C THE SQUARE Of OPPOSITION

r 1,+.

15.

Some tropical islands are wonderful vacation getaways.

No green vegetables are vitamin-deficient foods.

C. THE SqUARE OE OPPOSITION,\t have seen that the four types of categorical proposition forms differ in qualit,:uantity, or both. Opposition occurs whenever two categorical proposition forms:ave the same subject and predicate classes but di$er in quality, quantity, or both.{nd so far, we have been concerned only with understanding the structure of these

:ropositions. We have not considered their logical consequences. Ifthey are taken as

:rue or false, what can we conclude?The first relationship we will look at is called contradictories, which is a pair of

rropositions in which one is the negation ofthe other (they have opposite truth values).l-hrs occurs when we recognize that it is impossible for both propositions to be truerr both to be false at the same time. Contradictory categorical statements differ from:ach other in both quantity and quality. For example,

(1) Atl jnterstate highways are projects buj[t with taxpayers' money.(A-proposition)Some interstate highways are n0t projects buitt with taxpayers'money.(0-proposition)

Can both ofthese propositions be true (or false) at the same time? Think about it.-Ihe

answer is No. Ifthe first proposition is true, then the second must be false. Ifallevery) interstate highways are proiects built with taxpayers' money, then there cannot

l.e even one that is lrof built with taxpayers' money. Likewise, ifthe second proposi-:ion is true, then the first must be false. Ifthere is at least one interstate highway that:s notbuilt with taxpayers' money, then it cannot be true that all ofthem are built with:axpayers' money.

What happens if the first proposition is false? Well, then the second proposition,rould have to be true. Ifnot every interstate highway is built with taxpayers'mone,:hen there must be at least one that is r?ofbuilt with taxpayers' money. Likewise, iftheiecond proposition is false, then the Iirstmustbe true.Ifthere is not even one interstaterighway that is rzof built with taxpayers' money, then it must be true that all of them:re built with taxpayers' money.

For any two propositions to be truly contradictories, one ofthemhas to be true and:he otherhas to be false. Aswe saw for propositions (1) and (2), A- and O-propositions:re contradictories.

E- and I-propositions are contradictories, too:

(3) No interstate highways are projects built with taxpayers' money.(E-proposition)

(4) Sorne interstate highways are projects buitt with taxpayers' money.(I-propositio n)

Opposition When twostandard form categorrcalpropositions refer tothe same subjecr andpredrcate classes, butdiffer in qualit, quantit,orboth.

Contradictories lncategorical logic, pairsof propositions inwhichone is the negation ofthe

(2)

174 CHAPTER 5 CATIGORICAL PROPOSlT]ONS

Ifthe 6rst proposition is true then the second must be false, and vice versa. Also, ifthe first proposition is false then the second must be true, and vice versa.

We can display the resulls discussed so far as a bare-bones square. It is our flrst stepin building what we shall call the sqaa re oJ opposition:

Ihe orrot,s indicate thecontradictory pairs:

A_0E-I

Controdictaries

Figure 5.1 spells out what that means. Since A- and O-propositions are contradictor)they should have opposite values for quantitl, qualit, and distribution. So should thecontradictory propositions E and I.

Figure 5.1

A: All S are P.

UniversalAllirmativeSubject term distributeclPredicate term undistributed

I: Some S are P.

ParticularA EirrnativeSubject term undistribttedPredicate lerm u[drstributed

Contrddtttorie\

E: No S are P.

