Slides on conditionals, disjunction, validity, soundness, modus ponens, modus tollens, chain argument, disjunctive syllogism, and dilemma

1. Soundness
A Standard of Critical Thinking

2. Review
Logical negation, often
expressed in English by ‘not’,
is true when the component
claim is false, false when the
component claim is true. It is
symbolized by ‘~’ and has the
logical form ~P.
Logical conjunctions, often
expressed in English by
‘and’, is true when the
component claims it joins are
true, otherwise it is false. It is
symbolized by ‘&’. It’s logical
form is P & Q.

3. Review
Contradiction, a special form
of conjunction in which a claim
and its negation are joined—
they are always false. The
logical form of a contradiction
is P & ~P.
The Principle of
Noncontradiction, states that
no thing can, at the same
time and in the same
manner, both have and not
have the same property.

4. Review
The Standard of
Consistency—accept only
those beliefs which are
consistent with each other
and any accessible
evidence.
Reductio ad ridiculum,
appealing to ridicule (making
fun of an opposing view)
rather than providing
reasons against it—it is a
fallacy.
Equivocation, to use a
term ambiguously or
vaguely in an argument—it
is a fallacy.

5. Review
double negation—any
even number of negations
cancel each other out.
to prove a conjunction
false prove that one of the
component claims is false.
to evaluate by
contradiction—isolate the
subject and predicate,
generate lists of things that
fall under each, stopping
when you determine that
they are not identical.
proof by
counterexample—Choose
an item that is not on both
lists, explain how the
definition says it should
be, then explain why it is
not, indicate the
inconsistency, and reject
or revise the definition.

6. Proof by
Counterexample
A Method for Reasoning with
Contradictions
Line of
Reasoning
An explanation showing
that the definition should
be true of a specific
example (thing or event).
Reject the
original
definition
Original definition.
Another Line of
Reasoning
Another explanation
showing that the definition
is not true of the same
example.

7. Reductio ad
absurdam
Reductio ad absurdam
Indirect Proof, Proof by
Counterexample
Line of
Reasoning
1. Claim
2. reasons
3. conclusion
6. P & ~P
Another Line of 4. other reasons
Reasoning 5. other conclusion
7. Rejection

8. The Logical Form of a
Reductio
Reductio ad absurdam, Indirect
Proof, Proof by Counterexample
1.claim
2.reasons
3.conclusion
4.other reasons
5.other conclusion
6.contradiction
7.rejection

9. Review
to avoid equivocating—
define key terms by giving
them one (to disambiguate)
clear (to avoid vagueness)
meaning.
to avoid equivocating—use
the Principle of Charity to
settle on the best
interpretation, whether
normative or descriptive.
to avoid reductio ad
ridiculums—use the Principle
of Sufficient Reason and
attempt to provide reasons
for each claim.

10. The
Conditional
Logical
Complexity

11. A Logically Simple
Truth
Two logical
possibilities
Sarah attends
Stanford.
1
True
2
False
Given that
we’ve filled in
the indices,
made the
ceteris
paribus
explicit, and
defined key
terms.

12. Combining Logically Simple
Truths
Four states of
affairs (states)
or possible
worlds
Sarah attends
Stanford.
Sarah goes into debt.
1
True
True
2
True
False
3
False
True
4
False
False

13. Logical
Conjunction
Sarah attends Stanford
AND
Sarah goes into debt
1
True
True
True
2
True
False
False
3
False
False
True
4
False
False
False

14. Logical
Conditional
IF
Sarah attends
Stanford
THEN Sarah goes into debt
1
True
?
True
2
True
?
False
3
False
?
True
4
False
?
False

15. Logical
Conditional
IF
Sarah attends
Stanford
THEN Sarah goes into debt
1
True
True
True
2
True
?
False
3
False
?
True
4
False
?
False

16. Logical
Conditional
IF
Sarah attends
Stanford
THEN Sarah goes into debt
1
True
True
True
2
True
False
False
3
False
?
True
4
False
?
False

17. Logical
Conditional
IF
Sarah attends
Stanford
THEN Sarah goes into debt
1
True
True
True
2
True
False
False
3
False
True
True
4
False
True
False

18. Logical
Conditional
IF
Sarah attends
Stanford
THEN Sarah goes into debt
1
True
True
True
2
True
False
False
3
False
True
True
4
False
True
False

19. Logical
Conditional
Taking ‘if’ seriously
If Sarah goes to Stanford then she will incur debt.
≠
If Sarah will incur debt then she goes to Stanford.
Either Sarah will incur debt or she goes the Stanford.
Both Sarah will incur debt and she goes the Stanford.
=
=
Either Sarah goes the Stanford or she will incur debt.
Both Sarah goes the Stanford and she will incur debt.

20. Logical
Conditional
Order Matters
Antecedent
Consequent
If Sarah goes to Stanford then she will incur debt.

21. The Case of Iffy Advice
Sarah Scatterleigh weighed her options. She could
transfer to Stanford, which had a stronger program for
her major and a better track record of placing
graduates into the job market. But Stanford cost quite
a bit more than the school she was presently
attending, Jefferson University. She sought advice
from her friend, Johnny Nogginhead, musing that If I
go to Stanford then I’ll go into debt.
But, replied Johnny, You don’t go to Stanford.
I know, said Sarah, I said If I go to Stanford….
But you don’t, retorted Johnny, you go to Jefferson!
I never said I didn’t, said an exasperated Sarah, I
know I don’t go to Stanford, my point is that going to
Stanford might mean going into debt.
Why didn’t you just say that, said Johnny.

22. Logical Interpretations of ‘if’
In a certain sense, ‘if’ means the antecedent isn’t
true.
If Sarah goes to Stanford then she will incur debt.
Either Sarah doesn’t go to Stanford OR she does
AND will incur debt.
It is NOT that Sarah could go to Stanford and not
incur debt.

