Concerning the Laws of Contradiction and Excluded Middle

studien\seminarTEXT Th verydoubtful. following considerations, isthedominant viewatth Although this sa it:"All thesepropositions put Wittgenstein forexampl ofthought, school One prominent asotherlogicalprinciples. thought arenowgenerallyassignedthesame status revers a complete is today widely held thereforelay no clai principles, andcould Russell's premis theyunnecessaryas only were precedents, constructedpostulatesetsin theaccustomed forlogical reverence without thefield entering Mathematicians appear bythe outlook waschanged developmen withthe change lawsof thoughtwasboundto theseso-called attitudetoward The traditional principles. logical thanother important more has which opinion, majority represent the thattheyarepsychologicallimitations orverbalconventionsdonot change, they areabstractandmeaningless, that Theusual an arbitraryverbalconvention. who,whiledecryinghisdialecticasmadness, reducetheprincipleto philosophers, than thosecontemporary moreseriously ofcontradiction tookthelaw Hegel acertainsense ofcontradiction.In ofthelaw oneinterpretation which issimply philosophywas Hegelian doctrines ofthe thatitisnotworthstating." insignificant middle, asusuallyinterpreted, is"so whilethelawofexcluded is contradictory, of contradiction Thelaw andmeaningless thananyotherproposition. more empty all threeincontempt.Thelawof identity deliberateviolationof upona erected systems opini dissenting There havebeenmany self-evident. principlestheyare transcendental them, theirprioritywasc the otherprincipleswithoutcircularitya thesela Since to otherlogicalprinciples. ofthoughtthan laws totheso-called importance greater Tradition usuallyassigns Excluded Middle Concerning theLaws the laws of thought have no the lawsofthoughthave * [ V. J.McGILL published in: Philosophy ofScience, (1939) 196-211. 6 Principia Mathematica * ] onsidered well established. wellestablished. onsidered

proof and need none, that as universal constitutive or or constitutive none,thatasuniversal need proof and t of the postulational meth t ofthepostulational thattheycouldbededucedfrom otherlogical of Contradictionand nd all deductions appeared tomakeuseof deductionsappeared nd all I On the other hand, one of the most basic hand,oneofthemostbasic Ontheother ws could apparently not be deduced from from bededuced couldapparentlynot ws al of the traditional view. The laws of of Thelaws traditionalview. al ofthe es, but it appeared in Whitehead and in Whitehead itappeared but es, which the three laws did not occur.Not not lawsdid which thethree e present time, it seems in view of the inview of the itseems e presenttime, ons, of course, and many impressive andmanyimpressive ons, ofcourse, isemptyandmeanin they arestaticandinconsistentwith ance ofnon-Euclidiangeometries. objections to the laws of thought that ofthoughtthat objections tothelaws e fact that the laws of thought do not ofthoughtdonot the laws e factthat m to priority. This opinion which is is opinion which This m topriority. held them to be prior to,andhence, tobeprior held them that reality cannot be contradictory, contradictory, that realitycannotbe e, reduces them all to tautologies. As As e, reducesthem alltotautologies. y the same thing. That is, nothing." thing.Thatis,nothing." y thesame these laws. Hegel, for example, held forexample, laws.Hegel, these Generally, itwasheldthat od, just as the geometers' thegeometers' as od, just gless, apparently Sommer-Edition 2004 Sommer-Edition

studien\seminarTEXT assertion sign.) Theoretically, it is unnecessary ever to give a definition: we might wemight itisunnecessary evertogiveadefinition: Theoretically, assertion sign.) thisreason (For volition, notaproposition. what theysymbolize. Moreover, itisnot thesymbols, with andnot wholly says,"isconcerned Russell "A definition," inthetheorems. is contained often pointout,they authors themselves Definitions in propositions. toexcludeself-contradictory but drop ourpremises ofinfere rule well. The as middle excluded substitution servetoexcludeviolationsof p butnotinonepl ~p forpinbothplaces inp Thus, which poccurs. substitute ~pinoneplaceunlesswealso allowed tosubstitute~pfor everyoccurre onceinapropositionweareproving, we are proposition poccursmorethan up thepossibilityofthiskindcircular substitution, whicharenotexplicitlystatedin of Therules wouldoccurinthesystem. propositions self-contradictory what followedfromatrueproposition theasserti insures the ruleofinference isthesa *3.35,which to provethetheorem, follows "What ofinference, Thus therule aruleofinference. upon it,as proving apropositionweareusingthesame Carrolldemonstrated, cannot be,asLewis and premises inthearguments, propositionistrue"arenotputdownas from atrue ru Since beproved. to among thetheorems science,therulesofinfe in noother step were stillmanyloopholesforintuitive suggests,there logic),butasJohnson theexceptionofFrege's had everdone(with requiredfo an papermoreofthepremises of The importance falls firstprinciples enumerable thewh suppositionthat removed, thenthe using theverysame formulae thatwepropo formulae from anenumerated explicitly. We can,therefore, havenoa just thosefundamentallogicalprincipl by implicitly astobeacceptedwithoutdemonstration,isgoverned put forward ne ofderiving thattheprocedure namely, attempt atits solution presentsafundame "isperhaps "This Problem," Johnsonsays, intheproofs. involved ofasy theformal premises appear among ≡ V.J. McGill McGill V.J. ~p, which contradicts the definition ofequi thedefinition contradicts which ~p, Principia Mathematica Principia Mathematica Principia ≡ p(pismateriallyequivale setoffirst principles to the ground." (Vol. I,p.223.) theground."(Vol. to

