Section 3.1 Statements, Negations, and Quantified Statements

Section 3.1 Statements, Negations, and Quantified Statements Objectives 1. Identify English sentences that are statements. 2. Express statements using symbols. 3. Form the negation of a statement 4. Express negations using symbols. 5. Translate a negation represented by symbols into English. 6. Express quantified statements in two ways. 7. Write negations of quantified statements. 1/17/11 Section 3.1 1

Statements A statement is a sentence that is either true or false, but not both simultaneously. London is the capital of England William Shakespeare wrote the last episode of The Sopranos. Commands, questions, and opinions, are not statements because they are neither true or false. Titanic is the greatest movie of all time. (opinion) Read pages 23 57.(command) If I start losing my memory, how will I know? (question) 1/17/11 Section 3.1 2

Using Symbols to Represent Statements In symbolic logic, we use lowercase letters such as p, q, r, and s to represent statements. Here are two examples: p: London is the capital of England q: William Shakespeare wrote the last episode of The Sopranos. 1/17/11 Section 3.1 3

Example 1 Forming Negation The negation of a statement has a meaning that is opposite that of the original meaning. The negation of a true statement is a false statement and the negation of a false statement is a true statement. Form the negation of each statement ( two different ways): a. Shakespeare wrote the last episode of The Sopranos. Shakespeare did not write the last episode of The Sopranos. It is not true that Shakespeare wrote the last episode of The Sopranos. a. Today is not Monday It is not true that today is not Monday. Today is Monday 1/17/11 Section 3.1 4

Example 2 Expressing Negations Symbolically Let p and q represent the following statements: p: William Shakespeare wrote the last episode of The Sopranos. q: Today is not Monday Express each of the following statements symbolically: a. Shakespeare did not write the last episode of The Sopranos. ~p b. Today is Monday ~q 1/17/11 Section 3.1 5

Quantified Statements Quantifiers: The words all, some, and no (or none). Statements containing a quantifier: All poets are writers. Some people are bigots No math books have pictures. Some students do not work hard. 1/17/11 Section 3.1 6

Equivalent Ways of Expressing Quantified Statements Statement All A are B Some A are B An Equivalent Way to Express the Statement There are no A that are not B There exists at least one A that is a B Example All poets are writers. There are no poets that are not writers. Some people are bigots. At least one person is a bigot. No A are B All A are not B No math books have pictures. All math books do not have pictures. Some A are not B Not all A are B Some students do not work hard. Not all students work hard. 1/17/11 Section 3.1 7

Negation of Quantified Statements The statements diagonally opposite each other are negations. Here are some examples of quantified statements and their negations: 1/17/11 Section 3.1 8

Example 3 The mechanic told me, All piston rings were replaced. I later learned that the mechanic never tells the truth. What can I conclude? Solution: Let s begin with the mechanic s statement: All piston rings were replaced. Because the mechanic never tells the truth, I can conclude that the truth is the negation of what I was told. The negation of All A are B is Some A are not B. Thus, I can conclude that Some piston rings were not replaced. I can also correctly conclude that: At least one piston ring was not replaced. 1/17/11 Section 3.1 9

3.2 Compound Statements and Connectives Objectives 1. Express compound statements in symbolic form. 2. Express symbolic statements with parentheses in English. 3. Use the dominance of connectives. 1/17/11 Section 3.2 1

Simple and Compound Statements Simple statements convey one idea with no connecting words. Compound statements combine two or more simple statements using connectives. Connectives are words used to join simple statements. Examples are: and, or, if then, and if and only if. 1/17/11 Section 3.2 2

Example 1 And Statements If p and q are two simple statements, then the compound statement p and q is symbolized by p q. Let p and q represent the following simple statements: p: It is after 5 P.M. q: They are working. Write each compound statement below in symbolic form: a. It is after 5 P.M. and b. It is after 5 P.M. and they are working. they are not working. 1/17/11 Section 3.2 3

Common English Expressions for p q Symbolic Statement English Statement Example: p: It is after 5 P.M. q: They are working. p q p and q It is after 5 P.M. and they are working. p q p but q It is after 5 P.M., but they are working. p q p yet q It is after 5 P.M., yet they are working. p q p nevertheless q It is after 5 P.M.; nevertheless, they are working. 1/17/11 Section 3.2 4