UnivcrsalNegatiYe

Subject term distributedIrredicate term distribu ted

I: Some S are not P.ParticularNegativeSubject ierm undistributedPredicate tern] distributed

Let s see ifyou have grasped the idea ofcontradictories. Are the follotving two propositions contradictoriesl

(5) AIL zoos are places where aIimals are treated humanely. (A-propositron)(6) N0 zoos are places where anjmals are treated humaneiy. (E-proposltion)

Ifthe first proposition is true, then the second must, ofcourse, be false. Likewise, ifthe second is true, then the first must be false. However, ifyou guessed that they are

contradictoriesr Iou would be wrong. To see this, consider whatwould happer.r ifthe 6rstproposition were false. In that case, must the second proposrtion be true? Again/ thinkabout it. Ifit is false that 'A11 zoos are places where animals are treated humanely," mustit be true that "No zoos are places where animals are treated humanely"? The answer isNo because there mrghtbe one or more zoos where animals are treated humanely. Sincethis is possible, it would make the second proposition false, too. Since contradictorypropositions cannot both be false at the same time, we have shown that propositions (5)

and (6) are not contradictories, There[ore, A- and E-propositions are nol contradictories.

C. THE SqUARE OE O?POSITION a75

.rf,mPle ofcontraries:

(7) Al[ hurricanes(8) No hurricanes

This example reveals a newtype (6) can-

:,.tboth be true at the same time, Pairs of

- - "p"*tr"rf-t"-tng this particul another

are storms formed in the Atlantic 0cean (A-proposition)

ir. ,tor., formed in the Attantic 0cean (E-proposition)

iithe first proposition is true/ then the second must

e; both canbe false at the same time, because it is P

e ibrmed in theAtlantic Ocean This analysis revea

:-.ntraries. We can add this information to our square of oPPosition:

Contraries

contraries Pairs ofpropositions thatcannotbo!hbe true at the same

time,but canbothbefalse at the same time.

Subcontraries Pairs oIproposrtions that cannot

bothbe false at the same

time, but can both be

truej also, ifone is false,

then the other mustbe

:;-1ai

The flip side ofcontraries are subcontraries, which cannot both be false at the same

_re,butcanbothbetlueatthesametime.Also,ifoneisfalse,thentheothermust:E rrue. The next two ProPositions are subcontraries:

(0-proPosition)

ContrariesnE

Controdictories

Controdictories

Subcontraries

a'.- 1-:

176 CHAPTER 5 CATEGORICAL PROIOSITIONS

subaltehation The

relationshrp between a

universal proposition(referred to as thesrperdlrelr) and itscorresponding particularproposition (referred toas the irrdlrcrr)

A: All 5 are P

E:NoSarePcorresponds tocorresponds to

We need one linal relationship in order to complete the square of opposition. Sub-

alternation is the relationship between a universal proposition (referred to as the si.

peraltern) and its corresponding particular proposition (referred to as the subaltern

flere are two kinds ofcorresponding propositions:

I: Sone S are P.

0: Some S are not P.

Ifthe universal proposition ofa pair is true, then its corresponding particular wil,also be true. For example, if it is true that 'A11 modern holidays are greeting-cardcompany creations," then it is also true that "Some modern holidays are greeting-card

company creations." Likewise, if it is true that "No modern holidays are greeting-

card company creations/" then it is also true that "Some modern holidays are not

greeting card company creations." However, the reverse does not hold. That is, ifthe particular proposition of a pair is true, then its corresponding universal mightbe true-or it might be false.

Here is an example ofsubalternation:

(11) All musical instruments are difficult thjngs to master. (A-proposition)(12) Some musicaI instruments are difficutt things to master. (I-proposition)

Ifthe universal afiirmative categorical proposition (A) is true, then its corresponding

particular (I) will be true, too. However, we can see that even ifproposition (12), the

particular afirmative (I) is true, then its corresponding universal (A) might be true or

false. These same results hold [or the categorical propositions E and O. For example:

(13) No musical instrurrents are difficutt things to rnaster. (E-proposition)(14) Some musicaI jnstruments are n0t difficult things to master. (0-proposition)

As before, ifproposition (13)/ a universal negative (E) is true, then its correspondingparticular (O) will be true, too. However, we can see that even ifproposition (14), the

particular negative (O) is true, then its corresponding universal (E) might be true orfalse.

