Critical Thinking 6-1

B⊃(C⊃R)

If Bill O'Reilly spins the news, then if Chris Matthews fights back, then Rachel Maddow tells it straight. B = Bill O'Reilly spins the news. C = Chris Matthews fights back. R = Rachael Maddow tells it straight.

(B⊃C)⊃R

If Bill O'Reilly's spinning the news implies that Chris Matthews fights back, then Rachel Maddow tells it straight. B = Bill O'Reilly spins the news. C = Chris Matthews fights back. R = Rachael Maddow tells it straight.

[(C • J) ≡ K] ⊃ ∼(B ∨ S)

If Christina Aguilera's singing soul and Justin Timberlake's singing pop are sufficient and necessary conditions for Kelly Clarkson's singing rock, then neither Beyonce nor Shakira will sing rap. C =Christina Aguilera sings soul. J =Justin Timberlake sings pop. K =Kelly Clarkson sings rock. B =Beyoncè will sing rap. S =Shakira will sing rap.

(E•A)⊃(C∨S)

If Electronic Arts and Activision Blizzard sell extremely violent video games, then either Colt or Smith & Wesson can expect increased profits. E =Electronic Arts sells extremely violent video games. A =Activation Blizzard sells extremely violent video games. C =Colt can expect increased profits. S =Smith and Wesson can expect increased profits.

M⊃[P⊃(D⊃S)]

If Maria Cantwell promotes alternative energy, then if Patty Murray supports wilderness areas, then Dianne Feinstein's advocating gun control implies that Susan Collins does so, too. M =Maria Cantwell promotes alternative energy. P =Patty Murray supports wilderness areas. D =Dianne Feinstein advocates gun control. S =Susan Collins advocates gun control.

M⊃[N⊃(S•T)]

If Monsanto poisons the entire human race, then if Nestlé buries us in bottles, then Smithfield and Tyson load us up with antibiotics. M =Monsanto poisons the entire human race. N =Nestle buries us in bottles. S =Smithfield loads us up with antibiotics. T =Tyson loads us up with antibiotics.

S⊃(B∨M)

If Saks promotes gift cards, then either Macy's or Bloomingdale's puts on a fashion show. S = Saks promotes gift cards. B = Bloomingdale's puts on a fashion show. M = Macy's puts on a fashion show.

(L∨R)⊃∼(G∨B)

If either Lockheed Martin or Raytheon floods the world with weapons of war, then neither General Dynamics nor Boeing reduces the likelihood of armed conflict. L = Lockheed Martin floods the world with weapons of war. R = Raytheon floods the world with weapons of war. G = General Dynamic reduces the likelihood of armed conflict. B = Boeing reduces the likelihood of armed conflict.

P⊃I

Israel's abandoning its settlements is a necessary condition for the Palestinian's declaring an end to hostilities. I =Israel abandons its settlements. P =Palestinians declare an end to hostilities.

I≡P

Israel's abandoning its settlements is a sufficient and necessary condition for the Palestinians' declaring an end to hostilities. I =Israel abandons its settlements. P =Palestinians declare an end to hostilities.

I⊃P

Israel's abandoning its settlements is a sufficient condition for the Palestinians' declaring an end to hostilities. I =Israel abandons its settlements. P =Palestinians declare an end to hostilities.

∼[I•(S∨P)]

It is not the case that both Iran gives up its nuclear program and Sudan or Pakistan combats terrorism. I =Iran gives up its nuclear program. S =Sudan combats terrorism. P =Pakistan combats terrorism.

∼[(L∨K)•(T∨M)]

It is not the case that both Lindsey Vonn or Kevin Durant plays professional tennis and Tom Brady or Matt Ryan plays professional basketball. L =Lindsey Vonn plays professional tennis. K =Kevin Duran plays professional tennis. T =Tom Brady plays professional basketball. M =Matt Ryan plays professional basketball.

∼[(W⊃T)•(C⊃K)]

It is not the case that both Walmart's jeopardizing worker safety implies that Target does and Costco's treating workers decently implies that Kmart does. W =Walmart jeopardizes worker safety. T =Target jeopardizes worker safety. C =Costco treats workers decently. K =Kmart treats workers decently.

∼[H∨(A•I)]

It is not the case that either Hezbollah will renounce violence or al-Qaeda and ISIS will be defeated. H =Hezbollah will renounce violence. A =al-Qaeda will be defeated. I =ISIS will be defeated.

