How to Think: Introduction to Logic, Lecture 6 with David Gordon - Mises Academy
Lecture 6Hypothetical and DisjunctiveSyllogisms
Hypothetical Syllogisms• In a hypothetical syllogism, one or both ofthe premises are hypotheticals, i.e., “if”propositions.• In a pure hypothetical syllogism, bothpremises are hypotheticals
Pure Hypothetical Syllogisms• An example of a pure hypotheticalsyllogism:• If wishes are horses, beggars will ride• If beggars ride, donations to charity will rise• If wishes are horses, donations to charitywill rise
Mixed Hypothetical Syllogisms• A mixed hypothetical syllogism has onehypothetical premise and one categoricalpremise.• If wishes are horses, beggars will ride• Wishes are horses• Beggars will ride.• (H. W. B. Josephs objection to the majorpremise)
Modus Ponens• This mixed hypothetical syllogism is moreimportant than the pure hypothetical syllogism.• In mathematical logic, the two forms of the mixedhypothetical are the most important principles ofreasoning.• The first of these is modus ponens: If a, then b; a,therefore, b.• Our example with wishes and horses is in thepattern.
Modus Tollens• The other basic type of mixed hypotheticalis 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 andModus Tollens• The principle behind modus ponens and modusponens is exactly the one we have already coveredfor the categorical syllogism.• If the premises of a syllogism are true, then theconclusion is true. This corresponds to modusponens• If the conclusion of a syllogism is false, at leastone of the premises is false. This corresponds tomodus tollens
More on Hypotheticals• A hypothetical proposition identifies a sufficientcondition: If a, then b.• In other words, the occurrence of a is sufficient tomake b true.• If wishes are horses, beggars will ride. This saysthat wishes’ being horses is sufficient for the truthof “beggars will ride”.• This does not say that a is necessary for the truthof b. It’s left open whether one can have b withouta• Maybe beggars can ride even if wishes aren’t
Two Fallacies• Failure to realize this point leads to two fallacies.• If wishes are horses, beggars will ride; wishes arenot horses; therefore beggars will not ride• This is a fallacy because the hypothetical just tellsus that wishes’ being horses is sufficient forbeggars to ride. We can’t conclude that theabsence of this state of affairs will prevent beggarsfrom 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. Thefirst premise doesn’t say that only if wishesare horses will beggars ride
Sufficient and NecessaryConditions• Denying the antecedent and affirming theconsequent make the same mistake. They mistakea sufficient condition for a necessary condition. Ifa is a necessary condition for b, then b cannotoccur without a• “If a, then b” says that a is a sufficient conditionfor b. How do we say that a is a necessarycondition for b?• ‘If b, then a” states a necessary condition. Thissays that whenever b occurs, a occurs: b won’toccur unless a does.• Suppose “if a, then b” and “if b, then a” are both
Example of Affirming theConsequent?• It is sometimes claimed that physical science restson 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 isnt correct, unless the scientist claims thatthe truth of the results logically imply that thetheory is true. Instead, he can say that the resultsconfirm the theory.
Indicative and SubjunctiveConditionals• The type of hypothetical, or conditional, we havediscussed 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 withsubjunctive conditionals, not just indicativeconditionals
Subjunctive Conditionals• The study of subjunctive conditionals has becomea big topic in modern logic. They can be verytricky.• An indicative conditional can be true while asimilar-sounding subjunctive conditional is false.• Here is a famous example: “If Oswald didn’t killKennedy, somebody else did.” So long asKennedy was killed, this is true.• “If Oswald hadn’t killed Kennedy, somebody elsewould have” may well be false. Here, we areassuming that in the actual world, Oswald killedKennedy and saying that if, contrary to fact, he
When Are Counterfactuals True?• The truth conditions for counterfactualconditionals are often hard to determine and thereisn’t an accepted analysis of them.• One influential approach is due to David Lewis. Itrelies on the notion of “possible worlds”.• Here we start with the world as it actually is andimagine that it is changed in various ways. Eachsuch change is a “possible world”. Some changesdon’t change the actual world very much. Theseare called “close possible worlds”. In Lewis’sanalysis, the counterfactual is true if the indicativeconditional in the closest possible worlds where
The Conditional Fallacy• The conditional fallacy arises when one fails totake account of all the effects of a counterfactualconditional• John Rawls says that a plan of life is rational if itis a plan that you would adopt if you were actingwith full deliberative rationality.• In other words, “I’m not someone who acts withfull deliberative rationality now. But if I were,what would I decide to do?”