Subalternation gets more interesting ifwe ask what happens when one member ofa corresponding pair 1s false. On the one hand, ifthe universal proPosition of a Pairis false, then its corresponding particular partner could be true or false. For exam-

ple, ifit is false that'All honor students are hard workers" (A), then the proposition"Some honor students are hard workers" (I) could be either true or false. Similarly, ifit is false that "No honor students are hard workers" (E), then the propositron "Some

honor students are not hard workers" (O) could be either true or false. However, the

reverse does not hold. That is, ifthe particular proposition ofa pair ofcorrespondingpropositions is false, then its corresponding universal must be false as well. Here is

an example:

(t5) All musicaL instruments are difficutt things to master. (A-proposition)(16) Some musical instruments are difficult things to master. (I-proposition)

Ifproposition (15), a universal affirmative (A) is false, then its corresponding par-

ticular (I) could be either true or false. However/ we can see that ifproposition (16), the

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F

s

s

tc

L

al

P\S(

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5ub-

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C THE SQUARE OE OPPOSITION -177

particular amrmative (I), is false, then its corresponding universal (A) must be false,too. These same results hold for E- and O-propositions, Here is an example:

(17) No musicaI instruments are difficutt things t0 master. (E-propositjon)(18) Some musical instruments are not difficult things to master. (0-proposition)

Asbefore, ifproposition (17), auniversal negative (E) is false, then its correspondingparticular (O) could be either true or false. But, once again, we can see thatifpropJ-sition (18), the particular negative (O) ls false, then its corresponding universal fo,)must be false, too.

We can now complete the square of opposition:

SubaLternotion Subolternotion

TF TF0Subcontrories

Let's try it out and see where it takes us. Our discussion has included the creationand analysi are arguments that contain only onepremise. (A emise are calledmediate arguments,)\ow suppo "All clowns are scary people" (e). Ifso, can we go around the square ofopposition and say something about each oftheremaining three categorical proposition forms? We can. The proposition, ..No

clownsare scary people" (E), is the contrary of the original proposition (A). Since contrariescannot both be true at the same time, the proposition, "No clowns are scary people,,E), is false.Also, since theproposition/ "Some clowns are not scarypeople,, (6)-, is ihe

.ontradictory of the original proposition (A),, too, is false. The remaining proposi-tior, "Some clowns are scary people" (I), is th subaltern ofthe original propositionA), and so it is true.Now let's try the opposite truth value. What if the proposition .All clowns are

scary people" (A) is false? The contrary of this proposition is ,.No clowns are scary

people" (n). And going around the square, we determine that it could be true or false,5o its truth value is undetermined. However, the proposition "Some clowns are notscary people" (O), the contradictory of the original proposition (A), must then be:rue. The remaining proposition, "Some clowns are scarypeople,, (I), the subalternof the original proposition (A), might be true or false, so its truth value is therefore:.:ndetermined.

Immediate argumentAn argument that hasonlyonepremise.

Mediate argumentAnargument that has morethan one premise

Figure 5.2

178 CHAPTI R 5 CATEGORlCAL PROPOSITIONS

I]e

I. Use your understandrng of the square of opposition to determine the correct

ansl\,er.

l. The contradictorT of"No football players are opera singers" rs:

ia) All footballplayer\ are oPera \inge15'

(b) Some football players are opera srngels'

(c) Some footballplayers are not opera srngers'

Answer: (b) is correct. Since "No footballplayers are opera siDgers" is an E-proPosition

its contradictory must be an I-ProPosition, which is answer (b) The correctanswer can-

not be (a) because it is an A-Pl oPosition, which is the conrrarl ofan E-propositlon Also,

(c) is not correct because rt is an O-proposition, which is the subaltern ofan E-proposition

2. Are the following two ProPosition s contrat'iesl

A1l yo-yos are toys better left untouched.

