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1. Lecture 6
Hypothetical and Disjunctive
Syllogisms

2. Hypothetical Syllogisms
• In a hypothetical syllogism, one or both of
the premises are hypotheticals, i.e., “if”
propositions.
• In a pure hypothetical syllogism, both
premises are hypotheticals

3. Pure Hypothetical Syllogisms
• An example of a pure hypothetical
syllogism:
• If wishes are horses, beggars will ride
• If beggars ride, donations to charity will rise
• If wishes are horses, donations to charity
will rise

4. Mixed Hypothetical Syllogisms
• A mixed hypothetical syllogism has one
hypothetical premise and one categorical
premise.
• If wishes are horses, beggars will ride
• Wishes are horses
• Beggars will ride.
• (H. W. B. Joseph's objection to the major
premise)

5. Modus Ponens
• This mixed hypothetical syllogism is more
important than the pure hypothetical syllogism.
• In mathematical logic, the two forms of the mixed
hypothetical are the most important principles of
reasoning.
• The first of these is modus ponens: If a, then b; a,
therefore, b.
• Our example with wishes and horses is in the
pattern.

6. Modus Tollens
• The other basic type of mixed hypothetical
is modus tollens.
• The form here is: If a, then b; not b;
therefore, not a.
• If wishes are horses, beggars will ride.
• Beggars will not ride
• Wishes are not horses

7. More on Modus Ponens and
Modus Tollens
• The principle behind modus ponens and modus
ponens is exactly the one we have already covered
for the categorical syllogism.
• If the premises of a syllogism are true, then the
conclusion is true. This corresponds to modus
ponens
• If the conclusion of a syllogism is false, at least
one of the premises is false. This corresponds to
modus tollens

8. More on Hypotheticals
• A hypothetical proposition identifies a sufficient
condition: If a, then b.
• In other words, the occurrence of a is sufficient to
make b true.
• If wishes are horses, beggars will ride. This says
that wishes’ being horses is sufficient for the truth
of “beggars will ride”.
• This does not say that a is necessary for the truth
of b. It’s left open whether one can have b without
a
• Maybe beggars can ride even if wishes aren’t

9. Two Fallacies
• Failure to realize this point leads to two fallacies.
• If wishes are horses, beggars will ride; wishes are
not horses; therefore beggars will not ride
• This is a fallacy because the hypothetical just tells
us that wishes’ being horses is sufficient for
beggars to ride. We can’t conclude that the
absence of this state of affairs will prevent beggars
from riding
• This fallacy is called denying the antecedent

10. Affirming the Consequent
• Here is the other fallacy:
• If wishes are horses, beggars will ride
• Beggars will ride
• Therefore, wishes are horses
• This is called affirming the consequent. The
first premise doesn’t say that only if wishes
are horses will beggars ride

11. Sufficient and Necessary
Conditions
• Denying the antecedent and affirming the
consequent make the same mistake. They mistake
a sufficient condition for a necessary condition. If
a is a necessary condition for b, then b cannot
occur without a
• “If a, then b” says that a is a sufficient condition
for b. How do we say that a is a necessary
condition for b?
• ‘If b, then a” states a necessary condition. This
says that whenever b occurs, a occurs: b won’t
occur unless a does.
• Suppose “if a, then b” and “if b, then a” are both

12. Example of Affirming the
Consequent?
• It is sometimes claimed that physical science rests
on affirming the consequent
• Scientists reason in this way, it is claimed:
• If my theory is true, we will observe certain results
• We observe these results
• Therefore, my theory is true.
• This isn't correct, unless the scientist claims that
the truth of the results logically imply that the
theory is true. Instead, he can say that the results
confirm the theory.

