MODULE 4

Using the following schema: . J = John jumps R = Ray runs K = Kyle sings . Translate the following statement into propositional logic: . If John jumps and Kyle sings, then Ray runs. . (J ⦁ K) ⊃ R (J v K) ⊃ R R ⊃ (K ⦁ J) R ⊃ (K v J)

(J ⦁ K) ⊃ R

Using the following schema: . J = John jumps R = Ray runs K = Kyle sings . Translate the following statement into propositional logic: . Assuming Ray runs or Kyle sings, then John jumps and Kyle sings. (J ⦁ K) ⊃ (R v J) (R v K) ⊃ (J vK) (R v K) ⊃ (J ⦁ K) (R ⦁ K) ⊃ (J v K)

(R v K) ⊃ (J ⦁ K)

Using the following schema: . J = John jumps R = Ray runs K = Kyle sings M = Mike climbs P = Pam plays guitar . Translate the following argument into propositional logic: . If Pam plays guitar, then Mike climbs and John jumps. If Mike climbs and John jumps, then Ray runs. Ray is not running. Thus, Pam is not playing guitar. . 1. (M ⦁ J) ⊃ P 2. (M ⦁ J) ⊃ R 3. ~R //~P 1. P ⊃ (M ⦁ J) 2. (M ⦁ J) ⊃ R 3. ~R //~P 1. (M ⦁ J) ⊃ P 2. R ⊃ (M ⦁ J) 3. ~R //~P 1. P ⊃ (M ⦁ J) 2. R ⊃ (M ⦁ J) 3. ~R //~P

1. P ⊃ (M ⦁ J) 2. (M ⦁ J) ⊃ R 3. ~R //~P

Using the following schema: . J = John jumps R = Ray runs K = Kyle sings P = Pam plays guitar . Translate the following argument into propositional logic: . It is not the case that both Ray runs and Kyle sings. If John jumps, then both Ray runs and Kyle sings. If Pam plays guitar or Kyle sings, then John jumps. Thus, neither Pam plays guitar nor Kyle sings. . 1. ~(R ⦁ K) 2. J ⊃ (R ⦁ K) 3. (P v K) ⊃ J // ~(P v K) 1. ~(R ⦁ K) 2. J ⊃ (R ⦁ K) 3. (P v K) ⊃ J // ~P v ~K 1. ~R ⦁ ~K 2. J ⊃ (R ⦁ K) 3. (P v K) ⊃ J // ~(P v K) 1. ~R ⦁ ~K 2. (R ⦁ K) ⊃ J 3. J ⊃ (P v K) // ~(P v K)

1. ~(R ⦁ K) 2. J ⊃ (R ⦁ K) 3. (P v K) ⊃ J // ~(P v K)

Categorical Syllogism

A syllogism in which each statement begins with one of the words "all," "no," or "some."

Determine the valid form being used in the following argument: . Bob Marley sings the song "Put it On." The Pixies sing the song "Gigantic." Therefore, Marley sings "Put it On" and the Pixies sing "Gigantic." Addition Conjunction Simplification Modus Ponens

Addition WRONG Conjunction

Determine the invalid form being used in the following argument: . Assuming that Paul plays bass, then George plays guitar. George is playing guitar. hence, Paul is playing bass. Affirming the Consequent False Conversion Denying the Antecedent Undisturbed Middle Term

Affirming the Consequent

Determine the invalid form being used in the following argument: . Assuming that Ska is good music, then the Heptones are a solid band. The Heptones are a solid band. Thus, Ska is good music. Affirming the Consequent Denying the Antecedent Disjunctive Fallacy Undistributed Middle Terms

Affirming the Consequent

Determine the invalid form being used in the following argument: . Robert plays the blues if Desmond plays ska. Desmond is not playing ska. Thus, Robert is not playing the blues. Denying the Antecedent Affirming the Consequent False Conversion Undisturbed Middle Term