No yo-yos are toys better left untouched.

3. Are the following two ProPositions subcontrariesl

Some contact lenses are gas-permeable obiects'

Some contact lenses are not gas-permeable objects

4. TrueorFalse: lnthe square ofopposition, two contl adictory cateSorical ProPosi-

tions can both be false at the same time.

5. True or False: In the square ofopposition, t$'o contrary categorical propositions

canboth be false at the same time

II. Use your understandlng of the square of opPosition to determi ne the correct

answer: a, True, b. False, or c. Undetermined.

l. Ifit is false that "some implants are easily detectable obiects," then the proPosi

tion "No imPlants are easily detectable obiects" must be:

Answet a. True. The first is an I-propositiol1, and if it is false, its contradictory E-

proposition must be true.

2. Ifit is false that "some implants are easily detectable objects," thet.r the proposi'

tion "No implants are easily detectable objects" must be:

3. If it is false that "Some games are crazy inventions," then the proposition 'A11

games are crazY inventions" must be:

4. Ifit is true that "some games are crazy inventions," then the proposition 'Al1

games are crazY inventions" must be:

5. If it is true that "No games are crazy inventions," then the proposition "Some

games are not crazy inventions" must be:

6. Ifit rs false that "No games are crazy inventions," then the ProPosition "Some

games are not crazy inventions" must be:

180 CH APTER 5 CATIGORICAL PROPOS]TIONS

Conversion Anilnrr1ediate;rrgumentcreated by irterchnDgingihe subjeci and predicateierms of3 givencategorical propositior.

18. Ifan I-proposition is./alse, then you can conclude that the O-proposition would be:

19. Ifan O-proposition is ir4e, then you can conclude that theA-propositionwould ber

20. lfan 0-proposition rs lrae, thenyou can conclude that the E-propositionwouldbe:

21. lfan O-propositron is trre, then you can conclude that the I-propositionwould be:

22. lfan O-proposition is/alse, ther.ryou can conclude that theA-propositionwouldbe:

23. Ifan O-proposition is/alse, thenyou can conclude that the E-propositionwouldbe:

24. Ifan O-proposrtron is./alse, then you can conclude that the I-proposition would be:

D. CONIVERSION, OBVERSION, .A.ND

CONTRAPOSITIONThe square of opposition is a surprisingly powerful tool. Cor.rsider next some specialcases of immediate algument-and howwe get from one to another.

ConversionAn immediate argument can be created by ir.rterchanging the sublect and predicateterms ofa given categorical p(oposition, a process called conversion. The proposition$/e start urith is called the convertend, and it becomes the premise of the argument.fle propositionwe end up rvith after applying the process ofconversion rs called theconuerse, and rt becomes the conclusion ofthe argument. For exan-rple, consider thesetwo propositions:

Cor!,erte rd: E-proposition: No beer connercia[s are sublleadvertiseflerLs.

No subtle ad,rertlseireris arc beerE-p roposition:con nerciaIs.

The second proposition can be validly inferred from tire lirst. This can be understoodifyou recall that E-propositions completely separate the subject and predicate classes.

So ifno beer commercials are subtle advertisements, then ofcourse no subtle advertrse-

ments can be beer commercials. The conversion works for E-propositions, The same

iclea holds true lor I-proposrtrons:

Co lrverien d: I proposjtiof:

I proposjtio!r:

Sone texLboo (s a e e rtertair rg

diverslo rs.

Sofre efLertajning diversior,: arelextbool<s.

Recall that I-propositions make the subject and predicate classes overlap to some

degLee; this means that if some textbooks are entertaining diversions, ther-r, ofcourse,sonre entertaining diversions must be textbooks. Therefore, conversron works forI-propositions.

