LOGIC, MEANINGAND COMPUTATION

Essays inMemoryofAlonzoChurch

Edited by

C. ANTHONY ANDERSONUniversity of California, Santa Barbara, U.S.A.

and

MICHAEL ZELENYPTYX, Los Angeles, California, U.S.A.

...."KLUWER ACADEMIC PUBLISHERS

DORDRECHTI BOSTON I LONDON

JOHN CORCORAN

SECOND-ORDER LOGIC

Abstract.This expositoryarticlefocuses onthefundamental differencesbetweenfirst-orderlogicand second-orderlogic. It employs second-order propo­sitionsand second-order reasoning in a naturalway to illustrate the fact thatsecond-orderlogic is actually a familiarpartof ourtraditionalintuitive logicalframeworkandthatit is notan artificialformalismcreatedby sp ecialists for tech­nical purposes. To illustratesome of the main relationships between first-orderlogic and second-order logic, this paper introduces basic logic, a kind of zero­orderlogic,whichis m or erudimentarythanfirst-order and whichis transcendedby first-orderin the same way thatfirst-orderis transcended by second-order.The heuristiceffectiveness and the historicalimportance of second-order logicare reviewedin thecontext of thecontemporarydebate over the legitimacyofsecond-order logic. Rejectionof second-orderlogic is viewed as involvingradicalrepudiationof partof ourscientifictradition. Buteven if genuine logic comes tobe regardedas excludingsecond-orde r reasoning, which is a real possibility, itseffect iveness as a heuristic instrumentwillremain and its importan cefor under­standingthehistoryof logicand m athematics willnotbe diminished. Second­orde r logic may some day be gone, but it will never be forgotten. Technicalformalisms have been avoidedentire ly in aneffortto reachan inte rdisciplinaryaudience, butevery efforthas been made to limit the inevitable sacrifice of rigor.

No matterwhathumanactionyou consider,if everyonedoes it to everyonedoing it tothem, theneveryonehas it done tothem by everyone to whomtheydo it. For example, if everyoneteacheseveryonewho teaches them,theneveryone istaughtby everyonetheyteach.Likewise, if everyonehelpseveryonewho helpsthem, theneveryoneis helpedby everyonetheyhelp. Thesame holds for"encourages", "hinders","supports", "opposes", "ignores" ,andtherest.

Each oftheabovepropositionsis actuallya tautology, a propositionimpliedby its ownnegation. In fact, each ofthemcan be proved to betrueby logicalreasoningalone; e.g., by deducingthem from theirown negations. Sinceevery propositionin thesame form as atautologyis againa tautology, adiscourseformallysimilarto thatexpressedaboveobtainsin every universeof discourse,notjustin theuniverseof humans.

In metalogic,forexample, we often discuss theuniverseof propositionsinso far as various logicalrelationsareconcerned. By a logical relation I meanrelationssuch asimplication, consequence, contradiction, compatibility, in­dependence,etc. More specifically, I mean whatare calledbinaryrelationson theuniverse ofpropositions.If R indicatessuch arelationand ifa andbare eachindividualpropositions, thenaRb canbe used toexpressthepropo-

61

C. Anthony Anderson and M. Zeleny (eds.), Logic. Meaning and Computation, 61-75.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

62 JOHN CORCORAN

sit ionthatthefirstpropositiona is relatedby R to the secondpropositionb. It is notexcluded,of course,thata andb arethesame proposition.Forexample, everypropositionimplies itselfandsome butnoteverypropositioncontradictsitself.

Now logicalrelationsarecertainlynotactions. Saccheri's Postulatecontra­dicts theParallelPostulatebutthereis no actionthatSaccheri's Postulatecouldperform. Nevertheless, as we havejustseen, relation verbs functiongrammaticallyin certaincontextsin a mannersimilarto thefunctionof ac­tion verbs. The relationverbs significantin theuniverseof humansincludethe following:outweighs,outlives,succeeds (inseveralsenses), precedes (inseveral senses), equals (in many senses), and manyothers. The actionverbssignificant intheuniverse ofhumansinclude the following: calls (inatleastone sense), serves (inat leastone sense), teaches,commands, obeys, andmany others.

In normalEnglishsome of thelogicalrelationsareexpressedby relationverbs, as we have seen. For example,implicationis expressedby 'implies'and contradictionis expressedby 'contradicts'. However, some ofthemareexpressedby relation nouns. For example, consequenceis expressedby therelationnoun 'consequence'.The law oftransitivityof consequenceis thateveryconsequenceof aconsequenceof apropositionis againa consequenceofthatproposition. Moreover,thereare logicalrelationsexpressedby relationadjectives. Compatibilityand independenceare expressedby 'compatible'and 'independent'.Aristotle's fundamentallaw ofcompatibilityof truthisthatevery twotruepropositionsarecompatiblewith eachother. Using 'inde­pendent' in themost widelyacceptedsense we can saythateverypropositionwhich isindependentof a givenpropositionis neitherimplied by norcontra­dictedby the givenproposition, and conversely, everypropositionwhich isneitherimplied by norcontradictedby a givenpropositionis independentofthe givenproposition.