13. Indicative and Subjunctive
Conditionals
• The type of hypothetical, or conditional, we have
discussed so far is called an indicative conditional.
It says, “if a is the case, then b is the case
• A subjunctive, or counterfactual, conditional says.
“If a were the case, then b would be the case.”
• We can use modus ponens and modus tollens with
subjunctive conditionals, not just indicative
conditionals

14. Subjunctive Conditionals
• The study of subjunctive conditionals has become
a big topic in modern logic. They can be very
tricky.
• An indicative conditional can be true while a
similar-sounding subjunctive conditional is false.
• Here is a famous example: “If Oswald didn’t kill
Kennedy, somebody else did.” So long as
Kennedy was killed, this is true.
• “If Oswald hadn’t killed Kennedy, somebody else
would have” may well be false. Here, we are
assuming that in the actual world, Oswald killed
Kennedy and saying that if, contrary to fact, he

15. When Are Counterfactuals True?
• The truth conditions for counterfactual
conditionals are often hard to determine and there
isn’t an accepted analysis of them.
• One influential approach is due to David Lewis. It
relies on the notion of “possible worlds”.
• Here we start with the world as it actually is and
imagine that it is changed in various ways. Each
such change is a “possible world”. Some changes
don’t change the actual world very much. These
are called “close possible worlds”. In Lewis’s
analysis, the counterfactual is true if the indicative
conditional in the closest possible worlds where

16. The Conditional Fallacy
• The conditional fallacy arises when one fails to
take account of all the effects of a counterfactual
conditional
• John Rawls says that a plan of life is rational if it
is a plan that you would adopt if you were acting
with full deliberative rationality.
• In other words, “I’m not someone who acts with
full deliberative rationality now. But if I were,
what would I decide to do?”

17. The Conditional Fallacy
Continued
• Suppose that I frequently decide things
impulsively and this gets me into trouble. I’m
trying to decide whether I should see a therapist
about this.
• According to Rawls, I should ask, Would someone
who was fully deliberatively rational see a
therapist in this situation?
• But if I were fully deliberatively rational, I
wouldn’t need to se a therapist. I wouldn’t have
the problem. The conditional fallacy here is that if
Rawls’s counterfactual conditional were true, it
would change the original situation. What would

18. Another Example of the
Conditional Fallacy
• According to Roderick Chisholm, it’s reasonable
to believe something if it would be reasonable for
you to believe it if your concerns were purely
intellectual
• Suppose you want to know whether you should
believe, “My concerns are purely intellectual”,
meaning “My concerns now are purely
intellectual”
• If I follow Chisholm’s suggestion, I will believe
this is true, because if my concerns were purely
intellectual, I would believe they were. But they
aren’t now, so I shouldn’t believe it.

19. Reduction of Hypothetical
Syllogisms
• A modus ponens syllogism can be changed to a
modus tollens and a modus tollens can be changed
to a modus ponens
• “If a, then b, a; therefore b” can be changed to “if
not b, then not a”; a; therefore b”. We interchange
the antecedent and consequent of the hypothetical,
and then negate both.
• “If a, then b; not b; therefore not a” can be
changed to “If not b, then not a; not b; therefore
not a” Again, we exchange the antecedent and
consequent of the hypothetical and negate both.

20. Can a Hypothetical Syllogism Be
Changed to a Categorical
Syllogism?• If wishes are horses, beggars will ride
• Wishes are horses
• Beggars will ride
• It would seem that this could be changed to
• The situation in which wishes are horses is a
situation in which beggars will ride
• The situation in which wishes wishes are horses is
a situation that is true
• The situation that beggars will ride is true
• Joyce thinks that this change conceals the real
relationship. One proposition is conditional on

21. Two Kinds of Disjunction
• Disjunctions such as “A or B” can be interpreted
in two ways.
• Exclusive disjunction means “A or B, but not
both”. E.g., All animals are either one-celled or
many-celled. An animal can’t be both one-celled
and many-celled.
• Inclusive disjunction means “A or B or both”.
E.g., “Either all men are mortal or Obama is the
President”
• Inclusive disjunction is the standard usage in
modern logic.