Or Statements The connective or can mean two different things. Consider the statement: I visited London or Paris. This statement can mean (exclusive or) I visited London or Paris but not both. It can also mean (inclusive or) I visited London or Paris or both. Disjunction is a compound statement formed using the inclusive or represented by the symbol. Thus, p or q or both is symbolized by p q. 1/17/11 Section 3.2 5

Example 2 Translating From English to Symbolic Form Let p and q represent the following simple statements: p: The bill receives majority approval. q: The bill becomes a law. Write each compound statement below in symbolic form: a. The bill receives majority approval or the bill becomes a law. Solution: p q b. The bill receives majority approval or the bill does not become a law. Solution: p ~q 1/17/11 Section 3.2 6

If -Then Statements The compound statement If p, then q is symbolized by p q. This is called a conditional statement. The statement before the is called the antecedent. The statement after the is called the consequent. This diagram shows a relationship that can be expressed 3 ways: All poets appreciate language. There are no poets that do not appreciate language If a person is a poet, then that person appreciates language. 1/17/11 Section 3.2 7

Example 3 Translating From English to Symbolic Form Let p and q represent the following simple statements: p: A person is a father. q: A person is a male. Write each compound statement below in symbolic form: a. If a person is a father, then that person is a male. b. If a person is not a male, then that person is not a father. 1/17/11 Section 3.2 8

Common English expressions for p q Symbolic Statement English Statement Example p: A person is a father. q: A person is a male. p q If p then q. If a person is a father, then that person is a male. p q q if p. A person is a male, if that person is a father. p q p is sufficient for q. Being a father is sufficient for being a male. p q q is necessary for p. Being a male is necessary for being a father. p q p only if q. A person is a father only if that person is a male. p q Only if q, p. Only if a person is a male is that person a father. 1/17/11 Section 3.2 9

If and Only If Statements Biconditional statements are conditional statements that are true if the statement is still true when the antecedent and consequent are reversed. The compound statement p if and only if q ( abbreviated as iff ) is symbolized by p q. 1/17/11 Section 3.2 10

Common English Expressions for p q. Symbolic Statement English Statement Example p: A person is an unmarried male. q: A person is a bachelor. p q p if and only if q. A person is an unmarried male if and only if that person is a bachelor. p q q if and only if p. A person is a bachelor if and only if that person is an unmarried male. p q p q p q If p then q, and if q then p. p is necessary and sufficient for q. q is necessary and sufficient for p. If a person is an unmarried male then that person is a bachelor, and if a person is a bachelor, then that person is an unmarried male. Being an unmarried male is necessary and sufficient for being a bachelor. Being a bachelor is necessary and sufficient for being an unmarried male. 1/17/11 Section 3.2 11

Statements of Symbolic Logic Name Symbolic Form Common English Translations Negation ~p Not p. It is not true that p. Conjunction p q p and q, p but q. Disjunction p q p or q. Conditional Biconditional p q p q If p, then q, p is sufficient for q, q is necessary for p. p if and only if q, p is necessary and sufficient for q. 1/17/11 Section 3.2 12

Example 4 Symbolic Statements with Parentheses Let p and q represent the following simple statements: p: She is wealthy. q: She is happy. Write each of the following symbolic statements in words: a. ~(p q) It is not true that she is wealthy and happy. b. ~p q She is not wealthy and she is happy. c. ~(p q) She is neither wealthy or happy. 1/17/11 Section 3.2 13

Expressing Symbolic Statements with Parentheses in English Symbolic Statement Statements to Group Together English Translation (q ~p) ~r q ~p If q and not p, then not r. q ( ~p ~r) ~p ~r q, and if not p then not r. Notice that when we translate the symbolic statement into English, the simple statements in parentheses appear on the same side of the comma. 1/17/11 Section 3.2 14

Example 5 Expressing Symbolic Statements with Parentheses in English Let p, q, and r represent the following simple statements. p: A student misses lecture. q: A student studies. r: A student fails. Write each of these symbolic statements in words: a. (q ~ p) ~ r If a student studies and does not miss lecture, then the student does not fail. b. q (~p ~r) A student studies, and if the student does not miss lecture, then the student does not fail. 1/17/11 Section 3.2 15

Dominance of Connectives If a symbolic statement appears without parentheses, statements before and after the most dominant connective should be grouped. The dominance of connectives used in symbolic logic is defined in the following order. 1/17/11 Section 3.2 16