on of non-self-contradictory propositions. It It propositions. on ofnon-self-contradictory also merit close attention since, as the asthe since, attention meritclose also rence employed in proving theorems are are theorems inproving employed rence ace only, for in that case we should have have weshould only,forinthatcase ace convey more important information than information convey moreimportant w formulae fromw formulae been thosewhichhave es which it is our aim to formulate es whichitisouraimtoformulate r its proofs than any comparable system system r itsproofsthananycomparable owedmuchtothe true or false, being the expression of a expressionofa beingthe true orfalse, from a true propositionistrue",used atrue from ole logical system is based upon a few isbaseduponafew system ole logical les of inference suchas:"What follows the law of contradiction and the law of the lawof contradictionandthelawof nce of p.Butwear se to derive. If this objection cannot be be objectioncannot Ifthis se toderive. were ever afalseproposition,then were ever stem does not mean that they are not not theyare stem doesnotmeanthat me thinginsymbolic form. Ingeneral, s and concealed premises. In logic, as s andconcealedpremises.Inlogic, ssurance that,inexplicitlyderiving ity intheproofs. Forexample,ifa substitute ~p in every otherplaceinsubstitute ~pinevery nce, likewise,noton definitions are not preceded by the are notprecededbythe definitions of purely technical interest,andthe of purelytechnical there is always a possibility that in there isalwaysapossibilitythat principle, or one logically dependent dependent principle, oronelogically ntal, if not insuperable, difficulty: ifnotinsuperable,difficulty: ntal, valence. It is clear that the rules of valence. Itisclearthattherules Contradict Principia Mathematica we arenotsurreptitiously nt top)wecansubstitute ion and theEx fact thatitputdown e notpermittedto ly enables us to ly enablesusto cluded MId cluded , also open open , also dle 2 studien\seminarTEXT can test the truth of all of the elementary propositions.Whencan testthetruthofall oftheelementary truth tablesare the true. With thelogicalconstants defined in fals theyarenotboth when pq,except wh every pairofpropositions, truth possibilitieswhichtheyallow.Impl ofthe interms disjunction,etc.maybedefined asimplication, constants such this method others whohavefollowed with this conclusion.Thoseacquainted of truth-table methodofprovingpropositions priority of the argueforthe time, they circularity bededuced without thought can sufficient theyare at issue.Takentogether These considerationsarenot asRussellclaims.innocent tothesubject,andthat theyarenotas conveniencesunrelated typographical indication thatthedefinitions of implication, leads ofmaterial definition It willbeseeninwhatfollows that nor couldtheirsystemhaveretainedth fromthelaw contradiction principle,Whiteheaddeductionofthelaw DeMorgan's andRussell's p Without thedefinitionofconjunction, ofcrucial importance. areoften butaboutthings.Definitions about words, equivalence, intoanassertedpropositio equivalence, | theorem, means ofthe merely says, is verbal,Russell definition Mathematica inth beingintroduced axioms are Thisprocedure,whileveryus definitions. does inhisarticlesthe Insteadofstartingoffana definitions. very doubtful, in view of the fact that the factthatthe ofthe inview very doubtful, theironlyserviceisalso,ofcourse, cumbersome expressionsandcomplicationsis conv aremere typographical definitions istrueifsense thedefinition theauthorsareconsistent.Russell'sassertionthat shortforanother,moreanal be usedas assert isthatin reasonable more assertions atall.Itis propositions,theydonotmake elementary subject-matter,i.e., about the be trueorfalse.Russellseems toconclude wouldse claims, they expressions, asRussell to andgiveclaritydefiniteness conveyinformation since definitions Threecomments conveniences." typographical are"nopartofoursubjec definitions not needtodefine definition orto ( always use thedefiniens instead,andthus Principia Mathematica V.J. McGill McGill V.J. containsmore primitivepropositi Principia Mathematica , vol.I,FirstEdition,p.12.)He  .p of excluded middle could not have been accomplished, of excludedmiddle couldnothavebeenaccomplished, e, while conjunction only ho only e, whileconjunction new, buttheyareseldom,I Formulaire de mathématiques de Formulaire ⊃ p,anydefinition canbe

e guiseof definitions, andthat to suppose with G. E. Moore that what they they withG.E.Moorethatwhat to suppose a certain expression, the definiendum, will will thedefiniendum, expression, certain a e samedegreeof duality introduce itasaprimitive ideabecause thedefinitionof conjunction,likethe Principia Mathematica ⋅ to important paradoxes, which is another paradoxes,whichisanother to important n of the system, into a proposition not propositionnot n ofthesystem, intoa new subject with new axioms, as Peano asPeano axioms, subjectwithnew new will understand that the variouslogical thatthe understand will q=~(~p laws of thought. An examination of the ofthe ofthought.Anexamination laws eful, leaves open the possibility that new eful, leavesopenthepossibilitythatnew ytical expression, the definiens. In this Inthis thedefiniens. expression, ytical transition to all new topics is effected by by newtopicsiseffected all transition to the writings of Post, Wittgenstein and the writingsofPost,Wittgenstein terms of trueand that because they do not make thatbecausetheydonot assertion wholly dispensewith thedefiniendum." and no part of the subject,andyet,by ofthe nopart and todisprovethenotio ication canbedefined asholdingfor eniences and that the avoidance of of thattheavoidance eniences and t, but are, strictly speaking, mere t, butare,strictlyspeaking,mere en p is true and q false, disjunction, en pistrueandqfalse, from other principles. At the same same Atthe fromotherprinciples. em to make an assertion; and thus to to andthus anassertion; em tomake Contradict Principia Mathematica can bemade at thispoint.First, ons than the authors believe. A ons thantheauthorsbelieve.A ∨ ~q)Df., which is one form of isoneform ~q)Df.,which alsoremarksthathedoes lds when p and q are both lds whenpandqare think, appliedtothepoint ion and theEx , Russell begins with begins , Russell false possibilities,we transformed into an transformed intoan and completeness. andcompleteness. n that the laws of laws n thatthe are not mere mere arenot cluded MId cluded reinforces Principia Principia dle 3 studien\seminarTEXT power. "importance" onewouldsuppose systems, Scientific Method thananyoftheothers."( make thesemoreimportant, ormorecertain, fr bederived principles oflogiccould whyCo difficulttounderstand is therefore It arepriorandmoreimportant. laws thesetwo that toimply Thisseems purpose. benecessaryhereandnootherpr would elements The havesomevalueorother. differentvaluesandthatall only onevalue,orsign,andtorulethatp~phave allowpinitsvariousoccurrences itwouldalwaysbenecessaryto truth-tables This,itseems tome,impossible todo. Inthe orinformally. formally prior, either up benecessarytoset importance, itwould thesetw provethat needed. Ifonewereto are middle andexcluded of contradiction thetruth-table evidence. In thedefinitions, substitution, of inference, one.Therules isaverytechnical logicfrom otherlogicalprinciples contemporary which thelawof contradictionandthela here isavariantofthelawcontradicti Principia) thesamesubstitutionsmustbe withalmo isthecase (which a proposition proofmakesuseofanotherpr Secondly, the involved consistent ones.Thisprincipleis isusedtoexcludese rule ofinference basisofgeneralconsider argued aboveanthe circularity isalsoinvolvedinthe is ofcontradiction other words,thelaw in bedemonstrated. Here, value isexcluded,cantheconclusion,~(pand~p), everyone cansee.Onlywhenthepossibilit truth-tabl itself.Whenthe contradiction thelawof unstatedprincipleisinvolved,namely, realize thatanother excluded middle,DeMorgan's provedin is itself.Thelaw contradiction even truth-table method.Thisisrendered propositionsof elementary for theprovingof thelaw secondis whilethe contradiction, has asign,eithertrueorfalse. Thefi sign.Th same the proposition, fromhaving thepropositionspand~p,whentheyoccurinsame and several occurrences propositionpfrom thesame ha One prevents twometalogical principl set up,however, V.J. McGill McGill V.J. , p. 182.) "Importance" is a relative term. In relation to deductive todeductive relation In isarelativeterm. , p.182.)"Importance" proofs of the two-value logi two-value proofs ofthe principleand"Identity",