22. Modus Ponendo Tollens
• An animal is either single-celled or many-
celled
• Protozoa are single-celled
• Therefore, protozoa are not many-celled
• This type of inference is valid only when
exclusive disjunction is used.

23. Modus Tollendo Ponens
• Either Obama is the President or I am the
President
• I am not the President
• Therefore, Obama is the President
• This is valid whether the disjunction is exclusive
or inclusive
• Either Obama is the President or Mises was a
Keynesian
• Mises was not a Keynesian
• Therefore, Obama is the President
• Even though it’s false that Mises was a Keynesian,

24. Dilemmas
• A dilemma has two premises
• One of them is a compound hypothetical
proposition. Each part of the compound
hypothetical leads to an undesirable conclusion
• The other premise is a disjunction that says that
one of the parts of the compound hypothetical is
true.
• The adversary cannot avoid the undesirable
conclusion
• Joyce distinguishes different kinds of dilemma,
but we don’t need to go into this

25. The Barbershop Paradox
• In a village, there is a barber who shaves all
and only those who don’t shave themselves.
Does the barber shave himself?
• This isn’t a genuine paradox. We can show
why it isn’t by analyzing it as a dilemma.

26. Paradox Dissolved
• If the barber shaves himself, then he doesn’t shave
himself; (He shaves only those who don’t shave
themselves) and if the barber doesn’t shave
himself, then he shaves himself. ( He shaves all
those who don’t shave themselves)
• Either the barber shaves himself or he doesn’t
shave himself.
• Whatever the barber does leads to a contradiction
• Thus, there couldn’t be such a barber. We have a
proof that a barber of this description couldn’t
exist. This is why the barbershop paradox isn’t a
real paradox.

27. Responding to Dilemmas
• Joyce distinguishes three ways of responding to a
dilemma
• One is to take one or more of the “horns”
(alternatives) of the dilemma and show that the
bad consequences aren’t involved
• Another is to show that some other alternative
from those considered in the dilemma is possible.
This alternative doesn’t involve an undesirable
alternative.
• This is called escaping between the horns

28. The Third Alternative
• This response to the dilemma constructs a
counter dilemma. This takes the same
alternatives as the original dilemma and
shows that they are fatal to the original
argument.

29. The Litigiosus
• The is a famous example. Protagoras trained
Euathlus in rhetoric. Half of his fee was payable
when Euathlus won his first lawsuit. After he
finished his course, Euathlus wasn’t involved in
any lawsuits and didn’t pay
• Protagoras sued Euathlus. He constructed this
dilemma. “Either the court decides in my favor, or
it decides against me. If it decides in my favor, I
win and Euathlus has to pay. But if I lose,
Euathlus has won the suit, and by the terms of our
agreement, he has to pay. Thus, whether I win or
lose the suit, Euathlus has to pay.”

30. The Counter Dilemma
• Euathlus responded with a counter dilemma
• “If I lose the suit, then by the terms of the
agreement, I don’t have to pay; and if the court
decides in my favor, then I don’t have to pay. In
either case, I don’t have to pay”
• Joyce doesn’t think that there is a clear solution to
this puzzle, but in fact it can be solved.
• Neither the dilemma nor the counter-dilemma can
be accepted. Both rely on inconsistent criteria. We
can either decide according to the terms of the
agreement or according to the decision of the
court, but not both. If we go by the decision of the

31. Solution
• To solve the puzzle, we should consider the terms
of the agreement. Protagoras will lose the case,
because Euathlus hasn’t yet won a case.
• But once he loses, he can start a new suit. This
time he should win, because Euathlus has won a
lawsuit. By losing a case, Protagoras can bring
about the situation in which he will be paid.
• A provision of the U.S. Constitution says that the
representation of a state in the Senate can’t be
changed without its consent. This provision, it is
further stated,cannot be amended.
• Can this provision itself be amended? It can be