6.3 Truth Tables for Propositions

Renowned economist Harold Carlson makes the following prediction: "The balance of payments will decrease if and only if interest rates remain steady; however, it is not the case that either interest rates will not remain steady or the balance of payments will decrease."

Carlson's prediction is false (self-contradictory).

Eric Carson sums up his beliefs about God as follows: "God exists if and only if either life is meaningful or the soul is not immortal. God exists and the soul is immortal. If God exists, then life is not meaningful."

Eric's beliefs are inconsistent; therefore, they do not make sense.

[G ⊃ (N ⊃ ~G)] • [(N ≣ G) • (N v G)]

F F F F T T T T T F F F F T T T T F F F T T T T F T F F self-contradictory

(~K ⊃ H) ≣ ~(H v K)

F T F F T F T F F T T T F F T T F F T F self-contradictory

Antonia Martinez, who is running for the state senate, makes this statement: "Either a tax reduction is feasible only if both educational costs do not increase and the welfare program is abolished, or a tax reduction is feasible and either the welfare program will not be abolished or educational costs will increase."

Martinez's statement is tautologous; therefore, she has told us nothing about taxes, educational costs, or welfare.

A high school principal made this statement to the school board: "Either music is not dropped from the curriculum or the students will become cultural philistines; furthermore, the students will not become cultural philistines if and only if music is dropped from the curriculum."

Music is not dropped from the curriculum, and the students will become cultural philistines.

(S⊃R) • (S • ~R)

T F F F F F T T T F F F T F F T self-contradictory

[(C⊃D) • ~C] ⊃ ~D

T F F T F F F F T T T T T F F T T T T T contingent

(G⊃G) ⊃ G

T T T F contingent

N ⊃ (N⊃N)

T T T T tautologous

[(U • (T v S)] ≣ [(~T v ~U) • (~S v ~U)]

T T F F F F F F F F T T F F F F F T T F T T F T T F F F F F F F F T T F T T T F F T F F T T T F T T F T F F T T T T T T F T F T T T T F T T F F F T T T T T T T self-contradictory

[(Q⊃P) • (~Q⊃R)] • ~(P v R)

T T F T F F T T T F T F F T F F F T F F T F F F T F T F T T T T F F T T F T F F F T T T T T F F T T F T F F T F self-contradictory

[(E⊃F) ⊃ F] ⊃ E

T T T F T T T T F T F T contingent

(M⊃P) v (P⊃M)

T T T F T T T T F T T T tautologous

[(Z⊃X) • (X v Z)] ⊃ X

T T T T F F T T T T T T T F F T tautologous

[X ⊃ (R⊃F)] ≣ [(X⊃R) ⊃ F]

T T T T T F F T T F T T T F T T T T F T T T T T T T F F T F T T T T T T T F T F contingent

[(H⊃N) • (T⊃N)] ⊃ [(H v T) ⊃ N]

T T T T T T T T T T T T F F F T T F F F T T T F T T T T T T T T T T F T T F F T T F T T T T F T tautologous

{[(G • N) ⊃ H] • [(G ⊃ H) ⊃ P]} ⊃ (N⊃P)

T T T T T T T T T F T F T F T F F F T T T T F F F T T F F T T T T T T F T F T F T T F T T F T T T F T T F T T T F T T T T T T F T F T F T F F T T T T T T F T F T F T F F T T T T T T F T F T F T T F T T T T T T F T F T F T T tautologous

[(F v E) • (G v H)] ≣ [(G • E) v (F • H)]

T T T T T T T T T T T T T F T T T T F T T T F F T F F F T T T T F T T T T T F F F F T T T T F T T T F F T F F F T T T T T T F T T T T T T F T T T F F F F T F F T F F F F F T T F F F F F T T F F F F F T T F F F F F F T F F F contingent

Christina and Thomas are having a discussion about their plans for the evening. Christina: "If you don't love me, then I'm certainly not going to have sex with you." Thomas: "Well, that means that if I do love you, then you will have sex with me, right?"

Thomas is not correct; the two statements are not logically equivalent.

Two astronomers are discussing supernovas. Dr. Frank says, "Research has established that if a supernova occurs within ten light-years of the Earth, then life on Earth will be destroyed." Dr. Harris says, "Research has also established that either a supernova will not occur within ten light-years of the Earth or life on Earth will not be destroyed."

Yes A supernova will not occur within 10 light years of the earth.

Two stockbrokers are having a discussion. One claims that Netmark will introduce a new product if and only if both Datapro cuts its workforce and Compucel expands production. The other claims that Datapro will cut its workforce, and Compucel will expand production if and only if Netmark introduces a new product.

Yes Datapro will cut back its work force.

Automotive expert Frank Goodbody has this to say about Japanese imports: "If Mitsubishi is the sportiest, then both Toyota is the most trouble-free and Isuzu is not the lowest priced. If Isuzu is the lowest priced, then both Toyota is not the most trouble-free and Mitsubishi is the sportiest."

Yes Isuzu is not the lowest priced.

Cindy, Jane, and Amanda witnessed a bank robbery. At trial, Cindy testified that Lefty did not enter the bank, and if Howard pulled a gun, then Conrad collected the money. Jane testified that if Howard did not pull a gun, then Lefty entered the bank. Amanda testified that if Conrad collected the money, then Howard pulled a gun.

Yes Lefty did not enter the bank, Howard pulled a gun, and Conrad collected the money.

Nicole Evans expresses her philosophy as follows: "If the mind is identical to the brain, then personal freedom does not exist and humans are not responsible for their actions. If personal freedom does not exist, then the mind is identical to the brain. Either humans are responsible for their actions or the mind is not identical to the brain. If personal freedom exists, then humans are responsible for their actions."

Yes The mind is not identical to the brain, personal freedom exists, and humans are responsible for their actions.

G • (E v P)

contradictory

H ≣ ~G

contradictory

Q ⊃ (KvF)

contradictory

R v ~S

contradictory

(E⊃C) ⊃ L

does not apply consistent

R • (Q v S)

does not apply consistent

~K⊃L

does not apply consistent

F • M

does not apply inconsistent

N • (A v ~E)

does not apply inconsistent

W ≣ (B • T)

does not apply inconsistent

Z • (C≣P)

does not apply inconsistent

H • (K v J)

logically equivalent

M ⊃ (K⊃P)

logically equivalent

~A ≣ X

logically equivalent

~D v B

logically equivalent