23. Logical Interpretations of
‘if’
IF
THEN
Sarah goes into debt
1
True
True
True
2
True
False
False
3
Truth
Values
match line
for line
(across all
possible
worlds)
Sarah attends Stanford
False
True
True
4
False
True
False
Sarah doesn’t attend
Stanford
OR
(she
does
AND
goes into debt)
1
False
True
True
True
True
2
False
False
True
False
False
3
True
True
False
False
True
4
True
True
False
False
False
Either
It’s NOT that (Sarah attends Stanford
AND
doesn't go into debt)
1
True
True
False
False
2
False
True
True
True
3
True
False
False
False
4
True
False
False
True

24. Logical conditional, often
expressed in English by
‘if…then….’, is true when the
antecedent is true and the
consequent is false, otherwise it
is true. It is symbolized by ‘⊃’.
It’s logical form is P ⊃ Q.

25. Logical Form of
Conditionals
IF P THEN Q
P⊃Q
P→Q
P
⊃
Q
True
True
True
2
True
Fals
e
False
3
False
True
True
4
False
True
False
1

26. to prove a conditional
false
Prove that the antecedent is
true while the consequent is
false.

27. to interpret conditionals
logically
translate it as ‘Either not p or (p
and q)’ or ‘It not that (p and not
q)

28. indicators for
conditionals
a. if p then q
b. q if p
c. p only if q
d. not p unless q
e. supposing p, q
f. imagine p ... q
g. assuming p, q
h. all p are q
i. whenever p, q
j. when p, q

29. Validity
A Standard of Critical Thinking

30. Argument, a set of claims in
which some claims (premises)
are offered to show the truth (or
falsehood) of another claim (the
conclusion).
A line of reasoning.

31. Argument
s
Lines of reasoning
If it is a mammal then it gives live birth.
It lays eggs.
If it is red then it has color,
So it’s not a mammal.
if it has color then it emits or reflects a wavelength of light,
thus if it is red then it emits or reflects a wavelength of light.
If anything is a dog then it is a mammal.
If anything is a mammal then it is an animal.
which proves that if anything is a dog then it is an animal.
When water is heated to 212° it boils.
It’s not boiling, which demonstrates it
hasn’t been heated to 212°.
If living pigeons didn’t all come from
rock pigeons then they must have
come from other kinds of pigeons.
There are no other kinds of pigeons.
This established they all come from
rock pigeons.
All dogs are mammals.
All mammals are animals.
Hence all dogs are animals.
When a government abuses rights it ought to be removed.
The king abuses rights and so he ought to be removed.

32. indicators for
premises
a. as
b. as shown by
c. because
d. deduce from
e. derive from
f. finally, the last reason
g. first, second, third,… next
h. follows from
i. for
j. inasmuch as
k. indicated by
l. is the reason that
m.it is the case that
n. may be deduced from
o. may be derived from
p. may be inferred from
q. one reason being…
r. since
s. the fact that
t. the reason

33. indicators for
conclusions
a. as a result
b. consequently
c. demonstrates
d. entails
e. establishes
f. hence
g. I conclude that
h. implies
i. in conclusion
j. infer
k. it follows that
l. justifies
m.means
n. proves
o. shows
p. so
q. then
r. therefore
s. thus

34. Sets of
Claims Order
Are Without Any Particular
Sarah’s beliefs
The alarm did not go off.
Today is either Tuesday or
Thursday.
She will recognize her
teacher.
The class meets in the same
She has chemistry today.
room.
Today is Monday.
She will recognize her classmates.
She went to the right room.
She’s not dreaming.
She is late.

35. Argument
s therefore)
Have Order (‘∴’ means
1.If it is a mammal then it gives live birth.
2.It lays eggs.
1.If it is red then it has color,
3.∴ It’s not a mammal.
2.if it has color then it emits or reflects a wavelength of light,
3.∴ If it is red then it emits or reflects a wavelength of light.
1. If anything is a dog then it is a mammal.
2. If anything is a mammal then it is an animal.
3. ∴ If anything is a dog then it is an animal.
1.When water is heated to 212° it
boils.
2.It’s not boiling,
3.∴ It hasn’t been heated to 212°.
1.If living pigeons didn’t all come
from rock pigeons then they must
have come from other kinds of
pigeons.
2.There are no other kinds of
pigeons.
3.∴ They all come from rock pigeons.
1. All dogs are mammals.
2. All mammals are animals.
3. ∴ All dogs are animals.
1.When a government abuses rights it ought to be removed
2.The king abuses rights and
3.∴ He ought to be removed.

36. Validity, if the premises are true
then the conclusion is true.

37. Validity, either the premises are
false, or they are true and so it
the conclusion.
…it is not possible that the
premises are true while the
conclusion is false.

38. Validity
Another Emergent
Property
Wetness emerges as a
property of water when
hydrogen and oxygen are
properly combined—though
neither are wet themselves.
In a similar manner, validity
emerges when claims are
properly structured into an
argument.

39. Validity
Versus Truth
Validity
Truth
Applies to whole
arguments
Applies to claims, both
simple and complex
Does not apply to
claims
Does not apply to
arguments
As Technical Terms

40. Valid
Arguments
Premises
1
If anything is a dog then it is mammal.
2
If anything is a mammal then it is an
animal.
∴3
If anything is a dog then it is an animal.
Conclusion

41. Valid
Arguments
�
animals
�
Here, by premise 1, no dog can
be at the bottom of the blue (it is
outside of mammals). By premise
2 no mammal can be at the
bottom of the green (it is outside
of the animals). So there is no
place left for a dog to be.
�
mammals
�
�
dogs
�
�
�
�
✔
1
If anything is a dog then it is mammal.
�
✔
2
If anything is a mammal then it is an
animal.
✔
∴3
If anything is a dog then it is an animal.

