GENERAL DIRECTION: Read the following topics and answer the exercises Hpothetical !llogis"s
Hypothetical syllogisms syllogisms are different from standard syllogisms and thus have their own rules. In a hypothetical syllogism the first premise (or major proposition) presents an uncertain condition ("if A, then B") or a prolem ("either A or B"! " and # cannot oth e true") which must then e properly resolved y the second premise so that a valid conclusion can follow. #he resolution of the prolem is always in the form of affirmation or denial. In this article, the three types of hypothetical syllogism we will cover are the conditional syllogism, the disjunctive syllogism, and the conjunctive syllogism. The Conditional !llogis"
#he major proposition of this syllogism presents a conditional argument to the effect that if one thing is true, then another is also true. $or e%ample, "If this is li&uid nitrogen, then its temperature is more than ' degrees elow ero." #he truth of the antecedent (the first statement) implies or estalishes the truth of the conse&uent (the second statement). It follows that (*) if the first thing is not true++that is, is denied ("#his is not li&uid nitrogen")++then no conclusion can e drawn aout the second thing, and () if the second thing is not true or is denied ("#his is not more than ' degrees elow ero"), then neither is the first true, since the second thing would e true if the first had een true. $urther, if the second thing is true, that does not of itself prove the first to e true++the antecedent proves the conse&uent true, ut the conse&uent cannot prove the antecedent true (as in this case, other sustances have temperatures elow minus ', li-e li&uid hydrogen). #hus we have two valid and two invalid forms of the conditional syllogism. #he valid forms are If A, then B And A! therefore B (Affirm antecedent! then affirm conse&uent) If A, then B But not B! therefore not A (/eny conse&uent! then deny antecedent) #he invalid forms and their respective fallacies are If A, then B And B! therefore A (A0&) Fallacy of Affirming Affirming the Consequent Consequent (A0&) If A, then B 1ot A! therefore therefore not B Fallacy of Denying Denying the Antecedent Antecedent (/A) (/A) As you can see, it is the order of of affirmation or denial that determines the validity of the procedure. Affirming the conse&uent first conse&uent first or or denying the antecedent first antecedent first is is the invalid method. #he conditional syllogism can e a simple three+term argument such as "If that man is the mailman, the mail has arrived! and that man is the mailman, so the mail has arrived." 2r it can e a more comple% argument with more than three terms. #he form of the syllogism can e e%tremely loose, with a "then" statement in front of or ehind an "if" statement! and the conclusion statement may use either "therefore" or "then," "thus," "so," or "hence." 1ote also that a negative statement can e an affirmation, as if you were to say, "I affirm that the towel is not in in my loc-er." In such a case, to -eep the negative would e to affirm the
statement and to drop the negative would e to deny the affirmation of it. #his yields twelve more forms for the conditional syllogism. $or instance If not A, then not B But B! therefore A (valid) If not A, then not B But A! therefore B (/A) If not A, then B But not B! therefore A (valid) Here are some e%amples of the various forms •
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If life is a struggle, then I am fully involved in life, since I3m struggling to understand logic. And life is a struggle, so I am fully involved in life. (valid) If 0indy went to the each, she got sun+ urned. But she didn3t get sunurned, so she must not have gone to the each. (valid) If arah comes in late, we will have to start over. I see we are starting over, so arah came in late. (A0&) /. If this is 4ilson3s oo-, it contains an essay y wift. But this is not 4ilson3s oo-, so it doesn3t contain a wift essay. (/A) If this movie is not aout horses, then I will watch it. But I will not watch it, so it is aout horses. (valid) $. If we do not use premium gasoline, the engine will ping under acceleration. And the engine is pinging under acceleration, so we are not using premium gasoline. (A0&)
Exercise #
/etermine the validity of the following *. If you li-e molasses, grandma will uy you a ottle. But you don3t li-e molasses, so grandma will not uy you a ottle. . #he tires must e replaced if the wear indicators are showing. #he tires must e replaced. #herefore, the wear indicators are showing. 5. If you were self+motivated, you would e a good student. But you aren3t self+motivated, so you aren3t a good student. '. If 6eorge wanted a richer, fuller life, he would read good literature. Ah, I see he is reading some good literature now. He must want a richer, fuller life. 7. If lettuce is on sale today, ally will ma-e grinder sandwiches. ally is not ma-ing grinder sandwiches! therefore, lettuce is not on sale today. 8. If Birnam 4ood moves to /unsinane Hill, then 9aceth is in troule. 9aceth is indeed in troule, so Birnam 4ood has moved to /unsinane Hill. :. If Breenthorpe didn3t get drun- all the time, the people would vote for him. But he does get drun- all the time, so the people won3t vote for him.