Principia lf-contradictory propositions, and to include propositions,andtoinclude lf-contradictory laws of contradiction and excluded middle middle andexcluded laws ofcontradiction om thetraditionalthree,thatwouldnot on. Itseemsfairlyclearthatthesensein w of excluded middle can be deducedin be can middle w ofexcluded rst principle, of course, is the law of isthelaw principle,ofcourse, rst madeineachcase.Theruleemployed inciples by themselves would serve the the wouldserve bythemselves inciples e method of proof is used, however, e method isused,however, ofproof used to prove itself. That a kind of of kind Thata itself. usedtoprove of excluded middle. Both are necessary Botharenecessary middle. of excluded in the proof of the law of contradiction. ofcontradiction. theproofoflaw in es universally govern the possibilities. governthepossibilities. es universally clearerbytheexam a two-value logic in which they are not not logicinwhichtheyare atwo-value o principles have no priority orspecial o principleshavenopriority st all of the elemen could only mean priority or deductive or deductive onlymeanpriority could hen and Nagel say that "Even if other hen andNagelsaythat"Evenifother always employed, and few othersare few and always employed, proof of the law of contradiction was lawofcontradictionwas proofofthe Principia Mathematica and the truth-table proofs all furnish proofsallfurnish thetruth-table and e other insures that every proposition proposition thatevery otherinsures e inciple. Ifpoccursmorethanoncein y of p and ~phavingthesametruth ving bothatrueandfalsesigninits ations. First,thereis Contradict Principia Mathematica Principia and manyreadersmaynot cs, for example, the laws cs, forexample,the ion and theEx tary propositions of propositions tary ple of the law of lawof ofthe ple thefactthat by the law of bythelawof cluded MId cluded Logic and Logic bythe dle 4 studien\seminarTEXT Metaphysics while thelattercanbedeniedwith w beasserted can law ofcontradiction Aristotle onhasheldthattheyareinde shouldbeequivalent, middle, excluded is thatpreciselythesetwotheorems, th section of beeq wearediscussingshould the systems are false.Thereis,ofcourse,nothingsurp trueorboth bothare is,either That equivalent. middle are lawofexcluded and the important thingisthat,if theusualassu The usedinthesubstitionsornot,isamatterofconvenience. negation is theprinciple ofdouble andwhether isoptional agivensystem, primitive in Whether disjunction,conjunctionoreven assumed of doublenegationbeing thehelp with ofcontradiction from thelaw th implication, ofmaterial system of the ThusinBoolean principle. De Morgan's canbereplacedbyp,theonelawfollows ~~p of substitutionsandthreeprimitivepropos andconjunctiona ofimplication definitions It involves manymore assumptions. replaced inapropositioncannotbe ~~p se themuch orindependence principles is disputedquestionoftheirdependence thatthe thetheory Closely boundupwith 1 principles,""aprinciple certain ofall proveth isusedto middle law ofexcluded the laws is reversedand orderofthe the asprimitive, than conjunction,istaken Principia Mathematica andaccordingly, notbothfalse, true andare both statesthatpand~parenot thisformula sense.Sointerpreted, its exclusive por middleinasingleformula, excluded 136) and in Lewis's system of strict imp ofstrict system Lewis's 136) andin ( ofmaterialimplication and Langfordsversionofthesystem Morgan's helpofDe the with contradiction mi logicthelawofexcluded Boole-Schröder the Thusin theother. theonewithoutimplying toassert it impossible axiomsorrule definitions, and the other antheothe systems, two-value In modern two lawsaredistinctbutthattheyin andmostofhi Aristotle them, and between as thelawofexcludedmiddle. Yetit

V.J. McGill McGill V.J. . Aristotle,[ Categoriae Categoriae Principia Mathematica ( 1 Γ ] and many others put the law of contradiction and the law of andthelawof putthelawofcontradiction ] andmanyothers 10, 13a,37. 10, , 3),Aristotledescribesthelaw itself, the procedureisdiffer itself,the

