1. If the butler shot Jones, then he knew how to use a gun.
If the butler was a former marine, then he knew how to use a gun.
The butler was a former marine.
∴ The butler shot Jones. [Use S, K, and M.]
1. Invalid *
1 (S ⊃ K) *
2 (M K) *
3M*[∴S*
4 asm: ∼S *
5 ∴ K—{from 2 and 3}
M, ∼S, K
2. If virtue can be taught, then either there are professional virtue-teachers or there are amateur
virtue-teachers.
If there are professional virtue-teachers, then the Sophists can teach their students to be
virtuous.
If there are amateur virtue-teachers, then the noblest Athenians can teach their children to be
virtuous.
The Sophists can’t teach their students to be virtuous and the noblest Athenians (such as the
great leader Pericles) can’t teach their children to be virtuous.
∴ Virtue can’t be taught
2. Valid
1 (V ⊃(P v A))
2 (P ⊃ S)
3 (A ⊃ N)
4 (∼S •∼N)
I ∴∼V
5 ┌ asm: V
6 │ ∴ ∼S—{from 4} *
7 │ ∴ ∼N—{from 4} *
8 │ ∴ (P v A) {from 1 and 5}
9 │ ∴ ∼P {from 2 and 6}
10│ ∴ ∼A {from 3 and 7}
11└ ∴ A—{from 8 and 9} *
12 ∴ ∼V from 5, 10 contradicts 11
3. It would be equally wrong for a sadist (through drugs that would blind you but not hurt your
mother) to have blinded you permanently before or after your birth.
If it would be equally wrong for a sadist (through such drugs) to have blinded you permanently
before or after your birth, then it’s false that one’s moral right to equal consideration begins at
birth.
If infanticide is wrong and abortion isn’t wrong, then one’s moral right to equal consideration
begins at birth. Infanticide is wrong.
∴ Abortion is wrong
3. Valid *
1E*
2 (E ⊃ ∼R) *
3 ((I • ∼A) ⊃ R) *
4I*[∴A*
5 ┌ asm: ∼A *
6 │ ∴ ∼R—{from 1 and 2} *
7 │ ∴ ∼(I • ∼A)—{from 3 and 6} *
8 └ ∴ A—{from 4 and 7} *
9 ∴ A—{from 5; 5 contradicts 8}
4. If you hold a moral belief and don’t act on it, then you’re inconsistent.
If you’re inconsistent, then you’re doing wrong.
∴ If you hold a moral belief and act on it, then you aren’t doing wrong
Invalid
1. ((M • ∼A) ⊃ I)
2. (I ⊃W)
I ∴ ((M • A) ⊃∼W)
3. asm: ∼((M • A) ⊃∼W)
4. ∴ (M • A) from 3
5. ∴ W from 3
6. ∴ M from 4
7. ∴ A from 4
W, M, A
5. If Socrates escapes from jail, then he’s willing to obey the state only when it pleases him.
If he’s willing to obey the state only when it pleases him, then he doesn’t really believe what he
says and he’s inconsistent.
∴ If Socrates really believes what he says, then he won’t escape from jail.
5. Valid *
1 (E ⊃ W) *
2 (W ⊃ (∼R • I)) * [ ∴ (R ⊃ ∼E) *
3 ┌ asm: ∼(R ⊃ ∼E) *
4 │ ∴ R—{from 3} *
5 │ ∴ E—{from 3} *
6 │ ∴ W—{from 1 and 5} *
7 │ ∴ (∼R • I)—{from 2 and 6} *
8 └ ∴ ∼R—{from 7} *
9 ∴ (R ⊃ ∼E)—{from 3; 4 contradicts 8}
6. Either Socrates’s death will be perpetual sleep, or if the gods are good then his death will be
an entry into a better life.
If Socrates’s death will be perpetual sleep, then he shouldn’t fear death.
If Socrates’s death will be an entry into a better life, then he shouldn’t fear death.
∴ Socrates shouldn’t fear death.