D, CONVERSION, OBVERSION, AND CONTRAPOS1TION 181

S rrictly speaking, conversion does not work for A-propositions We can see this in

-e following examPle:

, -.iverten d:

_ Irverse:

A-proposition:

A-proposition:

- : nvertend: A-proposition:

-:.responding I-ProPosition:

rarticular:,: rverse: I-Proposition:

0-proposition:0-propositio n:

Att spam e'mailings are invasions of

your home.

A[[ invasions of Your home are sPam

e-maitings.

Alt spam e-mailings are invasions of

you r home.

Some spam e-maiLings

are invasions of Your home

Some invasjons of Your home are

spam e-maitings.

Some vegetabtes are not carrots.

Some carrots are not vegetables.

I: Some P are S. (b.Y liraiafion)

E: NoPareSI, Some P are S

(Cotlversion is nat vdlid)

Conversion bylimitation we 6rstchange a universalA-proposition into itscorresponding ParticularI proposition, and thenweuse the Process ofconverslon on the Iproposition

CIearly, A-propositions cannot be directly converted However' we can use what

.= kno* to -rk" "onversion work in a limited way The idea of subaltetnation tells

',t e have created a valid argument. The square of opposition enabled us to use sub-

L:.rnation to make conversion work for A-Propositions'

the finaltype ofproPositionto consider for conversion is O-propositions Let'slook

:: :his set of proPositions.

-i.s rre can plainly see, the conversion does not work for O-proPositions ThePremise is'

,: :..urse, trire because manyvegetables, such as Potatoes, celery, spinach' and so forth

;:: ;ertainly not ca.rots; boi the co,,clusion is false: every carrot is indeed a vegetable'

r:.rchange the subject and Predicaie

Predicate

A: All S are P

E: NoSarePI: Some S are P

O: Some S are not P.

fbversion-- recond type of immediate argument is formed by (1) changing the quality of the

=.n prop*r,rort ,nd (2) by re-placing the Predicate term with its complement 'Ihe

1A2 CHAPTER 5 CATTGORICAL PROPOSlTlONS

ComplementThe se! ofobjects thai do Llot belong

to n given class

ObversionAD immedirteargurneni formed bY

changing the qualltY ofthc givcr ProPosition,an({ then rePl:rcing the

predicate term widl itscomPlenreLlt

Contrapositionl\icreplscc the subjcct ierm

of n giver ProPositjonwith the comPlement

ofrts predicate term,and then rePlace thc

predicate term oIthegiven proPosiiion 1\'ith

the complement ofiis

0bverlend: A-proposition: All iackhammers are u/eapons'

0bverse: E-proposilion: No jackhammers are non-u/eaporls

therelore valid:

0 bve rte n d:

0bverse:

0bvertend:0 b ve rse:

0bverte n d:

O bv e rse:

t- p ro posltl on:

A-propositiof:

l-proposrtio n:

0- proposition:

0-propositto n:

I-proposition:

No comedians are brain surgeons

All conredians are non-brain

surge0ns.Some athletes are overpaid egojsts

Sorne athletes are not non overpaij

e g oist s.

Sorne pets are not Lovable aninrals.

5 ne p"r5 o o 'ol - o d. .e o l: 1ol(.

A:AllS are P

E No S are P

I: Some S are P.

O: Some S are Lrot P.

E: No S are rlor-P

A, AII S are non-P.

O, Some S are not non P

I: Some S are non-P

ContrapositionThe final type of immediate argument to consider, contraposition' is formed by ap-

next pair of ProPositions:

Given proposition: A-ProPosition:

Contrapositive: A-ProPOsition:

ALt pencils are ink free writitrgto o ls.

AIL non ink free,,^i riting tools are

non-pencils.

-r.d&l

:L+

![@

lr

!-!.