, for example, are equivalent. Whatissurprising areequivalent. , forexample,

in the procedure ofsubstitution. procedure in the which every one must have who understands which everyonemusthavewhounderstands ithout asserting the law of excluded middle middle lawofexcluded assertingthe ithout mptions are made, the law of contradiction mptions aremade, thelawofcontradiction II e law of excluded middle follows directly middlefollows ofexcluded e law s of substitution, are adopted which make make which areadopted s ofsubstitution, lication, where thesame where lication, out denying the former. Thusin the out denyingtheformer. pendent, and, more particularly, that the and,morethat the pendent, particularly, dependent, that neither implies the other. theother. thatneitherimplies dependent, would seem easy enough to distinguish easyenoughtodistinguish would seem algebra orLewisandLangford'sversion algebra r hand, the one is regularly derived from from derived oneisregularly r hand,the for an important body of opinion from for animportant bodyofopinion involves at leastthefollowing:The involves at not ~p, disjunction being understood in in understood being not~p,disjunction e lawof contradictionandthelawof of De Morgan's principle, the principle theprinciple ofDeMorgan's principle, principleandthesameistrueinLewis e law of contradiction. Likewise, since since Likewise, lawofcontradiction. e uivalent. Any two theorems of the first uivalent. Anytwotheoremsofthefirst rising in the fact that two theorems in rising inthefact thattwotheoremsin s followers have held not only that the followershaveheldnotonlythat s nd the rule of inference, the procedure nd theruleofinference,procedure laws of thought are prior totheother ofthoughtareprior laws states the law of contradiction as well aswell ofcontradiction statesthelaw Sheffer's stroke function is taken as as strokefunctionistaken Sheffer's itions (*I.2,*I.3,*I.6).If,however, by p,theproofisalsolonger,and ddle follows directly from the law of ddle followsdirectlyfrom thelawof from theotherdirectlybymeans of Contradict of contradiction as "the most as"the of contradiction ent. Sincedisjunction,rather ion and theEx Symbolic Logic Symbolic proof is used. In proofisused. cluded MId cluded inter dle , p. , p. 5 studien\seminarTEXT A "not (AorB)"and"(not In generalitappearsthat"neitherAnorB" admits of twointerpretations, namely: orbothtrue. are bothfalse contended and qarefalse,whiletheformer disjunction, thelatternaturallymaintained disjun ofexclusive theupholders between princi assuming DeMorgan's heis asthelaw ofexcludedmiddle, aswell thelawofcontradiction sense) states Aristotle andLeibnizstatethatpor~p(w explicitly stateduntilthenineteenthcen =~band~w. principle: ~(borw) words,theargume(~b and~w).Inother x iswhiteandnotwhite.Thisonlyfollows, What Aristotlesayshereisthatif xis white) istrueof it,andthis two thesecondof and isnotwhite;for itis itfollowsthat forthen itisneitherblacknorwhite, insaying shall bewrong is ofthecontraries one subject, ofwhich can betweencontraries "No intermediate Metaphysics here.[ us not concern Scientific Thought This argumentof Aristotlewasrevive tothatwhich the rulewhichapplies thecaseofthat (determinately). Forin one anddenial affirmation that ofan things inthefuture, itfails. "Itistherefor whereindetermination enters,asinthecaseof good,but lawholds actual, the about tojudgments notapply does middle ( another place andfalse,assuming oftrue the definitions Metaphysics middle ofexcluded attitude towardthelaw one anything thatis"and"whichevery 3 2 rejected. ButAristotle,it Morgan'sprincipleortheprin course, De of unless isasserted,theothermustbe ifonelaw For wehaveseenthat mostfund isthe the lawofcontradiction ready torejecttheuniversalityofla "Philosophische Bemerkungen zu mehrwert See"OnMr. Broad's Theory ofTime" by R.M.Blake, V.J. McGill McGill V.J. Vol. xxiii, 1930. Fascicule i-3, pp. 51-77. 51-77. pp. i-3, Fascicule 1930. xxiii, Vol. Comptes Rendus des Séances de la Sociétédes desSéancesde Rendus Comptes Philosophy of Science Broad's view, and"Are Some Propositions Neith , Ross Edition, (K.1063b) he states that: hestates (K.1063b) Edition, , Ross ( Γ De Interpretatione , 7)heapparentlywantstoprove andmorerecentlybyJ. 3 ] What interestsusinthisco (April, 1936)for a criticism of

and notB)".Common is the contradictory ofwhite." thecontradictory is is clear,acceptedbothth ple. Likewise, in the course of the long controversy inthecourseoflongcontroversy Likewise, ple. , Ch. 9, 19a) he argues , Ch.9,19a)heargues

exists actually doesnotholdgood." actually exists white (w)andxisnot Although De Morgan's principle was not DeMorgan'sprinciplewasnot Although w ofexcludedmiddlewh which existspotentially,butnotactually, amental and indubitable of all principles. ofallprinciples. andindubitable amental d some agoby years mustknowwhoknows e plain," he says, "that it is not necessary "thatitisnotnecessary e plain,"hesays, tury, itwasutilized here disjunction is taken in the exclusive intheexclusive istaken disjunction here ction and the defenders of non-exclusive ofnon-exclusive defenders andthe ction should be true and the otherfalse betrueandthe should nt assumes variationofDeMorgan's nt a , of course, the law of contradiction, in in lawofcontradiction, the , ofcourse, predicated. If the subject is white we subjectiswhite predicated.Ifthe ciple of double negation (~ ~p = p) is ofdoublenegation(~~p=p)is ciple bepredicatedofoneandthesame that p or q is true except whenbothp that porqistrueexcept to terms wehaveput that p or q is true except when pandq when trueexcept porqis that however, if x is ~(b or w) implies x is xis xis~(borw)implies however, if Ł igen Systemen desAussagenkalküls" in Sciences etdesLettre the future. With respect towhatis is very different. Although in the the Althoughin isverydifferent. ukasiewicz.[ er True Nor False?" by CharlesTrue NorFalse?" A. Baylis er Contradict the law of excluded middle from thelawofexcluded language passes from the one to oneto the from language passes Ł ukasiewicz. nnection isthatAristotle Mind ese principles. Thus in Thusin ese principles. that the law ofexcluded law that the 2 (1925)for adiscussion of ] Its logical forcedoes ] Itslogical ion and theEx (black orwhite)then long before. When s Varsovie, Classe III, Classe III, s Varsovie, gether (black and gether (blackand ile assertingthat C. D. Broad in in C. D.Broad anything." His anything."His cluded MId cluded dle 6 studien\seminarTEXT numbers of the form 2 numbers oftheform "All certainpropositionssuchas: that waythecontention Inthesame welcome. ofcontradic correct, torejectthelaw excluded middlepossibleandattractive, ar and false.Broad,E.T.Bellothers thatj thedubiousproposition false entails false)contenti plausible (if that acertainman isneitherbaldnornon- other.Theassertion theviolationof the ofoneinvolves theviolation equivalent, thetwolawsare If oneiscaughtonhornofthedilemma. alternative is erroneous,thattheyare,infact,(mat th ourintuition It may ofcourse,that be, abetterposition. wouldappeartobein merely one, otherhand,in Thedialectians,anthe made. have seen,thetwolawsstandorfall t aswe Yet, toquestionthelawofcontradiction. tendency None ofthem showany think thelawstandsin to restrictthislaw;andfinally, ther Brouwer, whocontendthat than thosewhichdo social scienceswhichviolatethelawof arethos There arguments. different employ of ofthelaw The enemies other. not the which would dispense with with dispense would which nogeneralinterest as therestofus,and clingto They contrary. Onthe De Morgan. noobjection,sofaras have negation, and ofabandoni noidea have its generality) toeliminate th the peoplewhoareprepared In eithercasethedenialoflawinvol principle only DeMorgan's areequivalentif middle ofexcluded andthelaw lawofcontradiction this case,the its usualform, ~~p=p,thewholematter, as theprincipleofdoublenegationistakenin If atleastbyimplication. have denied athingwhic areobviouslyequivalent, middle usual assumptionsaremade, thelawof andthe bothareretained, Yetif wouldbenecessary. alterations inlogicalsystems of dualitycharacteristic modernlogica and inevitable.Bothprinciplesareintuitivelycertain,alsorequiredfor aform automatically. Theacceptanceof the otherwithoutapause,andacceptsDeMorgan's principleinthisform This isonehornof thedilemma. If the false. allpr that assumptions, common given the contradiction are equivalent in any syst are equivalentinany contradiction V.J. McGill McGill V.J. not (Dewey); there arethe math theway ofanyadequateexplan 2n+9 +1 are factorable" are neither true nor false implies, areneither truenorfalseimplies, factorable" +1 are on that judgments about the future are neither true or neithertrueor on thatjudgmentsaboutthefutureare the best way to avoid the paradoxes of Mengenlehre is ofMengenlehre paradoxes toavoidthe bestway the and double negation are granted. negationaregranted. and double excluded middle hail from different disciplines and disciplines middlehailfromdifferent excluded it. Theyseeadvantagesin theprincipleof doublenegationisalsonatural