∼[(T•M)∨(D•S)]

It is not the case that either Tiger Woods and Maria Sharapova play professional football or Drew Brees and Serena Williams play professional baseball. T =Tiger Woods plays professional football. M =Maria Sharapova plays professional football. D =Drew Brees plays professional baseball. S =Serena Williams plays professional baseball.

(K•B)≡(E•N)

Katie Couric and Brooke Baldwin report international news if and only if Erica Hill and Norah O'Donnell cover political developments. K =Katie Couric reports international news. B =Brooke Baldwin reports international news. E =Erica Hill covers political developments. N =Norah O'Donnell covers political developments.

M•(E∨D)

Mariah Carey sings pop and either Elton John sings rock or Diana Krall sings jazz. M = Mariah Carey sings pop. E = Elton John sings rock. D = Diana Krall sings jazz.

M⊃(L•B)

Mercedes will introduce a hybrid model only if Lexus and BMW do. M = Mercedes will introduce a hybrid model. L = Lexus will introduce a hybrid model. B = BMW will introduce a hybrid model.

(∼P•∼C)⊃M

Mercury is a planet given that both Pluto and Ceres are not. M =Mercury is a planet. P =Pluto is a planet. C =Ceres is a planet.

(P⊃N)•[(C•A)⊃J]

Nassau's advertising sizzling night clubs is a necessary condition for Puerto Vallarta's offering luxury hotels; moreover, Cancun's having turquoise waters and Acapulco's promising powder-white beaches is a sufficient condition for Jamaica's offering reggae music. N =Nassau advertises sizzling night clubs. P =Puerto Vallarta offers luxury hotels. C =Cancun has turquoise waters. A =Acapulco promises power-white beaches. J =Jamaica offers reggae music.

∼(F∨M)

Neither Ferrari nor Maserati makes economy cars. F = Ferrari makes economy cars. M = Maserati makes economy cars.

∼(J•P)

Not both Jaguar and Porsche make motorcycles. J = Jaguar makes motorcycles. P = Porsche makes motorcycles.

∼(F∨M)

Not either Ferrari or Maserati makes economy cars. F = Ferrari makes economy cars. M = Maserati makes economy cars.

Compound Statement

One that contains at least one simple statement as a component

R ⊃ M

Reese Witherspoon wins best actress only if Martin Scorsese wins best director. R = Reese Witherspoon wins best actress.M = Martin Scorsese wins best director.

(T • B) ∨ (∼Z•∼K)

Tiffany and Ben Bridge will release an emerald collection unless Zales and Kay do not. T =Tiffany will release an emerald collection. B =Ben Bridges will release an emerald collection. Z =Zales will release an emerald collection. K =Kay will release an emerald collection.

T≡∼(R∨C)

Tommy Hilfiger celebrates casual if and only if neither Ralph Lauren nor Calvin Klein offers street chic. T = Tommy Hilfiger celebrates casual. R = Ralph Lauren offers street chic. C = Calvin Klein offers street chic.

A • ∼N

Translate the following statement into symbolic form using capital letters to represent an affirmative English statement. Arizona has a national park but Nebraska does not. A = Arizona has a national park. N = Nevada has a national park.

H • B

Translate the following statement into symbolic form using capital letters to represent an affirmative English statement. Both Harvard and Baylor have medical schools. H = Harvard has a medical school. B = Baylor has a medical school.

∼C

Translate the following statement into symbolic form using capital letters to represent an affirmative English statement. Cartier does not make cheap watches. C = Cartier makes cheap watches.

S ∨ T

Translate the following statement into symbolic form using capital letters to represent an affirmative English statement. Either Stanford or Tulane has an architecture school. S = Stanford has an architecture school. T = Tulane has an architecture school.

C⊃L

Translate the following statement into symbolic form using capital letters to represent an affirmative English statement. If Chanel has a rosewood fragrance, then so does Lanvin. C = Chanel has a rosewood fragrance. L = Lanvin has a rosewood fragrance.

M≡P

Translate the following statement into symbolic form using capital letters to represent an affirmative English statement. Maureen Dowd writes incisive editorials if and only if Paul Krugman does. M = Maureen Dowd writes incisive editorials. P = Paul Krugman writes incisive editorials.

P⊃A

Translate the following statement into symbolic form using capital letters to represent an affirmative English statement. The Packers' winning most of their games implies that Aaron Rodgers is a great quarterback. P = The Packers win most of their games. A = Aaron Rodgers is a great quarterback.