Using the Dominance of Connectives Statement Most Dominant Connective Highlighted in Red Statements Meaning Clarified with Grouping Symbols Type of Statement p q ~r p q ~r p (q ~r) Conditional p q ~r p q ~r (p q) ~r Conditional p q r p q r p (q r) Biconditional p q r p q r (p q) r Biconditional p ~q r s p ~q r s (p ~q) (r s) p q r p q r The meaning is ambiguous. Conditional? 1/17/11 Section 3.2 17

Example 6 Using the Dominance of Connectives Let p, q, and r represent the following simple statements. p: I fail the course. q: I study hard r: I pass the final. Write each compound statement in symbolic form: a. I do not fail the course if and only if I study hard and I pass the final. ~p (q r ) b. I do not fail the course if and only if I study hard, and I pass the final. (~p q) r 1/17/11 Section 3.2 18

Section 3.3 Truth Tables for Negation, Conjunction, and Disjunction Objectives 1. Use the definitions of negation, conjunction, and disjunction. 2. Construct truth tables. 3. Determine the truth value of a compound statement for a specific case. 1/17/11 Section 3.3 1

Truth Tables Negation p ~p T F F T Disjunction p q p q T T T T F T F T T F F F Negation (not): Opposite truth value from the statement. Conjunction (and): Only true when both statements are true. Disjunction (or): Only false when both statements are false. Conjunction p q p q T T T T F F F T F F F F 1/17/11 Section 3.3 2

Example 1 Using the Definitions of Negation, Conjunction, and Disjunction Let p and q represent the following statements: p: 10 > 4 This is a true statement q: 3 < 5 This is a true statement Determine the truth value for each statement: a. p q Since both are true, the conjunction is true. b. ~ p q Since p is true then, ~p is false, the conjunction is false. c. p q Since both are true and a disjunction is only false when both components are false, then this is true. c. ~p ~q Since both are false, the disjunction is false. 1/17/11 Section 3.3 3

Example 2 Constructing Truth Tables Construct a truth table for ~(p q) Step 1: First list the simple statements on top and show all the possible truth values. Step 2: Make a column for p q and fill in the truth values. p q p q T T T T F F F T F F F F 1/17/11 Section 3.3 4

Example 2 continued Step 3: Construct one more column for ~(p q). The final column tells us that the statement is false only when both p and q are true. p q p q ~(p q) T T T F T F F T colleges. F T F T F F F T tautology. tautology? For example: p: Harvard is a college. (true) q: Yale is a college. (true) ~(p q): It is not true that Harvard and Yale are A compound statement that is always true is called a Is this a NO 1/17/11 Section 3.3 5

Example 3 Constructing a Truth Table Construct a truth table for (~p q) ~q. p q ~p ~p q ~q (~p q) ~q T T F T F F T F F F T F F T T T F F F F T T T T 1/17/11 Section 3.3 6

Constructing a Truth Table with Eight Cases There are eight different true-false combinations for compound statements consisting of three simple statements. 1/17/11 Section 3.3 7

Example 4 Constructing a Truth Table with Eight Cases Construct a truth table for the following statement: I study hard and ace the final, or I fail the course. Suppose that you study hard, you do not ace the final and you fail the course. Under these conditions, is this compound statement true or false? Solution: We represent our statements as follows: p: I study hard. q: I ace the final. r: I fail the course. 1/17/11 Section 3.3 8

Example 4 continued Writing the given statement in symbolic form: The completed table is: The statement is True. 1/17/11 Section 3.3 9

Example 5 Determining Truth Values for Specific Cases We can determine the truth value of a compound statement for a specific case in which the truth values of the simple statements are known, without constructing an entire truth table. Substitute the truth values of the simple statements into the symbolic form of the compound statement and use the appropriate definitions to determining the truth value of the compound statement. 1/17/11 Section 3.3 10

Example 5 continued Use the information in the circle graphs to determine the truth value of the following statement: It is not true that freshmen make up 24% of the undergraduate college population and account for more than one-third of the undergraduate deaths, or seniors do not account for 30% of the undergraduate deaths. 1/17/11 Section 3.3 11

Example 5 continued Substitute the truth values for p, q, and r that we obtained from the circle graphs to determine the truth value for the given compound statement. ~(p q) ~r This is the given compound statement in symbolic form. ~(T T) ~F Substitute the truth value obtained from the graph. F T true. Replace T T with T. Conjunction is true when both parts are T Replace F T with T. Disjunction is true when at least one part is true. We 1/17/11 conclude that the given statement Section is 3.3 true. 12