e are the philosophers, such as Broad, who e arethephilosophers,suchasBroad,who tion aswell,aresultwhichneitherwould em, inadmissible consequences follow. The follow.The consequences inadmissible em, ogether ifthecustomaryassumptionsare is evoked by the idea of creating a system bytheideaofcreatingasystem is evoked erially) equivalent. When one adopts this oneadopts When equivalent. erially) at the two laws are logically independent independent arelogically at thetwolaws excluded middle areoftenmore middle excluded useful ves the denial of the other. But many of thedenialofother.Butmany ves , who find the rejection of the law of of , whofindtherejectionoflaw law ofexcludedmidd I know, to thevery I know,to ng the law of contradiction or double ordouble ofcontradiction ng thelaw udgments about the future are both true arebothtrue udgments aboutthefuture opositions of the kind are both true and are bothtrue kind ofthe opositions contradiction and the law of excluded andthelawofexcluded contradiction bald impliesthathe the law of contradiction as tenaciously tenaciously as ofcontradiction thelaw e lawofexcludedmiddle(ortorestrict l systems.Ifeither so far as they deny both laws, and not not and theydenybothlaws, sofaras e who argue that classifications in the e whoarguethatclassificationsin we have seen, is much simplified. In ismuch wehaveseen, simplified. e thus obliged if our argument is is e thusobligedifourargument h most logicians, from Aristotle on, from Aristotle h mostlogicians, Contradict excluding the one law, but but law, theone excluding ation of the fact of change. ofthefactchange. ation ematicians inthecircleof ion and theEx obvious principle of obvious principleof isboth.Aristotle's isomitted drastic le andthelawof cluded MId cluded dle 7 studien\seminarTEXT would meet the requirements, sincethepa therequirements, meet would the lawof contradictionand The Heyting logic[ The Heyting three-value logic. oftheprinciples "analogues" logic. Itwouldbemorecorrect(but whichfailinthethree-value butrathertheir analogues it isnotthesetwolaws lawofc middle, butalsothe of excluded areso negation, disjunctionandconjunction Ł doubtful extremely itis Unfortunately, didnot lawofcontradiction which the the paradox.Foritiscommonlyheldthat The questionnowariseswhet andemployment understanding form the from but itself, system formal all thatafalsepropositionimplies theorem of astheparadoxes status same it hasthe becomesAristotle. It systems.In clearerinmodernformal have beendiscussingcouldbediscovere havesu however,to beinteresting, would Ifp encouraging. paradox, sinceinthiscasep primitive propositionsmentioned above(* in appear that either DeMorgan'sor~~p which thelawof contradictionisreta theelabor tobe wouldappear contradiction to retain De Morgan's principle,double to retainDeMorgan's bothpand~pared when doesthelaw does nothold,butneither ofthede use found formaking that thetwolaws,when three-value system of make thesame assumptions,areinthe Morgan's principleand~~p=p),somo thelaw upholding middle(while excluded insofarasherestrictedthelawof contradiction became involvedina as Aristotle develops.Just seen,acontradiction isdenied.Inthiscase,aswehave equivalence ofthedilemmaisreachedif other horn 4 appears tocome solvin muchcloserto A.Heyting, "Die formalender in Regeln ukasiewicz-Tarski three-value system, fo three-valuesystem, ukasiewicz-Tarski V.J. McGill McGill V.J. Mathematical Monthly Also seeOrrin Frink's interesting discussion, "NewAlgebras ofLogic," 47-58. AkademiePreussischen der Wissenschaften, Ph Principia Mathematica Principia ⊃ 4 ], which gives formal expressi ], whichgivesformal p could not be proved, some pcouldnotbeproved, Ł ukasiewicz-Tarski seems to add weight to the contention thecontention toaddweight seems ukasiewicz-Tarski , Vol. XLV,No. 4,April,1938., Vol.

= porsomealternativeprin usual assumptionsaremade, usual oubtful (1/2). Far from resolving the paradox, this this paradox, the (1/2).Farfromresolving oubtful her three-valueandmulti-valuelogicsdonotresolve omit thelawof excludedmi ⊃ oflogicaloperations. finitions, and this would i finitions, andthiswould p could not be proved. But the prospects are not not pcouldnotbeproved.Buttheprospectsare

two value logic holding or failing in the holdingorfailing in valuelogic two thesimple omission ofonethethree ined, while lawof excludedmiddle, and also more cumbersome) tospeak of the also morecumbersome) same position.Theonly ontradiction. Strictly speaking, of course, ofcourse, Strictlyspeaking, ontradiction. negation and other needed propositions, negation andotherneededpropositions, g our paradox. For in this system the the thissystem g ourparadox.Forin of contradition. Both are doubtful (1/2) doubtful Bothare of contradition. dern logicians and philosophers, if they they if logiciansandphilosophers, dern there aremulti-valuelo ch a system developed. The paradox we asystemch developed.Theparadoxwe al system inrelationtoourcommon d, Ithink,inmost ofthelogicsince imply the law of excluded middle. middle. thelawofexcluded imply r example, the law of excluded middle middle lawofexcluded the r example, tuitionistischen Logik," Sitzungsberichte der radox would not appear in a system in notappearinasystem would radox propositions. It does not arise from the propositions. Itdoesnotarisefromthe implication, such as, for example, the the implication, suchas,forexample, defined as to exclude not only the law law toexcludenotonlythe as defined ation of aworkablesystem of logicin that suchasystem the other course is taken,ifthe theothercourseis I.2, *I.3, *I.6) would remove the I.2, *I.3,*I.6)wouldremove ysikalicsh-mathematische Klasse, 1930, p. of contradiction and assuming De of contradictionandassumingDe Contradict on totheideasofBrouwer, other waywouldhavetobe other ciples, isomitted. Itmight are inextricable. In order Inorder are inextricable. nvolve greatchanges.It ddle. If thiswereso,it ion and theEx Principia Mathematica gics whichassert exists.Inthe way outof this cluded MId cluded American dle 8

