3.1
Statements and Quantifiers
Symbols A proposition or statement is a declarative sentence that can be classified as true or false, but not both. Propositions can be joined by logical connectives such as and and or. We can write logical statements in terms of symbols. We represent the statements by letters such as p, q and r and we use the following symbols for and, or, and not. Connective
Symbol
Type of Statement
∧ ∨ ∼
and or not
Conjunction Disjunction Negation
Example 1. Let p represent the statement ”She has green eyes” and let q represent the statement ”He is 48 years old”. Translate each symbolic compound statement into words. 1. ∼ q 2. p ∨ q 3. p ∧ ∼ q 4. ∼ p ∧ ∼ q 5. ∼ (p ∨ ∼ q) Negations To negate a statement we can use the following table. This table works in both directions because ∼ (∼ p) = p Statement
Negation
all do
some do not not all do
some do
none do all do not
Example 2. Find the negation of the following statements. 1. Every dog has its day. 2. No rain fell in southern California today. 3. Some books are longer than this book. 4. Some people have all the luck. 5. Everybody loves Raymond.
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3.2
Truth Tables and Equivalent Statements
A Truth Table is a way to check all possible outcomes. This will allow us to check the ”Truth” of any statement. Conjunctions: A conjunction is a statement of the form ”p and q” (p ∧ q). We say that p ∧ q is True if p and q are true and it is False in all other cases. p and q p
q
p∧q
T T F F
T F T F
T F F F
Disjunction: A disjunction is a statement of the form ”p or q” (p ∨ q). We say p ∨ q is False if p and q are false and it is True in all other cases. Statement 1: I have a quarter or I have a dime. Statement 2: I will drive the Ford or the Nissan to the store. p or q p
q
p∨q
T T F F
T F T F
T T T F
Negation: A negation is a proposition of the form ”not p” (∼ p). We say ∼ p is True if p is False. not p p
∼p
T F
F T
Example 3. Let p represent ”5 > 3” and let q represent ”6 < 0”. Find the truth value of p ∨ q and p ∧ q.
Example 4. Suppose p is false, q is true, and r is false. What is the truth value of the compound statement ∼ p ∧ (q ∨ ∼ r)
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Constructing truth tables We will always construct our truth tables for two statements in this format: p
q
Compound Statement
T T F F
T F T F
Example 5. Construct the truth table for (∼ p ∧ q)∨ ∼ q. p
q
T
T
T
F
F
T
F
F
∼p
(∼ p ∧ q)
∼q
(∼ p ∧ q)∨ ∼ q
Example 6. Construct the truth table for p ∧ (∼ p ∨ ∼ q). p
q
T
T
T
F
F
T
F
F
p ∧ (∼ p ∨ ∼ q)
We will always construct our truth tables for three statements p, q and r in this format: p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
Compound Statement
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Example 7. Construct the truth table for (∼ p ∧ r) ∨ (∼ p ∧ ∼ q). p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
(∼ p ∧ r) ∨ (∼ p ∧ ∼ q)
Alternative Method for Constructing Truth Tables Example 8. Construct the truth table for ∼ p ∨ (p ∧ q).
p
q
T
T
T
F
F
T
F
F
∼p
∨
(p
∧
q)
Example 9. Construct the truth table for (p ∨ q) ∨ (r ∧ ∼ p). p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
(p
∨
q)
4
∨
(r
∧
∼ p)
Equivalent Statements Two statements are equivalent if they have the same truth value in every possible situation. They have the same truth tables. Example 10. Are the following statements equivalent? ∼ p ∨ ∼ q and ∼ (p ∧ q).
p
q
T
T
T
F
F
T
F
F
∼p
∼q
(p ∧ q)
∼p∨∼q
De Morgan’s Laws: For any statements p and q. ∼ p ∨ ∼ q ≡∼ (p ∧ q) and ∼ p ∧ ∼ q ≡∼ (p ∨ q)
5
∼ (p ∧ q)
3.3
The Conditional
Conditional: A conditional statement is a compound statement that is of the form if p then q. We say p is the antecedent and q is the consequent. We write p → q. Sometimes we read this as ”p implies q”. Example 11. If you study hard then you will get an A. p → q is False when p is true and q is false. It is True in all other cases. The truth table for p → q is p→q p
q
p→q
T T F F
T F T F
T F T T
IMPORTANT The use of the conditional connective in NO WAY implies a cause-and-effect relationship. Any two statements may have an arrow placed between them to create a compound statement. For example: If I pass math 151, then the sun will rise the next day. Special Characteristics of the Conditional Statement 1. p → q is false only when the antecedent is true and the consequent is false. 2. If the antecedent is false, then p → q is automatically true. 3. If the consequent is true, then p → q is automatically true. Example 12. Write True or False for each statement. Here T represents a true statement and F represents a false statement. 1. T → (6 = 3) 2. (5 < 2) → F 3. (3 = 2 + 1) →) T
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Example 13. Fill in the truth table for (∼ p →∼ q) → (∼ p ∧ q) p
q
T
T
T
F
F
T
F
F
∼p
∼q
(∼ p →∼ q)
(∼ p ∧ q)
(∼ p →∼ q) → (∼ p ∧ q)
Example 14. Fill in the truth table for (p → q) → (∼ p ∨ q) ∼p
p
q
T
T
T
F
F
T
F
F
(p → q)
(∼ p ∨ q)
(p → q) → (∼ p ∨ q)
Tautology: A statement that is always true no matter the truth values of the components is called a tautology. Here are some others: p ∨ ∼ p, p → p, (∼ p ∨ ∼ q) →∼ (q ∧ p). Negation of a Conditional Show that ∼ (p → q) ≡ p ∧ ∼ q. p
q
T
T
T
F
F
T
F
F
∼q
(p → q)
∼ (p → q)
p∧∼q
The negation of (p → q) is p ∧ ∼ q (p → q) is equivalent to ∼ p ∨ q
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3.4
More on the conditional
Converse, Inverse and Contrapositive Direct Statement: p→q Converse: q→p Inverse: ∼ p →∼ q Contrapositive ∼ q →∼ p
(If (If (If (If
p, then q) q, then p) not p, then not q) not q, then not p)
Example: Direct Statement: If you stay, then I go. Converse: If I go, then you stay. Inverse: If you do not stay, then I do not go. Contrapositive: If I do not go, then you do not stay. Fill in the following truth table p
q
T
T
T
F
F
T
F
F
p→q
q→p
∼p
∼q
∼ q →∼ p
∼ p →∼ q
Which statements are logically equivalent?