IIIL

.i,llfl

D CONVERSlON, OBVERSION, AND CONTRAPOSITION

The first proposition places every pencil in the class of in

-:-.trapositiue if this ProPosition claims that anything that

:: ..1 is also a non-Pencil, and surely this must be correct So

l-fropositions.Let's try a different pair ofpropositions:

0-proposition: So non-cuddLythingsnot non-hairY

creatures'

1A3

:'ren proposition:

-:ntrapositive:

:: hairy creatures. The argument is valid'

E-propositions are more problematic:

.'ven proposition:

-lntrapositive:

E-proposition:E-proposition:

No gori[[as are [ions.

No non-[ions are

non-gorittas.

Even ifthe first proPosition is true, the second can still be false This maybe hard to

=. "i n,,,, i* a",e analysis revears th" *"T:i*",f"'X.t"T#H?i::',;.,',:,1'*

j. at lhe same time, outside the class ofgoril

r.., ""iitt A* "ot

a gorilla So clearly the contrapositive doesn t work here'--**r.r,-"

""r, do iomething similar to what we did for A-propositions' and once

;; ;; ;l; ." the square of oiposition Contraposition of E-propositions can be

:nderstood as valid if we use contraposition by limitation: subalternation is used to

.Jg" tt "

oniu"rral E-propositioninto its corresPonding particular O-proposition'

,i. tf;"n "pply

th" ,egula, frocess of forming a contraPositive to this O-ProPosition'

riat "r"'"ir".ay

rnivn that this proc"'s *o-'k' on O-p'opositions') The cornpleted

:rocess looks like this:

Given proposition: E-proposition: |o eorl!ta.;,1re-lions'

i.rr"ri""il.gp.ni.,l,t' O-proposition: Somegoritla-s-arenottions'

i".ur'p"ri,i"i,' o-proposition: t"J"..:;:;lii.1: "'.,",

The square of opposition has thus enabled us to make contraposition work for E-

:ropositions by again using subalternation'

contrapositionby limitationSubalternation is used

to change a universal

E-proposrtion into its

corresponding ParticularO-proposition. We

then apply the regularprocess offorming a

contraPosilrve to this O-

propositron.

] 84 CHAPTER 5 CATEGOR]CAL PRO!OSITIONS

The plocess involr'ed in contraPosition do es not work for I-propositions (for reasons

similar to rvl.ry conversion did uot work for O-proposilions) Let's look at this set of

propositioDs:

i,i,/. i I :,li-,, ir l

t- )t\i tat)l: t tt:.

l -ir ,r . r, ttrr :

l-, ! ,lrt ! il r:

'.,,; |,. ,Lllll;: i r: laril

,r.a !.:,rtt,,tlr:i,!.t,iri, r! ,Lr i,,l r',lr!l' .lr l

r lj _lii .r :i,

A. All nor P rrc non S.

O: Some non'P a, e Dot noLl S (l'-r li''lldlior,(Co,in-r|oJ,f Io,r is rof ld/rd)

O: Some non'P are rlotron S

obverse, and contraPositive of tlle

subsequer1t ill-rn-rediate arguments

The Erst propositlon is tr r.re (some people fail to register to vote) but the second is clearl'v

falr". (\\'e'hore r",luced the phrase "ron '10'

-registeled vo[ers" to simPly "registel ed vo t-

ers.") So contrapositiol does r.tot yield valid immediate algurnents for I-propositions

Step l:Slritch ihe subjeci ind Prcdicite icrnls'

Stc; 2: Replice borh tilc subject an(l Predicatc terLns lviih their tet m complements'

A, A11S arc P

E NoSarcPI: Somc S rrc P

O. Some S are not P

For each of ihe following, provide the converse,

given proposrtioll. Also state r'vhether any of the

are not vrlicl,

1. Some games ofchance are sucker bets'

A. Converse:

B. Obverse:

C. ContraPositiYe:

Ansurers:

z. Some sandwiches are not meatythings

A. Converser

R. Obverse:

C. ContraPositlve:

No bats are vegetarians.

A. Converse'

B. Obverse,

C. Contrapositive:

Convelse: Some sucker bets are games of chance