studien\seminarTEXT existences inprocess of transition. Sinceal the meaninglessness of theprincipleof ....An offallacy onesort than anyother philosophical disc in fallacious reasoning middle a principle ofexcluded directly be or can "The notionthatpropositionsare position verystrongly: their argumentsandinstances.Professor Dewey inhisrecent perhapsnoneedtorepeat andthereis writers, by many ordenied been questioned aretobefo cuts orDedikind dichotomies element phaseofbisgreaterinthesamesense thananyelement of or the continuuminto dividing aprocessofchangeinto onlywhen,itis and when, law isvalid dichotom whether depend uponthequestion thelawofexcluded whether The question out ortowork well, as contradiction to restrictthelawofexcl andlogicians whoareready iscorrect,thatthepresentphilosophers our reasoning butboth.Indeedifone not one, ofthis of itwhichdonot,thepropriety Heyting's andth leadstoparadoxes system ismuchmodern logicalsystems, reduced.) idea, asaseparateprimitive introduced propositi unusually longlistofprimitive thereareco plane geometry.(Moreover, notonlysome, butallindir disqualifies analogue) withtheconseque its negation(orrather, ofdouble principle the byabandoning only achieved thatthisresultis however, middle Itisinterestingtoobserve, cannot. of excluded can lawofcontradiction analogue ofthe less withrespecttothequalitywhich dialecticia other,and the restrictionof areequivalent the dialectic.Iftheselaws (assuming DeMorgan'sprincipl The equivalenceofthelawcontra the commonview. haveseen,thisconclusioncontradicts But,aswe involves thedenialofformer. isalso middle occurs thelawofexcluded thelawofcontradiction Ineverylogicinwhich are thetwolawsindependent. excludes thelawof excludedmiddlewhile thatthissystem tohold impossible itappears analogues, butonlytheir system, it isalsoremembered thatthelawsof are thereforeobligedtoconcludethatin V.J. McGill McGill V.J. two segments uded middle mustbeprepared middle uded nce that new paradoxes arise. For example, the system thesystem Forexample, arise. paradoxes nce thatnew isdenied,theothermust e and~~p=p)isof some interestinrelationto a a

andnon- and a new system which evades theparadox. evades which system a new is changingthananyelementin III two-value logic do not occur in Heyting's inHeyting's donotoccur logic two-value diction and the law of excluded middle middle andthelawofexcluded diction b interpretation couldbe example sometimes put sometimes example mplications. The Heyting logic needs an an mplications. TheHeytinglogicneeds und everywhere in society and nature has has and nature insociety und everywhere excluded middleisits law whiletheanalogueof proved, be no systemwithwhic such that every element or phase in orphase element suchthatevery ect proofs, even the common proofs of proofs eventhecommon proofs, ect pplies is probably the source of more isprobablythesourceof more pplies the restrictionofoneisasreasonable ourse andinmoralsocialinquiries ns can scarcely be blamed for rejecting ns canscarcelybeblamedforrejecting and the duality which distinguishes andthedualitywhichdistinguishes Sincethetruth-valueinterpretationof possible to find a mathematical point point possible tofindamathematical ere areotherinterestinginterpretations l existencesareinprocess of changeit asserted, whilethedenialoflatter ies occuruniversallyintheworld.The retaining thelawof contradiction.We middle applies universally seems to middle seemsto appliesuniversally ons, each logical constant must be mustbe logicalconstant ons, each , inandof themselves,suchthatthe a , ortomakeaDe Contradict be denied. It follows, if Itfollows,if be denied. to restrict the law of torestrictthelawof ion and theEx h weareacquainted dikind cutdividing questioned.When inapplicability to forwardtoshow Logic cluded MId cluded b andevery putsthe a . That . That dle a is 9 studien\seminarTEXT conditions bethesame.Bothareused appear to statuswithouttheotherlosingit ontological couldnotlose one the then assumptions, usual the granted are equivalent, than thatthecanonexpressesacondition is concludedthattheprincipletotallyin Indeed, if ourcontentioniscorrect,the Indeed, if This iswhattheargumentinthispaperwo ontologically. law ofcontradiction Professor Deweyconcludesaccordinglythatitisalso impossible tointerpretthe lawof cannot knowinadvancethatthe middle holdsforallpropositions.Andthesamelaw ofexcluded reasonwe the that wecannotknowinadvance eachother, propositions, pand~p,contradict thatanytwo ofinquiry knowinadvance wecannot but onlyheuristic.Since onthisinterp The lawofexcludedmiddle, state.Theobjec intermediate transitional transitional state,istobegthequestion bysayingthatitiseithersolid,liquidorina liquid. Toavoidthisdifficulty freezing andoficethatismelting, itcannot nature and society is utterlydifferent. nature andsocietyis question. Thecustomary attitude Itissubjectiv process ofclarification. processofdenyingdichotomies down. The dichotomiesor originalthinkingbegins, assumption thatnon-Aristo practice, ofcourse,theamenities of thecustomarylaws expressions answerto beindou Weshouldstill change ofname. couldbegainedbyamere anything if satisfy thelawsoflogic,butitisdoubtful definiti bya ofcourse, be eliminated, which areneithertruenorfalse,andbot this isnotpossibleacertainrangeof processes upinthe canbeset cut Dedikind a satisfiedwhen, middleare excluded One possibleviewofthismatter wouldbe correspond with guidingorheuristicprinciples remain mere satisfied weresatisfied inagivenfield. woul ofthelaws not clearwhatthestatus inanyinstance. It isalso orhavenotuptodatebeensatisfied, cannot besatisfied, the world,heappearstoholdthatcondi whereitdoesapply.Althoughher examples give species,hefailsto thecaseofbiological forexample, as, does notapply, instances recognizes Althoughhe account. precisely atthispointthatitbecomes (i.e.whenthe theother also satisfy V.J. McGill McGill V.J. to be thefacts of thisfield? satisfied,andbotharesometime telian propositions do not occur. But when contention whencontention But occur. donot propositions telian toobjectivedialectic, todialecticasaprocessin