Invalid
1.(P v (G ⊃ B))
2. (P ⊃ ∼ F)
3. (B ⊃ ∼ F)
I ∴ ∼F
4. asm: F
5. ∴ ∼P from 2 and 4
6. ∴ (G ⊃ B) from 1 and 5
7. ∴ ∼B from 3 and 4
8. ∴ ∼G from 6 and 7
F, ∼P, ∼B, ∼G
7. If predestination is true, then God causes us to sin.
If God causes us to sin and yet damns sinners to eternal punishment, then God isn’t good.
∴ If God is good, then either predestination isn’t true or else God doesn’t damn sinners to
eternal punishment.
Valid
1. ( P ⊃ C)
2. ((C • D) ⊃∼G)
I ∴ (G ⊃( ∼P v ∼ D)
3. ┌ asm: ∼(G ⊃(∼P v ∼D))
4. │ ∴ G from 3
5. │ ∴ ∼( ∼P v ∼D) from 3
6. │ ∴ P from 5
7. │ ∴ D from 5
8. │ ∴ C from 1 and 6
9. │ ∴ ∼( C • D) from 2 and 4
10.└ ∴ ∼C from 7 and 9
11. ∴ (G ⊃(∼ P v ∼ D)) from 3, 8 contradicts 10
8. If determinism is true, then we have no free will.
If Heisenberg’s interpretation of quantum physics is correct, some events aren’t causally
necessitated by prior events.
If some events aren’t causally necessitated by prior events, determinism is false.
∴ If Heisenberg’s interpretation of quantum physics is correct, then we have free will.
Invalid
1. (D ⊃ ∼ F)
2. (H ⊃ E)
3. (E ⊃ ∼D)
I ∴ ( H ⊃ F)
4. asm: ∼(H ⊃ F)
5. ∴ H from 4
6. ∴ ∼F from 4
7. ∴ E from 2 and 5
8. ∴ ∼ D from 3 and 7
H, ∼F, E, ∼D
9. Government’s function is to protect life, liberty, and the pursuit of happiness.
The British colonial government doesn’t protect these.
The only way to change it is by revolution. If government’s function is to protect life, liberty, and
the pursuit of happiness and the British colonial government doesn’t protect these, then the
British colonial government ought to be changed.
If the British colonial government ought to be changed and the only way to change it is by
revolution, then we ought to have a revolution.
∴ We ought to have a revolution
Valid
1. G
2. ∼B
3. O
4. ((G • ∼B) ⊃ C)
5. ((C • O) ⊃R )
I∴ R
6┌ asm: ∼R*
7 │ ∴ ∼((C • O) from 5 and 6
8 │ ∴ ∼C from 3 and 7
9 │ ∴ ∼(G • ∼B) from 4 and 8
10└ ∴ B—{from 6 and 11} from 1 and 9
11. ∴ R from 6, 2 contradicts 10
10. The apostles’ teaching either comes from God or is of human origin.
If it comes from God and we kill the apostles, then we will be fighting God.
If it’s of human origin, then it’ll collapse of its own accord.
If it’ll collapse of its own accord and we kill the apostles, then our killings will be unnecessary.
∴ If we kill the apostles, then either our killings will be unnecessary or we will be fighting God
1. Valid *
1 (G ∨ H) *
2 ((G • K) ⊃ F) *
3 (H ⊃ C) *
4 ((C • K) ⊃ U) * 1[ ∴ (K ⊃ (U ∨ F)) *
5 ┌ asm: ∼(K ⊃ (U ∨ F)) *
6 │ ∴ K—{from 5} *
7 │ ∴ ∼(U ∨ F)—{from 5} *
8 │ ∴ ∼U—{from 7} *
9 │ ∴ ∼F—{from 7} *
10 │ ∴ ∼(G • K)—{from 2 and 9} *
11 │ ∴ ∼(C • K)—{from 4 and 8} *
12 │ ∴ ∼G—{from 6 and 10} *
13 │ ∴ H—{from 1 and 12} *
14 │ ∴ C—{from 3 and 13} * 1
15 └ ∴ ∼C—{from 6 and 11} *
16 ∴ (K ⊃ (U ∨ F))—{from 5; 14 contradicts 15}
11. If materialism (the view that only matter exists) is true, then idealism is false.
If idealism (the view that only minds exist) is true, then materialism is false.