Affirming the Consequent WRONG Denying the Antecedent

Determine the invalid form being used in the following argument: . If the Browns win the Super Bowl, then Joe will be happy. Therefore, if Joe is happy, then the Browns won the Super Bowl. False Conversion Undisturbed Middle Term Affirming the Consequent Denying the Antecedent

Affirming the Consequent WRONG False Conversion

Determine the invalid form being used in the following argument: . All punk bands play the beat fast. The Yeah, Yeah, Yeahs play the beat fast. Thus, the Yeah, Yeah, Yeahs, are a punk band. Denying the Antecedent Affirming the Consequent Undisturbed Middle Term False Conversion

Affirming the Consequent WRONG False Conversion WRONG

Determine the invalid form being used in the following argument: . Some comedians are funny and Bill Hicks is a comedian. Thus, Bill Hicks must be funny. Denying the Antecedent Undistributed Middle Term Affirming the Consequent Modus Tollens

Affirming the Consequent WRONG Undistributed Middle Term

Determine the valid form being used in the following argument: . Jimi Hendrix jams the guitar and MCA drops dope rhymes. Thus, Hendrix jams guitar and MCA drops dope rhymes, or Ice Cube writes fabulous lyrics. Addition Conjunction Disjunctive Syllogism Constructive Dilemma

Conjunction WRONG Constructive Dilemma WRONG

Determine the valid form being used in the following argument: . If Taylor writes her own lyrics, then Kanye interrupts her speech. If Dr. Dre is an audio engineer, then The Sugar Hill Gang wrote some of the first Hip Hop songs. As it turns out, Taylor writes her own lyrics or Dr. Dre is an audio engineer. Thus, either Kanye interrupts Taylor's speech or The Sugar Hill Gang wrote some of the first Hip Hop songs. Constructive Dilemma Pure Hypothetical Syllogism Disjunctive Syllogism Undisturbed Middle Term

Constructive Dilemma

Determine the valid form being used in the following argument: . All reggae music puts the emphasis on the third beat. Toots and the Maytals play reggae music. Thus, Toots and the Maytals put the emphasis on the third beat. Categorical Syllogism Constructive Dilemma Pure Hypothetical Syllogism Disjunctive Syllogism

Constructive Dilemma WRONG Categorical Syllogism

Determine the invalid form being used in the following argument: . If you drink milk, then you will consume calcium. You did not drink milk, thus you did not consume calcium. Denying the Antecedent Affirming the Consequnt Disjunctive Fallacy Modus Tollens

Denying the Antecedent

Determine the valid form being used in the following argument: . Either Dylan sings or Hendrix strums the guitar. Hendrix is not strumming the guitar. Thus, Dylan is singing. Disjunctive Syllogism Modus Tollens Modus Ponens Addition

Disjunctive Syllogism

Constructive Dilemma

Either A or B. If A, then C. If B, then D. So, either C or D.

Disjunctive Syllogism

Either P or Q Not P Therefore Q

Using the following schema: . J = John jumps R = Ray runs K = Kyle sings . Translate the following statement into propositional logic: . If John jumps, then Kyle sings or Ray runs. J ⊃ (K ⦁ R) J ≡ (K ⦁ R) (J v K) v R J ⊃ (K v R)

J ⊃ (K v R)

Using the following schema: .J = John jumps R = Ray runs K = Kyle sings Translate the following statement into propositional logic: If John Jumps, then Kyle sings and Ray runs. J ⊃ (K ⦁ R) J ≡ (K ⦁ R) J ⊃ (K v R) K ⊃ (J ⦁ R)

J ⊃ (K ⦁ R)

Using the following schema: . J = John jumps R = Ray runs K = Kyle sings . Translate the following statement into propositional logic: . Ray runs if Kyle sings. K ⊃ R R ⊃ K K ≡ R R ≡ K

K ⊃ R

Determine the valid form being used in the following argument: . If Hendrix jams on the guitar, then the Pixies rock out. Hendrix is jamming on the guitar. Hence, the Pixies rock out. Modus Tollens Modus Ponens Disjunctive Syllogism Categorical Syllogism

