Lecture 6

Hypothetical and Disjunctive Syllogisms

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

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

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)

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.

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

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

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 horses

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

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

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

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.

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

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

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

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?”

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

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.

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.

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

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.

• Note that “A disjunction is either exclusive or

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.

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,

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

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.

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.

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

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.

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.”

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

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