Principia Mathematica Intro-2

CHAPTER II THE THEORY OF LOGICAL TYPES Tno theory of logical types, to be explained in the present Chapter, recomrnended itself to us in the first instance by its ability to solve certain contradictions,of which the one best known to mathematiciansis Burali-X'orti's concerning the greatest ordinal. But the theory in question is not wholly dependentupon this indirect recomnrendation:it has alsoacertain consonance rvith comuron sensewhich makes it inherently credible. In what follows, we shall therefore first set forth the theory on its own accbunt, and then apply it to the solution of the contradictions. I. The Vicious-Circle Pdnaipla An analysis of the paradoxesto be avoided shorvsthat they all result from rr,certain kind of vicious circle*. The vicious circles in question arise lrom supposingthat a collection of objects may contain memberswhich can only be rlefinedby meansof the collectionas a whole. Thus, for example,the collection of propositions will be supposed to contain a proposition stating that " all prop n. Thus cardinalsg,:eaterthan n but not greater than 2" exist ru nlrPliedto classesofclasses,but not as applierl to classesof individuals, so tlrrtl,rvhatevermay be supposedto be the number of individuals,there will bc . r rrl,cnce-theorems which hold for higher types but not for lower types. Even Ir, r,', however,so long as the n.mber of individuals is not asserted,but is rrr,'r','lyir.ssumed hypothetically,rvemayreplace the type of individuals by any ,t,lr*r' t,ype,provided we make a correspondingchange in all the other types ,'r'r'rrrrirrg in the samecontext. That is, we rnay give the name',relativein,lrvrrlrrals"to the members of an arbitrarily chosen type r, and the narne " r,,lrrl,iveclassesof individuals" to classesof,,relative individuals,,,and so on. 'l'lrrrxs. long as only hypotheticalsa.e concerned, in which existence-theorems li'r' ,rr. bypeare shown to be implied by exisbence-theorems for another,only r r'lrr/ rr. types are relevant even in existence-theorems.Thi s applies also to case-s rvfr,'rr'l,lrc hypothesis(and therefore the conclusion)is usserted,, provided the ,r'r'r.r'l,i(). holds for any type, however chosen. For example,any type has at l,'rrnl... rncmber; henceany tlae rvhichconsistsof clnsses,ofwhateverorder, lrrrrrrrt,l.rrst two members. But the further pursuit of thesetopics must be left t,' l,lr,,lrrrly of the work.