If mental events exist, then materialism is false.
If materialists think their theory is true, then mental events exist.
∴ If materialists think their theory is true, then idealism is true.
Invalid
1.(M ⊃ ∼I)
2. (I ⊃ ∼ M)
3. ( E ⊃ ∼ M)
4. (T ⊃ E)
I ∴ (T ⊃ I)
5. asm: ∼( T ⊃ I)
6. ∴ T from 5
7. ∴ ∼ I from 5
8. ∴ E from 4 and 6
9. ∴ ∼ M from 3 and 8
T, ∼I, E, ∼M
12. If determinism is true and cruelty is wrong, then the universe contains unavoidable wrong
actions.
If the universe contains unavoidable wrong actions, then we ought to regret the universe as a
whole.
If determinism is true and regretting cruelty is wrong, then the universe contains unavoidable
wrong actions.
∴ If determinism is true, then either we ought to regret the universe as a whole (the pessimism
option) or else cruelty isn’t wrong and regretting cruelty isn’t wrong (the “nothing matters”
option).
Valid
1. (( D • C) ⊃ U)
2. ( U ⊃ O)
3. (( D • R) ⊃ U)
I ∴ (D ⊃( O v (∼ C • ∼R)))
4 ┌ asm: ∼(D ⊃ (O ∨ (∼ C •∼R ))) *
5 │ ∴ D from 4
6 │ ∴ ∼(O ∨ (∼C •∼R)) from 4
7 │ ∴ ∼O from 6
8 │ ∴ ∼( ∼C • ∼ R) from 6
9 │ ∴ ∼U from 2 and 9
10 │ ∴ ∼(D • C) from 1 and 9
11 │ ∴ ∼(D • R) from 3 and 9
12 │ ∴ ∼C from 5 and 10
13 │ ∴ R from 8 and 12
14 └ ∴ ∼R from 5 and 11
15. ∴ (D ⊃( O v ( ∼C • ∼R))) from 4, 13 contradicts 14
13. If a belief is proved, then it’s worthy of acceptance.
If a belief isn’t disproved but is of practical value to our lives, then it’s worthy of acceptance.
If a belief is proved, then it’s not disproved.
∴ If a belief is proved or is of practical value to our lives, then it’s worthy of acceptance.
Invalid
1.(P ⊃ W)
2. (( ∼D • V) ⊃ W)
3. (P ⊃ ∼D)
I ∴ ((P v V)⊃ W)
4. asm: ∼((P v V) ⊃ W)
5. ∴ (P v V) from 4
6. ∴ ∼W from 4
7. ∴ ∼ P from 1 and 6
8. ∴ ∼( ∼D • V) from 2 and 6
9. ∴ V from 5 and 7
10. ∴ D from 8 and 9
∼W, ∼P, V, D
14. If you’re consistent and think that stealing is normally permissible, then you’ll consent to the
idea of others stealing from you in normal circumstances.
You don’t consent to the idea of others stealing from you in normal circumstances.
∴ If you’re consistent, then you won’t think that stealing is normally permissible.
Valid
1. ((C • N) ⊃ Y)
2. ∼ Y
I ∴ ( C ⊃ ∼N)
3. ┌asm: ∼(C ⊃ ∼N)
4. │∴ C from 3
5. │∴ N from 3
6. │∴ ∼(C •N) from 1 and 2
7. └ ∴ ∼N from 4 and 6
8. (C ⊃ ∼N) from 3, 5 contradicts 7
15. If the meaning of a term is always the object it refers to, then the meaning of “Fido” is Fido.
If the meaning of “Fido” is Fido, then if Fido is dead then the meaning of “Fido” is dead.
If the meaning of “Fido” is dead, then “Fido is dead” has no meaning.
“Fido is dead” has meaning.
∴ The meaning of a term isn’t always the object it refers to.