Truth Tables for the Conditional and Biconditional Objectives 1. Understand the logic behind the definition of the conditional. 2. Construct truth tables for conditional statements. 3. Understand the definition of the biconditional. 4. Construct truth tables for biconditional statements. 5. Determine the true value of a compound statement for a specific case. 1/17/11 Section 3.4 1

Truth Tables for Conditional Statements p q antecedent consequent If p then q. A conditional is false only when the antecedent is true and the consequent is false. Conditional p q p q T T T T F F F T T F F T 1/17/11 Section 3.4 2

Example 1 Constructing a Truth Table Construct a truth table for ~q ~p p q ~q ~p ~q ~p T T F F T T F T F F F T F T T F F T T T 1/17/11 Section 3.4 3

More on the Conditional Statement You can reverse and negate the antecedent and consequent, and the statement s truth value will not change. If you re cool, you won t wear clothing with your school name on it. If you wear clothing with your school name on it, you re not cool. 1/17/11 Section 3.4 4

Example 2 Proving an Implication Implications: Conditional statements that are Tautologies. Construct a truth table for [( p q) ~ p] ~q (p q) [( p q) ~p] p q p q ~p ~p ~q T T T F F T T F T F F T F T T T T T F F F T F T 1/17/11 Section 3.4 5

Biconditional Statements p q p if and only if q: p q and q p True only when the component statements have the same value. Truth table for the Biconditional p q p q T T T T F F F T F F F T 1/17/11 Section 3.4 6

Example 3 Determining the Truth Value of a Compound Statement You receive a letter that states that you have been assigned a Super Million Dollar Prize Entry Number -- 665567010. If your number matches the winning preselected number and you return the number before the deadline, you will win $1,000,000.00. Suppose that your number does not match the winning pre-selected number, you return the number before the deadline and only win a free issue of a magazine. Under these conditions, can you sue the credit card company for making a false claim? 1/17/11 Section 3.4 7

Example 3 continued Solution: Assign letters to the simple statements in the claim. 1/17/11 Section 3.4 8

Example 3 continued Now write the underlined claim in the letter in symbolic form: Substitute the truth values for p, q, and r to determine the truth value for the letter s claim. (p q) r (F T) F F F T Our truth-value analysis indicates that you cannot sue the credit card company for making a false claim. 1/17/11 Section 3.4 9

The Definitions of Symbolic Logic Negation ~: not The negation of a statement has the opposite meaning, as well as the opposite truth value, from the statement. Conjunction : and The only case in which a conjunction is true is when both component statements are true. Disjunction : or The only case in which a disjunction is false is when both component statements are false. 1/17/11 Section 3.4 10

The Definitions of Symbolic Logic continued Conditional : if-then The only case in which a conditional is false is when the first component statement, the antecedent, is true and the second component statement, the consequent, is false. Biconditional : if and only if The only cases in which a biconditional is true are when the component statements have the same truth value. 1/17/11 Section 3.4 11

Section 3.5 Equivalent Statements, Variations of Conditional Statements, and De Morgan s Laws Objectives 1. Use a truth table to show that statements are equivalent. 2. Write the equivalent contrapositive for a conditional statement. 3. Write the converse and inverse of a conditional statement. 4. Write the negation of a conditional statement. 5. Use De Morgan s laws. 1/17/11 Section 3.5 1

Equivalent Statements Equivalent compound statements are made up of the same simple statements and have the same corresponding truth values for all true-false combinations of these simple statements. If a compound statement is true, then its equivalent statement must also be true. If a compound statement is false, its equivalent statement must also be false. A special symbol is used to show two statements are equivalent. 1/17/11 Section 3.5 2

Example 1 Showing that Statements are Equivalent Show that p ~q ~p ~q Solution: construct a truth table and see if the corresponding truth values are the same: p q ~q p ~q ~p ~p ~q T T F T F T T F T T F T F T F F T F F F T T T T 1/17/11 Section 3.5 3

The Contrapositive of the Conditional Statement p q A conditional statement and its equivalent contrapositive: p q ~q ~p The truth value of a conditional statement does not change if the antecedent and consequent are reversed and both are negated: ~q ~p is called the contrapositive of the conditional p q. 1/17/11 Section 3.5 4

Example 2 Writing Equivalent Contrapositives Write the equivalent contrapositive for: a. If you live in Los Angeles, then you live in California p: You live in Los Angeles. q: You live in California. If you live in Los Angeles, then you live in California. p q If you do not live in California, then you do not live in Los Angeles. ~q ~p 1/17/11 Section 3.5 5