One reasonfor reviewing the various waysthatlogicalrelationsare ex­pressed inEnglishis topointoutwhatallcreativewritersalreadyknow, viz.thatknowledge of theconventionalrules of EnglishshouldenhancebutnotinhibitEnglishwriting. For example, my very firstsentenceuses the plu­ralpronoun'them' as coreferentialwith 'everyone',which issingular. Evenworse, fromthepoint of view ofconventionalrules, is my use ofthefiller'you consider'.The propositionbeing expressedis not apredictionof whatwillhappenif you considersomething.The propositionis notaboutyou perse atall.The sentenceexpressesa general proposition predicatinga certaincomplexpropertyof everyactionon the universe ofhumans. The phrase'nomatterwhathumanactionyou consider'is justa heuristicallyeffective wayofexpressinga universalquantifier. From a logicalpointof view the followingwould dojustas well: 'everyhumanactionis one suchthat', 'everyhumanactionis one where', 'wit h everyhumanaction', etc. At anyrate,a sentencethatviolatesthe conventionalrules of Englishapplicableto the expression

SECOND-ORDER LOGIC 63

of a givenpropositionis sometimes neverthelessa perfectlyacceptableandeffective way ofexpressingthatveryproposition.

Whethera sentenceis an acceptableand effectiveexpressionof a givenpropositionis a matterof howreaderstakeit, andnot amatterofconventionsestablishedin thepast. Now wearereadyto presenta discourseobtainingin theuniverseof propositionsandformallysimilarto theone whichbeganthisessay.

No matterwhatlogicalrelationyou consider,if every givenpropositionbears it to every proposition bearingit tothegiven proposition,thenev­ery givenpropositionis borneit by everypropositionthegiven propositionbearsit to. For example, if everypropositioncontradictseverypropositioncontradictingit, theneverypropositionis contradictedby everypropositionit contradicts. Likewise, if everypropositionimplies everypropositionim­plying it,theneverypropositionis implied by everypropositionit implies.The same holds for "is aconsequenceof', "is compatiblewith", "is logicallyequivalentto", "is independentof', "is a contradictoryoppositeof', and therest.

The propositionsexpressedin theabove paragrapharealltautologiesandtheyare all laws of logic.The propositionsin thefirstparagraphofthisessayare alltautologiesbutnone ofthem are laws of logicbecausetheyare notabouta logicalsubject-matter. The proposition"Everypropositionimplieseverypropositionimplyingit" isabouta logicalsubject-matterbutit is nota law of logicbecauseit is false. For example, "Everypropositionis true"implies "Everyfalsepropositionis true", butnot conversely.The proposition"Everypropositioncontradictseverypropositioncontradictingit" is a lawof logic, ofcourse,butit is not a tautologybecauseit is inthesame formas a propositionconsideredjustabove and found to be false. By a law oflogic Imean a truepropositionabouta logicalsubject-matter, e.g., aboutpropositions,aboutarguments,aboutargumentations, etc.

The twoproperties,being tautologousandbeing a law of logic, areorthog­onal in thesensethateach ofthefourcombinationsofthetwo is exemplified.We have seen abovethatsome butnoteverytautologyis a law of logicandthatsome butnoteverynon-tautologyis a law of logic.Thereis much con­fusionconcerningthiselementarypoint. Some butnotall oftheconfusionis more or lessdeliberatelynurturedin theserviceof variousdogmas which,happily,are waning inpopularity.

The proposition,"Every propositioncontradictsevery propositioncon­tradictingit", is thelaw ofsymmetry (or reciprocity)of contradictionand"Some propositionimplies some propositionnot implying it in return"isthelaw of non-symmetry(or non-reciprocity)ofimplication. "No contradic­toryoppositeof a contradictoryoppositeof a propositionis a contradictoryoppositeof thatproposition"is the law ofantitransitivityof contradictoryopposition. A contradictory opposite of apropositionis, of course, apropo­sitionlogicallyequivalentto thenegationof thatproposition.For example,

64 JOHN CORCORAN

"Some truepropositionis nottautologous"and "Not everytruepropositionis tautologous"are bothcontradictoryoppositesof "Everytruepropositionis tautologous".In orderto avoid confusion itshouldbe notedthat,althoughevery twopropositionsthatarecontradictoryoppositesof eachothercontra­dicteach other,not every twopropositionsthatcontradicteachotherarecontradictoryopposites. To takean extremeexample, "No propositionim­plies itself'contradicts"Some propositionimplies everyproposition".Thesame exampleillustratesanotherpointthatclarifiesthingsand helps to avoidconfusion, viz. thatalthoughno truepropositioncontradictsa truepropo­sition,some falsepropositionscontradictfalsepropositions. In fact, somefalsepropositionscontradictthemselves. Thus althoughno twocontradict­ing propositionsarebothtrue, some twocontradictingpropositionsarebothfalse. In such cases, i.e., when twocontradictingpropositionsarebothfalse,theyare notcontradictoryoppositesbecauseevery twocontradictoryoppo­siteshave differenttruth-values.

