation: not, it is not the case that, no (~p) • The negation of a true statement is a false statement. The negation of a false statement is a true statement. • Conjunction, conjuncts: and, yet, but, although, however, nevertheless, even though (p & q) • A conjunction is true if and only if both of its conjuncts are true.

47. Compound Statements • Disjunction, disjuncts: or, either … or …, unless (p v q) • A disjunction is true except when both of its disjuncts are false. • Conditionals, antecedents and consequents: if … then …, … only if …, provided that, on the condition that (pq) • The if-clause of a conditional is the antecedent; the then-clause if the consequent. • A conditional is true except when its antecedent is true and its consequent is false.

48. Compound Statements • Biconditionals: … if and only if …, … just in case that … (pq) • A biconditional is true if and only if the statements the statements flanking the ‘if and only if’ have the same truth value: both true or both false. • .