Variations of the Conditional Statement Name Symbolic Form English Translation Conditional p q If p then q Converse q p If q then p Inverse ~ p ~ q If not p, then not q Contrapositive ~ q ~p If not q, then not p Conditional and Contrapositive are equivalent. Converse and Inverse are equivalent. 1/17/11 Section 3.5 6

Example 3 Writing Variations of a Conditional Statement Write the converse, inverse, and contrapositive of the following conditional statement: If it s a lead pencil, then it does not contain lead. (true) Solution: Use the following representations: p: It s a lead pencil. q: It contains lead. 1/17/11 Section 3.5 7

Example 3 continued Now work with p ~q to form the converse, inverse and contrapositive. Then translate the symbolic form of each statement back into English. 1/17/11 Section 3.5 8

Example 4 The Negation of a Conditional Statement The negation of p q is p ~q. This can be expressed as: ~(p q) p ~ q Write the negation of: If too much homework is given, a class should not be taken. Solution: p: Too much homework is given q: A class should be taken The symbolic form is p ~q 1/17/11 Section 3.5 9

Example 4 continued The negation of p ~q is p ~(~ q) which simplifies to p q. Translating into English: Too much homework is given and a class should be taken. 1/17/11 Section 3.5 10

1. ~(p q) ~p ~q De Morgan s Laws 2. ~(p q) ~p ~q Proof of the first law is shown in the Truth Table below. p q p q ~(p q) ~p ~q ~p ~q T T T F F F F T F F T F T T F T F T T F T F F F T T T T 1/17/11 Section 3.5 11

De Morgan s Laws and Negations 1. ~(p q) ~p ~q The negation of p q is ~(p q). To negate a conjunction, negate each component statement and change and to or. 2. ~(p q) ~p ~q. The negation of p q is ~p ~q. To negate a disjunction, negate each component statement and change or to and. 1/17/11 Section 3.5 12

Example 5 Negating Conjunctions and Disjunctions Write the negation for each of the following statements: a. All students do laundry on weekends and I do not. b. Some college professors are entertaining lecturers or I m bored. 1/17/11 Section 3.5 13

Section 3.5 Arguments and Truth Tables Objectives 1. Use truth tables to determine validity. 2. Recognize and use forms of valid and invalid arguments. 1/17/11 Section 3.6 1

Arguments An Argument consists of two parts: Premises: the given statements. Conclusion: the result determined by the truth of the premises. Valid Argument: The conclusion is true whenever the premises are assumed to be true. Invalid Argument: Also called a fallacy Truth tables can be used to test validity. 1/17/11 Section 3.6 2

Example 1 The Menendez case Did Eric and Lyle Menendez kill their parents to get the inheritance or as a result of years of abuse? Premise 1: If children murder their parents in cold blood, they deserve to be punished to the full extent of the law.. Premise 2: These children murdered their parents in cold blood. Conclusion: Therefore, these children deserve to be punished to the full extent of the law. 1/17/11 Section 3.6 3

Example 1 continued The Menendez case Solution p: Children murder their parents in cold blood. q: They deserve to be punished to the full extent of the law. Write the argument in symbolic form: Premise 1: p q If children under their parents in cold blood, they deserve to be punished to the full extent of the law. Premise 2: p These children murdered their parents in cold blood. Conclusion: q Therefore, these children deserve to be punished to the full extent of the law. 1/17/11 Section 3.6 4

Example 1 continued Rewriting as a conditional statement and forming the truth table: [(p q) p] q This conditional statement is a tautology. This argument is called direct reasoning and is valid. [(p q) p] p q p q (p q) p q T T T T T T F F F T F T T F T F F T F T 1/17/11 Section 3.6 5

Testing the validity of an Argument with a Truth Table 1. Use a letter to represent each simple statement in the argument. 2. Express the premises and the conclusion symbolically. 3. Write a symbolic conditional statement of the form: [(premise 1) (premise 2) (premise n)] conclusion, where n is the number of premeses. 4. Construct a truth table for the conditional statement in step 3. 5. If the final column of the truth table has all trues, the conditional statement is a tautology and the argument is valid. If the final column does not have all trues, the conditional statement is not a tautology and the argument is invalid. 1/17/11 Section 3.6 6

Example 2 Determining Validity with a Truth Table Determine if the following argument is valid: I can t have anything more to do with the operation. If I did, I d have to lie to the Ambassador. And I can t do that Solution: We can express the argument as follows: Henry Bromell If I had anything more to do with the operation. I d have to lie to the Ambassador. I can t lie to the Ambassador. Therefore, I can t have anything more to do with the operation. 1/17/11 Section 3.6 7