A contmdiction (or a self-contradiction)is a propositionthatcontradictsitself, i.e., thatimplies its ownnegation.Every contradictionis a contra­dictoryopposite of a tautologyand everytautologyis a contradictoryop­positeof a contradiction.A propositionis said to becontmdictory (or self­contradictory)if it is acontradiction.Everytwocontradictorypropositionscontradicteach otherbut no twocontradictorypropositionsare contradic­toryopposites of eachother. The expression'two contradictoryproposi­tions'means "twopropositionseach of which isself-contradictory"whereas'twocontradicting propositions'means "twopropositionscontradictingeachother"which, in view of thesymmetricalnatureof "contradicts" , amountsto "twopropositionsone of whichcontradictstheother".

In ordinarytechnicalEnglish,'is contraryto' and 'is a contraryto' areambiguous. Sometimes, "contradicts"is meantand sometimes "is a contra­dictoryopposite of' is meant. Surprisingly,the ambiguitydoes not seemto betroublesome.However, in former times logicians hadattacheda thirdtechnicalmeaningthatdid lead toconfusion. Two propositionswere said tobe contraries (sc of eachother)if theycontradicteachotherbuttheir nega­tionsdo notcontradicteachother. For example, "Everynumberis prime"and "Everynumberis non-prime" arecontraries. It is easy to provethatevery twocontradictingpropositionsthatarenotcontradictoryoppositesarecontraries,andvice versa. In modernlogic,'contrary'is rarelyused intheobsoletetechnicalsense.

We have had occasionjustnow tostateseveral laws of logic and to men­tion(or talkabout)several laws of logic. As indicatedabove, by a law oflogic I mean atruepropositionabouta logicalsubject-matter(propositions,arguments,argumentations,etc.). The most basic laws of logic arethelawsof excludedmiddle, non-contradiction, and truthandconsequence: "Everypropositionis eithertrueor false", "Nopropositionis bothtrueand false"and "Everypropositionimplied by atruepropositionis true".

SECOND-ORDER LOGIC 65

The laws of logic, in fact allpropositionsaboutlogicalsubject-matterarein some sensesecond-level (or meta-level) propositionsin thesensethattheyare aboutthingsthatarethemselvesaboutthings(usually, of course, non­logicalthings). Some people, eitherignorantof or inoppositionto logicaltradition, call suchpropositions'second-order' .This is not how 'second­order'is used in this essayalthoughsome second-levelpropositionsare alsosecond-order. A propositionis classified asbasic, first-order, second-order,etc., not on the basis ofwhatit isaboutbutratheron thebasis of its logicalstructure. Thevery firstpropositionof this essay issecond-order. Thesecondpropositionis first-order.Everypropositionto the effectthatone namedpropositionis in a mentionedlogicalrelationto anothernamed propositionis basic, e.g., "Saccheri'sPostulatecontradictsthe ParallelPostulate". Itwill become obviousthatthetwo properties,being second-level and beingsecond-order,areorthogonal.

The basic propositions, veryroughlyspeaking, are thosewithoutcommonnouns. It is perhapseasiest todescribethe basic propositions of arithmetic(BPA) . Actually,insteadof describingthe BPA outright,it isconvenienttodescribethebasic sentences of arithmetic (BSA) and thento saythattheba­sic propositionsofarithmeticarethepropositionsexpressedby the basic sen­tenceswhenthesentencesare understoodin theirintendedinterpretations.Now, thesubstantivesof thebasic sentencesof arithmeticare exclusivelynumerals (number-names)in thewide sense: 'zero','one', 'two', 't hree',... ,'zeroplus one', 'zero plustwo', . .. , 'two plus (zero timesone)', . ... AmongthenumeralsI alsointend: 'two-squared','two-cubed', etc. The atomic sen­tencesof arithmeticinclude, inthefirst place, all so-calledequations: 'oneplus one is two', 'one plus two isone',etc., in otherwords anysentencein thepatternnumeral is numeral. The 'is' here, of course, isintendedto expressnumericalidentitywhich is oftenimproperlycalledequality andexpressedby'equals'. Next we havethesentencesthatnormallyattributea qualityto anumber, e.g., 'one iseven','one is odd', 'two isprime', 'five isperfect';and soon. Next we have thesentencesthatnormallyrelateone numbertoanother,e.g., 'two exceedsthree','threedivides two', etc.These includetheidenti­ties (orequalities,equations)alreadymentioned. Next we have thesentencesthatnormallyindicatethatthreenumbersare in aternaryrelation,e.g., 'twois betweenone andthree', etc.

Thereare alsoquaternaryrelationalsentences,e.g., 'one is tothreeasthreeis tonine'. And so on.Anythingofthissortis countenancedas long asthereare no common nouns. Evencommon nouns are allowed as long astheyareunderstoodas nominalizedadjectives(e.g., 'two is aprime' means "twois prime") or asnominalizedrelatives(e.g., 'two is a divisor of four' means"two divides four"), etc. Once theatomicbasic sentencesofarithmetichavebeen determined,the basicsentencescan be defined as the so-calledtruth­functionalcombinationsofatomicsentences,theatomicsentencespluswhatcan beobtainedfrom atomic sentencesby negations,conjunctions, disjune-

66 JOHN CORCORAN

tions, conditionals,bi-conditionals, etc. It should be explicitlymentionedin thisconnectionthatpassives (or converses) ofbinaryrelationverbs areagainbinaryrelationalverbs and thussentencessuch as 'two isdivided byfour', 'two is exceeded by four', etc. are included. Likewise includedaresentencesinvolving the so-called modifiedrelationverbs : 'properlydivides','immediatelyprecedes', 'immediately exceeds', etc.