42. Valid
Arguments
✔
1
If anything is a dog then it is mammal.
✔
2
If anything is a mammal then it is an
animal.
✔
∴3
If anything is a dog then it is an animal.
So this argument is valid

43. Valid
Arguments
Premises
If:
hectagon means
1,000,000 sided, and;
chiliogon means 1,000
sided, and;
megagon means 100
sided;
then the conclusion
would have to be true.
1
If anything is a hectagon then it has more sides
than a chiliogon.
2
If anything is a chiliogon then it has more sides
than a megagon.
∴3
If anything is a hectagon then it has more sides
than a megagon.
Conclusion

44. Valid
Arguments
1
If anything is a hectagon then it has more sides
than a chiliogon.
2
If anything is a chiliogon then it has more sides
than a megagon.
∴3
If anything is a hectagon then it has more sides
than a megagon.
So this argument is valid

45. Valid
Arguments
But:
megagon means
1,000,000 sided, and
hectagon means 100
sided (chiliogon does
mean 1,000 sided)
so the premises are in
fact false.
✘
✘
1
If anything is a hectagon then it has more
sides than a chiliogon.
2
If anything is a chiliogon then it has more
sides than a megagon.
∴3
If anything is a hectagon then it has more
sides than a megagon.

46. Valid
Arguments
✘
If anything is a hectagon then it has more
sides than a chiliogon.
2
If anything is a chiliogon then it has more
sides than a megagon.
∴3
✘
1
If anything is a hectagon then it has more
sides than a megagon.
But this argument is valid, because if the premises
were true then the conclusion would be true too.

47. Valid
Arguments
✘
If anything is a hectagon then it has more
sides than a chiliogon.
2
If anything is a chiliogon then it has more
sides than a megagon.
∴3
✘
1
If anything is a hectagon then it has more
sides than a megagon.
Test it by replacing ‘hectagon’, ‘chiliogon’ and
‘megagon’ with ‘triangle’, rectangle’, and ‘octogon’.

48. Valid
Arguments
Both Arguments have the Form:
1. P ⊃ Q
2. Q ⊃ R
3. ∴ P ⊃ R

49. Invalid
Arguments
✔
1
If anything is a camel then it has four legs.
✔
2
If anything is a pig then it has four legs.
✘
∴3
If anything is a pig then it is a camel.
This argument is invalid, because even if the
premises are true then the conclusion is not.

50. Invalid
Arguments
This Argument Has the Form:
P⊃R
Q⊃R
∴P⊃Q
It is not a valid form.

51. to determine
validity
check to see if the form of the
argument fits one of the valid
patterns.

52. Validity: The Test
Yes
If it is…
…then it is invalid
Is it possible for the premises
to be true while the conclusion
is false?
No
If it is not…
…then it is valid

53. Validity: A Quick
Check
Yes
If it does…
…then it is valid
Does the argument have a valid form?
No
If it does not…
…then it is invalid

54. Soundness
A Complex Standard of Critical Thinking

55. Soundness, valid arguments
with true premises.

56. Arguments
arguments
valid arguments
sound
arguments
A Taxonomy
1. If anything is a camel then it is a has humps.
2. Thor has no humps.
3. ∴ Thor is not a camel.
1. If anything is a camel then it has four legs.
2. If anything is a pig then it has four legs.
3. ∴ If anything is a pig then it is a camel.
1. If anything is a dog then it is mammal.
2. If anything is a mammal then it is an animal.
3. ∴ If anything is a dog then it is an animal.
1. If anything is a hectagon
then it has more sides than a
chiliogon.
2. If anything is a chiliogon then
it has more sides than a
megagon.
3. ∴ If anything is a hectagon
then it has more sides than a
megagon.
1. If anything is a pig then it is a quadruped.
2. Trakr is a quadruped.
3. ∴ Trakr is a pig.
1. If anything is wild then it is free.
2. Peter is not wild.
3. ∴ Peter is not free.
1. If anything is a dog then it is has four legs.
2. If anything is a cat then it is has four legs.
3. ∴ If anything is a dog then it is a cat.

57. Soundness
Yes
Are the premises
true?
Yes
it is sound
Is the argument valid?
No
No
it is unsound

58. to determine
soundness
check to see if the form of the
argument fits one of the valid
patterns, then check to see if
the premises are true.

59. Validity
Some Common Forms
Chain
Arguments
1. If anything is a dog then it is mammal.
2. If anything is a mammal then it is an animal.
3. ∴ If anything is a dog then it is an animal.
Modus
Ponens
1. When a government abuses rights it
ought to be removed.
2. The king abuses rights .
3. ∴ He ought to be removed.
Modus
Tollens
1. If it is a mammal then it gives live birth.
2. It lays eggs.
3. ∴ It’s not a mammal.

60. Chain
Argument
A Common Form
Chain
Arguments
1.If anything is a dog then it is mammal.
2.If anything is a mammal then it is an animal.
3.∴ If anything is a dog then it is an animal.

61. Chain Argument
The Parts of a Chain Argument
If anything is a dog then it is mammal.
1.Conditional Premise
2.Conditional Premise
3.Conditional Conclusion
If anything is a mammal then it is an animal.
∴ If anything is a dog then it is an animal.

