logic & critical thinking HW 3a-e

Instructions: Choose the symbolic notation translation of the following statements that captures as closely as possible the logical structure of each statement. If it does not walk like a duck and does not talk like a duck, then it's not a duck.

( ~ W · ~ T ) ⊃ ~ D

Translate the following statements into symbolic form by using logical operators and uppercase letters to represent the English statements. If you have a good credit score and no debt, then you can buy a house. Let G = you have a good credit score, D = you have debt, and H = you can buy a house.

(G • ~ D ) ⊃ H It's important to bracket off the conjunction so that it can function as the antecedent of the conditional

Instructions: Choose the symbolic notation translation of the following statements that captures as closely as possible the logical structure of each statement. If Carly works hard and is not distracted, then she will get done on time.

(W · ~ D ) ⊃ T

Instructions: Use your knowledge of truth tables to determine if the following sets of statements are contradictory, consistent, or inconsistent. S = U | S v U

Consistent

Instructions: Use your knowledge of truth tables to identify the following statements as contingent, tautology, or self-contradiction. P v Q

Contingent

In ordinary language the words "only if" typically precedes the antecedent of a conditional.

False

The horseshoe symbol is used to translate a conjunction.

False

The main operator cannot be the negation operator.

False

Two (or more) statements are consistent when they have at least one line on their respective truth tables where the main operators are false.

False

Two statements that have opposite truth values on every line of their respective truth tables are logically equivalent.

False

Two truth-functional statements that have identical truth tables are contradictory statements.

False

Determine whether the truth table is correct for the following compound proposition. ​ PQ P v (P ⊃Q tt t...T...t tf t...T...t ft t...T...t ff f...F...f

Incorrect There is a mistake in the fourth line under the conditional.

A simple statement is one that does not have any other statement as a component.

True

Truth values 3 Instructions: If P is true, Q is false, R is true, and S is false, determine the truth value of the following: [ P v ( Q v R ) ] ⊃ ( S v P )

True

Truth values 3 Instructions: If P is true, Q is false, R is true, and S is false, determine the truth value of the following: S v ~ Q

True

[ (M ν P) ⊃ (Q ν R) ] ν (S · ~ P)

v In this case, it is the third "ν" connecting [ (M ν P) ⊃ (Q ν R) ] with (S · ~ P). This is question 17 in the book.

Instructions: Choose the symbolic notation translation of the following statements that captures as closely as possible the logical structure of each statement. It is not the case that either Lee Ann or Mary Lynn is a vegetarian.

~ (L v M)

Instructions: Use your knowledge of truth tables to identify the following statements as contingent, tautology, or self-contradiction. P · ~ P

Self-contradiction

A conjunction is a compound statement that has two distinct statements (called conjuncts) connected by the dot symbol.

True

A disjunction is a compound statement that has two distinct statements (called disjuncts) connected by the wedge symbol.

True

A statement form is a pattern of statement variables and logical operators.

True

Truth values 3 Instructions: If P is true, Q is false, R is true, and S is false, determine the truth value of the following: R · ~ S

True

A compound statement has at least three simple statements as a component.

False

An exclusive disjunction is where both disjuncts can be true at the same time.

False

An inclusive disjunction is where both disjuncts cannot be true at the same time.

False

In ordinary language, the word "if" typically precedes the consequent of a conditional.

False

Instructions: Use your knowledge of truth tables to identify the following statements as contingent, tautology, or self-contradiction. ( P · ~ P ) v Q

Contingent

Instructions: Use your knowledge of truth tables to identify the following statements as contingent, tautology, or self-contradiction. P · S

Contingent

Instructions: Use your knowledge of truth tables to determine if the pairs of statements are logically equivalent. ~ S v ~ R | ~ ( S · R )

Logically equivalent

(P · Q) ν ~ R

v Remember that the operator that has in its range the largest component or components in a compound statement is the main operator. This is question 9 in the book.

~ Q · P

x Remember that the operator that has in its range the largest component or components in a compound statement is the main operator. This is question 21 in the book.

If you have neither a bad credit score nor debt, then you can buy a house. Let B = you have a bad credit score, D = you have debt, and H = you can buy a house.

~(B v D ) ⊃ H Pay very close attention to how the parentheses affect the negation

You can buy a house with a good interest rate if and only if you have good credit and no debt. Let G = you have good credit, I = good interest rate, D = you have debt, and H = you can buy a house.

( H · I ) ≡ ( G · ~ D ) "If and only if" is your clue that the statement is a biconditional, which means that the complex statements on either side are bound by parentheses. In addition, "with" is translated using a conjunction.

