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It is obvious at once that ancestor must be capable of definition in terms of 'parent' but until Frege developed his generalized theory of induction, no one could have defined 'ancestor' precisely in terms of 'parent'(imp26)

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Frege was so disturbed by Russell's paradox that he gave up the attempt to deduce arithmetic from logic, to which, until then, his life had been mainly devoted(mpd58)

Traditional logic regarded the two propositions "Socrates is mortal" and "All men are mortal" as being of the same form. Peano and Frege showed that they are utterly different in form. This confusion obscured not only the whole study forms of judgment and inference, but also the relations of things to their qualities, of concrete existence to abstract concepts, and of the world of sense to the world of Platonic ideas. The philosophical importance of the advance which they made is impossible to exaggerate(okew50)

Frege's system is philosophically very superior to its more convenient rival in Giuseppe Peano. He invariably defines expressions for all values of the variable, whereas Peano's definitions are often preceded by a hypothesis. He has a special symbol for assertion, and he is able to assert for all values of x a propositional function not stating an implication, which Peano's symbolism will not do(pm501)

Frege regards Functions- and in this Russell agrees with him- as more fundamental than predicates and relations; but he adopts concerning functions the theory of subject and assertion which we discussed and rejected(pm505)

Frege does not seem to have clearly disentangled the logical and linguistic elements of naming: the former depend upon denoting, and have a much more restricted range than Frege allows them(pm510)

Frege was the first to devise a general notation of quantification, using auxiliary variables in the modern fashion. So important was this step that we might indeed look upon Frege, rather than Boole, as the founder of modern logic. The present notation, easier to print than Frege's, is from Whitehead and Russell. The pronominal character of the variable was clear to Peano(ml71)

Frege, in 1879, was the first to axiomatize the logic of truth functions and to state formal rules of inference(mol89)

Peano's methods, though adequate to the same purposes, differed in essential respects and were not as elegant as Frege's. Whitehead and Bertrand Russell's notation is more expedient typographically than Frege's, but Frege's exposition was more precise(slp18)

Frege did little with infinite numbers(slp31)

Frege had, even before the discovery of Russell's paradox, the theory of levels of attributes hinted at above; an anticipation, to some degree, of the theory of types. If in response to Russell's paradox Frege had elected to regiment his classes in levels corresponding to those of his attributes, his overall solution would have borne considerable resemblance to that in Principia(slp149)

Frege was bound to stress all three failures of extensionality, for he treated general terms and sentences as naming classes and truth values; all failures of extensionality became failures of substitutivity of identity(wo151)