Basic logic is the logic of basicpropositions. Basic logic isconcernedfun­damentallywith thequestionof how wedeterminethevalidity orinvalidityof anargumentwhose premisesandconclusion are exclusively basic propo­sitions. As you know, anargumentis determinedto be valid by giving aderivation(or adeduction)of its conclusion from its premises. This meansgiving anextendeddiscourse,normallymuch longerthanthepremises-plus­conclusionwhich showsstep-by-stephow theconclusion can be seen to betruewere the premises true.The rules for making up thesederivationsareobtainedby lookingatwhatpeople dowithbasic propositions when theyare reasoning correctly. Inorderto deduce from any set of basic premisesany basic conclusionthatactuallyfollows, itis sufficient to use rules from avery small set. These includetheusual rules ofpropositionallogic,theruleof substitutionof identities,therule of conversion (the activeand passiveare interdeducible)and the logical axioms ofidentity("one is one", etc.).To show thata given basic conclusion does notfollowfrom a given basicpremise-set, it is sufficient toproducea counterargument, i.e., a conclusionanda premise-settogetherin thesame form and having false conclusionandtruepremises. For example, to showthattheargumenton the left below isinvalid itis sufficient to noticethattheargumenton theright is inthesameform andhas truepremises and false conclusion.

Two is notthree.

Threeis not two plus two.

? Two is not two plus two.

One is not two.

Two is not one times one.

? One is not one times one.

The argumenton therightis obtainedin threesteps fromtheargumenton theleft.First'one' is substitutedeverywhere for 'two' on theleft. Theninthe "new" leftargument(in which'two'no longer occurs),'two' issubstitutedeverywherefor'three'.Then in the "second new" leftargument,'times' issubstitutedfor'plus'.

Strictlyspeakingan argument (more properly,premise-conclusion argu­ment) is a twopartsystem composed of a set ofpropositionscalledthepremise-set and a singlepropositioncalled theconclusion. To representor express anargumentwe use anargument-textwhich is a list ofsen­tences (notpropositions)followed by a single sentence somehowmarkedasthe conclusion-sentence. Some logic books use a line above the conclusion­sentence,butit is easier and less messy to use aquestion-markas above.The methodoutlinedabove oftransformingoneargument-textintoanother

SECOND-ORDER LOGIC 67

argument-textin such a waythattheargumentrepresentedby thesecondisin thesame form astheargumentrepresentedby thefirst works only whentheargument-textsarewrittenin a so-calledlogically perfect language in whichtheoutergrammaticalform ofthesentencesmirrorsexactlytheinnerlogicaiform ofthepropositions. Whenlogical issues areimportant,thelanguageinquestionis regimented (normalized) so thatit becomes logicallyperfect(orapproximatelyso). This is why I write'two plus (zero times one)' insteadof'twoplus zerotimes one'. Logicianstypicallygo immediatelyto asymboliclanguagecarefullyconstructedto be logicallyperfectbutformany purposes,especiallythatofexposition,thismethod,thoughvirtuallyessentialfor somepurposes,can becounter-productive.

Basic logiccanbe calledfinite logic becauseevery finite invalidargumentof basic logic isrefutableby a counterargumentwhose propositionshavereference only to a finitenumberof individuals.By a finiteargumentI meanan argumenthaving only a finitenumberof premises and by referenceonlyto a finitenumberof individualsI mean not onlythatthepropositionsreferonly to finitelymanyindividuals(which is obvious)butalsothatthefunctionsreferredto are all defined on one andthesame finiteuniverseof discourse.

By theway,thisincludestheso-calledzero-premisearguments(argumentshavingthenullpremise-set)which are valid when and only whentheconclu­sion is atautology.Some examplesfollow.

? One is one.

? If one is twothentwo is one.

? If (if one is not twothenone is two)thenone is two.

? If one exceeds twothentwo is exceeded by one.

It follows fromwhatwassaid abovethatevery basicpropositionthatis nota contradictionis in thesame logical form as atruebasic propositionhavingreferenceonly to a finitenumberof objects. This means thatamong thebasic propositionsthereare noso-calledinfinity propositions. In orderfor apropositionto be aninfinity proposition it isnecessaryandsufficientthatit benon-contradictoryandfor everypropositionin thesame formhavingreferenceonly to a finitenumberof individualsto be false. Inotherwords aninfinitypropositionis a propositionexpressedby a sentencewhich is "satisfiable"only in infinite universes ofdiscourse.

Basic logic covers most ofthearithmeticreasoningdone by school chil­dren,all ofthelogic "done" bycomputers(thoughin a sensecomputerscansimulate finitestretchesofhigherlogics),andmuch ofthelogic onthenormalaptitudetest.