62. Chain Argument
The antecedents and consequents of
the premises link up as in a chain.
The Structure of a Chain
Argument
1.If anything is a dog then it is a mammal.
2.If anything is a mammal then it is an animal.
3.∴ If anything is a dog then it is an animal.
The conclusion has the
same antecedent as the first
premise…
The conclusion has the
same consequent as the
last premise…

63. Chain
Arguments
Also called ‘hypothetical syllogisms’
1. P ⊃ Q
2. Q ⊃ R
3. ∴ P ⊃ R

64. Chain
Arguments
Can have indefinitely many
premises
1.
2.
3.
4.
5.
P⊃Q
Q⊃R
R⊃S
S⊃T
∴P⊃T

65. to calculate the number of possible
worlds
raise two to the power of the number
of claims being evaluated, here there
are three: P, Q, & R
3
2
=2•2•2=8

66. Proving Chain Arguments
Valid
P
Q
Q
R
P
Q)
(Q
R)
(P
1 True
True
True
True
True
True
2 True
True
True
False
True
False
3 True
False
False
True
True
True
4 True
False
False
False
True
False
5 False
True
True
True
False
True
6 False
True
True
False
False
False
7 False
False
False
True
False
True
8 False
False
False
False
False
False
(P
⊃
⊃
R
⊃
R)
Step One: Assign values to the simplest atomic claims, P, Q, & R

67. Proving Chain Arguments
Valid
(P
⊃
Q)
(Q
⊃
R)
(P
⊃
R)
1 True True True
True True True
True True True
2 True True True
True False False
True False False
3 True False False
False True True
True True True
4 True False False
False True False
True False False
5 False True True
True True True
False True True
6 False True True
True False False
False True False
7 False True False
False True True
False True True
8 False True False
False True False
False True False
Step Two: Determine the values of the next simplest or
molecular claims.

68. Proving Chain Arguments
Valid
(P
⊃
Q)
1 True True True
(Q
✔
⊃
R)
True True True
(P
✔
⊃
R)
True True True
2 True True True
True False False
True False False
3 True False False
False True True
True True True
4 True False False
False True False
True False False
5 False True True
6 False True True
✔
True True True
True False False
✔ False
False ✔ False
✔ False
True True
False True False
✔ False
False ✔ False
7 False True False
True True
True True
8 False True
True
True False
Step Three: Determine if there is a possible world where the
premises are both true while the conclusion is false.

69. Proving Chain Arguments
Valid
There is no possible world where the premises are true while the
conclusion is false.
(P
⊃
Q)
1 True True True
(Q
✔
⊃
R)
True True True
(P
✔
⊃
R)
True True True
2 True True True
True False False
True False False
3 True False False
False True True
True True True
4 True False False
False True False
True False False
5 False True True
6 False True True
✔
True True True
True False False
✔ False
False ✔ False
✔ False
True True
False True False
✔ False
False ✔ False
7 False True False
True True
True True
8 False True
True
True False
So Chain Arguments are valid.

70. An Unnamed
Fallacy
There is a possible world where the premises are true while the
conclusion is false.
(P
⊃
R)
1 True True True
(Q
✔
2 True False False
3 True True True
6 False True False
R)
True True True
✔ False
True True
False True False
✔
True True True
True False False
✔ False
False ✔ False
⊃
(P
✔
True False False
4 True False False
5 False True True
⊃
Q)
True True True
True True True
✘
True False False
True False False
✔ False
True True
False True True
✔ False
False ✔ False
7 False True True
True True
True False
8 False True
True
True False
So arguments of this form are invalid.

71. Modus
Ponens
A Common Form
modus ponens
1.When a government abuses rights it ought to
be removed.
2.The king abuses rights .
3.∴ He ought to be removed.

72. Modus Ponens
The Parts of a Modus Ponens
Argument
1.Conditional Premise
2.Premise Affirming the Antecedent of the Conditional
3.Concluding the Consequent of the Conditional
When a government abuses rights it ought to be removed.
The king abuses rights .
∴ He ought to be removed.

73. Modus Ponens
The Structure of a Modus Ponens
Argument
A conditional premise.
1. When a government abuses rights it ought to be removed.
2. The king abuses rights .
3. ∴ He ought to be removed.
A premise which affirms the
antecedent of the conditional premise.
The conclusion is the consequent of the conditional premise.

74. Modus
Ponens
Also called ‘Affirming the
Antecedent’ and ‘Conditional
Elimination’
1. P ⊃ Q
2. Q
3. ∴ P

75. Modus
Ponens
Can be extended by Chain
Argument
1.
2.
3.
4.
5.
P
P⊃Q
Q⊃R
R⊃S
∴S

76. to calculate the number of possible
worlds
raise two to the power of the number
of claims being evaluated, here there
are two: P & Q
2
2
=2•2=4

77. Proving Modus Ponens
Valid
P
P
Q
Q
(P
(P
Q)
Q
1
True
True
True
True
2
True
True
False
False
3
False
False
True
True
4
False
False
False
False
⊃
Step One: Assign values to the simplest atomic claims, P & Q

78. Proving Modus Ponens
Valid
(P
(P
⊃
Q)
Q
1
True
True
True
True
True
2
True
True
False
False
False
3
False
False
True
True
True
4
False
False
True
False
False
Step Two: Determine the values of the next simplest or
molecular claims.

79. Proving Modus Ponens
Valid
(P
(P
✔
⊃
Q)
True
True
True
Q
✔
1
True
True
2
True
True
False
False
False
3
False
False
True
True
True
4
False
False
True
False
False
Step Three: Determine if there is a possible world where the
premises are both true while the conclusion is false.

80. Proving Modus Ponens
Valid
There is no possible world where the premises are true while the
conclusion is false.
(P
(P
✔
⊃
Q)
True
True
True
Q
✔
1
True
2
True
True
False
False
False
3
False
False
True
True
True
4
False
False
True
False
False
So Modus Ponens Arguments are valid.
True

81. An Attendant Fallacy: Affirming the
Consequent
There is a possible world where the premises are true while the
conclusion is false.
(P
1
True
2
False
3
True
4
False
✔
✔
⊃
Q)
True
True
True
True
(Q
False
False
False
True
True
False
True
False
P
✔
True
True
✘
False
False
So arguments which affirm the consequent are invalid—and so such
arguments are fallacies

82. Modus
Tollens
A Common Form
modus tollens
1.If it is a mammal then it gives live birth.
2.It lays eggs.
3.∴ It’s not a mammal.

83. Modus Tollens
The Parts of a Modus Tollens
Argument
1.Conditional Premise
2.Premise Denying the Consequent of the Conditional
3.Concluding the Denial of the Antecedent of the Conditional
If it is a mammal then it gives live birth.
It lays eggs.
∴ It’s not a mammal.