Complete the proof: 1. [(M v N) ⊃ (C ∙D)] v (G ∙ D) 2. ~G/∴ (M v N) ⊃ (C ∙ D)

3. ~G v ~D 2, add 4. ~(G∙D) 3, DEM 5. (M v N) ⊃ (C ∙ D) 1,4, DS

Complete the Proof 1. M⊃(J v K) 2. (G v T)⊃M 3. G /∴ J v K

4. G v T 3, add 5. M 2,4, MP 6. JvK 1,5, MP

Complete the Proof: 1. A v B 2. C ⊃ ~B 3. C / ∴A

4. ~B 2,3, MP 5. A 1,4, DS

Instructions: Use your knowledge of truth tables to determine if the following sets of statements are contradictory, consistent, or inconsistent. A v ~ B | ~ A v B

Consistent

Instructions: Use your knowledge of truth tables to determine if the following sets of statements are contradictory, consistent, or inconsistent. P ⊃ Q | P v ~ Q

Consistent

Instructions: Pick out the premises and conclusion for each of the following arguments. Argument 3 Paris is called the "City of Lights." Las Vegas is also called the "City of Lights." So, there must be at least two cities with the same nickname. There must be at least two cities with the same nickname.

Conclusion

Instructions: Choose the symbolic notation translation of the following statements that captures as closely as possible the logical structure of each statement. My house is painted red and my car is painted green.

H • C

Instructions: Use your knowledge of truth tables to determine if the following sets of statements are contradictory, consistent, or inconsistent. ~ A · B | A · ~ B

Inconsistent

Determine whether the truth table is correct for the following compound proposition. ​ A B C (AvB) & C t t t T t t f T t f t F t f f F f t t T f t f F f f t F f f f F

Incorrect There are mistakes in the second and third lines

Instructions: Choose the symbolic notation translation of the following statements that captures as closely as possible the logical structure of each statement. Both Lee Ann and Mary Lynn are vegetarians.

L • M

If "~ (X v Y)" is true, then can one of the disjuncts be true?

No The statement negates the possibility that either disjunct is true

Instructions: Use your knowledge of truth tables to determine if the pairs of statements are logically equivalent. P = Q | ( P ⊃ Q ) v ( Q ⊃ P )

Not logically equivalent

Instructions: Pick out the premises and conclusion for each of the following arguments. Argument 2 Soy bean curd has no taste. The fat in hamburgers is what gives them their great taste. The fat in pizza is what gives it great taste. Food without fat tastes bland. Soy bean curd has no fat. Soy bean curd has no fat.

Premise

Instructions: Pick out the premises and conclusion for each of the following arguments. Argument 3 Paris is called the "City of Lights." Las Vegas is also called the "City of Lights." So, there must be at least two cities with the same nickname. Paris is called the "City of Lights."

Premise

Instructions: Pick out the premises and conclusion for each of the following arguments. Argument 4 Paris's Eiffel Tower is three times as tall as the one in Las Vegas. The Luxor Pyramid in Las Vegas is half the size of the original in Egypt. The Statue of Liberty in New York is four times the one in Las Vegas. Thus, every object in Las Vegas is smaller than in other cities. Paris's Eiffel Tower is three times as tall as the one in Las Vegas.

Premise

Instructions: Use your knowledge of truth tables to identify the following statements as contingent, tautology, or self-contradiction. P v ~ P

Tautology

(R ⋅ ∼ R) ⊃ (S ν ∼ S)

The statement is a tautology The truth table reveals that the main operator is necessarily true. this is question 17 in book

∼ (R ⋅ ∼ R) ν ∼ (S ν ∼ S)

The statement is a tautology The truth table reveals that the main operator is necessarily true. this is question 9 in book

The compound statement called biconditional is made up of two conditionals: one indicated by the word "if" and the other indicated by the phrase "only if."

True

If "X · ~ Y" is true, then which of the following is correct?

Y must be false The only way for a conjunction to be true is when both conjuncts are true. Since Y is negated, in order for the negated Y to be true, Y must be false.

If "X v Y" is true, then can one of the disjuncts be false?

Yes A disjunction can be true when only one of the disjuncts is true

L ⊃ (~ P ⊃ Q)

⊃ In this case, it is the first ⊃ symbol. Remember that the operator that has in its range the largest component or components in a compound statement is the main operator. This is question 25 in the book.

~ K ⊃ ~ P

⊃ Remember that the operator that has in its range the largest component or components in a compound statement is the main operator. This is question 13 in the book.

Identify the main operator in each of the following WFFs. Choose the correct answer. L ⊃ ~ P

⊃ Remember that the operator that has in its range the largest component or components in a compound statement is the main operator. This is question 5 in the book.