In a sense,first-order logic (FOL) begins when wegeneralizebasic propo­sitions. In fact, it is notstretchingthingsto saythatbasic tautologiesaretautologiesbecause theyare instancesof first-ordertautologies.When youprove a basictautologyyou feelthatyou have notexhaustedyourreasoning

68 JOHN CORCORAN

in thatdirection. To illustratethisI will give a basictautologyandthengivefour first-ordergeneralizations.

? If threeexceeds twothentwo is exceeded bythree.

? Everynumberexceedingtwo is anumberthattwo isexceededby.

? Everynumberthatthreeexceeds is exceeded bythree.

? Everynumberexceedinga givennumberis a numberthegiven num­ber is exceeded by.

? Everynumberthata givennumberexceeds is exceeded bythegivennumber.

We areinclinedto thinkthatthebasic tautologyis logicallyderivedfromits generalization, e.g., that"if threeexceeds twothentwo isexceededbythree"is truebecause "everynumberexceedingtwo is onethattwo is ex­ceeded by" istrue... thusemphasizingthefactthattheformeris no peculiar­ity ofthree.Likewise weareinclined tothinkthatthelattergeneralizationistruebecauseofthetruthof its generalization,viz. "Everynumberexceedingan arbitrarynumberis one thatthearbitrarynumberis exceededby" ...thusemphasizingthatno peculiarityof two is involved.

The first order sentences of arithmetic (FOSA) arethesentences obtain­able fromthebasic sentencesby quantificationand takingtruth-functionalcombinations. It is importantthattheseoperationsaretakenrecursively, e.g.,a basic sentencecan be generalizedand thencombined withothergeneral­izationsby truth-functionalcombinationsandthengeneralizedagainbeforetakingfurthertruth-functionalcombinations. The first-order propositions ofarithmetic (FOPA) arethepropositionsexpressedby thefirst-ordersentencesinterpretedin theusual way. Below is anexampleof one ofthesimplestvalidargumentsin first-orderlogic.

Everynumberis eithereven or odd.

No numberis bothevenandodd.

Everynumberwhich isodd is one whosesquareis odd.

? Everynumberwhosesquareis even isitselfeven.

Thereis a radicalincreasein expressivepower offirst-orderlanguagesascomparedto basiclanguages.Forexample,eventhefirstpremise in theaboveargumentimplies infinitelymany basic consequencesbutit is not implied byany numberof its basicconsequences,not even by all ofthemtogether. Theidea thata generalizationis logicallyequivalentto theset of itssingularinstancesis butone ofthefallaciesthatis to beconfrontedby thoseseekingto reducefirst-orderlogic to basic logic. Below are a few ofthesingularinstancesof thepropositionunderdiscussion.

SECOND-ORDER LOGIC

One is eitherodd or even.

Two is eitherodd or even.

Threeis eitherodd or even.

69

As mentionedabove basic logic issometimes called finite logic becauseeach of itsconsistent(or non-contradictory)propositionsis finitelysatisfiable.This is no longertrueof first-orderlogic. Indeed, theconjunction of thefollowing twopropositionsis notsatisfiablein any finiteuniverseof discourse.

Zero isnotthesuccessor of anynumber.

Every twonumberswhicharesuccessorsrespectivelyof distinctnum­bers arethemselvesdistinct.

It is known, however,thateveryfirst-orderpropositionwhich isconsistentis satisfiablein a countableuniverse of discourse. In fact, everyconsistentfirst-orderpropositionthatis notsatisfiablein a finiteuniverseof discourseis, like the aboveconjunction,satisfiablein theuniverseof naturalnumbers.For thisreason,first-orderlogic can be calledcountable logic.

Justas every validbasic argumentis deducibleusing a small set of ax­ioms and rules of inference, thesame is trueof validfirst-orderarguments.This means thatas faras knowledge ofvalidityof first-orderargumentsisconcerned,humanknowing faculties are equal tothetask.Theso-called prin­ciple of sufficiency ofreason,viz. thateverytruepropositioncan be known tobe true,can be shown to be false.Humanfaculties of knowingtruthare notequal tothetaskof knowingtruth-truthoutrunsknowledge. Withvalidityof first -order aryuments, reasonis sufficient--everyvalidfirst-orderargumentcan be known to be valid.Whetherevery validargument(whatevertheor­der) can be known to be valid is aquestionof considerablecomplexityandwell beyondthescope of thiselementaryexposition.

Thereis anothermuch lessimportantfactaboutfirst-orderand basic logicthatis worthmentioning. Forthiswe have to divide the logicalconceptsintopositive and negative . Withoutgoing into thedetails, let me saythatthereare nosurpriseshere. "Every", "Some" , "Is", "And", "Or", "If ' , etc. arepositive. "Not", "No", "Distinct","Nor", etc. are negative. The resultisthateverycontradictoryfirst-orderpropositioninvolves atleastone negativelogicalconcept.

Justas we motivatedthetransitionfrombasic logic tofirst-orderlogic byreflecting onthefactthatthe reasoningused toestablisha basic tautologyseems strongerthanneeded forthatpurpose and indeed is sufficient (orvirtuallyso) to establishallgeneralizationsof thebasic tautology,we usethe same sortof insightto transcendfirst-orderlogic.Considerthe followingfirst-orderpropositions.