84. Modus Ponens
The Structure of a Modus Tollens
Argument
A conditional premise.
1. If it is a mammal then it gives live birth.
2. It lays eggs.
3. ∴ It’s not a mammal.
A premise which denies the
consequent of the conditional
premise.
The conclusion is the denial of the antecedent of the conditional premise.

85. Modus
Tollens
Also called ‘Denying the
Consequent’
1. P ⊃ Q
2. ~Q
3. ∴ ~P

86. Modus
Tollens
Can be extended by Chain
Argument
1.
2.
3.
4.
5.
P⊃Q
Q⊃R
R⊃S
~S
∴ ~P

87. to calculate the number of possible
worlds
raise two to the power of the number
of claims being evaluated, here there
are two: P & Q
2
2
=2•2=4

88. Proving Modus Tollens
Valid
~Q
P
Q
~P
(~Q
(P
Q)
~P
1
False
True
True
False
2
True
True
False
False
3
False
False
True
True
4
True
False
False
True
⊃
Step One: Assign values to the simplest atomic claims, P & Q,
keeping track of negation.

89. Proving Modus Tollens
Valid
(~Q
(P
⊃
Q)
~P
1
False
True
True
True
False
2
True
True
False
False
False
3
False
False
True
True
True
4
True
False
True
False
True
Step Two: Determine the values of the next simplest or
molecular claims.

90. Proving Modus Tollens
Valid
(~Q
(P
⊃
Q)
~P
1
False
True
True
True
False
2
True
True
False
False
False
3
False
False
True
True
True
4
True
False
True
False
✔
✔
True
Step Three: Determine if there is a possible world where the
premises are both true while the conclusion is false.

91. Proving Modus Tollens
Valid
There is no possible world where the premises are true while the
conclusion is false.
(~Q
(P
⊃
Q)
~P
1
False
True
True
True
False
2
True
True
False
False
False
3
False
False
True
True
True
4
True
False
True
False
✔
✔
So Modus Tollens Arguments are valid.
True

92. An Attendant Fallacy: Denying the
Antecedent
There is a possible world where the premises are true while the
conclusion is false.
(~P
(P
⊃
Q)
~Q
1
False
True
True
True
False
2
False
True
False
False
True
3
True
False
True
True
4
True
False
True
False
✔
✘
False
True
So arguments which affirm the consequent are invalid—and so such
arguments are fallacies

93. Disjunction
Logical
Complexity

94. A Logically Simple
Truth
Two logical
possibilities
The coast is foggy.
1
True
2
False
Given that
we’ve filled in
the indices,
made the
ceteris
paribus
explicit, and
defined key
terms.

95. Combining Logically Simple
Truths
Four states of
affairs (states)
or possible
worlds
The coast is foggy.
The coast is sunny.
1
True
True
2
True
False
3
False
True
4
False
False

96. Logical
Conjunction
The coast is foggy.
AND
The coast is sunny.
1
True
True
True
2
True
False
False
3
False
False
True
4
False
False
False

97. Logical
Conditional
IF
The coast is foggy.
THEN
The coast is sunny.
1
True
True
True
2
True
False
False
3
False
True
True
4
False
True
False

98. Logical
Disjunction
The coast is foggy.
OR
The coast is sunny.
1
True
?
True
2
True
?
False
3
False
?
True
4
False
?
False

99. Logical
Disjunction
The coast is foggy.
OR
The coast is sunny.
1
True
?
True
2
True
?
False
3
False
?
True
4
False
False
False

100. Logical
Disjunction
The coast is foggy.
OR
The coast is sunny.
1
True
?
True
2
True
True
False
3
False
True
True
4
False
False
False

101. Logical
Disjunction
The coast is foggy.
OR
The coast is sunny.
1
True
True/False
True
2
True
True
False
3
False
True
True
4
False
False
False

102. Logical
Disjunction
The Ambiguity of ‘or’
Exclusive ‘or’
Inclusive ‘or’
Either the Giants win the division
Either the Giants make the
or the A’s do (but not both)
playoffs or the A’s do (or both)
Either heads or tails (but not
both)
Either by plane or by car (or
both)
Latin: aut
Latin: vel

103. Logical
Disjunction
Logic settles on an inclusive way
The coast is foggy.
OR
The coast is sunny.
1
True
True
True
2
True
True
False
3
False
True
True
4
False
False
False

104. Logical disjunction, often
expressed in English by
‘Either…or….’, is false when the
both components are false,
otherwise it is true. It is
symbolized by ‘V’. It’s logical
form is P V Q.

105. Logical Form of
Disjunctions
Either P OR Q
PVQ
P
V
Q
1
True
True
True
2
True
True
False
3
False
True
True
False
Fals
e
False
4

106. to prove a disjunction
true
Prove that one of the
component claims is true.

107. to interpret a
disjunction
specify if you are using it
inclusively or exclusively.

108. Tautolog
y
Putting Negation and Disjunction Together
Which claim is not a disjunction?
Hockey is better than basketball but it is not
better than basketball.*
Jupiter is bigger than Mars or it is not
Drinking milk is healthy or unhealthy.*
bigger than Mars.
New York either is or isn’t the largest city in the US.*
Same-sex schools are optimal unless same-se
Eleven is a prime number or
schools are less than optimal.
eleven is not a prime number.
Romeo and Juliette is a tragedy or it is not a tragedy.*
The child looks at the jellyfish or looks away from it*.
The jellyfish has tentacles—or not!
The music is loud or the
Either Jacqui thinks black is more alluring than pink
music is quiet.*
or she doesn’t.
The Constitution of the United States was adopted on either September 17,
1787 or July 4, 1776.*

109. Tautologie
s Tautology: P V ~P
The Logical Form of a
The square is white
V
The square is not white
1
True
?
False
2
False
?
True
Given that disjunctions are false when all component claims are
false, what is the truth value of this disjunction?