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#his syllogism presents two alternatives in an "either . . . or" form! one of the alternatives is for formal reasons assumed to e necessarily true, so that to deny one leaves the other as the only possiility. #he two possiilities, called disjuncts, are stated in the major premise! one is and must e denied in the minor premise! and the other is affirmed in the conclusion. #his is the valid form, which can e shown as follows >ither A or B 1ot A! therefore B (/eny first disjunct! affirm the second) >ither A or B 1ot B! therefore A (/eny second disjunct! affirm the first) #he opposite procedure of first affirming and then denying is, however, incorrect. >%cept where the memers are explicitly contradictory so that oth could not possily e true, the affirmation of one disjunct (in the minor premise) does not deny the other. $or e%ample, to say, ">ither the power is off or the ul is urned out! the power is off so the ul is not urned out," would e a fallacy, ecause, while we assume that one of the disjuncts is definitely true, oth might e true++we did not chec- the ul and so cannot e sure of its condition. ince the second disjunct has not een investigated, it cannot e denied y default. (4here the memers of the disjunct are contradictory, as in "#he plant is either alive or dead," the argument should, to avoid confusion, e changed into the conjunctive form of syllogism and wor-ed from there++see elow, section ?5.) #he fallacy, then, of first affirming one disjunct and then denying the other loo-s li-e this >ither A or B And A! therefore not B >ither A or B And B! therefore not A Fallacy of Affirming a Disjunct (A/) ome e%amples of valid and invalid forms •
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#his is either a dictionary or a chemistry oo-. It is not a dictionary, so it is a chemistry oo-. (valid) >ither the attery is dead or something is wrong with the starter. ither the oven does not wor- or I left out the a-ing soda. But the oven does wor-, so I left out the a-ing soda. (valid)
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>ither I studied disjunctive syllogisms or I am going to low this one. I did study disjunctive syllogisms, so I won3t low this one. (A/)
@ememer that, as in the third e%ample, to drop a negative is to deny the affirmation of it. A final note
/etermine the formal validity of the following disjunctive syllogisms. *. >ither $reentop is a croo-, or he is a very crafty individual. I -now he is very crafty, so he is not then a croo-. . >ither Bleps are nords or lots are nords. lots are not nords. #herefore, Bleps are nords. 5. >ither C is 1 or # is D. I just discovered that # is D. 4ell then, we -now that C can3t e 1. '. >ither we will have triulation in this world, or life will e uninterrupted liss. But life is not uninterrupted liss, so we will have triulation in this world. 7. >ither that statement is wrong or Eones is in error. But that statement is right, so Eones is in error. 8. >ither I am perfectly logical or you are logical. And since I am indeed perfectly logical, you are not logical. :. >ither you are in favor of our campus demonstration or you are a repressive fascist. And since you are against our demonstration, you are a repressive fascist. ;. I always write with either a all point pen or a pencil. #oday I decided to avoid pencils, so my letters this afternoon I have written with a all point pen. The Con$%ncti&e !llogis"
In the major premise of this syllogism two propositions, called conjuncts, are presented, oth of which cannot e true simultaneously. #he minor premise proceeds to affirm the true conjunct and the conclusion then denies the remaining one, which must e false y definition. #he valid form is A cannot e oth B and 0 A is B! therefore A is not 0 (Affirm the first conjunct! deny the second) A cannot e oth B and 0 A is 0! therefore A is not B (Affirm the second conjunct! deny the first)
1ow, we -now y definition that oth conjuncts cannot e true. But further, if we riefly loo- at an e%ample++". In our e%ample, rather than a mother or a father, the person++you++might not e a parent at all. #herefore, an attempt to affirm one conjunct (the remaining one) y first denying the other is an invalid procedure A cannot e oth B and 0 A is not B! therefore A is 0 A cannot e oth B and 0 A is not 0! therefore A is B Fallacy of Denying a Conjunct (/0j) >%amples of the forms • •
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A law cannot e oth variale and fair. #his law is variale, so it is not fair. (valid) I cannot oth go to a movie and finish my term paper. And I must finish my term paper. #herefore, I cannot go to a movie. (valid) He -new he couldn3t oth go to a movie and finish his paper, so he didn3t go to a movie. #hus he must have finished his paper. (/0j) $red cannot e oth a genius and a fool. And $red is certainly no genius. I guess that means he is a fool. (/0j)
$inal note As with other -inds of hypothetical syllogisms, the oppositions set up may have shortcomings of degree, representation, and so forth. #o say, "
/etermine the validity of the following conjunctive syllogisms. How well do the oppositions accord with realityF *.
Exercise )
In each case, name the type of syllogism involved (conditional, disjunctive, conjunctive) and then tell whether or not it is valid. If invalid, name the fallacy involved. *. lectricity is oviously getting to the power supply if the fan, which runs off the power supply, is running. And the power supply must e getting electricity since the fan is indeed running. '. #he vegetales will e free from harmful pests if they have een fumigated. #he inspector confirms that they are indeed free from harmful pests, so we can conclude that these vegetales have een fumigated. 7. #he patient cannot e oth completely healthy and have a lood pressure of more than *7*. #he patient is not completely healthy. o he must have a lood pressure of more than *7*. 8. >ither the witness is telling the truth or $rimpson is innocent. But the witness is lying so $rimpson is innocent. :. #he soup cannot have oth a salt content greater than five pounds per thousand gallons and a specific gravity of more than =;;. Goo- at the densitometer #he specific gravity of the soup is ==, so there must not e more than five pounds of salt per thousand gallons. ;. If this river is eing polluted y the impson factory, the fish will e dying. #he fish, however, are very healthy and alive. #hus, the river is not eing polluted y the impson factory. =. #he car is rapidly losing oil pressure. >ither the oil pump is failing or oil is lea-ing out. Hmm. #he dipstic- shows that the oil level is full. 4ell, then, the oil pump must e failing. *. #his machine cannot e in oth air conditioning mode and heating mode at the same time. I as-ed Eohn to chec- as he passed y if it was in heating mode and he said, "1o." o it must e in air conditioning mode. @eference Harrris, @oert. /eduction and Hypothetical yllogismsJ VirtualSalt. @etrieved $eruary *, *, from httpwww.virtualsalt.comthin-deduhypo.htm
