On these l,wo points, :rnd to a lesser degree on others, it has been found necessar5'tomake some sncrificc of lucidity to correctness 'Ihe sacrifice is, horvever,in the maiu orrly temporary: in each case, the notation ultimately adopted, though its runl rneaningis very complicated,ha^san apprrently simple rneaning rvhich, rrxcept at certain crucial points, can rvithout danger be substituted in t,hought for the real meaning. This :rppliesespeciallyto the theory of descriptive expressions(x14 and x30) and to the theory of classesand relations (x20 and x21). Of the above three purposes,the first and third often compel us to adopt rnethr-rds, definitions, and notations which are nrore conrplicated or more rlifficult than they would be if u'e had the secondobject alone in view. In the third place, the sysbernis specially framed to solve the partrdoxes rvhich, in recent years, have troubled students of symbolic logic and the t,heoryof aggregates it is believed that the theory of types, as set forth in what follows, leads both to the avoidance of contradictions, and to the rletectionof the precisefallacy rvhich has given rise to them. In the second place, it is framed with a view to the perfectly precise expression,in its symbols, of mathematical propositions: to securesuch expression,and to secureit in the simplest and rnostconvenientnotation possible,is the chief rnotive in the choiceof topics. In the first place, it aims at effecting the greatest possible anal.ysis of the ideas wiih which it deals and of the processesby which it conducts demonsbrabions, and at diminishing to the utmost the number of the undefined ideas and undemonstrated proposibions(called respectively primitiue ideas and primitiae propositions) fron'r which it starts. INTRODUCTION THe mathenratical logic which occupies Part I of the present work has been constructed under the guidance of three different purposes.
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