Alternative Forms of ”if p, then q” The conditional p → q can be translated in any of the following ways. If p, then q.
p is sufficient for q.
If p, q
q is necessary for p.
p implies q.
All p’s are q’s.
p only if q.
q if p.
Example: If you are 18, then you can vote. This statement can be written any of the following ways: You can vote if you are 18. You are 18 only if you can vote. Being able to vote is necessary for you to be 18. Being 18 is sufficient for being able to vote. All 18-year-olds can vote. Being 18 implies that you can vote. 8
Biconditional: The statement p if and only if q (abbreviated p if f q) is called a biconditional. It is symbolized p ↔ q and is interpreted as the conjunction of p → q and q → p. In symbols: (p → q) ∧ (q → p). It has the following truth table: p↔q p
q
p↔q
T
T
T
T
F
F
F
T
F
F
F
T
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3.5
Analyzing Arguments with Euler Diagrams
Logical Arguments An argument is said to be valid if the fact that all the premises are true forces the conclusion to be true. An argument that is not valid is invalid. It is called a fallacy. Example 15. Is the following argument valid? All amusement parks have thrill rides. Great America is an amusement park. Great America has thrill rides.
Example 16. Is the following argument valid? All people who apply for a loan must pay for a title search. Hillary Langlois paid for a title search. Hillary Langlois applied for a loan.
Example 17. Is the following argument valid? Some dinosaurs were plant eaters. Danny was a plant eater. Danny was a dinosaur.
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3.6
Analyzing Arguments with Truth Tables
Testing the Validity of an Argument with a Truth Table 1. Assign a letter to represent each component statement in the argument. 2. Express each premise and the conclusion symbolically. 3. Form the symbolic statement of the entire argument by writing the conjunction of all the premises as the antecedent of a conditional statement, and the conclusion of the argument as the consequent. 4. Complete the truth table for the conditional statement formed in part 3 above. If it is a tautology (always true), then the argument is valid; otherwise, it is invalid. Example 18. Is the following example valid? If the floor is dirty, then I must mop it. The floor is dirty I must mop it
p
q
T
T
T
F
F
T
F
F
p→q
(p → q) ∧ p
This pattern is called Modus Ponens. p→q p q
11
[(p → q) ∧ p] → q
Example 19. Is the following example valid? If a man could be in two places at one time, I’d be with you. I am not with you A man can’t be in two places at one time
p
q
T
T
T
F
F
T
F
F
p→q
∼q
(p → q)∧ ∼ q
∼p
[(p → q)∧ ∼ q] →∼ p
This pattern is called Modus Tollens. p→q ∼q ∼p Example 20. Is the following example valid? I’ll buy a car or I’ll take a vacation. I won’t buy a car. I’ll take a vacation
p
q
T
T
T
F
F
T
F
F
p∨q
∼p
(p ∨ q)∧ ∼ p
12
[(p ∨ q)∧ ∼ p] → q
This pattern is called The Law of Disjunctive Syllogism. p∨q ∼p q Example 21. Is the following example valid? If it squeaks, then I use WD-40. If I use WD-40, then I must go to the hardware store. If it squeaks, then I must go to the hardware store.
p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
p→q
q→r
p→r
(p → q) ∧ (q → r)
This pattern is called Reasoning by Transitivity. p→q q→r p→r
13
[(p → q) ∧ (q → r)] → (p → r)
Example 22. Is the following example valid? If my check arrives in time, I’ll register for the fall semester. I’ve registered for the fall semester. My check arrived on time.
p
q
T
T
T
F
F
T
F
F
p→q
(p → q) ∧ q
[(p → q) ∧ q] → p
This is called The fallacy of the converse. p→q q p Similar reasoning gives us The fallacy of the inverse. p→q ∼p ∼q An example might be: ”If it rains, I get wet. It doesn’t rain. Therefore I don’t get wet.”
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Example 23. Is the following example valid? If the races are fixed or the gambling houses are crooked then the tourist trade will decline. If the tourist trade declines then the police will be happy. The police force is never happy. Therefore the races are not fixed.
p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
p∨q
(p ∨ q) → r
∼r
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∼p
[(p ∨ q) → r∧ ∼ r] →∼ p