Simple Statement

Does not contain any other statement as a component

A∨S

Either Apple or Samsung makes smartphones. A = Apple makes smartphones. S = Samsung makes smartphones.

∼F∨∼M

Either Ferrari or Maserati does not make economy cars. F = Ferrari makes economy cars. M = Maserati makes economy cars.

(M•E)∨D

Either Mariah Carey sings pop and Elton John sings rock or Diana Krall sings jazz. M = Mariah Carey sings pop. E = Elton John sings rock. D = Diana Krall sings jazz.

(P∨S)•∼(P•S)

Either Pablo Picasso or Salvador Dali ignored perspective, but it is not the case that both of them did. P =Pablo Picasso ignored perspective. S =Salvador Dali ignored perspective.

∼R∨(M⊃P)

Either Rado does not make a sapphire watch or if Movado makes one then so does Pulsar. R = Rado makes a sapphire watch. M = Movado makes a sapphire watch. P = Pulsar makes a sapphire watch.

Consequent or Necessary condition

Event A is said to be this for event B whenever B cannot occur without the occurrence of A

Antecedent or sufficient condition

Event A is said to be this for event B whenever the occurrence of A is all that is required for the occurrence of B

Main operator

The operator (connective) in a compound statement that has as its scope everything else in the statement

Propositional Logic

A kind of logic in which the fundamental components are whole statements or propositions

Conjuction

A statement having a dot as its main operator; used to translate "and" "also" or "moreover"

Conditional or Material Implication

A statement having a horseshoe as the main operator; used to translate "if"... "then"..., "only if"

Negation

A statement having a tilde as its main operator; used to translate "not" or "it is not the case that"

Biconditional or Material Equivalence

A statement having a triple bar as its main operator; used to translate "if and only if"

Disjunction

A statement having a wedge as its main operator; used to translate "or" or "unless"

Well-Formed Formulas (WFF)

A syntactically correct arrangements of symbols

(B⊃C)•(K•E)

Chevron continues to pollute provided that British Petroleum does, but Koch Industries and Exxon Mobil deny climate change. C =Chevron continues to pollute. B =British Petroleum continues to pollute. K =Koch Industries denies climate change. E =Exxon Mobile denies climate change.

G ⊃ A

Armani will launch a leather collection given that Gucci rejects skinny models. A =Armani will launch a leather collection. G =Gucci rejects skinny models.

∼B∨W

Bill Gates does not support malaria research unless Warren Buffet does. B = Bill Gates supports malaria research. W = Warren Buffet supports malaria research.

∼J•∼P

Both Jaguar and Porsche do not make motorcycles. J = Jaguar makes motorcycles. P = Porsche makes motorcycles.

S⊃(B⊃C)

Cameron Diaz promotes environmental causes if Ben Affleck supports civil liberties, provided that Sean Penn opposes the death penalty. C =Cameron Diaz promotes environmental causes. B =Ben Affleck supports civil liberties. S =Sean Penn opposes the death penalty.

L ⊃ C

Chanel has a rosewood fragrance if Lanvin does. C = Chanel has a rosewood fragrance. L = Lanvin has a rosewood fragrance.

(S•N)∨J

Saturn has rings and Neptune is windy, or Jupiter is massive. S = Saturn has rings. N = Neptune is windy. J = Jupiter is massive.

S•(N∨J)

Saturn has rings, and Neptune is windy or Jupiter is massive. S =Saturn has rings. N =Neptune is windy. J =Jupiter is massive.

S•[N⊃(CvJ)]

South Korea supports diplomacy; but if North Korea starts a war, then either China or Japan will be involved. S =South Korea supports diplomacy. N =North Korea starts a war. C =China will be involved. J =Japan will be involved.

Connectives

Symbols used to connect or negate propositions in propositional logic

Operators

Symbols used to connect simple propositions in propositional logic

(T⊃P)⊃(U⊃A)

The Taliban's being defeated is a sufficient condition for Pakistan's winning the war on terror only if Afghanistan's securing its borders is a necessary condition for the UN's stopping the opium trade. T =The Taliban is defeated. P =Pakistan wins the war on terror. A =Afghanistan secures its borders. U =The U.N. stops the opium trade.

Conjuncts

The component in a conjunctive statement on either side of the main operator

Disjuncts

The component in a disjunctive statement on either side of the main operator