By Agustín Rayo
Este libro reune ensayos de los más grandes especialistas en el tema de los angeles 'cuantificación common' abordado desde todas sus dimensiones. Ellos no sólo se limitan a hablar sobre el tema sino que después del análisis que realizan, presentan avesadas tesis de cara a los problemas que se sucitan.
Lo recomiendo, pues, creo que si todavía no lo es, en el futuro se convertirá en un clásico.
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Extra info for Absolute Generality
38 Kit Fine The answer lies in the nature of the postulational method. For not every object can be postulated into existence. x∀y(y admires x)). And likewise, we cannot postulate an object which stands in the membership relationship to pre-existing objects. But this means that, once a universal set for a given domain has been introduced, no further objects that might be introduced can be among its members. Thus the membership—and hence identity—of the set will be fixed ‘for all time’, once it has been introduced.
Then (R) yields: (R∗ ) ∃y[∀x(x ∈ y ≡ ∼(x ∈ x))], which, by the reasoning of Russell’s paradox, leads to a contradiction. But the truth of (E) then shows that the original use of the quantifiers ∃ and ∀ was not absolutely unrestricted after all. Even though we have stated the argument for the particular case of sets, a similar line of argument will go through for a wide range of other cases—for ordinal and cardinal numbers, for example, or for properties and propositions. In each of these cases, a variant of the paradoxical reasoning may be used to show that the original quantifier was not absolutely unrestricted.
But for us, it is the other way round; the quantifier ∀+ is to be understood in terms of the quantifier ∀. It is through a prior understanding of the quantifier ∀ that we come to appreciate that there is a sense of the quantifier ∀+ in which it is correct to suppose that some one thing has the objects in the range of ∀ as members. Thus far from presupposing that the all-in-one principle is true, we presuppose that it is false! Of course, there is some mystery as to how we arrive at this new understanding of the quantifier.
Absolute Generality by Agustín Rayo