110. Tautologie
s Tautology: P V ~P
The Logical Form of a
The square is white
V
The square is not white
1
True
True
False
2
False
True
True
Tautologies are true in all possible worlds.

111. Tautolog
y Tautology: P V ~P
The Logical Form of a
P
V
~P
1
True
True
False
2
False
True
True
Tautologies are true in all possible worlds.

112. Tautology, a special form of
disjunction in which a claim and
its negation are joined—they
are always true. The logical
form of a tautology is P V ~P.

113. Tautolog
yPrinciple of Sufficient
The Logical Form of the
Reason
For every claim, give a reason why it is true or not true.
T
V
~T
1
True
True
False
2
False
True
True
The Principle of Sufficient Reason covers all possible worlds.

114. Controlling the
Question
Is drinking milk healthy for humans?
What are the healthiest drinks for humans?
What Constitutional rights should we
keep?
Has the Constitutional right to bear
arms outlived its usefulness?
Are single-sex schools better for education?
What is the best method of education?

115. Controlling the
Question
Open Questions
What are the healthiest
drinks for humans?
What Constitutional
rights should we keep?
What is the best
method of
education?
Yes-or-no Questions
Is drinking milk
healthy for humans?
Has the Constitutional
right to bear arms outlived
its usefulness?
Are single-sex
schools better for
education?

116. Open Questions
Are Topic or Theme Questions
What are the healthiest
drinks for humans?
What Constitutional
rights should we keep?
What is the best
method of
education?

117. A Topic Question: an open
question. Such questions
require disjunctive
reasoning to treat the
alternates.

118. Disjunctive Reasoning
Reasons supporting
1st alternate: this alternate or
refuting the others.
the question
Reasons supporting
2nd alternate: this alternate or
refuting the others.
.
.
.
Final
alternate:
.
.
.
Reasons supporting
this alternate or
refuting the others.
Which
alternate has
the better
reasons?

119. Disjunctive Reasoning
Water:
What are the
healthiest drinks
for humans?
Reasons supporting
water or refuting the
others.
Milk:
Reasons supporting
milk or refuting the
others.
.
.
.
.
.
.
Reasons supporting
Electrolyte
electrolyte solutions or
Solutions
refuting the others.
Which
alternate has
the better
reasons?

120. Validity
Some Common Forms Involving Disjunctions
Disjunctive
Argument
1. Either Kierkegaard can be a Christian or a
philosopher.
2. He cannot be a philosopher.
3. ∴ So he must be a Christian.
Simple
Dilemma
1. If Johnny’s friendship is for pleasure
then he is not a true friend.
2. If Johnny’s friendship is for utility then
he is not a true friend.
3. Either Johnny’s friendship is for
pleasure or utility.
4. ∴ Johnny’s friendship is not a true
friendship.
Dilemma
1. If existence precedes essence then
humanity is free.
2. If there is no God then we we alone can
justify ourselves, without excuse.
3. Either existence precedes essence or
there is not God.
4. ∴ Either humanity is free or is without
any justifications or excuses but those
they provide.

121. Disjunctive
Argument
A Common Form
Disjunctive
Argument
1.Either Kierkegaard can be a Christian or a
philosopher.
2.He cannot be a philosopher.
3.∴ So he must be a Christian.

122. Disjunctive
Argument
The Parts of a Disjunctive Argument
1.Disjunctive Premise.
2.Premise Denying one of the disjuncts of the Disjunction.
3.Concluding the remaining Disjunct.
Either Kierkegaard can be a Christian or a philosopher.
He cannot be a philosopher.
∴ So he must be a Christian.

123. Disjunctive
Argument
The Structure of a Disjunctive Argument
A disjunctive premise.
1. Either Kierkegaard can be a Christian or a philosopher.
2. He cannot be a philosopher.
3. ∴ So he must be a Christian.
A premise which denies one of the component
claims of the disjunctive premise.
The conclusion is the remaining component claim of the disjunctive premise.

124. Disjunctive
Argument
More commonly called ‘Disjunctive Syllogism’
and also called ‘modus tollendo ponens'
1. P V Q
2. ~Q
3. ∴ P
1. P V Q
2. ~P
3. ∴ Q
Can be run either
way

125. Disjunctive
Syllogism
Can be extended indefinitely
1.
2.
3.
4.
5.
PVQVRVS
~Q
~R
~S
∴P

126. to calculate the number of possible
worlds
raise two to the power of the number
of claims being evaluated, here there
are two: P & Q
2
2
=2•2=4

127. Proving Disjunctive Argument
Valid
~P
P
Q
Q
~P
(P
Q)
Q
1
False
True
True
True
2
False
True
False
False
3
True
False
True
True
4
True
False
False
False
V
Step One: Assign values to the simplest atomic claims, P & Q,
minding the negations.

128. Proving Disjunctive Argument
Valid
~P
(P
V
Q)
Q
1
False
True
True
True
True
2
False
True
True
False
False
3
True
False
True
True
True
4
True
False
False
False
False
Step Two: Determine the values of the next simplest or
molecular claims.

129. Proving Disjunctive Argument
Valid
~P
(P
V
Q)
Q
1
False
True
True
True
True
2
False
True
True
False
False
3
True
False
True
True
4
True
False
False
False
✔
✔
True
False
Step Three: Determine if there is a possible world where the
premises are both true while the conclusion is false.

130. Proving Disjunctive Argument
Valid
There is no possible world where the premises are true while the
conclusion is false.
~P
(P
V
Q)
Q
1
False
True
True
True
True
2
False
True
True
False
False
3
True
False
True
True
4
True
False
False
False
✔
✔
So Disjunctive Arguments are valid.
True
False

131. Simple
Dilemma
A Common Form
Simple
Dilemma
1.If Johnny’s friendship is for pleasure then he
is not a true friend.
2.If Johnny’s friendship is for utility then he is
not a true friend.
3.Either Johnny’s friendship is for pleasure or
utility.
4.∴ Johnny is not a true friend.