The Conditional FallacyContinued• Suppose that I frequently decide thingsimpulsively and this gets me into trouble. I’mtrying to decide whether I should see a therapistabout this.• According to Rawls, I should ask, Would someonewho was fully deliberatively rational see atherapist in this situation?• But if I were fully deliberatively rational, Iwouldn’t need to se a therapist. I wouldn’t havethe problem. The conditional fallacy here is that ifRawls’s counterfactual conditional were true, itwould change the original situation. What would
Another Example of theConditional Fallacy• According to Roderick Chisholm, it’s reasonableto believe something if it would be reasonable foryou to believe it if your concerns were purelyintellectual• Suppose you want to know whether you shouldbelieve, “My concerns are purely intellectual”,meaning “My concerns now are purelyintellectual”• If I follow Chisholm’s suggestion, I will believethis is true, because if my concerns were purelyintellectual, I would believe they were. But theyaren’t now, so I shouldn’t believe it.
Reduction of HypotheticalSyllogisms• A modus ponens syllogism can be changed to amodus tollens and a modus tollens can be changedto a modus ponens• “If a, then b, a; therefore b” can be changed to “ifnot b, then not a”; a; therefore b”. We interchangethe antecedent and consequent of the hypothetical,and then negate both.• “If a, then b; not b; therefore not a” can bechanged to “If not b, then not a; not b; thereforenot a” Again, we exchange the antecedent andconsequent of the hypothetical and negate both.
Can a Hypothetical Syllogism BeChanged to a CategoricalSyllogism?• 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 asituation in which beggars will ride• The situation in which wishes wishes are horses isa situation that is true• The situation that beggars will ride is true• Joyce thinks that this change conceals the realrelationship. One proposition is conditional on
Two Kinds of Disjunction• Disjunctions such as “A or B” can be interpretedin two ways.• Exclusive disjunction means “A or B, but notboth”. E.g., All animals are either one-celled ormany-celled. An animal can’t be both one-celledand many-celled.• Inclusive disjunction means “A or B or both”.E.g., “Either all men are mortal or Obama is thePresident”• Inclusive disjunction is the standard usage inmodern logic.
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 whenexclusive disjunction is used.
Modus Tollendo Ponens• Either Obama is the President or I am thePresident• I am not the President• Therefore, Obama is the President• This is valid whether the disjunction is exclusiveor inclusive• Either Obama is the President or Mises was aKeynesian• 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 hypotheticalproposition. Each part of the compoundhypothetical leads to an undesirable conclusion• The other premise is a disjunction that says thatone of the parts of the compound hypothetical istrue.• The adversary cannot avoid the undesirableconclusion• 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 alland only those who don’t shave themselves.Does the barber shave himself?• This isn’t a genuine paradox. We can showwhy it isn’t by analyzing it as a dilemma.
Paradox Dissolved• If the barber shaves himself, then he doesn’t shavehimself; (He shaves only those who don’t shavethemselves) and if the barber doesn’t shavehimself, then he shaves himself. ( He shaves allthose who don’t shave themselves)• Either the barber shaves himself or he doesn’tshave himself.• Whatever the barber does leads to a contradiction• Thus, there couldn’t be such a barber. We have aproof that a barber of this description couldn’texist. This is why the barbershop paradox isn’t areal paradox.
Responding to Dilemmas• Joyce distinguishes three ways of responding to adilemma• One is to take one or more of the “horns”(alternatives) of the dilemma and show that thebad consequences aren’t involved• Another is to show that some other alternativefrom those considered in the dilemma is possible.This alternative doesn’t involve an undesirablealternative.• This is called escaping between the horns
The Third Alternative• This response to the dilemma constructs acounter dilemma. This takes the samealternatives as the original dilemma andshows that they are fatal to the originalargument.
The Litigiosus• The is a famous example. Protagoras trainedEuathlus in rhetoric. Half of his fee was payablewhen Euathlus won his first lawsuit. After hefinished his course, Euathlus wasn’t involved inany lawsuits and didn’t pay• Protagoras sued Euathlus. He constructed thisdilemma. “Either the court decides in my favor, orit decides against me. If it decides in my favor, Iwin and Euathlus has to pay. But if I lose,Euathlus has won the suit, and by the terms of ouragreement, he has to pay. Thus, whether I win orlose 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 theagreement, I don’t have to pay; and if the courtdecides in my favor, then I don’t have to pay. Ineither case, I don’t have to pay”• Joyce doesn’t think that there is a clear solution tothis puzzle, but in fact it can be solved.• Neither the dilemma nor the counter-dilemma canbe accepted. Both rely on inconsistent criteria. Wecan either decide according to the terms of theagreement or according to the decision of thecourt, but not both. If we go by the decision of the
Solution• To solve the puzzle, we should consider the termsof 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. Thistime he should win, because Euathlus has won alawsuit. By losing a case, Protagoras can bringabout the situation in which he will be paid.• A provision of the U.S. Constitution says that therepresentation of a state in the Senate can’t bechanged without its consent. This provision, it isfurther stated,cannot be amended.• Can this provision itself be amended? It can be