Modus Ponens

Determine the valid form being used in the following argument: . If the Beastie Boys shake their rump, then RUN DMC wears Adidas. RUN DMC does not wear Adidas. Thus, the Beastie Boys do not shake their rump. Modus Ponens Simplification Modus Tollens Disjunctive Syllogism

Modus Tollens

Determine the valid form being used in the following argument: . Provided that Badfish plays ska, then Sublime is famous. If Sublime is famous, then they owe a debt of gratitude to the Skatalites. Therefore, provided that Badfish plays ska, they owe a debt of gratitude to the Skatalites. Modus Ponens Pure Hypothetical Syllogism Conjunction Constructive Dilemma

Pure Hypothetical Syllogism

Using the following schema: . J = John jumps R = Ray runs K = Kyle sings . Translate the following statement into propositional logic: . Ray runs only if Kyle sings. . K ⊃ R R ⊃ K R ≡ K K ≡ R

R ⊃ K

Determine the valid form being used in the following argument: . Fleetwood Mac wrote some solid tunes and The Velvet Underground kicks it hard. Thus, Fleetwood Mac wrote some solid tunes. Addition Pure Hypothetical Syllogism Conjunction Simplification

Simplification

sufficient condition

a condition that will bring about another event

necessary condition

a condition without which another event cannot occur

simple statement

a statement that does not contain any other statement as a component Here are some examples: Fast foods tend to be unhealthy. James Joyce wrote Ulysses. Parakeets are colorful birds. The bluefin tuna is threatened with extinction. Any convenient uppercase letter may be selected to represent each statement. Thus, F might be selected to represent the first, J the second, P the third, and B the fourth.

well-formed formulas (WFFs)

a syntactically correct arrangement of symbols In English, for example, the expression "there is a cat on the porch" is syntactically correct, but "Porch on the is cat a there" is not syntactically correct

argument form

an arrangement of statement variables and operators such that the uniform substitution of statements in place of the variables results in an argument

in the statement N ⊃ F, its components are called

antecedent (N) and consequent (F)

B ≡ R is called a

biconditional or biconditional statement and it expresses the relation of material equivalence if and only if, is equivalent to, sufficient and necessary condition for

The statement N ⊃ F is called a

conditional or conditional statement it expresses the relation of material implication. if . . . then, only if, implies, given that, in case, provided that, on condition that, sufficient condition for, necessary condition for (Note: Do not confuse antecedent with consequent!) The statement that follows "if" is always the antecedent, and the statement that follows "only if" is always the consequent Antecedents follow: If, assuming, provided Consequents follow: then, only if, it follows that

The statement D . C is called a

conjunction or conjunctive statement and, yet, but, however, moreover, nevertheless, still, also, although, both, additionally, furthermore

in the conjunctive statement, the components D and C are called

conjuncts

the statement P ⋁ E is called a

disjunction or disjunctive statement "or" and "unless."

in the disjunctive statement the components P and E are called

disjuncts

The statement ∼A is called a

negation not, it is not the case that, it is false that

compound statement

one that contains at least one simple statement as a component. Here are some examples: It is not the case that Al Qaeda is a humanitarian organization. Dianne Reeves sings jazz, and Christina Aguilera sings pop. Either people get serious about conservation or energy prices will skyrocket. If nations spurn international law, then future wars are guaranteed. The Broncos will win if and only if they run the ball. Using uppercase letters to stand for the simple statements, these compound statements may be represented as follows: It is not the case that A. D and C. Either P or E. If N then F. B if and only if R

propositional logic

the fundamental elements are whole statements (or propositions)

Using the following schema: . J = John jumps R = Ray runs K = Kyle sings . Translate the following statement into propositional logic: . It is not the case that both Kyle sings and John jumps. . ~ (K v J) ~ K ⦁ ~J ~ (K ⦁ J) ~K v ~J

~ (K ⦁ J)