132. Simple Dilemma
The Parts of a Simple Dilemma
Either Johnny’s friendship is for pleasure or utility.
1. Disjunctive Premise.
2. A Conditional Premise whose antecedent is one of the disjuncts of the Disjunctive
Premise and whose consequent is the same as the other Conditional Premise.
3. Another Conditional Premise whose antecedent is the other disjunct of the Disjunctive
Premise and whose consequent is the same as the other Conditional Premise.
4. Concluding the Consequent of the Conditional Premises.
If Johnny’s friendship is for pleasure then he is not a true friend.
If Johnny’s friendship is for utility then he is not a true friend.
∴ Johnny is not a true friend.

133. Simple Dilemma
The Structure of a Simple Dilemma
One antecedent is
a component of
the disjunction.
The other antecedent is the
other component of the
disjunction.
1. If Johnny’s friendship is for pleasure then he is not a true friend.
2. If Johnny’s friendship is for utility then he is not a true friend.
3. Either Johnny’s friendship is for pleasure or utility.
4. ∴ Johnny is not a true friend.
A disjunctive premise.
The conclusion is the consequent of the conditional premises.
Both Conditional
Premises share a
consequent.

134. Simple
Dilemma
Also called ‘Disjunctive Elimination’
1.
2.
3.
4.
PVQ
P⊃R
Q⊃R
∴R

135. Simple
Dilemma
Can be extended indefinitely
1.
2.
3.
4.
5.
PVQVR
P⊃S
Q⊃S
R⊃S
∴S

136. Destructive
Dilemma
Simple Dilemma combined with Modus
Tollens
1.
2.
3.
4.
~R
P⊃R
Q⊃R
∴ ~P V Q

137. to calculate the number of possible
worlds
raise two to the power of the number
of claims being evaluated, here there
are three: P, Q, & R
3
2
=2•2•2=8

138. Proving Simple Dilemma
Valid
P
Q
P
R
Q
Q)
(P
R)
(Q
1 True
True
True
True
2 True
True
True
3 True
False
4 True
(P
V
⊃
R
⊃
R
R)
R
True
True
True
False
True
False
False
True
True
False
True
True
False
True
False
False
False
False
5 False
True
False
True
True
True
True
6 False
True
False
False
True
False
False
7 False
False
False
True
False
True
True
8 False
False
False
False
False
False
False
Step One: Assign values to the simplest atomic claims, P, Q, & R

139. Proving Simple Dilemma
Valid
(P
V
Q)
(P
⊃
R)
(Q
⊃
R)
R
1 True True True
True True True
True True True
True
2 True True True
True False False
True False False
False
3 True True False
True True True
False True True
True
4 True True False
True False False
False True False
False
5 False True True
False True True
True True True
True
6 False True True
False True False
True False False
False
7 False False False
False True True
False True True
True
False True False
False True False
8 False False False
Step Two: Determine the values of the next simplest or
molecular claims.
False

140. Proving Simple Dilemma
Valid
(P
V
Q)
1 True True True
2 True True True
(P
✔
⊃
R)
True True True
5 False True True
✔
True False False
3 True True False ✔ True True True
4 True True False
(Q
True True
R)
True True True
R
✔
True False False
✔ False
True False False
✔ False
⊃
True True
False
✔
False True False
✔
True True True
True
True
False
✔
True
6 False True True
False True False
True False False
False
7 False False False
False True True
False True True
True
8 False False False
False True False
False True False
False
Step Three: Determine if there is a possible world where the
premises are both true while the conclusion is false.

141. Proving Simple Dilemma
Valid
There is no possible world where the premises are true while the
conclusion is false.
(P
V
Q)
1 True True True
2 True True True
(P
✔
⊃
R)
True True True
4 True True False
5 False True True
✔
True False False
3 True True False ✔ True True True
True True
R)
True True True
R
✔
True False False
✔ False
True False False
✔ False
⊃
(Q
True True
False
✔
False True False
✔
True True True
True
True
False
✔
True
6 False True True
False True False
True False False
False
7 False False False
False True True
False True True
True
8 False False False
False True False
False True False
False
So Simple Dilemmas are valid.

142. Dilemma
A Common Form
Dilemma
1.If existence precedes essence then humanity
is free.
2.If there is no God then we we alone can justify
ourselves, without excuse.
3.Either existence precedes essence or there is
no God.
4.∴ Either humanity is free or is without any
justifications or excuses but those they
provide.

143. Dilemma
The Parts of a Dilemma
Either existence precedes essence or there is no
God.
1. Disjunctive Premise.
2. A Conditional Premise whose antecedent is one of the disjuncts of the Disjunctive
Premise.
3. Another Conditional Premise whose antecedent is the other disjunct of the Disjunctive
Premise.
4. Concluding a Disjunction of the Consequents of the Conditional Premises.
If existence precedes essence then humanity is free.
If there is no God then we we alone can justify ourselves, without
excuse.
∴ Either humanity is free or is without any justifications or excuses but those they provide.

144. Dilemma
The Structure of a Dilemma
One antecedent is
a component of
the disjunction.
1.
2.
3.
4.
The other antecedent is the
other component of the
disjunction.
If existence precedes essence then humanity is free.
If there is no God then we we alone can justify ourselves, without excuse.
Either existence precedes essence or there is no God.
∴ Either humanity is free or is without any justifications or excuses but those
they provide.
A disjunctive premise.
The conclusion is a disjunction of the consequents of the conditional premises.