70 JOHN CORCORAN

No numberdivides exactlythenumbersthatdo not dividethemselves.

No numberprecedesexactlythenumbersthatdo notprecedethem­selves.

No numberexceedsexactlythenumbersthatdo not exceedthemselves.

No numberperfectsexactlythenumbersthatdo notperfectthemselves.

The relationof perfectingarises inconnectionwith theso-calledperfectnumbers. Everynumberhavingproperdivisors isperfectedonly bythesuc­cessor of the sum of itsproperdivisors. The othernumbers,viz. zero, oneandthe primenumbers,are notperfectedby anynumbersat all.Thus fouris perfectedby threesince two istheonlyproperdivisor of four.Butsix isperfectedby itself. In fact, as you may have seen already, everyperfectnum­ber perfectsitselfand, conversely, everynumberperfectingitselfis perfect.Now, thereason forintroducingtheperfectingrelationis to give anexampleof atautologyin thesame form asthefirstthreeof the above setbutnot asmathematicallytrivial.Thefirst of the abovepropositionsis mathematicallytrivialbecause zero, which istheonlynumberthatdoes not divide itself, isdivided by everyothernumber. The second istrivialbecauseeverynumberprecedesothernumbersbutnotitself.The thirdis trivialforsimilarreasons.

Now, as you know, each of the above can be deduced fromtheirownrespectivenegationsby familiar(but intricate)reasoning.The fact isthatthefollowingpremise-conclusionargumentis valid.

Some numberperfectsexactlythenumbersthatdo notperfectthem­selves.

? No numberperfectsexactlythe numbersthatdo notperfectthem­selves.

A deductionof thisargument,i.e., a deductionof itsconclusionfrom itspremise, can easily betransformedinto anindirectproof of its conclusion.The reasonthata deductionof a conclusionfrom thenull-premiseset is aproof (i.e., a deductionwhose premisesareknown to betrue)is becauseuni­versalpropositionswith null"subjects"are vacuouslytrue. Everymemberof thenull set of premises is known to betrue. . . therebeing nocounterex­amples.

Once one ofthesetautologieshas been proved to betrueby a deductionfrom thenull set ofpremises theothersare alsovirtuallyproved to betruealso. The reason for this is theprincipleof form fordeductions:every ar­gumentationin the same form as adeductionis againa deduction. Thus aproofof, say,thefourthcan beobtainedfrom a proofof, say, the first bysubstitutingin thelatterthe concept "perfects" for theconcept"divides". Soit is clearthatthereasoningestablishingone ofthefourvirtually establishesmuch more.

SECOND-ORDER LOGIC 71

Now we move tothe second-ordergeneralizationof theabove. Actually,thefollowingsecond-orderpropositionis atonce ageneralizationof each oftheabove four firstordertautologiesand, in a certainreasonablesense, theonlygeneralization.

No matterwhichnumericalrelationyou consider, no numberbearsitto exactlythenumbersthatdo notbearit tothemselves.

Once you have seenthatthisis trueyou will feelthatit is theground ofthetruthofthepreviousfourpropositions,e.g., thatthetruthof thefourthof themdependson nopeculiarityof theperfectingrelation.

My main pointin this essay isthatthereasoningin a given logic achievesmore thancan beexpressedin thatlogicandthatthetranscendingof a givenlogic by going to ahigherorderis one way ofreapingthefullfruitof one'sreasoningin a given logic. This vagueprincipleappliesnotjustto first-orderin relationto basic logicand to second-orderlogic inrelation to first-orderbutin generalto any logic inrelationto thenext lowerorder.

In basic sentences,thereareno common nouns. In first-ordersentences,thereare common nouns,butno "second-ordernouns" such as'property','relation','function' , etc.The presenceofnounsinevitablyandautomaticallyentailsthepresenceof quantifiersbecausenounsrequire articlesandarticlesexpressquantifiers. For example, thefollowingsentencesexpressthesameproposition.

Every falsepropositionimplies a trueproposition.

Everypropositionwhich is false implies somepropositionwhich istrue.

For everypropositionwhich is falsethereexistsa propositionwhich istrueand which isimplied by thefalseproposition.

The same phenomenoncan be exemplified intheuniverseof naturalnumbers(beginningwithzero).

Every oddnumberexceeds an evennumber.

Everynumberwhich isodd exceeds somenumberwhich is even.

For everynumberwhich isodd thereexistsa numberwhich is evenandwhich is exceeded bytheodd number.

Whenwe move tosecond-orderby addingsecond-ordernounswe alsoaddsecond-orderadjectiveswhose rangesof significance are thesecond-orderob­jectsdenotedby thesecond-ordernouns.Examplesofsecond-orderadjectivesare thefamiliartermsindicatingpropertiesof relations:reflexive,symmet­rical,transitive,dense, etc. The following aretypicalsecond-ordersentencesinvolving suchexpressions.

72 JOHN CORCORAN

Everyreflexiverelationrelateseveryobjectto itself.

Everyrelationthatrelateseveryobjectto itselfis reflexive.

