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Abhandlungen der Fries'chen aliquid analysis analytically true apparent variable assert avoids the paradoxes axiom of reducibility class of classes connotation and denotation containing a variable contradicting a constant decimals definable definition containing definition of false dicta potest significare dictory different universes difficulty doxes epsilon relation excluded expressed fallacy false and true fert judicium finite number Fries'chen Schule Grelling-Nelson Harold Chapman illegitimate implies its contradictory impredicable Insolubilia ipsam logical type merely nihil non-predicative function number of words Paradoxical propositions Paulus Venetus Poincare possible values Prantl predicative functions premise Principles of Mathematics proposition implies proposition is false proposition of order propositional functions propositionem propositions are true quod hoc range of significance restriction Russell's device Russell's theory Scholastic logic Socrates dicit falsum solution substitution symbolic statement term theory of types tion traditional logic true and false true or false truth type forms universe of discourse value that contradicts veram verum vicious circle Wycliffe's
Page 21 - ... values severally and individually. If this were not the case, no function could be apprehended at all, since the number of values (true and false) of a function is necessarily infinite and there are necessarily possible arguments with which we are unacquainted. What is necessary is not that the values should be given individually and extensionally, but that the totality of the values should be given intensionally, so that, concerning any assigned object, it is at least theoretically determinate...
Page 10 - This method of dealing with the problem appealed to a number of later writers, Johannis Majoris Scotus, Olkot, and Rosetus among them. It is much the same as Russell's device of the theory of types which depends on the principle of the vicious circle, namely that no term in a proposition can presuppose the proposition or have it as one of its possible values. The only Scholastic to make a serious criticism of this view was...
Page 5 - R does not have the relation R to S". Hence, giving the value T to both R and S," T has the relation T to T " is equivalent to " T does not have the relation T to T".
Page 10 - If we are not to allow a proposition to refer to itself we make a general proposition like "All propositions are true or false" exceptive. It becomes, "All propositions are true or false except this proposition.
Page 15 - I am lying", we must interpret him as meaning : "There is a proposition of order n which I affirm and which is false". This is a proposition of order n + 1 ; hence the man is not affirming any proposition of order n ; hence his statement is false, and yet its falsehood does not imply, as that of "I am lying" appeared to do, that he is making a true statement.
Page 8 - ... the last makes its own truth imply the falsity of a class of propositions to which it itself belongs, as does "All propositions are false.
Page 15 - The principle that the more widely distributed traits are the older is correct if properly used, he says, but it must not be used incorrectly. This seems to be equivalent to stating that it is true when it is true, and false when it is false, but gives us no inkling of when it is true and when it is false.
Page 9 - We are given to understand that though the form of our words may seem to refer to itself, nevertheless the thought that leads to the expression never has that character, but always refers to some object independent of it. The rule for avoiding the paradoxes becomes, "Pars propositions non potest supponere...