15. Invalid *
1 (A ⊃ B) *
2 (B ⊃ (F ⊃ M)) *
3 (M ⊃ ∼H) *
4 H * [ ∴ ∼A *
5 asm: A *
6 ∴ B—{from 1 and 5} *
7 ∴ (F ⊃ M)—{from 2 and 6} *
8 ∴ ∼M—{from 3 and 4} *
9 ∴ ∼F—{from 7 and 8} An “F” premise would make it valid
H, A, B, ∼M, ∼F
16. God is all powerful. If God is all powerful, then he could have created the world in any
logically possible way and the world has no necessity.
If the world has no necessity, then we can’t know the way the world is by abstract speculation
apart from experience.
∴ We can’t know the way the world is by abstract speculation apart from experience.
Valid
1. A
2. (A ⊃ (C • ∼N)
3. (∼N ⊃ ∼K)
c∴∼ K
4. ┌asm: K
5. │∴ ( C • ∼ N) from 1 and 2
6. │∴ C from 5
7. │∴ ∼N from 5
8. └ ∴ N from 3 and 4
9. ∴ ∼K from 4, 7 contradicts 8
17. If God changes, then he changes for the worse or for the better.
If he’s perfect, then he doesn’t change for the worse.
If he changes for the better, then he isn’t perfect.
∴ If God is perfect, then he doesn’t change.
Valid
1. (C ⊃(W v B))
2. (P ⊃ ∼W)
3. (B ⊃ ∼P)
I ∴ (P ⊃ ∼C)
4. ┌asm: ∼(P ⊃∼C)
5. │∴ P from 4
6. │∴ C from 4
7. │∴ (W v B) from 1 and 6
8. │∴ ∼B from 3 and 5
9. │∴ ∼ W from 2 and 5
10. └ ∴ Wfrom 7 and 8
11. (P ⊃ ∼ C) from 4, 9 contradicts 10
18. If belief in God has scientific backing, then it’s rational.
No conceivable scientific experiment could decide whether there is a God.
If belief in God has scientific backing, then some conceivable scientific experiment could decide
whether there is a God.
∴ Belief in God isn’t rational.
Invalid
1. (B ⊃ R)
2. ∼ D
3.( B ⊃ D)
I ∴ ∼R
4. asm: R
5. ∴ ∼B
∼D, R, ∼B
19. Every event with finite probability eventually takes place.
If the nations of the world don’t get rid of their nuclear weapons, then there’s a finite probability
that humanity will eventually destroy the world.
If every event with finite probability eventually takes place and there’s a finite probability that
humanity will eventually destroy the world, then humanity will eventually destroy the world.
∴ Either nations of the world will get rid of their nuclear weapons, or humanity will eventually
destroy the world.
Valid
1.E
2. (∼R ⊃ F)
3. ((E • F) ⊃H)
I ∴(R v H)
4. ┌asm: ∼(R v H)
5. │ ∴ ∼R from 4
6. │ ∴ ∼H from 4
7. │ ∴ F from 2 and 5
8. │ ∴ ∼(E • F) from 3 and 6
9. └ ∴ ∼ F from 1 and 8
10. (R v H) from 4, 7 contradicts 9
20. If the world isn’t ultimately absurd, then conscious life will go on forever and the world
process will culminate in an eternal personal goal.
If there is no God, then conscious life won’t go on forever.
∴ If the world isn’t ultimately absurd, then there is a God.
20. Valid *
1 (∼A ⊃ (F • C)) *
2 (∼G ⊃ ∼F) *[ ∴ (∼A ⊃ G) *
3 ┌ asm: ∼(∼A ⊃ G) *
4 │ ∴ ∼A—{from 3} *
5 │ ∴ ∼G—{from 3} *
6 │ ∴ (F • C)—{from 1 and 4} *
7 │ ∴ F—{from 6} *
8 │ ∴ C—{from 6} *
9 └ ∴ ∼F—{from 2 and 5} *
10 ∴ (∼A ⊃ G)—{from 3; 7 contradicts 9}
21. If it rained here on this date 500 years ago and there’s no way to know whether it rained
here on this date 500 years ago, then there are objective truths that we cannot know.