Everysymmetricrelationrelatesto eachotherevery twoobjectsone ofwhich itrelatesto theother.

Everyrelationthatrelatesto eachotherevery twoobjectsone of whichit relatesto theotheris symmetric.

Orthogonalityis a second-orderrelationbetweenproperties.In orderforone propertyto beorthogonalto anotherit is necessaryandsufficientthattherebe fourobjects,one havingbothproperties,one havingthefirstbutlackingthesecond, onelackingthe firstbut havingthe second, and onelackingboth.Theseexamplesshowthatmuch ofthisessay has beenwrittenusing asecond-orderlanguage.

Since basic logic is finiteand since first-orderlogic iscountable,neitheris adequatetoaxiomatizetheorieswhose universes ofdiscourseareuncount­able. The most familiarexamplesof suchtheoriesarecalculusandgeometry.Nowjustas first-orderlogic is not finite,second-orderlogic is notcountable.Thereare consistentsecond-orderpropositionswhich are notsatisfiableinany countableuniverse. One example is from Hilbert'saxiom set forthetheoryof realnumbers(which isfoundationalforcalculus).Anotheris fromVeblen's axiom set forEuclideangeometry.Naturally,second-orderlogic canbe calleduncountable logic.

First-orderlogic isnoteven adequateto axiomatizetheorieswhose uni­verses ofdiscoursearecountablyinfinite. The paradigmcase ofsucha theoryis numbertheory, orthearithmeticof naturalnumbers, which requirestheprinciple of mathematical induction (PM!) .

Everypropertybelongingto zeroandtothesuccessorof everynumberto which it belongs also belongs to everynumberwithoutexception.

In orderfor apropertyto belong to everynumberit is sufficient forthatpropertyto belong tothesuccessorof everynumberhavingit andalsothatzero have it.

Mathematicalinductionis thesecond-ordergeneralizationof each ofthefollowingpropositionswhich areamong its first-orderinstances.

If zero is evenand thesuccessorof every evennumberis even, theneverynumberis even.

If zero isperfectandthesuccessorof everyperfectnumberis perfect,then everynumberis perfect.

In first-orderaxiomatizationsofarithmeticPMI, induction,is replacedbytheinfinite set of itsfirst-orderinstances,a set which is insufficient to imply

SECOND-ORDER LOGIC 73

mathematicalinduction.In fact, nosetoftruefirst-orderpropositionsis suf­ficient to implyPMI andthereforenofirst-orderaxiomatizationofarithmeticadequatelycodifiesourknowledge ofarithmetic. Moreover,thegroundof ourknowledge oftheinstancesis ourknowledge ofPMI itself.Thus infinitelymanyof the so-calledaxioms offirst-orderarithmeticarenotaxiomaticin thetraditionalsense. Nevertheless,thereare able logiciansand mathematicianswho reject thetraditionalsecond-orderaxiomatizationsdue toDedekindandPeanoin favor offirst-orderaxiomatizationswhichdatefrom the1930's.

Justas second-orderlogic isnecessaryto fullyexploitfirst-orderreasoningas well as tounderstandthegroundoffirst-ordertautologies,likewise second­orderlogic isnecessaryto fullyexploitfirst-orderknowledge inarithmeticaswell as tounderstandthegroundof acceptanceof first-orderaxiomatiza­tionsof arithmetic. Even logicianswho rejectsecond-orderaxiomatizationsof arithmeticadmit theirhistoricimportanceand make heuristicand ped­agogical use of suchaxiomatizations.By theway, thesame thingmay besaid ofaxiomatizationsofsettheory,butthetechnicaldetailsinvolved in settheoryrequiredistinctionsandprincipleswhich gobeyondthescope ofthisessay.

In thecase of basic logic, as well asthatof first-orderlogic, a small setof simple rules ofinferencessuffices toenableevery validargumentto bededuced. This is no longerthecase withsecond-orderlogic. In fact, it is acorollarytothefamous G6delIncompletenessTheoremthatno simple set ofrules is sufficient forthispurpose. This means thattheprincipleof sufficiencyofreasonwhenappliedtosecond-ordervalidityis false. To beexplicit, thereare finite validargumentsin second-orderlogic whoseconclusionscan not bededuced(in a finitenumberofsteps using simple rules) fromtheirpremise­sets. This resultis known astheincompleteness of second-order logic.

Thereare logicians who feelthathumanreasoningmust be equalto thetaskof determiningthevalidityof validarguments.In most cases such lo­gicians areempiricisticallyorientedand are fully willing toacceptthefactthattherearetruepropositionsaboutthematerialuniversethatcannotbeknown to betrue. But theyfeelthatvalidityis intrinsicallyamenabletoanalytica priori methodsand, in particular,thatevery validargumentmustbe deducible. One wayoutof thisquandaryis to denythatsecond-orderlogic is really logic.