145. Dilemma
Also called ‘Constructive Dilemma’
1.
2.
3.
4.
PVQ
P⊃R
Q⊃S
∴RVS

146. Simple
Dilemma
Can be extended indefinitely
1.
2.
3.
4.
5.
PVQVR
P⊃S
Q⊃T
R⊃U
∴SVTVU

147. Destructive
Dilemma
Dilemma combined with Modus Tollens
1.
2.
3.
4.
~R V ~S
P⊃R
Q⊃S
∴ ~P V ~Q

148. Destructive
Dilemma
Dilemma combined with Modus Tollens
and a Tautology
1.
2.
3.
4.
~R V ~R
P⊃R
Q⊃R
∴ ~P V ~Q

149. to calculate the number of possible
worlds
raise two to the power of the number
of claims being evaluated, here there
are three: P, Q, & R
4
2
= 2 • 2 • 2 • 2 = 16

150. Proving Dilemma
Valid
P
Q
(P
V
P
Q)
(P
R
⊃
Q
R)
(Q
S
⊃
R
S)
R
S
V
S
1
True
True
True
True
True
True
True
True
2
True
True
True
True
True
False
True
False
3
True
True
True
False
True
True
False
True
4
True
True
True
False
True
False
False
False
5
True
False
True
True
False
True
True
True
6
True
False
True
True
False
False
True
False
7
True
False
True
False
False
True
False
True
8
True
False
True
False
False
False
False
False
9
False
True
False
True
True
True
True
True
10
False
True
False
True
True
False
True
False
11
False
True
False
False
True
True
False
True
12
False
True
False
False
True
False
False
False
13
False
False
False
True
False
True
True
True
14
False
False
False
True
False
False
True
False
15
False
False
False
False
False
True
False
True
16
False
False
False
False
False
False
False
False
Step One: Assign values to the simplest atomic claims, P, Q, R,

151. Proving Dilemma
Valid
(P
V
Q)
(P
⊃
R)
(Q
⊃
S)
R
V
S
1
True
True
True
True
True
True
True
True
True
True
True
True
2
True
True
True
True
True
True
True
False
False
True
True
False
3
True
True
True
True
False
False
True
True
True
False
True
True
4
True
True
True
True
False
False
True
False
False
False
False
False
5
True
True
False
True
True
True
False
True
True
True
True
True
6
True
True
False
True
True
True
False
True
False
True
True
False
7
True
True
False
True
False
False
False
True
True
False
True
True
8
True
True
False
True
False
False
False
True
False
False
False
False
9
False
True
True
False
True
True
True
True
True
True
True
True
10
False
True
True
False
True
True
True
False
False
True
True
False
11
False
True
True
False
True
False
True
True
True
False
True
True
12
False
True
True
False
True
False
True
False
False
False
False
False
13
False
False
False
False
True
True
False
True
True
True
True
True
14
False
False
False
False
True
True
False
True
False
True
True
False
15
False
False
False
False
True
False
False
True
True
False
True
True
16
False
False
False
False
True
False
False
True
False
False
False
False
Step Two: Determine the values of the next simplest or molecular
claims.

152. Proving Dilemma
Valid
(P
⊃
R)
(Q
⊃
S)
True
True
True
True
True
True
True
True
True
True
True
False
True
True
True
False
False
True
True
True
True
True
False
False
5
True
True
False
✔
True
True
True
6
True
True
False
✔
True
True
True
7
True
True
False
True
False
8
True
True
False
True
9
False
True
True
10
False
True
True
11
False
True
True
12
False
True
13
False
14
(P
V
Q)
R
V
S
1
True
True
True
True
True
True
2
True
True
False
True
True
False
3
True
True
True
False
True
True
4
True
False
False
False
False
False
✔
False
True
True
✔
True
True
True
✔
False
True
False
✔
True
True
False
False
False
True
True
False
True
True
False
False
False
True
False
False
False
False
False
True
True
True
True
True
True
True
True
False
True
True
True
False
False
True
True
False
False
True
False
True
True
True
False
True
True
True
False
True
False
True
False
False
False
False
False
False
False
False
True
True
False
True
True
True
True
True
False
False
False
False
True
True
False
True
False
True
True
False
15
False
False
False
False
True
False
False
True
True
False
True
True
16
False
False
False
False
True
False
False
True
False
False
False
False
✔
✔
✔
✔
✔
✔
✔
✔
✔
Step Three: Determine if there is a possible world where the premises are both true while the conclusion is false.

153. Proving Dilemma
Valid
There is no possible world where the premises are true while the conclusion is
false.
(P
⊃
R)
(Q
⊃
S)
True
True
True
True
True
True
True
True
True
True
True
False
True
True
True
False
False
True
True
True
True
True
False
False
5
True
True
False
✔
True
True
True
6
True
True
False
✔
True
True
True
7
True
True
False
True
False
8
True
True
False
True
9
False
True
True
10
False
True
True
11
False
True
True
12
False
True
13
False
14
(P
V
Q)
R
V
S
1
True
True
True
True
True
True
2
True
True
False
True
True
False
3
True
True
True
False
True
True
4
True
False
False
False
False
False
✔
False
True
True
✔
True
True
True
✔
False
True
False
✔
True
True
False
False
False
True
True
False
True
True
False
False
False
True
False
False
False
False
False
True
True
True
True
True
True
True
True
False
True
True
True
False
False
True
True
False
False
True
False
True
True
True
False
True
True
True
False
True
False
True
False
False
False
False
False
False
False
False
True
True
False
True
True
True
True
True
False
False
False
False
True
True
False
True
False
True
True
False
15
False
False
False
False
True
False
False
True
True
False
True
True
16
False
False
False
False
True
False
False
True
False
False
False
False
✔
✔
✔
✔
✔
✔
So Dilemmas are valid.
✔
✔
✔

154. Fallacies
A Relevant Fallacy

155. False Dilemma, to provide a
non exhaustive disjunction as a
premise—it is a fallacy.

156. to avoid false
dilemma
Provide an exhaustive list of
the possible answers to the
topic question, listing
explicitly those you may not
wish to treat.

157. Assignment
What does it mean to be ethical?
Ethics
How do you come to an ethical
decision?
What is hypocrisy?