The idea that dialectical, non-Aristotelian dialectical,non-Aristotelian ideathat The difficult to understand Professor Dewey's Dewey's tounderstandProfessor difficult e or Socratic dialectic which few would would e orSocraticdialecticwhichfew h true and false. These propositions could Thesepropositions could h trueandfalse. usual assumptionsaremade).Butitis are challenged and the assumption breaks breaks andtheassumption are challenged propositions wouldhavetobeadmitted tion is wholly sound an any other ground ground other tion iswhollysoundanany on of "proposition" requiring that they on of"proposition"requiringthat to be polite orroutinediscourseobligethe nd only when, a strict dichotomyor astrict nd onlywhen, conditions whichsatisfy onelaw,must applicable. For example, contradiction holds for all propositions contradiction holdsforallpropositions Would thelawsrestrictedtothisfield d be if the conditions they set up to be setuptobe they d beiftheconditions at issue:namely,determination of the retation, is not an ontological principle retation, isnotanontologicalprinciple bt a great deal of the time whether our our time whether bt agreatdealofthe as heuristic principles. Both specify Bothspecify principles. as heuristic of logic, whatever we call them. oflogic,whateverwecall In uld leadustoexpect.Ifthetwolaws and establishing new dichotomies is a isa dichotomies new and establishing , or would they not also describe and theynotalsodescribe , orwould too. The status of the two laws would twolaws ofthe too.Thestatus tions theysetuptobesatisfied either in which the law of excluded middle in whichthelawofexcluded ecognizes therelevanceoflawsto of nature and society, and that where where of natureandsociety,that thatthelawsof contradictionand satisfied." satisfied." be said that water is either solid or be saidthatwateriseithersolidor Contradict s satisfied and sometimes, and s satisfied not. ion and theEx of waterthatis cluded MId cluded dle 10 studien\seminarTEXT from in thecolleagues profession) Science;middle spentin the tenyearsof his car ISSN 1619-9324 1619-9324 ISSN I. profession. been hasever my knowledge, world. ThehypothesisthatAristotelianlogi transitionalstatesoftheobjective the describe propositionscan non-Aristotelian teethofmuch inthe evid inquiry, and clarification. Ontheotherhand,itseems of processes holdthatnatureimitates oursubjective absurd, ofcourse,to rulesoflogicas Aristotelian regard the Aristo overcomethe anhaving themselves society isa and ofnature processes propositions couldcorrectlydescribethedevelopmental III. II. as large sixtimes which isroughly profession, woul here named philosophers The thirteen Some remarks on theCVof V.J.McGill and otherspublished by John McCumber: also apply to me.) also applyto department,because Irealizedthat thatanyr new andrenewing ideas." (Iam including Col someand entrenchedisunderassault traditional idea tempest of holderthe ofpowerfeels whenits it is properlyfunctioningthan that certain again. Colodny taughtPitt atmore 25years. In1970, hewrote:more "Auniversity canneverbe against Pitt's him. administration then conducted its ownsixmonth inquiry, him andcleared testified that hehadmisquotedbeen andwasnot aCommunist. Thecommittee took no action He wascalledbeforetheHouseCommittee on accuse Pittsburgh, Colodnywas philosophy of science. In1961, while hewasin the History Department atthe University of California atBerkeley 1950.Accordingin to GeorgeReisch, he workedprimarilyin technical provide thisisimpossible, so faras somenon-Aris that set updichotomies, and hired at Michigan in1948.Upon hired atMichigan changed hisname andwrote to "Copi" Copilowish,as alogician, was"freeofMarxist bias" inCopilowish bothideas. life his andhis confessed to his colleagues. Frankena, William

V.J. McGill McGill V.J.

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a printable version may be obtained from [email protected] [email protected] from obtained be may version a printable The HonorRoll:Amer earnedadoctoratein history andphilosophy fromthe University of , JohnsHopkins; philosopher ofscienceand editor ofPhilosophy of concealed his previous membership hisprevious concealed to many writers who, tomanywriters d byaStateRepresentativeof commemorated in any way by way any in commemorated realizing that his deception w realizingthathis . Later employed at Rutgers. . Lateremployed Copyright 2004 vordenker.de vordenker.de 2004 Copyright a standard logic te a standardlogic

s at least a basis for furtherinquiry. for abasis s atleast eason for notconsideringhimeason aphilosopher would ed, provided the author and sources are cited eer working oneer a bookstore (personal statements mere verbal conventions. It would be Itwouldbe conventions. mereverbal telian viewof thewo odny here,eventhough he was in ahistory ring theMcCarthyEra faculty is accused of subversion, because then then ofsubversion,because accused is faculty Chair oftheDepartment ence, that sequences of dialectical, ofdialectical, sequences ence, that as it was in those days. None of them, to them,to None of as itwasinthosedays. very arbitrarytodenyinadvanceof ican Philosophers Philosophers ican totelian or dialectical logic applies in in logicapplies totelian ordialectical c applies in sofar c appliesin d correspond to about 78 in today's toabout78intoday's d correspond UnAmerican Activities(HUAC), where he Contradict xtbook (Hollinger 178179) xtbook (Hollinger in a Trotskyist groupwhenhewas aTrotskyist in

oddly enough, congratulate oddly enough,congratulate being a Communist sympathizer. sympathizer. Communist a being ould be discovered, Copilowish ould bediscovered,Copilowish the American philosophical the Americanphilosophical ion and theEx Hunter College, New York. York. New College, Hunter as itispossibleto rld, andwhooften , then certified that that certified then , by E. von Goldammer John McCumber cluded MId cluded

