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argument Arithmetic assertion asymmetrical relation axiom axiom of Archimedes belongs calculus called Cantor cardinal number Chapter class of classes class of terms class-concept collection compact series complex numbers concept concerning considered contained contradiction correlation defined definition denoted descriptive Geometry discussion distance distinction distinguish divisibility entities equal equivalent Euclidean space existence fact false finite number follows formal implication Frege given greater Hence hold ideal points identical implies indefinable infinite classes infinitesimal infinity integers involved kind Leibniz less limit logical constants logical product magnitude material implication mathematical induction means metrical notion null-class number of terms object obtained one-one relation ordinal Peano philosophical plane possible predicate premisses presupposed principle projective Geometry projective space properties propositional function prove quantities question rational numbers real numbers regard seems segments sense Socrates straight line stretch supposed Symbolic Logic theory transfinite transitive transitive relation true values variable zero
Page 3 - Mathematics is the class of all propositions of the form "p implies q" where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants.
Page 459 - It follows that an infinite series already elapsed is impossible and that, consequently, a beginning of the world is a necessary condition of its existence. And this was the first thing to be proved. As regards the second, let us take the opposite for granted.
Page 460 - For let us assume that compound substances did not consist of simple parts, then if all composition is removed in thought, there would be no compound part, and (as no simple parts are admitted) no simple...
Page 168 - greater than" is typical in this respect; it is conventional to argue that if A is greater than B, and B is greater than C, then A is greater than C.
Page 252 - As for the objection that space and time are quantities, or rather things endowed with quantity ; and that situation and order are not so: I answer, that order also has its quantity; there is in it, that which goes before, and that which follows ; there is distance or interval.
Page 222 - The ratio or proportion between two lines L and M, may be conceived three several ways; as a ratio of the greater L, to the lesser M; as a ratio of the lesser M, to the greater L; and lastly, as something abstracted from both, that is, as the ratio between L and M, without considering which is the antecedent, or which the consequent; which the subject, and which the object.
Page 9 - For example, it is always true that if p implies q and q implies r then p implies r, or that, if all a's are j3's and x is an a then x is a j8.
Page 42 - The study of grammar, in my opinion, is capable of throwing far more light on philosophical questions than is commonly supposed by philosophers. Although a grammatical distinction cannot be uncritically assumed to correspond to a genuine philosophical difference, yet the one is prima facie evidence of the other, and may often be most usefully employed as a source of discovery.
Page 469 - Change is the difference, in respect of truth or falsehood, between a proposition concerning an entity and the time T, and a proposition concerning the same entity and the time T', provided that these propositions differ only by the fact that T occurs in the one where T