If it didn’t rain here on this date 500 years ago and there’s no way to know whether it rained
here on this date 500 years ago, then there are objective truths that we cannot know.
There’s no way to know whether it rained here on this date 500 years ago.
∴ There are objective truths that we cannot know.
Valid
1. ((R • ∼K) ⊃O)
2. (( ∼R • ∼K) ⊃O)
3. ∼ K
I∴O
4. ┌ asm: ∼O
5. │ ∴ ∼(R • ∼ K) from 1 and 4
6. │ ∴ ∼( ∼R • ∼K) from 2 and 4
7. │ ∴ ∼ R from 3 and 5
8. └ ∴ R from 3 and 6
9. ∴ O from 4, 7 contradicts 8
22. If you know that you don’t exist, then you don’t exist.
If you know that you don’t exist, then you know some things.
If you know some things, then you exist.
∴ You exist.
Invalid
1. (K ⊃ ∼E)
2. (K ⊃ S)
3. (S ⊃ E)
I∴E
4. asm: ∼E
5. ∴ ∼S from 3 and 4
6. ∴ ∼ K from 2 and 5
∼E, ∼S, ∼K
23. We have an idea of a perfect being.
If we have an idea of a perfect being, then this idea is either from the world or from a perfect
being.
If this idea is from a perfect being, then there is a God.
∴ There is a God.
Invalid
1. I
2. ( I ⊃ (W v P))
3. (P ⊃ G)
I∴G
4. asm: ∼G
5. ∴ (W v P) from 1 and 2
6. ∴ ∼P from 3 and 4
7. ∴ W from and 6
I, ∼G, ∼P, W
24. The distance from A to B can be divided into an infinity of spatial points.
One can cross only one spatial point at a time.
If one can cross only one spatial point at a time, then one can’t cross an infinity of spatial points
in a finite time.
If the distance from A to B can be divided into an infinity of spatial points and one can’t cross an
infinity of spatial points in a finite time, then one can’t move from A to B in a finite time.
If motion is real, then one can move from A to B in a finite time.
∴ Motion isn’t real.
Valid
1. D
2. O
3. (O ⊃ ∼C)
4. ((D • ∼C) ⊃ ∼M)
5. (R ⊃ M)
6. ┌asm: R
7. │ ∴ ∼C from 2 and 3
8. │∴ M from 5 and 6
9. │ ∴ ∼(D • ∼C) from 4 and 8
10,└ ∴ C from 1 and 9
11. ∴ ∼R from 6, 7 contradicts 10
25. If the square root of 2 equals some fraction of positive whole numbers, then (we stipulate)
the square root of 2 equals x/y and x/y is simplified as far as it can be.
If the square root of 2 equals x/y, then 2 = x2/y2.
If 2 = x2/y2, then 2y2 = x2.
If 2y2 = x2, then x is even.
If x is even and 2y2 = x2, then y is even.
If x is even and y is even, then x/y isn’t simplified as far as it can be.
∴ The square root of 2 doesn’t equal some fraction of positive whole numbers.
25. Valid *
1 (F ⊃ (E • S)) *
2 (E ⊃ T) *
3 (T ⊃ T´) *
4 (T´ ⊃ X) *
5 ((X • T´) ⊃ Y) *
6 ((X • Y) ⊃ ∼S) * [ ∴ ∼F *
7 ┌ asm: F *
8 │ ∴ (E • S)—{from 1 and 7} *
9 │ ∴ E—{from 8} *
10│ ∴ S—{from 8} *
11 │ ∴ T—{from 2 and 9} *
12 │ ∴ T´—{from 3 and 11} *
13 │ ∴ X—{from 4 and 12} *
14 │ ∴ ∼(X • Y)—{from 6 and 10} *
15 │ ∴ ∼Y—{from 13 and 14} *
16 │ ∴ ∼(X • T´)—{from 5 and 15} *
17 └ ∴ ∼X—{from 12 and 16} *
18 ∴ ∼F—{from 7; 13 contradicts 17}