Incidentally,second-orderaxiomatizationsdo not evadetheincompletabil­ity ofarithmetic. First-orderaxiomatizationsare deficientbecausefirst-orderlanguagesare too weak toexpressourknowledge ofarithmeticeven thoughfirst-orderreasoningis adequateto first-ordervalidity. Withinsecond-orderthesituationis reversed.Second-orderaxiomatizationsare deficientbecausesecond-orderreasoningis too weak to deduce all ofthe consequencesofsecond-orderaxioms even thoughsecond-orderlanguageis strongenoughtoexpressourknowledge ofarithmetic. In fact,oursecond-orderarithmeticknowledge impliesabsolutelyeverytruesecond-orderarithmeticproposition

74 JOHN CORCORAN

even thosethatwe are powerless todeduce(using any givensimple set ofrules fixed inadvance).

Anotherphenomenonthatgives some logiciansdoubtsaboutsecond-orderlogic isexistenceofcontradictorypropositionsdevoid ofnegativelogical con­cepts. Recallthatin first-orderlogic everycontradictorypropositioninvolvesat least onenegativelogicalconcept. Below are twosecond-orderproposi­tionsthefirst of which istautologicalandthesecond of which iscontradictory,neitherof which involvenegativelogicalconcepts.

Everyobjecthas atleastone property.

Everypropertybelongsto atleastone object.

The reasonthatthesecondpropositionis self-contradictoryis thatit con­tradictsthefollowingtautology.

No objecthas thepropertyof beingdistinctfrom itself.

We have seenthatsecond-orderlogic differsradicallyfrom first-order.First-orderis a logic ofcountability;second-orderis a logic ofuncountabil­ity. First-orderis deductivelycomplete; second-orderis deductivelyincom­plete. Infirst-ordereverycontradictionis negative;in second-orderthereareself-contradictorypropositionswhich are exclusivelypositive. The above­mentionedhistoricexamplesofaxiomatizedsciencesremind us thathigher­orderreasoningis not arecentinnovationbut rathera featureof humanthoughthavinga longhistory. Moreover, it isnotthecase thatlogiciansstartedoutstudyingbasic logicand thenmoved on tofirst-orderand thentosecond-order,etc. Inthefirst place,Aristotle'slogic is afragmentof first­orderand fundamentalaspectsof basic logic were not to bediscoveredforsome centurieslater.In thesecond place, inmoderntimes higher-orderlogicswerestudiedbefore first-orderlogic wasisolatedas a systemworthyofstudyin its ownright.

AfterAristotle's logic had been assimilatedby laterthinkers,peopleemergedwho couldnot accepttheidea thatAristotle'slogic was not com­prehensive. These conservativelogiciansattemptedto "reduce"all logicallycogentreasoningto Aristotle'ssyllogisticlogic. Likewise, afterfirst-orderlogic had beenisolatedand had beenassimilatedby thelogiccommunity,people emergedwho couldnotaccepttheidea thatfirst-orderlogic was notcomprehensive.Theselogicians can be viewed not asconservativeswho wantto reinstatean outmodedtraditionbutratheras radicalswho wantto over­throwan establishedtradition. It remainsto be seenwhetherhigher-orderlogic will ever regainthedegree ofacceptancethatit enjoyedbetween1910and 1930. Buttherehas never been aseriousdoubtconcerningits heuristicandhistoricimportance. In fact, people who do not knowsecond-orderlogiccan notunderstandthemoderndebateover itslegitimacyandtheyarecut­off fromtheheuristicadvantagesof second-orderlogic. And, whatmay be

SECOND-ORDER LOGIC 75

worse,theyare cut-off from anunderstandingofthehistoryof logicandthehistoryof mathematics, andthusareconstrainedto havedistortedviews ofthenatureofthetwosubjects.As Aristotlefirst said, we do notunderstanda disciplineuntilwe have seen itsdevelopment. It is atruismthata person'sconceptions ofwhata discipline is and ofwhatit can become arepredicatedon aconceptionofwhatit has been.

ACKNOWLEDGEMENTS

This essayis based on atutorialthatI led at the OhioUniversityInferenceConference,October9-11,1986. I am indebtedtoProfessorRichardButricknot only fororganizingtheconferencebutalso fororganizingme. In theyearpriorto theconference hesuppliedme, by phone andin writing,withdozens ofquestions,hypotheses,suggestionsandrequestsfrom whichtheba­sic contentandgoals of mytutorialemerged. James Gasser, Woosuk Parkand Ronald Rudnicki helped withtheeditingand proof-reading. I am alsoindebtedto thestudentsand colleagueswithwhom I studiedsecond-orderlogic over the years, especially George Weaver,StewartShapiro, MichaelScanlanand EdwardKeenan. Verylittlein this essay is original. Most ofwhatis here isalreadyin thewritingsof Alfred Tarski, Leon Henkin, AlonzoChurch, Georg Kreisel and George Boolos. Almost everyhumanisticallyori­ented essay onmodernlogic isindebtedto Tarski,Church,andQuine. Theirtechnicalcompetence,theirobjectivity,theircreativityand, above all,theirconstantattentionto thehumanimportanceof logic are largelyresponsibleforpreserving,transformingand revitalizinga richtraditionthattracesbacktoAristotle. One sign ofthevitalityofthetraditionis thefactthatnot oneof theabove-mentionedlogicians agrees witheverythingwrittenin this essay,even when theheuristicover-simplificationsare emended.