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11 studien\seminarTEXT before HUAC latein 1950; hetalked about hi life working asaship'sthe scalerwaterfront Seattle on (Schrecker44f,104; also see case, Allenbecame anationalspokesman foracademic dishonest,their derelictin to duty and find andteac membershiptheir admitted in the Communist Pa wrote that Phillips andhis colleague JosephButterworth, an English professor, "by reasonof case ofthe entire war"(Schrecker94108, cold 320). University PresidentRaymond B. Allen inmanyin"in whatSchreckercalls January1950, most waysimportantthe academicfreedom Phillips Herbert XI. from colleague). tenure wasrevokedthe monthnext (Schrecker2052 and frankly" if called upon by alegislativemanaged to keephis committee.job,but Parry his issued onthe very day Parry his received subpoena-thatmembers faculty should "testify fully takethe Fifth Amendment, whichdid. This he unfortunately violatedUniversity'sthe policy- names accomplish ofotherpeople.Theonlywayto already stated publiclythat was willing while he totalkhimself,about would not he give the Parry William X. (Schrecker 236240). The Board of Trustees admitted that itsaction with respect to Moore hadbeen wrong in 1978 extended, itturned out, onlythe by faculty; Moore wasfired by the Board of inTrustees August. tolerance would help himwhenheappearedbe him "afanaticalMarxist,both in inthought theory practice." and He that Reed's reputation for offer from because BrooklynCollegewas rescinded StanleyIX. Moore fired from hislectureship(personal interview with colleagues). Sidney Hookcontacted newlyappointed Chancellor a lecturerhis careeras in philosophy at San Francisco State. During the student revolt of1968, McGill VIII. V.J. (Gardner 229, 268). reachedretiremen having Eventually, dismissed. he refusedto takeCalifornia the loyalty oath VII. JacobLoewenberg correspondence) fifty yearsbefore.Thechief witnesses wereiden public untilMay, 2002.At thattimeJudd finally saw the testimony that hadendedhis career managing thefamily's office in his junkyard. The Department, which considered himmost its promising instructor. Judd spent his working life refused tofurtherdiscusshispolitics.He hired byBoard ofRegents the above,"David (see VI. entire Philosophy department, the which careerofMorris ended Judd(q.v.; Schrecker249f) therefore kept his job; but the University Board ofRegents orderedaninvestigation into the V. him,in September 1953.His position wasrestored-in 1981 (Schrecker 209212). Fifth Amendment. Hewastried forthis, andwas cooperate with the Committee, hegaveonly his subpoenaed by HUACin February, 1953.Though the Temple administration encouraged him to IV. BarrowsDunham "UnAmerican ActivitiesCommittee"), he di While being investigated by the State of Washington's Canwell Committee state (the joined the Party, though he hadworked on behalf of the Loyalists during the Spanish Civil War. Rader Melvin XII. http://www.Washington.edu.research/showcase/1950a.html V.J. McGill McGill V.J.

David Hawkins Morris Judd wasanuntenured professor atUniversity the ofColorado. He told investigators , ofthe UniversitybeforeHUACin May Buffalo, appeared of1953.Hehad of was fired from Hunter College and movedand from wasfired College toSanFrancisco,wherehespent Hunter hadjoined the philosophy departmentReed Collegeinat Oregonafter ajob , aphilosophy professor at the University of Washington, denied ever having , a philosopher of science at the ofscienceat, aphilosopher University ofColorado,wassummoned wasfired from Philosophy the Department at the University ofWashington , chair ofthe philosophy department at Temple University, was , a Berkeley Hegel scholar, wasfire Hegel scholar, aBerkeley ,

mself (though not about others), and hadtenure. He fore HUAC in June, 1954. But the tolerationthe 1954.But HUAC inwas June, fore was fired overtheprotestsofPhilosophy scovered that the committee hadsuppressed apparentlythe only philosopher therebeso to acquitted in 1955.But Temple hadalready fired name,and home age, address beforetaking the tified as "A" and "B."tified (Schrecker250,personal as"A"and Regentsmakereport refusedtothe against him t age, Loewenberg was given emeritus status status Loewenbergwasgivenemeritus t age, Hawkins") that hewasnotaCommunist, but 07). It was later restored (personal testimony thiswithoutbeingcite one of his letters of recommendation called called oneofhislettersrecommendation ). rty ... (were) incompetent, intellectually intellectually incompetent, ...(were) rty h the truth" p.40). (CAF As aresult ofthis S. I.HayakawaandattemptedMcGill get to Red hunting. Phillips spent therestofhis Redhunting.Phillips Contradict d after 35 years of service because ofservice after 35years d ion and theEx d for contemptwas to d for cluded MId cluded dle 12 studien\seminarTEXT © JohnMcCumber invitations were but thesevictims], commemorate so ....PacificDivision,APA, future corrections toof this someinformation arelikely. Addition offurther namesmore is even introductorycomments. Thisknow sofa iswhat I to circulate free NB: Pleasefeel Ellen Schrecker.No/vory TowerNew York: Oxford University Press. 1986 19421953 SanJose:iUniverse. 2001 Hist Short A Moment: Shining One Jonas, Gilbert McCumber'sTimein Ditch the Philosophical Studies 102(2002) pp. 173181. David Hollinger "Religion, Ethnicity, andPoliti 1967 David P.Gardner,TheCalifornia Controversy Oath Washington Seattle: University ofWashington Press,1949 (citedas "CAF") and Academic TheRecordof Freedom: Communism matterreconsider the (Jonas,260264) efforts ofthe Philosophy departmentand the student body, Board the ofRegents to refused Communisthe wasnot a did he not knowwhat thewere. Inspite Party'softhe strenuous views differed hishisCommunist fromthe viewsthose Deanhow of Party, Wiggins repliedthat since research program the (at UniversityDecember, ofMinnesota),inwas firedaskedby 1951. When ForrestWiggins XIII. 38,96). York,theattemptfailed(Schrecker, home inNew Though the State ofWashington then attemptedthe to witness, extradite GeorgeHewitt, fromhis exonerating evidence, andthat one of the witn Sources V.J. McGill McGill V.J.

, the first African-American tobecome , thefirstAfrican-American this onthe Internet, but if you doIask you to include these attempted toorganize asessi

declined because of difficulty of travel. ofdifficultytravel. declinedbecause orythe Student of Federalist World Movement, r. Thought Ihave sourcesandhavecited them, esses against him given perjured testimony. cs in American Philosophy:cs in American Reflectionson Berkeley: University of California Press, Press, California University of Berkeley: the Tenure Cases at the University of the Universityof at theTenureCases Contradict a tenuretrack professor in a major atenuretrackprofessorin major on atits2003 ion and theEx cluded MId cluded meetings [to dle 13