Example 2 continued Step 1: Use a letter to represent each statement in the argument: p: I have more to do with the operation q: I have to lie to the Ambassador. Step 2: Express the premises and the conclusion symbolically. p q If I had anything more to do with the operation, I d have to lie to the Ambassador. ~q I can t lie to the Ambassador.. ~p Therefore, I can t have anything more to do with the operation. 1/17/11 Section 3.6 8

Example 2 continued Step 3. Write a symbolic statement: [(p q) ~q ] ~ p Step 4. Construct the truth table. The form of this argument is called contrapositive reasoning. It is a valid argument. p q p q ~q (p q) ~q ~ p [(p q) ~q ] ~ p T T T F F F T T F F T F F T F T T F F T T F F T T T T T 1/17/11 Section 3.6 9

Standard Forms of Arguments Valid Arguments Direct Contrapositive Disjunctive Transitive Reasoning Reasoning Reasoning Reasoning p q p q p q p q p q p ~q ~p ~q q r q ~p q p p r ~r ~p Invalid Arguments Fallacy Fallacy Misuse of Misuse of of the of the Disjunctive Transitive Converse Inverse Reasoning Reasoning p q p q p q p q p q q ~p p q q r p ~q ~q ~p r p ~p ~r 1/17/11 Section 3.6 10

Example 3 Determining Validity Without Truth Tables Determine whether this argument is valid or invalid: There is no need for surgery because if there is a tumor then there is a need for surgery but there is no tumor. Solution: p: There is a tumor q: There is a need for surgery. Expressed symbolically: If there is a tumor then there is need for surgery. p q There is no tumor.. ~p Therefore, there is no need for surgery. ~q The argument is in the form of the fallacy of the inverse and is therefore, invalid. 1/17/11 Section 3.6 11

Section 3.7 Arguments and Euler Diagrams Objective 1. Use Euler diagrams to determine validity 1/17/11 Section 3.7 1

Euler Diagrams Technique for determining the validity of arguments whose premises contain the words all, some, and no. Euler Diagrams for Quantified Statements 1/17/11 Section 3.7 2

Euler Diagrams and Arguments 1. Make an Euler diagram for the first premise. 2. Make an Euler diagram for the second premise on top of the one for the first premise. 3. The argument is valid if and only if every possible diagram illustrates the conclusion of the argument. If there is even one possible diagram that contradicts the conclusion, this indicates that the conclusion is not true in every case, so the argument is invalid. 1/17/11 Section 3.7 3

Example 1 Arguments and Euler Diagrams Use Euler diagrams to determine whether the following argument is valid or invalid: All people who arrive late cannot perform. All people who cannot perform are ineligible for scholarships Therefore, all people who arrive late are ineligible for scholarships. Solution: Step 1: Make an Euler diagram for the first premise. Step 2: Make an Euler diagram for the second premise. Since there is only one possible diagram, and it illustrates the argument s conclusion The argument is valid. 1/17/11 Section 3.7 4

Example 2 Arguments and Euler Diagrams Use Euler diagrams to determine whether the following argument is valid or invalid: All poets appreciate language. All writers appreciate language, Therefore, all poets are writers. Step 1. Make an Euler diagram for the first premise: Step 2: Make an Euler diagram for the second premise on top of the one for the first premise. Adding All writers appreciate language can be done in four ways: 1/17/11 Section 3.7 5

Example 2 continued Not all diagrams illustrate the argument s conclusion that All poets are writers. The first two diagrams do not. The argument is invalid! 1/17/11 Section 3.7 6

Example 3 Euler Diagrams and the Quantifier SOME All people are mortal Some mortals are students. Therefore, some people are students. Step 1: Make an Euler diagram for the first premise. All people are mortal. Step 2: Make an Euler diagram for the second premise on top of the one for the first premise. Some mortals are students. 1/17/11 Section 3.7 7

Example 3 continued The dot in the region of intersection shows that at least one mortal is a student. The diagram does not show the people and students circle intersecting with a dot in the region of intersection. Step 3. The argument is valid if and only if every possible diagram illustrates the conclusion of the argument. The arguments conclusion is: Some people are students. The diagram does not show the people circle and the students circle intersecting with a dot in the region of intersection. The conclusion does not follow from the premises. The argument is invalid. 1/17/11 Section 3.7 8